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November 2, 2009

Interview with Manin

Posted by John Baez

Try this interview with Yuri Manin:

Manin is most famous for his work on number theory and algebraic geometry. But he’s also famous for his work on noncommutative geometry, the self-dual solutions of the Yang–Mills equations, and much more. He was wide interests and erudition to spare, so it’s interesting to read his view on the history of mathematics, and its future. Check out his remarks on Feynman integrals as an ‘Eiffel tower hanging in the air with no foundation’, the role of Edward Witten in modern mathematics, and the trend towards the ‘homotopification’ of mathematics.

Posted at November 2, 2009 3:37 PM UTC

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Re: Interview with Manin

The renowned physicist and mathematician Freeman Dyson in his Gibbs lectures “Missed opportunities” (1972) has beautifully described many cases when “mathematicians and physicists lost chances of making discoveries by neglecting to talk to each other.” Especially striking for me was his revelation that he himself “missed the opportunity of discovering a deeper connection between modular forms and Lie algebras, just because the number theorist Dyson and the physicist Dyson were not speaking to each other.”

Anyone else suffer this kind of schizophrenia?

Posted by: some guy on the street on November 2, 2009 5:49 PM | Permalink | Reply to this

Re: Interview with Manin

Obligatory Correction. Technically speaking, the “split” in schizoid conditions is a split of the mind from reality, not a split within the mind.

But yes, I think the metaphorical “multiplicity” and “failure to communicate” between divers cognitive styles, even within the same person, is endemic to the post*modern mentality. And a lot of that is engrained into us by our not-so-integrated dysciplinary educations.

Posted by: Jon Awbrey on November 2, 2009 6:56 PM | Permalink | Reply to this

Re: Interview with Manin

Obligatory Correction. Technically speaking, the “split” in schizoid conditions is a split of the mind from reality, not a split within the mind.

Oh! Weird… thanks!

Posted by: some guy on the street on November 2, 2009 11:19 PM | Permalink | Reply to this

Re: Interview with Manin

I like this:

The Riemann Hypothesis, without a doubt, is a problem that Riemann originated within a program, although during the course of a century and a half, the narrow number theorists continued to look at it as a very important isolated challenge. I’m somewhat apprehensive that its first solution might be a proof using blunt analytic methods. It will receive every imaginable prize, the solution will be acclaimed in every newspaper in the world, and all of this will be misleading because the “right” solution should be given in a wider context, which we already know. We even know several approaches to a solution. Nevertheless, it is quite possible that the first solution will be a poor and uninteresting one.

Posted by: David Corfield on November 2, 2009 7:41 PM | Permalink | Reply to this

Re: Interview with Manin

It also reminds me of a comment about the Appel-Haken computer-aided proof of the four-colour theorem, that such a proof spoiled the field. Since it had been proved, brilliant mathematicians wouldn’t work on the 4CT because they then wouldn’t be first to prove it, but the proof was considered (at the very least!) extremely inelegant (even by one of its originators). Where then was the beautiful proof to come from?

I think that this view was a little short-sighted, because if someone came up with a computer-free proof of the 4CT today, they would be lauded nonetheless. However, for the Riemann Hypothesis, it is hard to see that if the second proof was the ‘right’ one, as it were, the authors would receive proportionally as much attention.

Posted by: David Roberts on November 2, 2009 10:53 PM | Permalink | Reply to this

Re: Interview with Manin

It’s worth noting that for both the 4-colour theorem and
classification of finite simple groups (where the first proofs were too big for an individual human to verify every step in the entire proof, one involving computers and one not) work has been done by other mathematicians to produce more manageable proofs. So whilst it’s subjective whether these are “brilliant” mathematicians, it’s certainly hasn’t inhibited all mathematicians from working on newer, more manageable proofs.

I guess one issue is that it’s clear irritant about these proofs was their “size”, whereas it’s conceivable that the first proof of the Riemann hypothesis could be manageably small but very ugly and lacking in simplifying structure.

Posted by: bane on November 3, 2009 7:18 PM | Permalink | Reply to this

Re: Interview with Manin

I think the intent of the (badly referenced) comment above is that the Appel-Haken proof didn’t give, at the time, much mathematical insight (this is what I gather, from my distant viewpoint). Since then, as you say, there have been of course different proofs, and work on graph-colouring type results has obviously not ground to a halt.

Posted by: David Roberts on November 3, 2009 11:10 PM | Permalink | Reply to this

Re: Interview with Manin

You were perfectly clear, I wasn’t :-)

I was merely pointing out that, contrary to the expectations of some people at the time of Appel-Haken, there’s empirical evidence from two proofs that there being an already existing a proof didn’t dissuade some mathematicians from working on other proofs and so there’s potential for a small-enough-for-humans proof to be discovered. My understanding is that the “simplest” currently known complete proof of 4CT is still too large for human verification. (As you mention, there’s some distinction between a proof being too big to need a computer and a proof being “insightful”, but the issue of deciding how “insightful” a proof is has, after reading Doron Zeilberger’s writings, become for me a debatable concept. So I’ll punt on the question of insightfulness of the proof.)

The grammar-scrambled last line is that in both of the proofs mentioned the motivation for looking for new ones is to reduce the size, so it doesn’t provide direct evidence for whether mathematicians would be incentivized to look for other proofs of RH should a small but ugly proof be the first one discovered.

Posted by: bane on November 3, 2009 11:50 PM | Permalink | Reply to this

Re: Interview with Manin

But, realistically, it’s normal for mathematicians to look for multiple proofs of interesting or important results, even when all the existing proofs are elegant or otherwise satisfactory. New proofs can either streamline existing proofs, or provide new approaches, possibly opening up new areas of research, or providing methods for proving entirely different theorems. Also, the fact that a statement has been proved can encourage people to look for a (new) proof, because they have the assurance that a search for a proof isn’t futile (or at least, not futile by virtue of the falsity of the claim, which is of course always a potential danger with unproven statements). And if the only existing proof is ugly, that just adds a further spur to produce something more beautiful.

Posted by: Tim Silverman on November 4, 2009 12:27 AM | Permalink | Reply to this

Re: Interview with Manin

TS: But, realistically, it’s normal for mathematicians to look for multiple proofs of interesting or important results, even when all the existing proofs are elegant or otherwise satisfactory. New proofs can either streamline existing proofs, or provide new approaches, possibly opening up new areas of research, or providing methods for proving entirely different theorems.

One test of a satisfying QED is whether it affords understanding.

A theorem prover that simply says, “Yup, it’s true”, or a genetic algorithm that finds a solution by flipping coins till the cows come home tends to leave one in the lurch as far as understanding goes.

This kind of understanding is always Janus-faced, settling one doubt only to raise new issues, not simply the sort of Procrustean research programme that kills inquiry in putting it to bed.

Posted by: Jon Awbrey on November 4, 2009 12:16 PM | Permalink | Reply to this

Re: Interview with Manin

That sort of “rating system” is fine (and I’d agree with) in the abstract, but tends to encounter difficulties when you try and say what qualifies as insightful.

For instance, which of the three proofs this blog post is more insightful? (The blog post discusses “simpler” proofs, but one can look at the proofs in terms of “insightfullness”.) In some ways the non-induction proof is more “unmotivated algebraic manipulation” or “rabbit out of the hat” (depending on your philosophical view) that might be produced automatically but it seems to me to reveal more interesting structure and provide more clues for investigating related phenomena. In contrast, the well-motivated “proof by induction” strikes me as not illuminating things as much. You may well argue that lots of human mathematicians would prefer the algebraic proof as well; all I’m saying is that I don’t think you can take the blanket position that proofs that involve “sophisticated ideas” that humans come up with are automatically more “insightful” than brute-force/computer-amenable proofs.

This is an excessively simple example, but it shows that the notion of “degree of insightfulness” isn’t crisply defined (and indeed that insightfulness sometimes gets mixed up with “cleverness which however doesn’t really convey meaningful insight”). Just to be clear, I’m not claiming insight doesn’t exist, just that it’s a tricky concept that deserves further thought.

Posted by: bane on November 4, 2009 3:55 PM | Permalink | Reply to this

Re: Interview with Manin

To echo the immortified David Letterman, it’s only an exhibition, not a competition, so I won’t be doing any wagering.

Any time we have grounds to trust a given algorithm or automaton, we’d be foolish to reject the bit of information that it gives. It’s just that some oracles have that Gypsy Rose Lee character of leaving something to be desired, and there are definite insights to be gained from admitting that desire into awareness.

Back in my Lisp and Pascal hacking days, recursion often gave me my first cut at solving a problem, even if sustained reflection on the efficiency of the program would force me to convert large segments of it to less recursèd iterations.

Posted by: Jon Awbrey on November 4, 2009 5:22 PM | Permalink | Reply to this

Re: Interview with Manin

I’m afraid that I sometimes don’t understand what understanding the words in your posts are supposed to inovke, such as in the post above. Just in case the first sentence is meant to suggest that I was being confrontational, let me say that I’m just trying to highlight an alternate viewpoint, namely that the intuitively appealing “it’s all about being insightful” view of proofs becomes much, much less clear when it’s considered in detail. I tend to say this whenever it’s raised, and in “provocative” terms, just because I think it’s an issue that deserves attention.

Posted by: bane on November 4, 2009 6:10 PM | Permalink | Reply to this

Re: Interview with Manin

I was just saying nolo contendere — that it’s not a contest between different styles of proof or programming, since all sensible people take what they can get from whatever source provides it.

I was speaking of the aims of understanding in the context of general research programs — this may be a different shade of meaning from the use of the label “insight proof” in scare quotes, I don’t know.

Posted by: Jon Awbrey on November 4, 2009 6:58 PM | Permalink | Reply to this

Re: Interview with Manin

it shows that the notion of “degree of insightfulness” isn’t crisply defined

Of course. To one, a proof of Galois’ main theorem on solvability via Artin, using Galois cohomology or something (not that I know details, this is just a rough example) is fantastic for providing insight into fields and extensions and so on. To another, it is obfuscating the end result, namely how can we describe solutions extractable by repeated roots.

Posted by: David Roberts on November 5, 2009 4:32 AM | Permalink | Reply to this

Re: Interview with Manin

Interesting, and of course what Manin says is a very important point, but I wonder why he chose the Riemann hypothesis as an example? Riemann tried to learn something about prime numbers. In the paper where he states what is now called his hypothesis as a side remark, he derives estimates for the number of primes less than a given magnitude (which is the title of the paper), and he uses (complex) analysis for this. This is pretty much what people working in this field are doing now, isn’t it? (Well, I only know about some of Connes work).

And I think he underestimates how much doing mathematics has changed during the last 150 years. Sure, you could publish Riemann’s paper today (after fixing some minor problems: It’s written in German, it’s handwritten, I can’t read his handwriting because the font that I learned in school differs considerably from the one that was used in Riemann’s time, and many of his formulations seem weird to modern eyes).

But that’s not my point: Today there are much much more professional mathematicians, the amount of essential material you have to learn has grown and the rate of exchange of information has increased tremendously, while the way humans think about mathematics hasn’t changed much. Today Riemann would probably not publish his paper right away, but try to google if Gauss did something similar already.

Posted by: Tim vB on November 3, 2009 9:33 AM | Permalink | Reply to this

Re: Interview with Manin

Tim vB wrote:

Interesting, and of course what Manin says is a very important point, but I wonder why he chose the Riemann hypothesis as an example? Riemann tried to learn something about prime numbers. In the paper where he states what is now called his hypothesis as a side remark, he derives estimates for the number of primes less than a given magnitude (which is the title of the paper), and he uses (complex) analysis for this. This is pretty much what people working in this field are doing now, isn’t it?

I think that’s the approach Manin is saying might lead to a “poor and uninteresting” proof of the Riemann Hypothesis.

A long time ago André Weil generalized the Riemann Hypothesis to function fields. He proved a special case while in prison, but later Grothedieck led a marvelous conceptual attack on the so-called Weil conjectures using the concept of étale cohomology — an attack completed by Deligne. It’s widely believed that a really good concept of ‘the field with one element’ would allow a proof of the Riemann Hypothesis along these conceptual lines. I imagine that this is what Manin would consider a good proof.

For more, read Looking for Fun by Lieven le Bruyn. If you read this, and understand it, you’ll see that the Riemann Hypothesis lies at the nexus of some of the most tantalizing dreams in mathematics…

Posted by: John Baez on November 3, 2009 4:33 PM | Permalink | Reply to this

Re: Interview with Manin

Thanks! I’ll do my best…
(The link to the Weil conjectures seems to be broken, but I guess it refers to the Wikipedia entry, right?)

Posted by: Tim vB on November 3, 2009 9:12 PM | Permalink | Reply to this

Re: Interview with Manin

Yes — and I’ve fixed that link. Thanks!

Posted by: John Baez on November 3, 2009 9:56 PM | Permalink | Reply to this

Re: Interview with Manin

Hmm. For a less enthusiastic, though still pretty sanguine, reaction, see this post by Emmanuel Kowalski and some of the comments.

Posted by: Yemon Choi on November 3, 2009 10:26 AM | Permalink | Reply to this

Re: Interview with Manin

Even though I’m fundamentally sympathetic to Manin’s comments, it’s obvious that the ‘programmatic approach’ can be overemphasized. It’s widely acknowledged, for example, that a key ingredient in Deligne’s ability to finish off the Weil conjectures was a certain pragmatism that went beyond Grothendieck. The specific mathematics is known as ‘the Rankin-Selberg method’ in analytic number theory.

Another good example is class field theory, which appears to me constantly on the verge of disorder. People refer to the ‘Langlands program’ or even ‘Langlands’ philosophy,’ but this is a case of a mathematical battleground where extensive intertwining of overarching philosophy and creative piecemeal engineering is the preferred mode of operation for the best people in the field. If you look at the great results like the local Langlands correspondence or the Sato-Tate conjecture, the proof *never* works in a philosophically optimal or premeditated way. It’s actually quite interesting to observe how many adjustments are introduced along the way to arrive at a target theorem.

The ability to put on hold an apparently insurmountable urge for harmonic order is an important component of mathematical progress.

Posted by: Minhyong Kim on November 3, 2009 5:30 PM | Permalink | Reply to this

Re: Interview with Manin

In my own work I’d rather have ‘harmonic order’ than ‘progress’ — and I’m not so egotistical as to think this personal decision will slow down the march of mathematics!

Posted by: John Baez on November 3, 2009 8:23 PM | Permalink | Reply to this

Re: Interview with Manin

You mean you haven’t realized we’re just cogs and wheels in a big machine? I’m shocked!

Posted by: Minhyong Kim on November 3, 2009 8:51 PM | Permalink | Reply to this

Re: Interview with Manin

Sometimes I feel a bit like this.

Posted by: John Baez on November 3, 2009 10:01 PM | Permalink | Reply to this

Re: Interview with Manin

Heh. But you still imagine yourself to be the (hapless) human.

Posted by: Minhyong Kim on November 4, 2009 12:33 AM | Permalink | Reply to this

Re: Interview with Manin

If you look at the great results like the local Langlands correspondence or the Sato-Tate conjecture, the proof *never* works in a philosophically optimal or premeditated way. It’s actually quite interesting to observe how many adjustments are introduced along the way to arrive at a target theorem.

But is your sense that ‘eventually’ we’ll have philosophically optimal proofs? I wonder if there’s a nice example from a century ago where the early decades see the piecemeal engineering you describe, and then more recently we can see how to do the work ‘properly’.

Posted by: David Corfield on November 3, 2009 9:00 PM | Permalink | Reply to this

Re: Interview with Manin

The answer to this is a resounding ‘Of course!’ For the above-mentioned results in the Langlands program, this is happening at a fearful pace even as I write. There are so many other examples, I hardly know where to begin. A nice example with a classical flavor might be Hilbert’s twenty-first problem concerning the existence of regular singular differential equations with prescribed monodromy. I’m not sure how many times it’s been proved. In his summary of Deligne’s beautiful version, Nick Katz famously quipped that ‘what’s worth proving once is worth proving many times.’

The famous Index Theorem for elliptic operators has a very instructive history in this regard. In the paper explaining their second proof, Atiyah and Singer emphasize the philosophical importance of the new approach, which they refer to as an ‘embedding proof’ modeled on Grothendieck’s proof of the Riemann-Roch theorem. Eventually there were ‘local’ proofs, path integral proofs, supersymmetric proofs, ad infinitum, until one was left with the impression that the whole thing was gradually being reduced to a triviality. Such reduction is good in many ways, as in Whitehead’s remark about the progress of civilization. But a very interesting and relevant episode occurred in the early 70’s that profoundly influenced subsequent developments. This was the ‘heat equation proof’ for Dirac operators by Atiyah, Patodi, and Singer. Most of the later refined proofs follow the pattern of this one. It is remarked there that the ‘miraculous cancellations’ that occur in the asymptotic expansion of the heat kernel had first been calculated through a ‘tour de force’ earlier by Patodi alone (for the Laplace-Beltrami operator). It goes without saying that Patodi’s horribly messy calculations were a crucial step in the streamlining that eventually took place.

I sometimes flatter myself that I have been working for a while on a ‘philosophically satisfactory’ proof of the Mordell conjecture. (This sentiment will quite likely strike Professor Faltings as laughable.) But the story of the Mordell conjecture is itself a nice illustration of messy serendipity. To an outsider, Faltings may appear to be an abstract arithmetic geometer like any other, but his style has always been quite distinct from the Grothendieck school. Of course he’s incredibly well-educated in the Grothendieck machinery and uses it inccessantly, but there’s always been an aspect of his approach to mathematics that could also only be described as ‘pragmatic.’ That he managed to prove the Mordell conjecture that had eluded the enormously productive French school can be attributed to this pragmatism coupled to tremendous technical prowess. One way of describing the awkwardness of the Mordell conjecture is exactly that equations over number fields have none of the tidy patterns that govern finite fields a la Weil-Deligne. To our present day understanding, number fields display exactly the kind of order ‘at the edge of chaos’ that arithmeticians find so tantalizing, and which might have repulsed Grothendieck. The kind of things Faltings was good at that were not entirely mainstream in Orsay at the time included technology like Diophantine approximation, heights and metrized line bundles, and geometric invariant theory, to mention just a few salient items. And then there was the p-adic Hodge theory that had in fact been a preoccupation of Grothendieck’s, but which had remained largely unwieldy up to that point compared, say, to l-adic cohomology. Faltings’ success in applying it to something ‘practical’ was a driving force in yet another process of streamlining, so that it’s now standard machinery in too many theorems to name (including results of Wiles, Harris-Taylor, etc.). This particular kind of interaction between theory and problem-solving, even though it’s well-known to practioners, has perhaps received insufficient attention from the commentators on the ‘two cultures.’ That is

theory–> resolution of important problems

is the subject of many conspicuous arguments, but the arrow

resolution of important problems–> mature development of theories

is equally important. Obviously, a more accurate picture would be a complicated graph with different amalgams along the nodes and multiple edges.

The collective arsenal of the mathematical community includes grand visions, brute force, wishful thinking, and even wrong proofs that lead to important developments. Some other essential ingredients that may go unnoticed include the ‘misguided program’ and the *accident.* My advisor Serge Lang has sometimes been accused of pointless generality. This is supposed to occur, for example, in his (great) book on Diophantine geometry. Part of the reason for this impression has also to do with his somewhat awkward stance vis-a-vis the Grothendieck school. Lang was a great fan of algebraic geometry in the style of Grothendieck, but never really mastered it well enough to use it with fluency. This imparted something of an archaic flavor to his mathematical style, even when he was theorizing in a grand fashion. The statement of the Mordell-Lang conjecture, for example, will strike many as rather odd, and meaninglessly general. However, I can’t resist recounting a story that combines into a single narrative the many different faces of accidental progress and quixotic programs. In the early 90’s I was briefly a colleague of the logician Udi Hrushovski. He was a person with very lively interests in all aspects of mathematics, and would frequently approach me with an array of questions about number theory, including the Weil conjectures and etale cohomology. However, we once had a conversation about the work of Angus Macintyre, strikingly ironical in retrospect. Macintyre was still on the faculty of Yale when I was contemplating studies there, and I had some interest in him as a possible advisor. So he happened be the only logician whose work I knew something about. When I inquired about Hrushovski’s opinion of it, the reponse was something like ‘It’s very nice, but somehow too applied. Macintyre always thinks about applications to algebra, whereas I’m much more of a pure model-theorist.’ (Even these days, Macintyre constantly urges young logicians to educate themselves in mainstream areas of mathematics.) Pure model theory here refers roughly to the classification of a more extensive collection of structures than most self-respecting mathematicians at the time would have cared a hoot about. One lazy afternoon, I had been browsing through a paper by Alexandru Buium in the common room, in which he used differential algebras for the proof of some geometric version of Mordell-Lang. (Such proofs, by the way, had evolved out of Manin’s slightly incorrect proof of the geometric Mordell conjecture.) At some point, I absent-mindedly dropped it on the table as I left the room. Hrushovski told me that he picked it up soon afterwards and suddenly realized that a significant portion of the structures there were familar to him. In particular, the differentially closed fields used by Buium, somewhat exotic to geometers, were standard among model-theorists and the ‘too general’ statement of the Mordell-Lang conjecture was strongly reminiscent of definable subsets of certain groups of finite Morley rank. Some of you will know that Hrushovski rapidly came up with a model-theoretic proof of the Mordell-Lang conjecture over function fields of positive characteristic, and ushered in a really new era of interaction between arithmetic geometry and model theory. In short, spectacular ‘applications,’ a great deal of entirely new theorizing, and freshly uncovered bits of harmony. Incidentally, Hrushovski’s proof, influential as it was, is still very hard to understand after nearly two decades of collective effort. My friend Anand Pillay has been devoting an inordinate amount of time to coming up with a more ‘philosophically satisfactory’ version, even though he dislikes philosophy on the whole (except perhaps in its Marxist incarnations).

My own feeling is that there is an abiding logic in all this, but at a much higher level than can be captured by familiar dichotomies.

Posted by: Minhyong Kim on November 4, 2009 12:22 AM | Permalink | Reply to this

Re: Interview with Manin

Amen.

The two-cultures question often reminds me of questions like whether seeing or hearing is more useful, or whether arms or legs are, and so on.

Posted by: James on November 4, 2009 4:01 AM | Permalink | Reply to this

Re: Interview with Manin

They are silly questions indeed. However, the questions of how better to coordinate seeing and hearing or arm and leg movements are not.

Minhyong writes

My own feeling is that there is an abiding logic in all this, but at a much higher level than can be captured by familiar dichotomies.

So that gives rise to a string of projects: capture the logic; discover whether or not it is unique to mathematics; examine whether its articulation allows for better communication between practitioners, and so on.

Posted by: David Corfield on November 4, 2009 8:40 AM | Permalink | Reply to this

Re: Interview with Manin

I agree entirely. Isn’t this the kind of process Lakatos was interested in? In any case, if you eventually have some insights on these issues, I’d be grateful to hear of them.

I’m rushing off to the station now. I’m going to Sheffield to talk to some higher category theorists!

Posted by: Minhyong Kim on November 4, 2009 9:56 AM | Permalink | Reply to this

Re: Interview with Manin

Lakatos’s great case study was VE+F=2V - E + F = 2 through the 19th century. I could characterise a lot of my work as taking this as a starting point and reacting to it. Obvious flaws included that Poincaré’s proof just emerged in the second chapter of Proofs and Refutations without a whiff of all that work by Riemann on his surfaces, abelian integrals, and so on. I was keen to show that progress doesn’t take place merely through attempts to prove a conjecture, i.e., that there’s also theory building to consider (chapter 7 and 8 here).

This led me on to look at debates concerning the introduction of entities (in my case groupoids) with no obvious advantage for problem-solving.

But I can see that it might be time to perform some dialectical reversal, and do the kind of study we’re talking about above. Colin McLarty has certainly looked at what might be seen from the purist’s point of view as the philosophically imperfect solution by Deligne of one of the Weil conjectures.

What would be interesting to know is what kind of trace is to be found in more philosophically satisfactory theoretical proofs left behind by their less satisfactory calculational predecessors.

Posted by: David Corfield on November 4, 2009 11:28 AM | Permalink | Reply to this

Re: Interview with Manin

Perhaps a subject like class field theory would be reasonable for a case study. It’s really full of things that required serious effort to clean up. One difficulty is that the formulations and proofs are still far from optimal.

Posted by: Minhyong Kim on November 6, 2009 2:33 PM | Permalink | Reply to this

Re: Interview with Manin

If I could repeat my point above

What would be interesting to know is what kind of trace is to be found in more philosophically satisfactory theoretical proofs left behind by their less satisfactory calculational predecessors,

I wonder if you have a sense of what would be found on this spectrum in the case of class field theory:

  • End 1: the ‘dirty’ moves (if I can use that term for what needs to be cleaned up) are merely psychologically important for giving us confidence that something is the case so that our hopes for a clean theory may be realised.
  • End 2: the ‘dirty’ moves contain the essence of what is needed for a clean version, and this essence must be carefully extracted.

Of course, mixtures of positions on the spectrum are possible.

Posted by: David Corfield on November 6, 2009 3:42 PM | Permalink | Reply to this

Re: Interview with Manin

I tend to think that End 2 is the more common phenomenon, if not the main thing with only a few exceptions. But I would have to work quite a bit in order to justify this thought. Here is one example that immediately comes to mind: The analytic continuation and the functional equation for the Dedekind zeta function of a number field. I believe it is common to think of Tate’s proof using Fourier analysis on adeles as the most elegant. However, Hecke’s original proof, which I would regard as much ‘dirtier,’ has been tremendously influential as well. Tate certainly credited Hecke very strongly, and gave the general impression of regarding his slick refinement as something of a swindle.

If you don’t mind a bit of egotism, I might mention an interesting case which is neither 1 nor 2, about which I can perhaps speak with some authority.

I managed to reprove that an equation of the form ax n+by n=cax^n+by^n=c with n4n\geq 4 has at most finitely many rational solutions. Now, I would hardly regard Faltings’ great proof as consisting of ‘dirty moves,’ but it might be argued that my approach is more streamlined. Furthermore, the main points are so different from Faltings’ approach that there’s no way to see End 2 as applying here. On the other hand, it does make important use of pp-adic Hodge theory. Although the parts I use have no obvious relation with what occurred in Faltings’, a main reason that the key tools were developed at all was because Faltings managed to use pp-adic Hodge theory so successfully. So this seems to be a complicated case.

We should also be aware that later proofs sometimes regress rather than progress. At least, mainstream opinion mostly regards Erdos and Selberg’s elementary proof of the prime number theorem as inferior to the zeta function approach of Hadamard and de la Vallee-Poussin. I remember hearing, I’m not sure from whom, that the Erdos-Selberg work was regarded as important enough for a Fields medals because Hardy thought it would lead to great insight on the Riemann hypothesis. This hasn’t come about yet.

Finally, an entirely different light on the question of what constitutes a good proof might be cast by the following quotation from Borel, speaking of Harish-Chandra:

‘He wrote to me that algebraic number theory was the most beautiful topic he had ever come across and that the sole consolation in his misery was his lecturing on class field theory…. This was indeed the kind of mathematics he had admired most: the main results are of great scope, of great aesthetic beauty, but the proofs are technically extremely hard.’

Harish-Chandra started out as a research student of Dirac, but eventually turned into a legendary figure in the representation theory of non-compact groups, creating many of the ideas that evolved into Langlands’ philosophy.

Posted by: Minhyong Kim on November 7, 2009 12:35 AM | Permalink | Reply to this

Re: Interview with Manin

“‘philosophically satisfactory’ proof of the Mordell conjecture”:

Do I understand correctly, that such a more satisfying approach gives more quantitative infos than Falting’s proof? Was Grothendieck’s ‘philosophical’ idea to prove finiteness theorems with the section conjecture about finitenesses only, or did he think at quantitative estimetes too?

Posted by: Thomas on November 4, 2009 11:19 AM | Permalink | Reply to this

Re: Interview with Manin

Yes, my wishful thought is that the most pleasant proof should also be the most practical one in the case of the Mordell conjecture.

I’m not sure what Grothendieck had in mind.

Posted by: Minhyong Kim on November 4, 2009 9:57 PM | Permalink | Reply to this

Re: Interview with Manin

i realized that if i don’t use this story now i’ll probably have to wait a long time for it to become near-relevant again. sometime in the 1980’s a professor at the university of buffalo sat through a colloquium lecture on morley’s conjecture and after it was over asked about the connections to fermat’s last theorem that he’d been hearing about; a chorus of voices from the audience reminded him that that was the mordell conjecture.

Posted by: james dolan on November 17, 2009 11:59 PM | Permalink | Reply to this

Re: Interview with Manin

I must add that Pillay was reading philosophy in college before moving into math.

Posted by: Javier on September 6, 2013 2:26 PM | Permalink | Reply to this

Re: Interview with Manin

There’s the case of Newton and calculus. Newton was a brilliant manipulator of expressions, but he also was acutely aware of the aesthetic and logical underpinnings of mathematics and was very bothered by the lack of an axiomatic foundation for calculus. So having discovered results by manipulations with no known sound basis, he forced them into the sounder but somewhat unnatural form of theorems in Euclidean geometry. This is said to have hampered the development of British mathematics, since it enabled students to get away with not learning the dubious manipulations that were necessary to make progress. On the continent, things were rather different.

Eventually, calculus was provided with a pretty axiomatic basis and the relevant concepts and philosophy to support it, but this didn’t come out of basing it on Euclidean geometry.

(I think I got this view from Morris Kline—if not him, then I’m not sure who. I’m also reading a very interesting book at the moment on the adoption of a “problem-solving” pedagogy in mathematics in Cambridge over the period about 1760–1860 (mainly prepared over 1800–1820 and completed 1820–1840). This is Masters of Theory by Andrew Warwick, U of Chicago Press 2003).

On the other hand, progress in elliptic integrals was slow until Abel and Jacobi independently realised that the “philosophically correct” approach was to work with elliptic functions instead.

(I think I got this story from a book on elliptic curves by McKean and Moll.)

Posted by: Tim Silverman on November 4, 2009 12:45 AM | Permalink | Reply to this

Re: Interview with Manin

Perhaps it would be interesting to look at a program fitting Manin’s criteria, but still in progress, like Mochizuki’s “inter-universal geometry” about the abc-conjecture and a generalization of anabelian geometry?

Posted by: Thomas on November 4, 2009 12:25 PM | Permalink | Reply to this

Re: Interview with Manin

Judging by this description of his work, Motizuki’s program would be a demanding case study.

Posted by: David Corfield on November 4, 2009 1:12 PM | Permalink | Reply to this

Re: Interview with Manin

Over on the category theory mailing list, André Joyal wrote:

Dear category theorists,

I invite everyone to read the interesting interview of Yuri Manin published in the November issue of the Notices of the AMS.

One the ideas discussed by Manin is that of a “pragmatic foundation” of mathematics as opposed to a “normative foundation” by logicists or constructivists. He attributes the former to Bourbaki.

I disagree.

The foundational framework of Bourbaki is very much in the tradition of Zermelo-Fraenkel, Godel-Bernays and Russell. I am aware that Bourbaki was more interested in the development of mathematics than in its foundation. My guess is that the foundation was too problematic to be given a prominent place in the treaty, not for logical reasons but for conceptual reasons. I claim that nobody truly understand set theory, even today! The emperor has no clothes! I mean that the hierarchy of infinite cardinals is so profoundly mysterious that it looks pathological. What is the value of a theory if it leads to meaningless problems and structures? Having no good answer to offer, Bourbaki decided to diminish the importance of foundation rather than leaving it open. It may explain why category theory was not incorporated in the foundation later.

In the interview, Manin also said that:

And so I don’t foresee anything extraordinary in the next twenty years. Probably, a rebuilding of what I call the “pragmatic foundations of mathematics” will continue. By this I mean simply a codification of efficient new intuitive tools, such as Feynman path integrals, higher categories, the “brave new algebra” of homotopy theorists, as well as emerging new value systems and accepted forms of presenting results that exist in the minds and research papers of working mathematicians here and now, at each particular time.

Any comments?

AJ

Posted by: John Baez on November 7, 2009 3:04 PM | Permalink | Reply to this

Re: Pragmatic Norms

People who call themselves “pragmatists”, philosophically speaking, have long recognized that pragmatism has both a high road and a low road, but a few on both roads have also seen that the roads are destined to converge in the end.

How long that takes is anybody’s guess.

But I do not think that any opposition between “normative” and “pragmatic” can withstand critical reflection on the true meanings of those words for very long.

Posted by: Jon Awbrey on November 7, 2009 5:56 PM | Permalink | Reply to this

Re: Pragmatic Norms

Those working from especially a Peircean pragmatic standpoint might tend to agree with your idea, Jon, that there seems very little sense in opposing ‘normative’ and ‘pragmatic’. Take, for example, this passage from a letter by Peirce to William James.

Peirce: “These three normative sciences [including logic conceived as semiotic] correspond to my three categories, which in their psychological aspect, appear as Feeling, Reaction, Thought… The true nature of pragmatism cannot be understood without them. It does not, as I seem to have thought at first, take Reaction as the be-all, but it takes the end-all as the be-all, and the End is something that gives its sanction to action. It is of the third category [i.e., Thought]. Only one must not take a nominalistic view of Thought as if it were something that a man had in his consciousness. Consciousness may mean any one of the three categories. But if it is to mean Thought it is more without us than within. It is we that are in it, rather than it in any of us” [CP 8.256].

On the other hand, it seems to me that the pragmatic maxim itself is not explicated within Peirce’s presuppositional classification of the sciences until the third branch or logic, namely, methodeutic (theoretical rhetoric) where it appears to be central to his theory of learning. In that (limited) sense the normative seems to ‘lead to’ the pragmatic, or, at least, to the pragmatic maxim and philosophical pragmatism.

Having said that, I yet tend to agree with your comment, Jon.

Posted by: Gary Richmond on November 7, 2009 7:23 PM | Permalink | Reply to this

Re: Pragmatic Norms

Thanks, Gary. Give me a few days to metabolize that feast of inferential symboly. In the mean time, my continuing interest in the issue led me to search Manin’s text for clues to the contrast that he or his translator drew between “normative” and “pragmatic”.

Here’s a sample of relevant passages:

Cantor’s theory of the infinite had no basis in the older mathematics. You can argue about this as you like, but this was a new mathematics, a new way to think about mathematics, a new way to produce mathematics. In the final analysis, despite the arguments, the contradictions, Cantor’s universe was accepted by Bourbaki without apology. They created “pragmatic foundations”, adopted for many decades by all working mathematicians, as opposed to “normative foundations” that logicists or constructivists tried to impose upon us.

What Bourbaki did was to take a historical step, just as what Cantor himself did. But this step, while it played an enormous role, is very simple — it was not creating the philosophical foundations of mathematics, but rather developing a universal common mathematical language, which could be used for discussion by probabilists, topologists, specialists in graph theory or in functional analysis or in algebraic geometry, and by logicians as well.

When “pragmatic foundations” of mathematics are made explicit, usually in several variants, the advocates of different versions may start quarreling, but to the extent that it all exists in the brains of the working generation of mathematicians, there is always something they have in common.

From reading that, I would guess that Manin means “pragmatic” in the sense of culturally embedded practices — bringing to mind Wilder’s theme on “Mathematics as a Cultural System” 1, 2, 3 — and, as far as “normative” goes, he seems to be using it more in the sense of programmatic principles that are perhaps too prematurely “prescriptive” than referring to norms with a solid grounding in practice and oriented toward the ultimate pragma of the given enterprise.

Posted by: Jon Awbrey on November 8, 2009 2:40 PM | Permalink | Reply to this

Re: Interview with Manin

“codification of efficient new intuitive tools, such as … the “brave new algebra” of homotopy theorists”

makes me puzzle, because I thought that is codified in e.g. Lurie’s articles. But I read only his survey on elliptic cohomology and some standart articles on symmetric spectra. Taking Y.I.Manin’s remark as indicator for having missed to notice something, I’d like to read what others think about that, esp. what the intuition on “brave new algebra” is.

Posted by: Thomas on November 7, 2009 9:23 PM | Permalink | Reply to this

Re: Interview with Manin

Answers here.

Posted by: Thomas on November 8, 2009 11:05 AM | Permalink | Reply to this

The Wand Chooses The Wizard

My favorite:

I must explain to you how I imagine mathematics. I am an emotional Platonist (not a rational one: there are no rational arguments in favor of Platonism). Somehow or other, for me mathematical research is a discovery, not an invention. I imagine for myself a great castle, or something like that, and you gradually start seeing its contours through the deep mist, and begin to investigate something. How you formulate what it is you’ve seen depends on your type of thinking and on the scale of what you have seen, and on the social circumstances around you, and so on.

Posted by: Jon Awbrey on November 8, 2009 6:48 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

It would be interesting to know how such a perception of mathematics is distributed among mathematicians (perhaps among non-mathematicians too?) and how that influences attitudes towards specific concepts and themes. E.g. guiding ideas like F_1, primes as knots, motives, anabelianistics, ‘yoga des … ’ which stayed ‘math scifi’ for a long time.

Posted by: Thomas on November 8, 2009 11:32 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I think even committed formalists sometimes feel this way. For example, Mac Lane (who might reject being called a ‘formalist’, but there was certainly a strong formalist strain in his thought) has said that in his work with Eilenberg on what are now known as Eilenberg-Mac Lane spaces, they had an abiding faith that such spaces must ‘exist’ somehow before they succeeded in constructing them. Perhaps Mac Lane would describe this instance of emotional Platonism as an intuition of a mathematical possibility of form, while not committing to belief in (Platonic) Form.

But I think you could also turn this around and say that even committed Platonists may experience an emotion of having invented something more than having discovered it. For example, if a student asks for an example of two noncommuting matrices, we may just toss something off quickly: invent a counterexample. (I think it would feel a little odd to say one ‘discovered’ it.) That’s a trivial example, but more complicated constructions of counterexamples may again feel rather more invented than discovered.

Thinking aloud here, a lot of mathematics is presumably ‘invented’ in the sense that mathematicians spend a great deal of time analyzing arguments and carefully isolating the precise hypotheses need to effect them, and then recording them as new sets of axioms which could be considered mathematical inventions. So for example, the precise conditions on the category of sets which enable the construction of a bicategory of relations is revealed, after some analysis, to be axioms as ensconced in the notion of regular category (the newly minted concept). Do we say that we ‘discovered’ regular categories? I guess you could say the analysis of relevant conditions and consequences constitutes an act of logical ‘discovery’ of some kind, but this kind of situation doesn’t seem to me packed with the same kind of emotional Platonism that I think Manin is talking about, even when the feeling of ‘rightness’ or ‘naturality’ of the conditions is strong. In other studies such as categories of motives, where the connection with logical analysis may be much more groping or tentative, perhaps the feeling of gradual discovery would be keener.

Posted by: Todd Trimble on November 8, 2009 9:26 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I don’t myself have a firm philosophical stance on the matter, but I don’t quite see what’s so inventive about regular categories as opposed to say, groups.

We can go a bit further and consider something like ‘chordates,’ which Wikipedia defines to be ‘animals which are either vertebrates or one of several closely related invertebrates.’ Given the slippery nature of classification, one can go on and on about the extent to which chordates form a natural category, and perhaps some people do. Such arguments might even be important in theoretical biology. However, no one doubts that an individual chordate like me exists (I hope). And one might argue that the extension of the concept is real enough, regardless of the its conceptual value. So for a mathematical Platonists, mathematical structures are as real as the animal kingdom, even with the usual arguments about the naturality of any particular named phylum.

Posted by: Minhyong Kim on November 8, 2009 11:07 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

While these are good points, my comments were less to do with any kind of coherent philosophizing than with descriptions of situations where one might feel inclined, on quasi-emotional grounds, to apply the words “invented” or “discovered”. I think for each of us, there could be occasions when one seems more fitting than the other, and others where it seems less.

As for regular categories vs. groups: I don’t claim any coherent conception as to what might separate the two. On quasi-emotional grounds, I felt I could make the rough point more effectively with regular categories, since the notion seems (I guess!) more the result of a logical conceptual analysis rather than the result of a revelation, say of the mathematical essence of Symmetry or something like that. In either case I’m oversimplifying matters greatly – please don’t take me too literally here.

Posted by: Todd Trimble on November 9, 2009 12:00 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I wonder if those who codified the definition of a group felt as you do towards regular categories. And if they did feel there was something inventive about their work, this might be contrasted with the hunt for a particular group, such as the monster simple group, which feels much more like an act of discovery. Similarly, given all that work on the foundations of algebraic topology, hunting for Eilenberg-Mac Lane spaces would feel like discovery.

But then, given all that has happened since the nineteenth century, don’t we feel inclined to read back that work on defining groups as a kind of discovery? Invention suggests to me a greater freedom; we might have illuminated our houses by devices other than light bulbs. Given the stage of advance of contemporary mathematics, it’s hard to imagine that we might have bypassed the group concept. I suppose we might have moved straight to the groupoid concept, and then have taken as a special case connected groupoids.

I have an article coming out in the next issue of the European Mathematical Society Newsletter on this theme. This continues a series by different authors in the June 07, June 08, and December 08 editions. The second of these is by Barry Mazur.

Posted by: David Corfield on November 9, 2009 10:32 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I thought it was well-known that groups were resisted in many ways as unnatural or unnecessary, as in the reference to Gruppenpest and all that.

To my mind, the reasonable statement of mathematical Platonism is

*The mathematical universe is as real as the physical universe.*

This makes it clear that any particular category within either realm can be argued about in entirely analogous fashion, as when one asks if elementary particles are real or if the category ‘cat’ is real. And then, to recapitulate an earlier point, does anyone know if the phylum ‘chordate’ is invented or discovered?

There is an interview somewhere with Chomsky where he dismisses the question of whether the mental structures he uses are real. If I recall, he makes two points:

(1) He’s mostly interested in their utility.

(2) They’re about as real as many entities in other natural sciences.

There, I guess he was mostly expressing unconcern over the non-uniqueness of theoretical structures explaining observed phenomena. The criticism had been brought up by Quine and Putnam.

Posted by: Minhyong Kim on November 9, 2009 11:00 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I thought it was well-known that groups were resisted in many ways as unnatural or unnecessary, as in the reference to Gruppenpest and all that.

Yes, but what does this show? The realist about groups says that those resisting were wrong, and have been shown to be wrong; the nominalist, that levels of acceptance ebb and flow.

Anyway, we’re coming up against the concept of ‘natural kind’, which I wrote about a while ago. The article probably needs some rewriting now.

Posted by: David Corfield on November 9, 2009 12:23 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

“Yes, but what does this show?”

A careful compilation of examples may show that mathematical existence is discussed in pretty much the same way as the existence of apparently mundane categories.

Posted by: Minhyong Kim on November 9, 2009 1:24 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Minhyong wrote:

To my mind, the reasonable statement of mathematical Platonism is

*The mathematical universe is as real as the physical universe.*

I guess there are several possible readings of that, but it sounds to me like a soft-core version of Platonism. To give a different kind of example: governments are quite ‘real’ as well. That’s not the same as asserting some immutable Platonic Ideal of ‘government’ which existed before any humans were around. It’s a question of modes of existence; a hard-core Platonist would maintain that such things as infinite sets really exist independent of life forms which contemplate them.

David wrote:

I wonder if those who codified the definition of a group felt as you do towards regular categories. And if they did feel there was something inventive about their work, this might be contrasted with the hunt for a particular group, such as the monster simple group, which feels much more like an act of discovery.

My own sense, in case it wasn’t already clear, is that there probably was some logical conceptual analysis prior to anyone’s writing down the group axioms. (Was it Cayley who first wrote them down?) Before that time one knew of permutation group structures as in Galois theory, but it took some stripping away of concrete particulars – the elements of groups don’t need to act on roots of equations or anything specific – and hence some analysis of what is left that is essential. I suppose that under another analysis, we might have been left with something like a torsor, where one posits a ternary operation which in the language of groups would be (x,y,z)xy 1z(x, y, z) \mapsto x y^{-1} z.

Posted by: Todd Trimble on November 9, 2009 2:07 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I don’t quite see what’s soft-core about that position. To reiterate: any particular category within either realm can be argued about in entirely analogous fashion. Thus, the same person might feel that groups and atoms are ‘really real,’ while feeling not quite so stubborn about the 2-category of categories or nation-states. Meanwhile, not even the most single-minded Platonist would maintain that every single definition is endowed with equal degrees of reality (or ‘naturality’).

Posted by: Minhyong Kim on November 9, 2009 6:02 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I meant ‘soft-core’ in the sense that even I, as someone fairly opposed to Platonist points of view, can find something to agree with there. :-) Mathematics is definitely real to me, at least in the sense that mathematical statements are not merely subjective – we can agree on them with extremely high certainty. And not only that: mathematics is incredibly vivid to me as well; it feels very real. (So I too have emotional-Platonist moments, as I was saying was true of many others who don’t consider themselves Platonists.)

To reiterate myself: I understood more hard-core Platonism as asserting that mathematics was something existing independently of comprehending intelligences, which to me is a wholly different proposition.

The idea of degrees of reality in mathematics, even if it is at the level of hard-to-explain feelings, would be something very interesting to explore more deeply. (I’m not saying I disagree with it at all – I just think it would make an interesting philosophic study. Maybe this should be part of the Math Overflow discussion as well.) I found it additionally interesting that you quickly followed it with [degrees of] ‘naturality’ – is there a close kinship in your mind?

Here are some follow-up questions whose answers I’d be interested in: would the most real or most natural thing for you be the (ahem) natural numbers (as in, “God made the integers, …”)? If so, does the long thread about the natural numbers depending on a background set theory affect such belief in any way?

Or would finite structures be more real than infinite structures? If so, would the number 2 be more real than an exponential tower of 2’s 100 stories high?

As for the 2-category of categories not seeming so real as groups: hmmph! :-) No, seriously, is that roughly your own feeling? That would be interesting to me, and if it’s so, I’d ask: is it because it seems more nebulous or more difficult to comprehend? (I’ll assume you didn’t mean less real because a naive treatment is subject to ‘paradoxes’.)

Posted by: Todd Trimble on November 9, 2009 8:07 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

“I meant ‘soft-core’ in the sense that even I, as someone fairly opposed to Platonist points of view, can find something to agree with there.”

I’m glad! My intention was exactly to come up with a formulation that would capture many things that a Platonist would desire, but sensible enough for anti-Platonists not to find threatening.

In particular, if we stick to the duality of discovery vs. invention, even a strong Platonist would acknowledge that some things in the mathematical universe are closer to inventions, in the same way that ‘the Soviet Union’ might be less inevitable than ‘cat,’ although both are (or were) real enough. The word ‘natural’ I intended to be somewhat more ambiguous (and hence, softer) than ‘real,’ to which it seems to be equivalent for the purposes of scientific discourse.

I’ll think a bit about your specific questions, even though I’m unlikely to have decent answers.

Posted by: Minhyong Kim on November 9, 2009 9:53 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Todd wrote:

That’s not the same as asserting some immutable Platonic Ideal of ‘government’ which existed before any humans were around.

I don’t like the word ‘before’ here, which seems to indicate temporal priority. Plato believed that Forms have a certain sort of priority over their manifestations, but what sort of priority? I think it has to be more of an ontological priority than a temporal priority. After all, to make any sense at all, Forms can’t be things that exist in spacetime, like rocks or people.

(If ‘ontological priority’ sounds too mysterious to some folks out there, let me talk less fancy: the idea is that some things can’t exist unless others do.)

It’s a question of modes of existence; a hard-core Platonist would maintain that such things as infinite sets really exist independent of life forms which contemplate them.

I like this better except for the example of ‘infinite sets’. I suppose Plato counts as a ‘hard-core Platonist’ if anyone does — but I suspect he considered the infinite or apeiron to be too murky and indefinite to count as a Form. A Form — or ‘idea’, which comes from a Greek root related to ‘see’ — was supposed to be clearly and vividly intelligible. The infinite is quite different.

There’s some quite subtle discussion of the finite and infinite in Plato’s Philebus, but it’s so alien that it takes a real effort to get a feel for what he’s trying to say. I don’t dare try to summarize it! But, I think that almost anything Plato wrote is more interesting than the straw-man ‘Platonism’ that many modern philosophers of mathematics like to criticize.

Posted by: John Baez on November 10, 2009 7:02 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

John wrote

I think that almost anything Plato wrote is more interesting than the straw-man ‘Platonism’ that many modern philosophers of mathematics like to criticize.

I think it’s got to the point where it is acknowledged that ‘Platonism’ in the philosophy of mathematics is not held to be a doctrine attributable to Plato. Consider it rather as a position framed in the twentieth century, articulated early on by Paul Bernays in Platonism in mathematics, and carried on by Gödel and others (see the Stanford Encyclopedia entry).

If you want some scholarly research on Plato’s philosophy of mathematics, read

  • McLarty C. (2005), ”Mathematical Platonism’ Versus Gathering the Dead: What Socrates teaches Glaucon’, Philosophia Mathematica 13(2): 115-134. (pdf)
  • Miles Burnyeat, ‘Plato on Why Mathematics is Good for the Soul’, here.
Posted by: David Corfield on November 10, 2009 9:35 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Paul Bernays in Platonism in mathematics

Bad link; try

Paul Bernays in Platonism in mathematics

Posted by: Toby Bartels on November 10, 2009 5:20 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Posted by: Eugene Lerman on November 10, 2009 7:52 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

John took me to task, saying

Todd wrote:

That’s not the same as asserting some immutable Platonic Ideal of ‘government’ which existed before any humans were around.

I don’t like the word ‘before’ here, which seems to indicate temporal priority.

Yes, you are quite right. The point I was trying to make is that even though things like “mathematics” and “government” are abstract things, there is still no such thing as government abstracted away from an actual human context (and no such things as mathematics abstracted away from human contexts, little green things aside). Is that a better way to put it? It’s a confession of faith (or non-faith) on my part: I don’t believe there is such a thing as, e.g., a Universal Government; governments are invariably founded upon live social interactions.

It’s a question of modes of existence; a hard-core Platonist would maintain that such things as infinite sets really exist independent of life forms which contemplate them.

I like this better except for the example of ‘infinite sets’. I suppose Plato counts as a ‘hard-core Platonist’ if anyone does — but I suspect he considered the infinite or apeiron to be too murky and indefinite to count as a Form. A Form — or ‘idea’, which comes from a Greek root related to ‘see’ — was supposed to be clearly and vividly intelligible. The infinite is quite different.

David has already made the basic point here, and you are both correct. I was applying the term informally, referring to a commonly held modern mathematical philosophy (or again ‘article of faith’ might be a better term). I wasn’t attempting to be exact in any scholarly sense.

Still, it’s good to keep in mind what Plato actually said! Maybe next time I won’t use the upper-case P.

Posted by: Todd Trimble on November 10, 2009 12:01 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Todd wrote:

The point I was trying to make is that even though things like “mathematics” and “government” are abstract things, there is still no such thing as government abstracted away from an actual human context (and no such things as mathematics abstracted away from human contexts, little green things aside). Is that a better way to put it?

Yes, that’s a better way to put the anti-Platonist position. I’m a Platonist myself, so I still don’t actually agree with you. But that’s okay.

As you can probably tell, I’ve got a few bees in my bonnet. One is that I think it’s better for anti-Platonists to confront Plato, or at least actual specific self-proclaimed Platonists, instead of beating a straw man riding a dead horse.

Another is that combining Platonism with a materialistic concept of existence creates a parody of Platonism that’s much too easy to refute. In this sort of Platonism, the Forms are said to exist chronologically ‘before’ people came along — presumably locked up in a glass display case in a museum on Mount Olympus?

Luckily the Wikipedia gets it right:

A Form is aspatial (outside the world) and atemporal (outside time). Atemporal means that it does not exist within any time period. It did not start, there is no duration in time, and it will not end. It is neither eternal in the sense of existing forever or mortal, of limited duration. It exists outside time altogether. Forms are aspatial in that they have no spatial dimensions, and thus no orientation in space, nor do they even (like the point) have a location. They are non-physical, but they are not in the mind. Forms are extra-mental.

Of course to a materialist or subjectivist concept of existence none of this makes any sense — but fine, that’s where the argument should start.

David wrote:

I think it’s got to the point where it is acknowledged that ‘Platonism’ in the philosophy of mathematics is not held to be a doctrine attributable to Plato. Consider it rather as a position framed in the twentieth century, articulated early on by Paul Bernays in Platonism in mathematics, and carried on by Gödel and others (see the Stanford Encyclopedia entry).

Okay — if these guys called themselves ‘Platonists’ and someone wants to read what they wrote and argue against it, that’s fine with me. I hope — and suspect — that their brand of Platonism is less silly than the brand most critics of ‘Platonism’ seem content to argue against. But the brand of Platonism I find most interesting is that espoused by Plato. It’s very subtle, it changed over the course of his thinking, and at its best it contained quite a lot of mysticism and humor.

Posted by: John Baez on November 10, 2009 7:15 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I’m a Platonist myself, so I still don’t actually agree with you. But that’s okay.

Of course, most mathematicians subscribe to some form of mathematical realism, at least on an emotional level, platonism (lower-case!) being the most prevalent form. Interestingly, although I have no firm data to back this up, I sense that most category-theorists are not platonists on the rational level, although I know of some who are.

I guess platonism seems to me to be a kind of metaphysics or religion. Gödel for example believed in the reality of the True universe of sets, and for him the continuum hypothesis had an absolute truth value (he believed it was false). What’s your feeling about that? Would you agree with him that there is a True universe of sets, and that CH has an absolute truth value? If not, what would mathematical realism mean for you specifically?

As you can probably tell, I’ve got a few bees in my bonnet.

Yes, I can tell. :-)

One is that I think it’s better for anti-Platonists to confront Plato, or at least actual specific self-proclaimed Platonists, instead of beating a straw man riding a dead horse.

I thought I was in fact addressing commonly held beliefs of those who would consider themselves platonists, but citing specific people is probably good advice.

Unless I’ve badly misunderstood, most of them would indeed hold mathematical truth to be eternal (despite your correction that Plato’s Forms are to be considered beyond space and time and contingency). My counter-position, although I am open to someone trying to convince me otherwise, is that there is no sense to “truth” if there is no witness to it. I don’t think I’m attacking a straw man here; I can even appreciate where such realists are coming from.

Keep in mind that the discussion from people like Manin is expressly not of a scholarly sort. It’s fine if you love Plato and his writings (I am not competent to discuss that myself), but I thought we were really discussing contemporary attitudes, which in itself is a very interesting and sometimes subtle discussion.

Posted by: Todd Trimble on November 10, 2009 10:23 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Todd wrote:

I guess platonism seems to me to be a kind of metaphysics or religion. Gödel for example believed in the reality of the True universe of sets, and for him the continuum hypothesis had an absolute truth value (he believed it was false). What’s your feeling about that? Would you agree with him that there is a True universe of sets, and that CH has an absolute truth value?

No, I don’t there’s just one true universe of sets — I think they’re all ‘true’ to some extent.

A bit more precisely: I think that all patterns exist, in a certain sense of ‘exist’ — but with some ‘shining more brightly’ or ‘occupying a more central location’ than others. And I think our physical universe is one of these patterns: it seems more real to us than others merely because we’re part of it.

If not, what would mathematical realism mean for you specifically?

I don’t really use the term ‘mathematical realism’ very much — certainly not in my heart of hearts. But I’m sympathetic to some philosophers of physics who call themselves ‘structural realists’. There’s something called the ‘pessimistic meta-induction’, namely the observation that every theory of physics so far has turned out to be wrong. And this leads some philosophers of physics to worry about whether theoretical structures that play a role in our current theories should actually be said to ‘exist’. And the structural realists say that even when our theories change, certain structural aspects of these theories are preserved.

And personally — now I’m drifting off into my own views! — I think these ‘structures’ are closely akin to Plato’s ‘Forms’. I think we encounter them both in mathematics and physics. And of course I’m fascinated by how the same structures turn up in both places. This leads me towards a kind of Platonism, or you could even say Pythagoreanism, in which patterns (e.g. mathematical patterns) rather than matter or experiences are the most real things there are.

From the Stanford Encyclopedia of Philosophy:

Scientific realism is the view that we ought to believe in the unobservable entities posited by our most successful scientific theories. It is widely held that the most powerful argument in favour of scientific realism is the no-miracles argument, according to which the success of science would be miraculous if scientific theories were not at least approximately true descriptions of the world. While the underdetermination argument is often cited as giving grounds for scepticism about theories of unobservable entities, arguably the most powerful arguments against scientific realism are based on the history of radical theory change in science. The best-known of these arguments, although not necessarily the most compelling of them, is the notorious pessimistic meta-induction, according to which reflection on the abandonment of theories in the history of science motivates the expectation that our best current scientific theories will themselves be abandoned, and hence that we ought not to assent to them.

[…]

According to Worrall, we should not accept standard scientific realism, which asserts that the nature of the unobservable objects that cause the phenomena we observe is correctly described by our best theories. However, neither should we be antirealists about science. Rather, we should adopt structural realism and epistemically commit ourselves only to the mathematical or structural content of our theories. Since there is (says Worrall) retention of structure across theory change, structural realism both (a) avoids the force of the pessimistic meta-induction (by not committing us to belief in the theory’s description of the furniture of the world) and (b) does not make the success of science (especially the novel predictions of mature physical theories) seem miraculous (by committing us to the claim that the theory’s structure, over and above its empirical content, describes the world).

John wrote:

One is that I think it’s better for anti-Platonists to confront Plato, or at least actual specific self-proclaimed Platonists, instead of beating a straw man riding a dead horse.

Todd wrote:

I thought I was in fact addressing commonly held beliefs of those who would consider themselves platonists, but citing specific people is probably good advice.

Maybe you were addressing commonly held beliefs… but I wonder who holds them! Personally I see people attacking Platonism a lot more often than defending it — so I start wondering who if anyone admits to holding the views being attacked. Even Manin seems to be attacking Platonism while admitting he believes it on an ‘emotional’ level:

I am an emotional Platonist (not a rational one: there are no rational arguments in favor of Platonism).

This make me start wanting to drag poor Plato out of his tomb and see what he thinks. And I like how it shakes people up — sort as if a mummy suddenly came to life.

Back when I was in grad school, I had some friends who enjoyed discussing philosophy. We started by reading through a lot of Heidegger’s works out loud, and discussing them. The later works of Heidegger got us very interested in the Pre-Socratics. Then we went to a class taught by Gadamer, and Gadamer told us that anyone interested in philosophy should start with Chuang-Tzu and the dialogues of Plato. I’d read Chuang-Tzu already, but not Plato. So we read through all the dialogues of Plato out loud, with lots of breaks for discussing what he meant. And it was really great! First of all, it’s fun to read a dialogue as an actual dialogue. It brings out the ironic humor that underlies a lot of what Socrates says. But second, we’d already gotten a weird spin on things from Heidegger, who claimed that later Catholic interpretations of Plato had gotten everything hopelessly wrong, trivializing some ideas that were actually quite deep. Regardless of whether Heidegger’s scholarship was sound (his version of Plato sounds a lot like Heidegger), this opened us up to looking at the dialogs in a more fresh way.

Posted by: John Baez on November 11, 2009 12:13 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

John,

Did you know that Heidegger was a Nazi?

Posted by: Richard on November 11, 2009 3:21 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

There are certainly a LOT of books to be burn if we want to maintain our philosopic traditions and catch up on Mao’s Cultural Revolution, those darn chinese tried to overtake us!

Posted by: J-L Delatre on November 11, 2009 5:35 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Richard wrote:

Did you know that Heidegger was a Nazi?

Yes, of course.

Posted by: John Baez on November 11, 2009 7:30 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

A bit more precisely: I think that all patterns exist, in a certain sense of ‘exist’

What counts as a “pattern” as opposed to just noise?
Or does every possible “blob of noise” exist in the same sense as your patterns?

but with some ‘shining more brightly’ or ‘occupying a more central location’ than others. And I think our physical universe is one of these patterns: it seems more real to us than others merely because we’re part of it.

Isn’t this choosing among patterns? (whatever the reason may be)

Posted by: J-L Delatre on November 11, 2009 5:54 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

J-L Delarte wrote:

What counts as a “pattern” as opposed to just noise? Or does every possible “blob of noise” exist in the same sense as your patterns?

Tricky questions!

I think every blob of noise counts as a pattern, and thus exists as such. But the more a pattern resembles a mere blob of noise, the less ‘central’ it is. In other words, it’s less likely to be connected in ‘interesting’ ways to ‘interesting’ patterns.

You may complain that the concept of ‘centrality’ or ‘interestingness’ is somewhat subjective. That’s probably true. I’m not sure there needs to be a completely ‘objective’ way of rating patterns. It’s not like school, where if you get really bad grades you get kicked out.

Posted by: John Baez on November 11, 2009 8:02 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

But the more a pattern resembles a mere blob of noise, the less ‘central’ it is. In other words, it’s less likely to be connected in ‘interesting’ ways to ‘interesting’ patterns.

Not so sure, if you compress some ‘interesting’ or ‘meaningful’ pattern the better the compression the more the output looks like random noise.
Sounds like the ‘meaning’ of the pattern is complemented (?) by the ‘meaning’ of the compression algorithm.

Posted by: J-L Delatre on November 11, 2009 8:40 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I’m a platonist but not a Platonist, and I subscribe to this:

No, I don’t there’s just one true universe of sets — I think they’re all ‘true’ to some extent.

A bit more precisely: I think that all patterns exist, in a certain sense of ‘exist’

but not this:

but with some ‘shining more brightly’ or ‘occupying a more central location’ than others.

or this:

I think our physical universe is one of these patterns

(Though, while not believing that some shine more brightly, I certainly believe that some are of more personal interest to me than are others.)

Just to reassure you that people like me exist (I’m not sure if you find that reassuring or not …).

We started by reading through a lot of Heidegger […] went to a class taught by Gadamer […] . I’d read Chuang-Tzu already […] read through all the dialogues of Plato […]

I never had the energy to get involved with Continental philosophy, and was always too much of a dilettante to read all of Plato’s dialogues out loud. Besides, the more one knows about a subject, the harder it is to hold strong opinions, and where’s the fun in that?

Posted by: Tim Silverman on November 11, 2009 12:29 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Tim wrote:

but not this:

but with some ‘shining more brightly’ or ‘occupying a more central location’ than others.

or this:

I think our physical universe is one of these patterns

Just to be clear, when I said ‘one of these patterns’ I simply meant to say the universe was a pattern. I wasn’t trying to say it’s one of the ‘central ones’ that ‘shine more brightly’. And I certainly wasn’t saying it seems real to us because of some quality of this sort. Sorry, I wasn’t being very clear!

A good example of what I consider a ‘central’ pattern might be the ring of integers, or the category of finite sets. Meaning: if you start studying patterns and how they’re related to each other, these are among the ones you keep coming back to. Sort of like towns located at major crossroads.

The physical universe might or might not be one of these ‘central’ patterns — I don’t think we understand it well enough yet to say. It certainly doesn’t seem as simple as the ring of integers! But it might turn out to have a simplicity and inevitability of a higher sort. Some of the physical laws we’ve stumbled upon hold out this hope… even though other things we know about physics seem to mock this hope. Physics is constantly tantalizing and frustrating.

But: I think the universe is a pattern, and that this pattern seems to ‘actually exist’ instead of ‘merely be a logical possibility’ simply because we are part of it.

Indeed, I think it makes perfect sense for every pattern to seem ‘actual’ as viewed from the inside — at least as long as the pattern is sufficiently complex to have a ‘view from the inside’ in which concepts like ‘actual’ play a role!

I never had the energy to get involved with Continental philosophy…

I wouldn’t have, either. It just sort of happened. I went to Princeton as an undergraduate, and had a good friend there who was a religion major. At the time, Continental philosophy was sort of banned from the philosophy department — Anglo-Saxon analytic philosophy reigned supreme! But it was allowed in the religion department. In his senior year, after I’d already gone to MIT, he took a course on phenomenology, and sent me letters with juicy Heidegger quotes like “Dasein is essentially an entity with Being-in…”, or “The nothing nothings” — quotes I didn’t understand at all, but which seemed somehow tantalizing.

I was then surprised to find that at MIT the mathematician Gian-Carlo Rota taught a regular course on phenomenology — in the mathematics department, since again the philosophy department wouldn’t countenance it. So, I signed up for it. And there I met a group of people who liked to read and discuss Heidegger every Friday night at a café called Au Bon Pain in Harvard Square. And since I was often lonely and bored, this sounded like fun. And it was. We wound up reading all of Being and Time out loud, and then various other works (it was around this time that a lot of Heidegger’s later course notes came out). And then the dialogues of Plato, which were downright easy and entertaining by comparison!

… and was always too much of a dilettante to read all of Plato’s dialogues out loud. Besides, the more one knows about a subject, the harder it is to hold strong opinions, and where’s the fun in that?

Well, it’s actually a lot of fun to read Plato out loud with friends, especially the dialogues that are really dialogues, instead of monologues like the Republic. One reason is that the Socrates character is constantly needling and teasing the others…

And we didn’t mainly discuss our opinons about philosophy, though there was plenty of that too. Our goal was mainly to figure out what the text meant, in a more or less cooperative spirit. While sipping coffee and eating cake with friends, this can be quite enjoyable.

Posted by: John Baez on November 12, 2009 1:41 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

“Our goal was mainly to figure out what the text meant” You read it in greek? My impression from the little greek I learned is that that language itself induces one to some ‘platonism’.

Posted by: Thomas on November 12, 2009 8:13 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Thomas Riepe wrote:

You read it in Greek?

No, alas, in English — but with a certain amount of care about how Greek philosophical terms tend to get imbued with later Latin meanings when translated into English.

It’s not me, but my wife Lisa, who really knows classical Greek and Chinese and studies their effects on the thinking of these two cultures!

Posted by: John Baez on November 12, 2009 7:59 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I was rather terse, and not very clear either. I should clarify a couple of things.

some ‘shining more brightly’ or ‘occupying a more central location’ than others

I understood what you meant. It’s just that I feel that ‘occupying a more central location’ is a rather subjective judgement. I admit it’s not impossible that there might be some objective way in which some things are more central, but my natural reaction is that the appearance of centrality is a subjective consequence of the fact that we think in a certain way. Given the universe we live in, I don’t think our way of thinking is likely to be unique or accidental, but across the entire platonic realm of all kinds of pattern, I feel very reluctant to assume it has some kind of special status. I guess I’m inclined to a sort of platonic copernicanism!

I think our physical universe is one of these patterns

I understood what you meant here, too: this is an idea that has occurred to me independently. But, I don’t think I agree with it. I readily admit that the universe has a pattern; I’m much more doubtful that it is a pattern. That’s possible, but it doesn’t feel right. However, this goes along with a number of other related philosophical problems (the problem of subjective awareness (i.e. “consciousness” or the mind-body problem), the subjective perception of the passage of time, the subjective perception of occupying a particular branch of the “many-worlds” basis of the state vector … ) where I feel there’s “something extra”, but of course I can’t demonstrate this, even in principle, so I might be wrong. But the countervailing view can’t be demonstrated even in principle, either, so I might be right. I’ve given up thinking about this: there doesn’t seem to be any possibility of further progress.

Well, it’s actually a lot of fun to read Plato out loud with friends

I did spend a lot of time philosophising with friends in my teens and again in my mid-twenties, but not really at university, where I was a bit isolated. However, I don’t think I’m very good at either absorbing ideas or expressing them in this sort of environment, so I tend to feel frustrated after a while, and want to go away and write things down which say what I really meant. Of course, I have learned some very interesting things in conversation, but more rarely than I would like. Usenet and blogs were a great boon to me in that way, since I absorb information a great deal better visually, even though it’s less fun than laughing together over a pint or two in the pub.

I quickly get bored with the interpretation of texts. The ratio of insights to pilpul is generally too low …

However, my main problem is that there are too many things to learn, and not enough time to learn them!

Posted by: Tim Silverman on November 12, 2009 1:29 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Tim wrote:

It’s just that I feel that ‘occupying a more central location’ is a rather subjective judgement. I admit it’s not impossible that there might be some objective way in which some things are more central, but my natural reaction is that the appearance of centrality is a subjective consequence of the fact that we think in a certain way. Given the universe we live in, I don’t think our way of thinking is likely to be unique or accidental, but across the entire platonic realm of all kinds of pattern, I feel very reluctant to assume it has some kind of special status. I guess I’m inclined to a sort of platonic copernicanism!

I’m fascinated by this issue, and I find my opinion vacillates on it. Today I believe something like this: there’s certainly some subjectivity involved in our opinion about what patterns — in math, say — are more centrally located than others. It’s not wholly subjective, but nor will it ever become fully objective in some quantifiably precise way — except using methods that involve subjective choices!

Right now I’m imagining an analogy like this: take a person and ask which of their blood vessels are more important than others. If a tiny capillary is clogged, little harm will be done; if the aorta is blocked they will quickly die. So, it doesn’t seem like a wholly subjective question. On the other hand, the body doesn’t have built-in markers labelling the blood vessels according to their level of importance! And there are cases where we could argue endlessly, or shrug our shoulders, about which blood vessel is more important than which.

I’m imagining the fabric of mathematics as vaguely analogous to this network of blood vessels.

But of course the analogy is imprecise. For one, it’s not so easy to remove a theorem and see if mathematics ‘dies’! For two, lots of people are doing lots of different things with mathematics, and somebody’s important theorem is someone else’s crummy little corollary.

But there still seems to be some level of agreement about, say, the number 24 being more interesting than the number 23.

(I deliberately picked a rather shocking example here, instead of 3.1415926535… versus 2.0304815424…, where I could probably get lots more people to agree.)

I readily admit that the universe has a pattern; I’m much more doubtful that it is a pattern.

Indeed, in saying that the universe is a pattern, I’m completely abandoning Platonism in favor of something more like Pythagoreanism — “the universe is number”. So, as far as standing up for Platonism goes, I’m doing a terrible job! It’s just my way of dodging the big question about the relation between a thing and its form. I don’t claim this is terribly well thought-out…

However, this goes along with a number of other related philosophical problems (the problem of subjective awareness (i.e. “consciousness” or the mind-body problem), the subjective perception of the passage of time, the subjective perception of occupying a particular branch of the “many-worlds” basis of the state vector… ) where I feel there’s “something extra”, but of course I can’t demonstrate this, even in principle, so I might be wrong.

Yes, all these problems are closely related. What makes me feel like me, what makes now feel like now, what makes this state of affairs feel real to us… I’ve concluded that the simplest solution is that every pattern feels real to itself: there’s no “something extra” required to make it really real.

I’ve given up thinking about this: there doesn’t seem to be any possibility of further progress.

Well, certainly the progress will be much slower than in, say, nn-categories or even quantum gravity — so I spend a lot less time thinking about this stuff… but still not none.

There are academic physicists who actually dare to write papers about these issues. Max Tegmark, who began by studying supernova-driven winds, now writes about things like the mathematical universe. I’m not sure if he’s making ‘progress’, but it’s certainly good that people can read his papers and think about them.

Why is it good? I’m not sure, but it’s certainly more fun to have encountered these ideas than to have never encountered them!

Posted by: John Baez on November 12, 2009 8:41 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

John said:

Yes, all these problems are closely related. What makes me feel like me, what makes now feel like now, what makes this state of affairs feel real to us… I’ve concluded that the simplest solution is that every pattern feels real to itself: there’s no “something extra” required to make it really real.

That’s certainly the simplest solution. The question is whether it’s the right solution. :-)

I said:

I’ve given up thinking about this: there doesn’t seem to be any possibility of further progress.

And John replied:

Well, certainly the progress will be much slower than in, say, nn-categories or even quantum gravity

Then, a propos of Tegmark’s papers on this sort of thing:

I’m not sure if he’s making ‘progress’, but it’s certainly good that people can read his papers and think about them.

Why is it good? I’m not sure, but it’s certainly more fun to have encountered these ideas than to have never encountered them!

I think it’s good to get to the point where we understand clearly what the issues are and what the various possible solutions look like. But at that point, I don’t think there’s much more we can do that each point at our preferred solution and say “That one! I think that one’s right!”

I used to think I could persuade people to my point of view, but experience has shown that’s unlikely to happen. I still have some hope that I can at least explain my point of view and understand other people’s, but even that is looking doubtful. But I do at least think there is some hope of progress there, and it is very interesting to get to understand how other people think.

Actually, I guess the most rewarding outcome I can get from these discussions is when I discover that a position that I see as obviously completely wrong has been constructed to deal with a very interesting problem that I hadn’t noticed or appreciated the importance of. Even if the position still seems like nonsense, I can learn a lot from discovering what makes it seem appealing to someone.

Also, it’s nice to talk to people who have actually thought hard about these issues. It’s not too hard to get people to express their personal metaphysics in the course of an argument, but usually that metaphysics is rather naively defended and historically uninformed, repeating elementary errors that were analysed to death centuries ago. (Well, I guess that doesn’t happen only in philosophy.)

Also, it’s nice to occasionally run into people who think more-or-less the same way I do. :-)

Posted by: Tim Silverman on November 12, 2009 11:07 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

No, I don’t there’s just one true universe of sets — I think they’re all ‘true’ to some extent.

Interesting. At least in discussions of foundations, it seems to me that “Platonism” usually refers to the position that there is one true universe of sets, though of course this may have nothing to do with Plato. For instance, this is from Kenneth Kunen’s “Set Theory:”

A Platonist believes that the set-theoretic universe has an existence outside of ourselves, so that CH is in fact either true or false (though at present we do not know which). From this point of view, the axioms of ZFC are merely certain obviously true set-theoretic principles. The fact that these axioms neither prove nor refute CH says nothing about its truth or falsity and does not preclude the possibility of our eventually being able to decide CH using some other obviously true principles which we forgot to list in ZFC.

Here is Kurt Gödel:

For if the meanings of the primitive terms of set theory… are accepted as sound, it follows that the set-theoretical concepts and theorems describe some well-determined reality, in which Cantor’s conjecture must be either true or false. Hence its undecidability from the axioms being assumed today can only mean that these axioms do not contain a complete description of that reality.

It is this view often ascribed to “Platonists,” in whose ranks Gödel must be included, that I have trouble seeing any justification for. Your position that all universes of sets are “true” to some extent avoids the obvious problems with this, but I have to wonder whether you haven’t just shifted the problem to the metatheory. When you say “they’re all ‘true’ to some extent,” in what sense do you mean “all”? Is there one true universe of universes of sets? Or are all possible universes of universes of sets true to some extent? Is there one true universe of universes of universes of sets?

Posted by: Mike Shulman on November 13, 2009 1:20 AM | Permalink | PGP Sig | Reply to this

Re: The Wand Chooses The Wizard

Mike wrote:

When you say “they’re all ‘true’ to some extent,” in what sense do you mean “all”?

I mean “all” in the strongest possible sense of “all”. Whatever pattern we can come up with, together with lots that are too complicated for us to comprehend — they all exist.

Is there one true universe of universes of sets?

Not just one…

Or are all possible universes of universes of sets true to some extent?

That’s more like it.

Is there one true universe of universes of universes of sets?

Not just one…

Posted by: John Baez on November 13, 2009 1:34 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Is there one true universe of universes of universes of sets?

Not just one…

Okay, that’s what I thought you’d say. So there are lots of universes of universes of … universes of sets. Presumably each universen^n is contained in a universen+1^{n+1}, so we have a tree of all levels of universes. This tree of universes is sort of a universeω^\omega. But presumably there isn’t just one true tree of universes—there are probably lots of them. So we have a universeω+1^{\omega+1} of universeω^\omegas, or rather lots of them. In fact, we should have universeα^\alphas for all ordinals α\alpha. But different universes of sets have different collections of ordinals. It seems like we should have one class of “universeα^\alphas for all α\alpha” for each universe of sets—but now we’ve just gotten started all over again.

Maybe you don’t have a problem with this, but it feels fishy to me. (-:

Posted by: Mike Shulman on November 13, 2009 3:09 AM | Permalink | PGP Sig | Reply to this

Re: The Wand Chooses The Wizard

If you can make up rules for consistently reasoning about something, I’ll grant it existence. But I sure wish you’d make up stuff that’s more useful than these endless bloody ‘universes of universes’. To me these are on par with Minhyong’s mathematical hairball.

Posted by: John Baez on November 13, 2009 3:57 AM | Permalink | Reply to this

multi

every theory of physics so far has turned out to be wrong.

In a discussion that spends so many careful words on the nature of reality this statement surprises me.

I don’t [think] there’s just one true universe of sets — I think they’re all ‘true’ to some extent.

A multiverse of sets, almost.

Since text-mode jokes are a dangorous thing, let me emphasize that this is meant jokingly. By the way of jokes: funny is the standout box leading this Wikipedia entry, which complains that the article doesn’t give enough citations for verifications. Also cute is the colorful figure that illustrates the notion of “bubble universes”.

But then, apart from their way of going about it, the people whose ideas are summarized on this entry are having a conversation not entirely unrelated to the one you are having here.

Posted by: Urs Schreiber on November 11, 2009 8:41 AM | Permalink | Reply to this

Re: multi

John wrote:

There’s something called the ‘pessimistic meta-induction’, namely the observation that every theory of physics so far has turned out to be wrong.

Urs wrote:

In a discussion that spends so many careful words on the nature of reality this statement surprises me.

That was just a lightning summary of the pessimistic meta-induction. Of course most physicists know it’s a bit more accurate to say “every theory except the latest best one has been superseded by the latest best one, and is thus ‘wrong in some ways’ — but many of these old theories are still right in some useful ways”. The philosophers are engaged in working out the details. A better summary of the debate is here.

Posted by: John Baez on November 12, 2009 3:09 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Todd said:

I guess platonism seems to me to be a kind of metaphysics or religion

Belief in the existence of the material universe is also a kind of metaphysics or religion. (Specifically: a kind of metaphysics; I don’t see what religion has to do with it. It’s not like I expect the natural number system to demand sacrifices or answer prayers).

Also:

My counter-position […] is that there is no sense to “truth” if there is no witness to it.

From my point of view, objections to the existence of the things witnessed, if valid, would equally be objections to the existence of the witnesses. Even more, in what sense could a witness be a witness (even assuming the witness existed at all) if they are not witnessing something that exists?

Posted by: Tim Silverman on November 11, 2009 11:26 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Belief in the existence of the material universe is also a kind of metaphysics or religion.

I don’t deny that for a moment. I also don’t affirm that I am a materialist.

(Specifically: a kind of metaphysics; I don’t see what religion has to do with it. It’s not like I expect the natural number system to demand sacrifices or answer prayers).

Taking religion in a broad sense, obviously, and not relegated to belief in a personal god directly involved in human affairs. I am not particularly well-read in philosophy, but think the God of Spinoza or of Einstein. Nor is religion limited to theistic beliefs (think Mahayana Buddhism in its more philosophical incarnations). For the purposes of present discussion, think of “belief in something Transcendent”.

From my point of view, objections to the existence of the things witnessed, if valid, would equally be objections to the existence of the witnesses. Even more, in what sense could a witness be a witness (even assuming the witness existed at all) if they are not witnessing something that exists?

“Existence” is an awfully slippery word. In mathematics we use the word in a formal sense (e.g., adjoint to pulling back); in agreeing to work with a particular axiomatic system which asserts, for instance, the existence of an infinite set, I don’t for a moment feel compelled to believe in an actual infinite set. Or in reading a work of fiction, I may sense the existence of a lurking danger. I’m maybe not sure what you’re saying, but it feels like there could be differing senses or modalities of “existence” being played with here.

I guess what I was saying is that I don’t really know what it means to say that, e.g., Fermat’s last theorem was “true” in prehistoric times. True for whom? An imaginary intelligent spectator?

(I am thinking of the word “truth” here as something applied to assertions or sentences, which are parts of a linguistic framework.)

Posted by: Todd Trimble on November 11, 2009 1:55 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Belief in the existence of the material universe is also a kind of metaphysics or religion.

I don’t deny that for a moment. I also don’t affirm that I am a materialist.

Now I’m curious and a little taken aback. It sounded like you thought “metaphysics” was a bad thing. Do you believe anything exists? (Witnesses, presumably ….) All those foundational beliefs are metaphysics. That makes them impossible to “establish” on firmer grounds but doesn’t make them false or unreasonable—difficult if not impossible to convey, maybe, but that’s different.

Taking religion in a broad sense, obviously, and not relegated to belief in a personal god directly involved in human affairs. I am not particularly well-read in philosophy, but think the God of Spinoza or of Einstein.

I would emphasize here that I don’t believe in any sort of god or God, except that you say:

Nor is religion limited to theistic beliefs (think Mahayana Buddhism in its more philosophical incarnations). For the purposes of present discussion, think of “belief in something Transcendent”.

I’m not entirely sure what you mean by “Transcendent” (or whether it’s different from “transcendent”), but I don’t think I believe in any kind of Buddhism or anything similar either. I’ll readily admit that a lot of different attitudes and beliefs can loosely be counted as “religious”, but the belief that (roughly) patterns are in some sense independent of, and in some sense prior to, their instantiations, does not strike me as religious in any sense, per se.

“Existence” is an awfully slippery word.

That is indeed a crucial point. “Existence” is one of those words (“real” and “God” are others) that people seem to use in an extraordinarily large number of very different ways (sometimes I wonder if everyone has their own private definition, virtually unrelated to anybody else’s). It’s always very difficult in these discussions to penetrate the fog of words to work out what people actually believe underneath, or indeed to convey what you believe to other people.

I don’t really know what it means to say that, e.g., Fermat’s last theorem was “true” in prehistoric times. True for whom?

Whereas I don’t really know what it means to say “True for whom?” or, generally, “True for X.” I might consider truth relative to a situation or context, but to make it relative to a person makes no sense to me. Unless it means, “Believed by X.” But then we have the more complicated task of understanding what the content of X’s alleged belief is and, separately, understanding X’s state of mind or attitude towards that content.

That is, if nobody has heard of Fermat’s Last Theorem, what is the thing that they haven’t heard of, and what is the sense in which it would be correct or incorrect for them to believe it? Just as, e.g., <doubleness> as a concept (and indeed <apple> as a concept) seems separate from having two particular apples sitting in front of me on the table, so the meaning or meanings of Fermat’s Last Theorem seems independent of any particular people who might be aware of it.

I’m not sure if any of this will make much sense to you.

Posted by: Tim Silverman on November 11, 2009 3:00 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Hi Tim –

I’ll just respond to some of this now. I’m sure other parts will need more time.

Can I just say that if I sounded dogmatic (not at all sure that I did), then please know these are live things I grapple with myself and I’m never too sure where I stand exactly.

Now I’m curious and a little taken aback. It sounded like you thought “metaphysics” was a bad thing.

No, no, no, no, no! I’m not saying that at all. It’s more along the lines of Laplace saying to Napoleon, “Sir, I had no need of that hypothesis.” Laplace may or may not have had religious convictions, but what they were particularly needn’t play a role in his natural philosophy.

I actually thought we were talking about platonism. Are we talking about something else now?

Do you believe anything exists?

Uhhmmm….

Can I assert my rights under the Fifth Amendment?

Sorry, the question seems very odd, and I don’t know what you’re getting at. I do wish to keep any religious beliefs such as I may have to myself. I guess I’ll allow that there are aspects of materialist philosophies that I have problems with, but perhaps that’s a discussion for another time.

Well, here’s one belief which I guess I don’t mind sharing: everything changes. So the person I am now is very different from the person I was years ago. For example, I would have considered myself a platonist once. And I may yet decide one day in the future that I am a platonist again (I don’t think that’s likely, but it’s in the realm of possibility I suppose). As I say, I’m not exactly sure where I stand on a lot of philosophic issues. I’m thinking aloud here.

I’m not entirely sure what you mean by “Transcendent” (or whether it’s different from “transcendent”)

As I understand it, Plato’s Forms would be an example of what I mean. Mathematical patterns from the point of view of modern-day mathematical platonists might be another. The idea being: that which is beyond space, time, and contingency. Not belonging to the sphere of birth and death. That sort of thing.

If you and I don’t agree on the definition of “religious”, then that’s okay – I don’t think it matters greatly for this discussion, and we might as well drop it. Probably we can agree more closely on “metaphysical”. But just speaking of people like Plato and Plotinus, it’s hard for me to agree with the sentiment that their philosophy didn’t have a strongly religious coloration (cf. Plotinus’ “the One, Intellect, and Soul”). And I personally see modern-day platonism, what I make of it anyway, as not too far removed along the continuum of beliefs. But maybe that’s just me.

The rest of your comment I’d prefer not to just dash off a reply to. It’s all good though.

Posted by: Todd Trimble on November 11, 2009 5:18 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Hi Todd,

Thanks for your reply! It was actually very helpful. I’ll just make a few comments.

It sounded like you thought “metaphysics” was a bad thing.

No, no, no, no, no!

Ah, that’s helpful. Usually when I get into arguments or discussions where words like “exist” and “real” (or “really exist”!) get bandied around, then “metaphysics” is used in a dismissive sense inherited, I think from the logical positivists, meaning, more-or-less, “empty, meaningless, dispensible”. (In fact, I do think there is a strongly dismissive strain in Laplace’s “Sir, I had no need of that hypothesis” that you quote, although perhaps you do not agree with me.)

So against that, I was observing that however little use one might have for “metaphysical” hypotheses in the course of an ordinary scientific or mathematical activity, they are unavoidable at the ultimate foundation of any intellectual endeavour, even if they only take the simple form of “The senses are a fairly reliable guide to (some aspects of) material reality” (or “The senses are highly inferior to reason”/”The senses are instruments of Satan” or whatever).

But it seems you understand this perfectly well already; I apologise for not seeing this before. I was probably biased by past experience.

I actually thought we were talking about platonism. Are we talking about something else now?

No, I also think we’re talking about platonism, even if we’re sometimes talking past each other.

Do you believe anything exists?

Sorry, the question seems very odd, and I don’t know what you’re getting at.

Yes, I’m sorry about that, it is odd. The reason I asked it is that I was so unclear about your position at that point that I found it difficult to ask a sensible question, so I deliberately asked a somewhat facetious question in the hope of dislodging something. Luckily it seems to have worked, without causing you too much annoyance. Perhaps I should have spent some time constructing a more precise and less provocative question; but that could easily have missed the target.

I do wish to keep any religious beliefs such as I may have to myself. I guess I’ll allow that there are aspects of materialist philosophies that I have problems with, but perhaps that’s a discussion for another time.

Fair enough.

(On “Transcendent”):

The idea being: that which is beyond space, time, and contingency. Not belonging to the sphere of birth and death. That sort of thing.

OK. I sort of have a mental category which sort of lines up with this, and I agree “transcendent” is a good word for this; but I get the impression from your phrasing that our attitudes towards transcendent things are rather different. There probably isn’t much I can say on this in detail, but perhaps I should at least say that, in my view, transcendence, and transcendent things, are really rather ordinary and commonplace; not objects of religious awe or anything like that. I’m not sure if that makes things any clearer for you.

If you and I don’t agree on the definition of “religious”, then that’s okay – I don’t think it matters greatly for this discussion, and we might as well drop it. Probably we can agree more closely on “metaphysical”. But just speaking of people like Plato and Plotinus, it’s hard for me to agree with the sentiment that their philosophy didn’t have a strongly religious coloration (cf. Plotinus’ “the One, Intellect, and Soul”). And I personally see modern-day platonism, what I make of it anyway, as not too far removed along the continuum of beliefs. But maybe that’s just me.

OK, I do have some things to say about this, which I’ll try to get out not too incoherently. I’m not sure how much we agree or disagree on the meaning of the word “religious” as opposed to disagreeing about the religious nature of platonism. I’ll say that while I am very interested in religion as an intellectual phenomenon, and am acquainted with (I think) religious feeling, I don’t really think I have religious beliefs. And in particular, I don’t think my mathematical platonism is religious. It’s metaphysical, but only for the comparatively trivial reason (as I see it) that all foundational beliefs (even anti-foundationalism!) are ipso facto metaphysical.

As to Plato, and the Neoplatonists, etc, yes, I agree, their motivation absolutely is religious, and in Plato’s case, at least, is moral and political and, for that matter, psychological, as well. They are steeped in religion, albeit rather different kinds of religion at different periods.

And for some modern platonists this is no doubt still true. But not for me and hence, presumably, not for some other modern platonists (I hardly imagine I’m unique …). My motivation really doesn’t start with anything religious at all, and the claim that it does kind of feels the way it would if someone said my belief in the Big Bang was motivated by some kind of (non-literalist) Biblical Creationism: it’s sort of annoying and utterly wrong. Well, it’s annoying because it’s utterly wrong. I’m really, really not a Platonist with a capital ‘P’. I’ve read enough genuine Platonists to be quite sure of this.

I’ve seen enough of the emotional affiliations of various participants in this sort of argument that it wouldn’t surprise me if I was quite unusual in this way (although, as I said, it certainly would surprise me if I was unique. Well, it would surprise me just on general principles, I guess.)

Don’t know if this clarifies things at all.

Posted by: Tim Silverman on November 11, 2009 7:53 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Hi again, Tim. No, I wasn’t put out by anything you said, and I understand your initial bafflement at what I was saying. I was keeping my cards close to my chest, as I do sometimes!

Some trivial observations:

Hard to say how dismissive Laplace was being. You could be right, but also he might have said it more nicely than it sounds to us now. My impression is that Laplace was a political smoothie, and could roll with whoever was in power, and I imagine he knew enough not to upset Napoleon too much!

On “religious mathematical platonists”: my impression is that there have been quite a few (in fact, I could hazard a guess that some are listening in now); to what degree someone takes his platonism religiously is perhaps dependent upon personality. I certainly take you at your word that platonism for you doesn’t carry religious connotations; Cantor on the other hand I imagine as someone in thrall to a sort of numinous awe, faced with the immensities of the infinite (even of just the countable ordinals! – one could easily feel a kind of religious terror at how enormous they can be). Maybe I see a platonist outlook as lending itself more easily to that sort of emotion, as opposed to say formalism which seems much more austere. That said, if we agree for now, just for the sake of clearer present discussion, that religion is concerned specifically with the numinous, then I’d happily withdraw from imputing a specifically religious significance to platonism (as opposed to Platonism).

By the way, I doubt that you are at all unusual in this respect. :-)

Posted by: Todd Trimble on November 11, 2009 9:16 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Thanks, Todd. Nothing really to add since I agree with what you say here. I think you may very well be right that Platonism has an affinity with religious emotion or attitudes.

I’ll take your word for it that I’m not unusual :-) I don’t have these discussions often enough to have good data on the sociology of philosophies.

Posted by: Tim Silverman on November 11, 2009 10:33 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

It’s perhaps worth noting that Edward Nelson, who is one of the most resolute finitists around (he thinks PRA is absurdly over-generous, and that exponentiation may not be total), is also a convinced Christian. In his essays, he seems to draw an analogy between something like Brouwer’s idea of the creating subject and God’s creative powers – so to him, Platonism is a rejection of the idea that man was made in God’s likeness.

Which is just to say that the religious impulse can interact with mathematical experience in rather complicated ways.

Posted by: Neel Krishnaswami on November 12, 2009 10:28 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

That’s very interesting, Neel!

Come to think of it, I think Doron Zeilberger (also a religious man) makes a similar point somewhere in his famous Opinions.

Posted by: Todd Trimble on November 12, 2009 2:25 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

In mathematics we use the word [existence] in a formal sense (e.g., adjoint to pulling back); in agreeing to work with a particular axiomatic system which asserts, for instance, the existence of an infinite set, I don’t for a moment feel compelled to believe in an actual infinite set. Or in reading a work of fiction, I may sense the existence of a lurking danger. I’m maybe not sure what you’re saying, but it feels like there could be differing senses or modalities of “existence” being played with here.

This is why I generally tend to stay out of philosophical arguments about the “existence” of mathematical objects (or about the nature of “existence” in general). My (admittedly limited) experience is that philosophers often seem to think the word “exists” should always mean the same thing, when of course (to my mind) the notion of existence depends on the context. Do unicorns exist? They don’t (presumably) exist in the sense that dogs exist, but the idea of a unicorn surely exists, in the same sense that the idea of a dog exists. But the idea of a dog has the property that it has physical realizations (i.e. it “exists” in the other sense of the word), whereas the idea of a unicorn doesn’t have that property. Neglecting the distinction between different meanings of “existence” leads to you absurdities like this.

Posted by: Mike Shulman on November 11, 2009 5:19 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Yes, a uniform conception of existence is disastrous. I like Michael Harris’s point in “Why Mathematics?” You Might Ask that if we want an example of a something that is mathematical, we should consider what is being talked about when X claims that Y stole his idea. Such a claim clearly makes sense. Indeed the worry is expressed here from time to time that ideas on nLab may be used unacknowledged for profit.

Mathematics seems to me to contain a lot of second and higher order thinking. Perhaps we might say that this tower of thinking rests ultimately on patterns of imagined, idealised, potential sensations. Since we are limited by our powers of thinking and by our imagination, we might take our mathematics to be a very human affair.

For me an interesting debate is between those on a spectrum, one end of which takes our mathematics to be the product of a specific primate, evolved on a specific planetary rock, whose intellectual activity has taken a specific erratic path to what we have today, and the other end of which takes our mathematics to have a much greater universality, where details of our particular condition are less important, where we’ve tapped into a space of all possible thought, possible for any intelligence.

Posted by: David Corfield on November 12, 2009 12:17 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

in agreeing to work with a particular axiomatic system which asserts, for instance, the existence of an infinite set, I don’t for a moment feel compelled to believe in an actual infinite set. Or in reading a work of fiction, I may sense the existence of a lurking danger.

I'm glad to see a comparison of mathematical and fictional worlds. I call myself a ‘fictionalist’, because I believe that the world of mathematics is, like the world of Star Wars or Arthurian legend, a fictional world that we consider and discuss because we find it interesting and useful. (Then again, I hold the same belief about reality.)

I guess what I was saying is that I don’t really know what it means to say that, e.g., Fermat’s last theorem was “true” in prehistoric times.

I would certainly say that FLT was true in prehistoric times, but I would mean operationally something that amounts to these examples:

  • Someday we are amazed to discover prehistoric artefacts that make it clear that people were raising natural numbers to powers of natural numbers much earlier than we thought, also summing them and recording equalities, despite apparently not recording natural language. It is reported in the press that one of these artefacts records a counterexample to FLT. I conclude that there is an error (possibly in the press report, possibly in the interpretation, possibly in the artefact).
  • Someday we are amazed to discover a physical process that depends on precise equalities between the sums of powers of natural numbers. It follows theoretically that some version of the process could only occur in correspondence with a counterexample to FLT, a result that I can confirm for myself. I conclude that, if the theory is right, then this version of the process has never occurred.
Posted by: Toby Bartels on November 11, 2009 5:25 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

“Fictionalism” sounds like something I might be comfortable with, but I don’t know of other declared fictionalists out there. Can you point me to something?

The closest I can recall is a marvelous essay by James O. Bullock, which appeared in the American Mathematical Monthly (vol. 102, no. 6, 1995), on mathematics as metaphor.

Posted by: Todd Trimble on November 12, 2009 2:36 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

“Fictionalism” sounds like something I might be comfortable with, but I don’t know of other declared fictionalists out there. Can you point me to something?

Well, there's a Stanford Encyclopedia article, but that's not a very good description of my opinion. For one thing, the article consistently uses the word ‘false’ in a way that I would not (and which I find rather foolish, in fact, even though the analytic philosophers would probably approve). And when it writes ‘It is important to note, however, that despite the name of the view, fictionalism does not involve any very strong claims about the analogy between mathematics and fiction.’, I expect that my views are more radical.

Actually, reading the article again in more detail, it appears that I may be a Neo-Meinongian and not a fictionalist at all. The problem is that the article is using the word ‘true’ in such a very strange way, and taking it to be the obvious default meaning too, that I can't be at all sure if the Neo-Meninongian objection to that usage is mine or not. It may be simpler just to say that all the people described in that article are crazy.

But the really important point is that I haven't read any of the primary references there. I came up with the term on my own.

Posted by: Toby Bartels on November 12, 2009 6:10 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Actually, the Stanford article on just plain fictionalism is much more attractive to me. I can even pinpoint, in Section 2.3, that I am a mathematical meta-fictionalist, which is more than I can do for most of the varieties of fictionalism in either article.

Posted by: Toby Bartels on November 12, 2009 6:24 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Todd wrote:

I guess what I was saying is that I don’t really know what it means to say that, e.g., Fermat’s last theorem was “true” in prehistoric times. True for whom?

I don’t know what this statement means either… but not because nobody was around back then. I just don’t know what it means to say that a mathematical statement is true at a particular time. It just seems like a category mistake. I understand what it means for a mathematical statement to be known to be true at a particular time, by a particular person. But that’s different.

In other words, the question “Was Fermat’s Last Theorem true in 300,000 BC?” makes as little sense to me as “Was the square root of 2 irrational at 8:37 pm last Tuesday?” — or, since I treat spacetime as a unified whole, “Is 2 + 2 equal to 4 under this big rock?” I just don’t think of mathematical statements as having time-dependent or location-dependent truth values.

In fact, I usually don’t think of any statement as having a spacetime-dependent truth value unless it contains ‘indexicals’ like ‘here’ or ‘now’, as in “Is it raining here now?” — or implicit indexicals, as in “Is it raining?”, which I’d treat as an abbrevation for something like “Is it raining here now?”

So, in a formalized language, I’d only think of a statement as having a truth value dependent on spacetime coordinates (t,x)(t, \vec x) if the statement actually contains some of these coordinates as free variables. So, “It is raining at (t,x)(t, \vec x)” is the sort of statement whose truth could depend on (t,x)(t, \vec x). But “There is no nontrivial integer solution of i 7+j 7=k 7i^7 + j^7 = k^7” is not.

Posted by: John Baez on November 12, 2009 2:15 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Yes, this is precisely my position too.

Posted by: Tim Silverman on November 12, 2009 1:35 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I think you say such things with such conviction because the platonic outlook really is seductive! I’ll put forth some arguments against them, even if they don’t gain any traction with you.

The basic points I’m trying to make are simple: (1) it is a category mistake to think that “truth” adheres to things “out there”. “Truth” is properly applied to assertions, to linguistic constructs. And language evolves. (2) An unobserved “truth”, truth in a vacuum if you will, is no truth at all.

While I can empathize with the position that linguistic constructs are extra-temporal, this is by no means clear-cut. Let me remind you of the idea of “stages of knowledge”. Truth of assertions becomes manifest at points in time. There are modalities involved. Borrowing an example from Toby: if we discovered an ancient artifact which we interpret as giving a counterexample to FLT, it would be proper to say something like, “with hindsight, we now know that there could be no correct counterexample.” The expression of knowledge with hindsight is the modality which is usually left unspoken.

If you were travel back in time and attempt to explain your FLT-based insight to that ancient reckoner, it would of course carry no weight; his language is not at that point of preparation. In order to convince him, you’d have to say instead, “Look, you forgot to carry the 1” or something similar – that, not FLT, would be the mutually observed truth.

(As for Toby’s other example, where we imagine FLT embedded somehow in the warp and woof of some physical theory: I don’t quite get it. Physical theories can only be falsified; they can never be proven true. So it can’t be the case that if the putative physical theory were false, then FLT would be false, or the contrapositive of that. Did I misunderstand your example, Toby?)

Anyway, you can reject this conception of truth and declare “category error!” all you like, but I think all that comes down to is stamping your platonist feet and insisting that when you have proved a new theorem, you have discovered something timeless. I don’t think there’s a fully rational basis for that. It’s a confession of your platonist faith.

Posted by: Todd Trimble on November 12, 2009 2:27 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

(As for Toby’s other example, where we imagine FLT embedded somehow in the warp and woof of some physical theory: I don’t quite get it. Physical theories can only be falsified; they can never be proven true. So it can’t be the case that if the putative physical theory were false, then FLT would be false, or the contrapositive of that. Did I misunderstand your example, Toby?)

It's supposed to be the converse of that: ‘I conclude that, if the theory is right, then this version of the process has never occurred.’. But maybe I should state that in the contrapositive: If the relevant version of the process has ever occured (even in the prehistoric past), then the theory is wrong (falsified).

Posted by: Toby Bartels on November 12, 2009 5:45 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Sorry, let me try again. You said, in giving this example,

It follows theoretically that some version of the process could only occur in correspondence with a counterexample to FLT

and I’m having trouble imagining how we would know the same process couldn’t be explained by a different theory which doesn’t involve FLT. In other words, the putative physical theory could be wrong for reasons other than FLT, which is a purely mathematical statement.

Posted by: Todd Trimble on November 12, 2009 7:25 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

You said, in giving this example,

It follows theoretically that some version of the process could only occur in correspondence with a counterexample to FLT

and I’m having trouble imagining how we would know the same process couldn’t be explained by a different theory which doesn’t involve FLT.

And I'm having trouble imagining how that would be relevant.

Let me start over from the beginning, only using the contrapositive to put falsification in its proper place. So suppose:

  • Someday we are amazed to discover a physical process that seems to depend on precise equalities between sums of powers of natural numbers. It follows theoretically that some version of the process could only occur in correspondence with a counterexample to FLT, a result that I can confirm for myself. I conclude that, if this version of the process has ever occurred, then the theory proposed to explain the process is wrong.

Of course, there may be other explanations for the process, but I'm considering this one that involves precise equalities between sums of powers of natural numbers. And if the FLT-prohibited version occurred in a prehistoric era, then the theory is falsified. If somebody objects that Fermat had not even formulated FLT then (much less that Wiles and Taylor had proved it), I would reply that FLT was already true in that prehistoric era. (And that was the utterance that you did not think would be meaningful.)

Posted by: Toby Bartels on November 12, 2009 8:52 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Oh, I see now. Thanks!

Posted by: Todd Trimble on November 12, 2009 9:17 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Todd wrote:

It’s a confession of your platonist faith.

By the way, I don’t call myself a ‘Platonist’ except when there’s a crowd of people arguing against Platonism and nobody for them to argue with. In other circumstances I’d portray my views quite differently — for example, sometimes I say I’m a Taoist, and in a crowd of rabid atheists I sometimes stick up for God. But I’m not just arguing for the sake of arguing. I really believe what I’m saying.

Posted by: John Baez on November 12, 2009 7:17 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

“Truth” is properly applied to assertions, to linguistic constructs.

If I understand this correctly, I reject it, precisely because it leads to the absurdities you describe (“language evolves”, therefore truth evolves).

Truth applies not to utterances but to their content (or, to speak a little more technically, to the propositions they express).

So, for instance, say at time t 0t_0, my fridge is white. And, as a standard English speaker, I utter the sentence “my fridge is white”, and it’s true. Now suppose that the language changes, so that the meanings of the word “white” and “black” change places. So at time t 1t_1, I can utter the sentence “My fridge is black” (or, to be perfectly clear, emit from my mouth the string of phonemes /mɑi fɹɪʤ ɪz blæk/) and it will be true.

But has my fridge changed colour? No. Was my utterance at time t 0t_0 really false, but I’ve only just discovered that? Or, even more mysteriously, has it retrospectively become false? Why would I say that? Isn’t it much simpler to say that “mɑi fɹɪʤ ɪz wɑiʔ” at t 0t_0 and “mɑi fɹɪʤ ɪz blæk” at t 1t_1 mean the same thing, and what they mean is, in fact, true?

Also, languages don’t just vary over time, but from speaker to speaker. Suppose a Frenchman tells me, “Votre frigo est blanc,” and I don’t speak French. Is his utterance not true? If I learn French, then does it become true? At what point does its truth value change? Wouldn’t it be easier to say he’s telling the truth, but I don’t understand what he’s saying? Suppose someone’s speaking with his mouth full, so I don’t hear him clearly when he says, “Your fridge is white”? Is he not telling the truth? Since he presumably knows perfectly well what colour my fridge is, is he lying? Telling lies and speaking with your mouth full are generally considered quite different faults!

Which brings us to another problem: beliefs are true or false, and beliefs aren’t necessarily linguistic. If I think that the bank is on the left of the pizzeria, and go in there to pay in a cheque, only to discover that it’s a launderette, because the bank is actually on the right of the pizzeria, I was suffering from a false belief, surely, even if I never uttered or imagined uttering the phrase “The bank is to the left of the pizzeria”. It is enough that the content of my belief can be rendered into propositional form.

Of course, the content of my beliefs can change—as they do, when I unexpectedly find myself in the launderette with a cheque in my hand—but to describe that, I’d rather say something like, “The content of my beliefs has changed”, rather than, “I have the same beliefs but the truth value that they had when I went into the launderette has retrospectively changed”. (Has the concept of a bank now changed into the concept of a launderette? If we can’t refer back to the meaning, how could we tell? And if we can refer back to the meaning, then it seems to me that we should.)

Of course, it is certainly very interesting to look at how beliefs, utterances, understanding etc correspond to or approximate the state of affairs that they are purportedly about, but truth seems to be a separate and much simpler issue.

I see now that John has also replied to this. I agree with what he said. (In fact, I think I’ve made very much the same argument at some time in the past.)

Posted by: Tim Silverman on November 12, 2009 7:25 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

But has my fridge changed colour?

No, it has always been grue. Or is it bleen?

Posted by: John Armstrong on November 12, 2009 7:57 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Truth applies not to utterances but to their content

Yes, certainly. Linguistic constructs are understood to have a semantic component, and that’s what I was referring to. Sorry if I wasn’t being clear about that.

It is enough that the content of my belief can be rendered into propositional form.

Yes, precisely.

Of course, the content of my beliefs can change—as they do, when I unexpectedly find myself in the launderette with a cheque in my hand—but to describe that, I’d rather say something like, “The content of my beliefs has changed”, rather than, “I have the same beliefs but the truth value that they had when I went into the launderette has retrospectively changed”. (Has the concept of a bank now changed into the concept of a launderette? If we can’t refer back to the meaning, how could we tell? And if we can refer back to the meaning, then it seems to me that we should.)

First of all, as I said, we agree: truth is applied to the meanings of statements or, as you said, to contents of beliefs that can be expressed in propositional form. Let’s put that behind us, because there’s no debate there.

Second, I don’t see what the problem is. It could dawn on one as follows, “Funny, I thought the pizzeria was supposed to be here, but [I see] now that can’t be right: this is a bank!” Of course the beliefs change, synchronously with what one thinks is true. One can relive the scene and think “well, that belief was wrong; why did I ever think that” or something, and I don’t have a problem with that, but of course it’s not an exact re-enactment because one has also injected the updated knowledge back into the scene.

I may have to end my part in the discussion soon, unfortunately – I have other things that need attending to!

Posted by: Todd Trimble on November 12, 2009 10:18 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Said Todd:

First of all, as I said, we agree: truth is applied to the meanings of statements or, as you said, to contents of beliefs that can be expressed in propositional form. Let’s put that behind us, because there’s no debate there.

Second, I don’t see what the problem is. It could dawn on one as follows, “Funny, I thought the pizzeria was supposed to be here, but [I see] now that can’t be right: this is a bank!” Of course the beliefs change, synchronously with what one thinks is true. One can relive the scene and think “well, that belief was wrong; why did I ever think that” or something, and I don’t have a problem with that, but of course it’s not an exact re-enactment because one has also injected the updated knowledge back into the scene.

Oh dear, I’m wondering if I’ve missed something obvious. I don’t see how the truth value of the proposition that the bank is on the left of the pizzeria (or whatever) is changing just because of changes in my beliefs or statements. And if you don’t think it is changing, then I’ve failed to understand what you mean for something to be true for someone or true at some time.

Or maybe this is because you think Fermat’s Last Theorem is a completely different sort of statement from “The bank is to the left of the pizzeria.” Yes, that would make sense. But then we are no further forward, alas.

Posted by: Tim Silverman on November 12, 2009 11:16 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Truth applies not to utterances but to their content

Unfortunately one never gets to communicate the content to others except by piling on ever more utterances (as in this thread).
The content is everyone’s private illusion, we are fooled in (sometimes) believing in intersubjective agreement about such matters because as human beings we obviously share a lot of preconceptions and mental habits but that does not give any guarantee about the “models” we attempt to talk about.
I am such a radical non-platonist that I don’t even believe that cats, molecules or triangles “exist” as objects in the world, these are only words with which we try to convey a summary of our inner experience of the flow of phenomenons, within which we (hopefully) find “interesting regularities” and that we expect others have noticed too.

Posted by: J-L Delatre on November 13, 2009 12:18 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Well, if you’re really such a skeptic, you’ll have to face the question in this story:

Chuang Tzu and Hui Tzu were strolling along the dam of the Hao River when Chuang Tzu said, “See how the minnows come out and dart around where they please! That’s what fish really enjoy!” Hui Tzu said, “You’re not a fish—how do you know what fish enjoy?” Chuang Tzu said, “You’re not me, so how do you know I don’t know what fish enjoy?”…

Posted by: Minhyong Kim on November 13, 2009 1:34 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I am on the side of Hui Tzu. He could have reasonably replied to

“You’re not me, so how do you know I don’t know what fish enjoy?”

with

“You and I are both humans. As social animals humans are very good at the theory of minds, as far as other humans go (those who were not didn’t leave descendants). But the evolutionary pressure to understand fish minds has been less severe. So there is a good chance I know (approximately) what you know and there is a good chance that you don’t know what fish enjoy.”

Of course, I don’t know Chuang Tzu and for all I know he spent years trying to empathize with fish. But it is still a valid point that his innate capacity of empathizing with humans is much greater than his capacity of empathizing with fish. Of course, I don’t know

Posted by: Eugene Lerman on November 13, 2009 9:38 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Eugene Lerman said:

As social animals humans are very good at the theory of minds, as far as other humans go (those who were not didn’t leave descendants). But the evolutionary pressure to understand fish minds has been less severe.

On the other hand, humans have been hunting and otherwise managing non-human animals for a long time, so there’s certainly been some pressure to understand non-human minds (or to develop a theory of mind that’s general and adaptable enough to cover a variety of different kinds of mind). And although people are sometimes guilty of undoubted anthropomorphisation, they also successfully keep pets and farm animals. So it’s not utterly implausible that Chuang Tzu knew something about what fish like.

In any case, if he wanted to learn what fish want, he could perhaps find this out by observing them. He wouldn’t necessarily need to know what it “felt like” to be a fish. Depending on what exactly he wanted to learn …

Also, I don’t think you can distinguish between accurate knowledge and false belief by what they “feel like”. I’m not a Cartesian Rationalist about knowledge of the material world.

Posted by: Tim Silverman on November 13, 2009 10:41 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

“humans are very good at the theory of minds” - last semester I used an anthropology seminar of a friend to ask around in some MPI’s about that, here my bloggyfied conclusion. Interestingly ca. 1-2% of the general population lack empathy and, acc. to experts, the social ladder selects against that. Should empathy be the root out of which emotional platonistic develops, ‘emotional platonism’ would be rarer among graduates than in the general population.

Posted by: Thomas on November 14, 2009 12:55 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I forgot to give this link for the statements on homo erectus. Aside the “artificial microenvironment”, as the archaeologists call their settlement, the bones with geometrical carvings are interpreted as indicating that they had even abstract concepts. IMO really awsome, if one takes their small and speachless brain into account. So, there may have existed homo erectus platonists.

Posted by: Thomas on November 14, 2009 2:33 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

A relatively serious reply to this would be:

Of course you know I don’t *really* know how fish feel. On the other hand, it’s equally obvious that I can get a pretty good sense of it, through judicious observation of animal habits, common biological heritage, etc.

Posted by: Minhyong Kim on November 13, 2009 10:57 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

You’re right. I was bluffing.

Posted by: Chuang Tzu on November 14, 2009 12:58 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

On the other hand, I’m pretty sure fish enjoy eating worms. What more is there to say?

Posted by: Chuang Tzu on November 14, 2009 1:30 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Even if the fish would enjoy eating them, do they know that they enjoy it?

Posted by: Thomas on November 14, 2009 2:11 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

J-L Delatre said:

I don’t even believe that cats, molecules or triangles “exist” as objects in the world, these are only words with which we try to convey a summary of our inner experience of the flow of phenomenons, within which we (hopefully) find “interesting regularities” and that we expect others have noticed too.

If you don’t believe that cats, molecules or triangles exist, how do you justify believing that “words”, “inner experience”, “the flow of phenomena”, “we”, “interesting regularities” or “others” exist? What are triangles if not “interesting regularities”? Suppose “we” are “cats”?

Posted by: Tim Silverman on November 13, 2009 10:43 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

To attempt to clarify my earlier comment, now that I have had more tea,

J-L Delatre said:

I am such a radical non-platonist that I don’t even believe that cats, molecules or triangles “exist” as objects in the world, these are only words with which we try to convey a summary of our inner experience of the flow of phenomenons, within which we (hopefully) find “interesting regularities” and that we expect others have noticed too.

As a preliminary: I think it’s generally agreed that cats and triangles may be different, so that reasons for rejecting the existence of cats are not necessarily reasons for rejecting the existence of triangles, and vice versa, so I’ll treat them separately.

Now: I can make sense of rejecting the existence of material objects like cats, while being some sort of solipsist, believing in the reality of your own experiences as entities in themselves. What I don’t understand is rejecting the existence of cats while accepting the existence of people. And, even more, I don’t understand rejecting the existence of people’s bodies, for which the evidence is generally considered fairly direct, while accepting the existence of other people’s (somehow disembodied) minds, for which the evidence is not only indirect, but specifically goes via observation of their bodies and other material effects. And even less do I understand rejecting minds, while accepting social constructions like words, language, and blog conversations, which are inferred via minds together with material evidence like sounds, ink and computer screens.

As I said before, I also don’t see how one can consistently reject triangles while accepting “interesting regularities” in the “flow of phenomena”.

Posted by: Tim Silverman on November 13, 2009 1:00 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

You are puzzled likely because you are a “true believer” in all the concepts mentionned above while I am a miscreant.
It is not that I do not see or use the same words and concepts as you and everyone else.
What I don’t believe is that there is a one only true definition of those concepts that could be used to sort out proper and improper uses of the words and concepts, be it for one’s own use (consistency of thoughts) or for intersubjective communication.
The Platonic Ideals fail us because there are not there to begin with and there is even no way that we could (safely) build them out of experience.
The persistence and consistence of our world views is only a bet which is most often won but which is not warranted, think of optical illusions, software bugs, errors in proofs, accidental medical malpractice etc…
So in this respect cats and triangles are just as suspicious.
Do you “believe in the electron”? then what about the Double-slit experiment?

Posted by: J-L Delatre on November 13, 2009 6:45 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

J-L Delatre wrote:

You are puzzled likely because you are a “true believer” in all the concepts mentionned above while I am a miscreant.

It is not that I do not see or use the same words and concepts as you and everyone else.

What I don’t believe is that there is a one only true definition of those concepts that could be used to sort out proper and improper uses of the words and concepts, be it for one’s own use (consistency of thoughts) or for intersubjective communication.

I think I would be less puzzled if you were a thoroughgoing sceptic, but you don’t appear to be.

I don’t think that I need to believe that there is a one only true definition of cats—let alone that I am in possession of such a definition—in order to believe that cats exist. I think I can get away with being moderately vague about what precisely I might mean by identifying some things as cats, and yet not be actually wrong in my belief that cats exist.

Of course, I might be wrong to believe that cats exist, but I think I’d have to be very wrong about a lot of things if that was the case. There’s certainly no guarantee that I’m not very wrong, possibly about everything. But I don’t think the fact that I don’t know everything that could possibly be known about cats is, in itself, a reason to say that I’m that wrong.

Posted by: Tim Silverman on November 13, 2009 7:33 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I think I can get away with being moderately vague about what precisely I might mean by identifying some things as cats, and yet not be actually wrong in my belief that cats exist.

I guess I see where our disagreement lies, it’s likely about the meaning of the word “exist”.
If the meaning of “exist” is only that “cat looking” things repeatedly show up before our eyes then we are actually in agreement.
If however you ascribe to the word cat some “essence” then we are not.

Posted by: J-L Delatre on November 13, 2009 8:54 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

J-L Delatre said:

If the meaning of “exist” is only that “cat looking” things repeatedly show up before our eyes then we are actually in agreement.

If however you ascribe to the word cat some “essence” then we are not.

I don’t ascribe an “essence” to the word cat, but I do think it is reasonable to suspect that the repeated appearance of “cat-looking” things (and other cat-related phenomena) in our experience reflects some stable aspect of an independent external world, about which we can gain some (incomplete, approximate) knowledge. So our notion “cat” isn’t a mere phantom, although we shouldn’t put too much weight on it either.

So we probably aren’t in perfect agreement, but we’re a lot closer than we appeared to be at first sight.

Which is nice. :-)

Posted by: Tim Silverman on November 13, 2009 9:09 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Well, you’re wrong. I *know* cats exist. And there’s no way you can know that I don’t know.

Posted by: Minhyong Kim on November 13, 2009 9:14 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Sayeth Minhyong:

I *know* cats exist. And there’s no way you can know that I don’t know.

Well, I guess on the internet, no one knows you’re an elf dreaming it’s a butterfly who *knows* that what fish really enjoy is observing the truth of Fermat’s Last Theorem …

but

I think I can admit I have no way of knowing for sure the phenomenal content of your subjective experiences, and yet still be able to judge the truth of your beliefs, should I by some means find out what they are.

Posted by: Tim Silverman on November 13, 2009 9:29 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Todd wrote:

Truth of assertions becomes manifest at points in time.

I have no problem with the idea that truth of a proposition becomes manifest to a particular person at a particular time. I just prefer to set up my conventions — and I regard them as conventions! — so that I don’t say the proposition ‘becomes true’ at this moment.

It’s very much like this: when I open the refrigerator and see a sandwich, I say ‘the sandwich became visible’ the moment I opened the door. I don’t say ‘it became true that the sandwich was in the refrigerator’ the moment I opened the door.

The reason is if I go down the other road, and say propositions become true when their truth becomes manifest to me, I’m led into complexities which I don’t see a way out of. And I don’t want to have to surmount all these complexities before I can flatly assert something like “2+2 = 4”, or “the statement 2+2 = 4 is true”.

Let me sketch some of these complexities.

First of all, there is a problem with saying “truth of assertions becomes manifest at points in time”. There are no ‘points in time’: in our universe the closest approximation is a point in spacetime.

When Wiles proved Fermat’s Last Theorem here on Earth, did its truth become manifest in the Andromeda Galaxy at the same moment? Alas, that question doesn’t even make sense: thanks to special relativity, there is no good notion of ‘simultaneity of distant events’. Even if we beamed out the good news at the speed of light, the mathematicians on Andromeda would only consider the theorem proved when the news reached them.

So now we’re getting pushed towards a setup where ‘truth travels no faster than light’. This means we need to take not only special relativity but also general relativity and not-yet-understood refinements like quantum gravity into account when discussing the location-depedent truth of a statement like 2+2=42 + 2 = 4.

But of course this physics stuff is only the beginning, and in some sense a red herring. You can beam out a proof at the speed of light, but presumably only when the proof is understood by a given ‘witness’ does a given theorem ‘become true’ for that witness. Presumably in your setup truth does not really occur ‘at a point in spacetime’, but ‘in the mind of a witness’. But what counts as a ‘witness’? And how well do they need to understand a proof for the theorem to become true?

It would seem arbitrary to have the truth value of a theorem for a particular witness jump discontinuously from 0 to 1 at some particular moment: this would require an utterly sharp distinction as to whether they followed a proof or not. I can list lots of examples that make the possibility of such a sharp distinction seem implausible. So, we’d need some subtler measure of truth. Of course, topos theory is very happy to deal with truth values that are more subtle than 0 and 1. But I’m far from a convinced that we know any way to keep track of how ‘the growth of truth’ occurs in the actual mental process of proving a theorem.

And of course we’d also need to understand the ‘decay of truth’ as someone forgets the proof of a theorem.

All of these complexities are real things, but I claim that it’s more convenient to treat them as complexities of cognitive psychology rather than complexities of the truth of statements like 2+2=42+2 = 4. Why? Because I don’t need to understand any of these complexities to correctly reason with statements like 2+2=42 + 2 = 4. I can separate these complexities off and deal with them later.

The world is a complicated and squishy mess. But it’s very useful to bracket off some of this mess when discussing mathematics.

For example: if I am interested in understanding precisely how well the truth of the Hahn–Banach theorem is manifest to me, I should expect to hire a psychologist and a mathematician to ask me lots of probing questions, and perhaps even undergo a brain scan while I’m answering them. But if I’m only interested in whether the Hahn–Banach theorem is true, it’s a marvelous simplification to say “Okay, for this I can just think about math”.

Posted by: John Baez on November 12, 2009 6:09 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I think you’re taking my using the word “time” much too literally! Well, it’s my fault for using the word. It would be safer to speak in terms of stages of knowledge, or instead of saying “points in time”, say “points of logical development”. There is a kind of arrow of time in the idea of stages of knowledge, and I meant temporality in that sense, not the sense of physical time that you were talking about. The bit about 300,000 BC is a complete red herring here.

Also, I’ll take this opportunity to rephrase something I said earlier.

An unobserved “truth”, truth in a vacuum if you will, is no truth at all.

Perhaps that is a metaphysical claim too, so let me put it this way instead: one can consistently assume such a statement, will full confidence that no one could ever refute it by actually producing a convincing example of an unobserved truth. For once it’s been recognized as being convincing, the truth has been observed! :-)

To give a snappy name for it, call it the “truth is emergent” doctrine. I’m not saying it’s “true”, I’m just suggesting it’s no less viable a doctrine than what platonism suggests.

Posted by: Todd Trimble on November 12, 2009 7:21 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Todd said:

An unobserved “truth”, truth in a vacuum if you will, is no truth at all.

But then who observes the observers? And who observes the observers of the observers?

Posted by: Tim Silverman on November 12, 2009 7:55 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

C’mon, there is no need for recursion. The truth may need observers to exist, but observers don’t need other observers to exist.

Posted by: Eugene Lerman on November 13, 2009 2:08 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Quoth Eugene Lerman:

C’mon, there is no need for recursion. The truth may need observers to exist, but observers don’t need other observers to exist.

Now I’m starting to wonder if you exist, Eugene Lerman! For can you prove to me that you are an “observer”? Is that proof more convincing than Wiles & Taylor’s proof of FLT?

Posted by: Tim Silverman on November 13, 2009 11:04 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Eugene wrote:

C’mon, there is no need for recursion. The truth may need observers to exist, but observers don’t need other observers to exist.

Hmm. Okay: every true theorem is already known by an elf specially devoted to that task. What, you’re complaining you don’t see those elves? That’s no problem: each elf can see itself perfectly fine! And yet you doubt they exist?

Posted by: John Baez on November 13, 2009 3:21 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Okay: every true theorem is already known by an elf specially devoted to that task. What, you’re complaining you don’t see those elves? That’s no problem: each elf can see itself perfectly fine! And yet you doubt they exist?

Isn't this what Bishop Berkeley eventually said about (what is ordinarily regarded as) the real world? Only he used a single God in place of this multiplicity of elves.

Posted by: Toby Bartels on November 13, 2009 6:11 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Eugene Lerman exclaimed:

The truth may need observers to exist, but observers don’t need other observers to exist.

To which John B riposted:

Okay: every true theorem is already known by an elf specially devoted to that task.

Whereat Toby observed:

Isn’t this what Bishop Berkeley eventually said […]?

Prompting the following faint recollection of a verse:

Dear Sir, Your inquiry delves
Not too deep: Wiles is e’er on the shelves,
And eternally proves
FLT’s truth ne’er moves—
Since observed by, Yours faithfully, Elves.

Posted by: Tim Silverman on November 13, 2009 8:58 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

You’re right that Berkeley’s God is a bit like my plethora of unobservable elves.

But my point — because I have the feeling that not everyone got it, even if you did — was this. Eugene seems to think that truths need to be observed to be true, but observers don’t need to be observed to count as observers. This might mean he’d happy with the idea of unobservable observers, like little elves, devoted to making sure that each truth is observed. But I was hoping he’s not happy with this idea, because this idea seems silly to me.

So if he’s not happy with it, I would like to know how exactly he wants to get around the problem that Tim raised:

But then who observes the observers? And who observes the observers of the observers?

One standard way to get around it is to say: “I know I exist. I’m willing to believe that most stuff I observe exists. And if I observe someone, and trust them, I’m willing to believe in stuff that they say they observe. And I’m even willing to believe in a bunch of stuff observed by people who they observe, and so on.”

In other words, “the buck stops with me.” Depending on how you take this, it could be a strong solipsist position or just a sensible practical attitude.

But Eugene — and Todd — never mentioned any idea like this. They seemed happy with the idea that truth was witnessed by some observer… even one not linked to them.

I can imagine a “mentalist” philosophy that believes primarily in the existence of minds, and secondarily in anything that any mind perceives. This makes as much — and as little! — sense to me as the “materialist” philosophy that believes primarily in the existence of matter. But while materialists are a dime a dozen, I haven’t met many “mentalists”, so I was wondering if Todd or Eugene were coming out in support of such a view.

Posted by: John Baez on November 15, 2009 10:31 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I guess there have been periods in my life when I would not have been unhappy with being considered a “mentalist”; these days I feel somewhat agnostic about such things. Or perhaps it’s more accurate to say that I feel confused about lots of stuff!

By the way, I left this thread temporarily because I was spending too much time thinking about it at the expense of other things. I actually appreciate you and Tim subjecting some things I said to scrutiny; it forced me to examine some things which would have been left unexamined. But I was getting the feeling it was going to take a long time to better elucidate the positions I had begun to espouse, and I wasn’t sure it was really worth the effort!

Posted by: Todd Trimble on November 15, 2009 11:54 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Thanks for explaining why you quit this thread, Todd. I’m glad to hear you didn’t leave in a huff. I can certainly understand your fear that doing justice to this subject would eat up too much time: it’s a huge quagmire — nay, a downright abyss! I, too, have limited time for such deep issues. I’m happy to shelve the discussion and continue where we left off in our next reincarnation.

Posted by: John Baez on November 16, 2009 5:50 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

John Baez

it’s a huge quagmire — nay, a downright abyss!

Yes but you (and a few others) are the culprits with your eagerness to conflate mathematical existence with what could be termed “ordinary” existence of figures and themes in the surrounding world of everyday experience.
These are two different kinds of existence, willing to actually reduce them to the same realm (or “connect” them via some pineal gland) IS the Platonic error which, as I said before has been poisoning mathematics and philosophy for more than two millenia.
The only purpose of objects is to be recognised as “the same” at different occurences within discourse such as to serve as carriers of properties, this is why they must appear as intemporal (Platonic!) and why an improper denotation (lack of definedness) fails to bring an object “into existence”.
While informal language allows to play loose with these two requirements mathematics cannot afford to disregard them on pain of inconsistency.

Posted by: J-L Delatre on November 16, 2009 7:21 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Could you suggest to us how mathematics might have been different had it not been so poisoned?

Posted by: David Corfield on November 16, 2009 8:20 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Mathematics would not have been so much different than it would have been more productive and avoided the waste of brain power of so many brilliant minds, the more brilliant the more entangled in that “flypaper” of irrelevant metaphysics.
The best example at hand is here at the Café, Arnold Neumaier FMathL, which I doubt will produce any kind of breakthrough not because it is “too ambitious” or will lack enough funding but because it is obviously bogged down from the very start in “sex of angels” arguments.
Though, since I think, as Chaitin showed, that the field of possible mathematics is open ended and at least countably infinite, does it really matter where we actually stand?
I dunno.

Posted by: J-L Delatre on November 16, 2009 1:42 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

FMathL

Bad link; this should be

FMathL

Posted by: Toby Bartels on November 17, 2009 5:43 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

This may have poisoned the philosophy of math
but I don’t see it intruding into the day to day practice of math

Posted by: jim stasheff on November 16, 2009 1:40 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Todd said:

I left this thread temporarily because I was spending too much time thinking about it at the expense of other things.

Yeah, I can sure relate to that. Thanks for taking the time to mull these things over with me. It’s been very stimulating!

Posted by: Tim Silverman on November 16, 2009 11:09 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

They seemed happy with the idea that truth was witnessed by some observer even one not linked to them.

I can’t speak for Todd and I am juggling too many balls.

I will stubbornly insist that a truth not witnessed by anyone is no truth at all for any practical purpose.

Well, you may asked what about a truth witnessed by someone? I am guessing this is what John and Tim are asking. I am not sure exactly what it means to “witness” truth but for the sake of the argument let’s assume we’re talking about someone who proved a theorem to her satisfaction. Is the theorem then true? Well, it depends. It depends on the person’s track record, on whether someone else with a great track record checked the proof, and so on. If it has been checked by a number of reputable mathematicians, then it’s more likely true than not. If, on the other hand, John and I haven’t even heard of it being proved, what do we then know? I would say that for us the truth of the theorem does not exist [I am sure I am going to get into trouble for saying this, but it’s late and I still have to prepare my classes].

Posted by: Eugene Lerman on November 16, 2009 2:11 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Eugene said both:

I will stubbornly insist that a truth not witnessed by anyone is no truth at all for any practical purpose.

and

I am not sure exactly what it means to “witness” truth

Indeed, this is one of the things that bothers me about your approach. I’m OK with some vagueness in our concepts of “knowledge” or “understanding”, which seem inherently quite psychological and complicated (as well as logical, physical and epistemological), but I don’t want that much vagueness to adhere to something as basic as “truth”.

Also, you seem to want not to have “true facts” that nobody knows. But since I don’t have any problem with this in the material world (I don’t know, offhand, the boiling point of mercury, but that doesn’t mean I think it doesn’t have a boiling point), I don’t see why it should be a problem in the mathematical world.

Also, what is going on when someone “proves” a theorem, if they are not discovering the truth of the theorem (within a given context)? When you say “It depends on the person’s track record”, what does the concept of a “track record” even mean if there is no observer-independent way to establish whether their work was right or wrong in the past?

Posted by: Tim Silverman on November 16, 2009 11:19 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Tim said:

I’m OK with some vagueness in our concepts of “knowledge” or “understanding”, which seem inherently quite psychological and complicated (as well as logical, physical and epistemological), but I don’t want that much vagueness to adhere to something as basic as “truth”.

I understand that many would not be comfortable with such a position. It is different than believing that all truth is relative or that it could be decided by a majority vote. But if truth is a social construct, then it has to come with a degree of uncertainty. And it often does. You probably know the examples I am about to trot out: Appel & Haken proof of the four color theorem, Hales’ proof of Kepler conjecture, the classification of finite simple groups.
Are we certain these theorems are true? How certain?

Posted by: Eugene Lerman on November 16, 2009 12:50 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Eugene said,

if truth is a social construct, then it has to come with a degree of uncertainty

And if truth is a carrot, it’s probably orange, but since I don’t see any reason to accept the premise, I don’t see any reason to accept the conclusion.

You probably know the examples I am about to trot out: Appel & Haken proof of the four color theorem, Hales’ proof of Kepler conjecture, the classification of finite simple groups.

No, I didn’t expect you to trot them out, because I simply don’t accept that the truth of a statement has any relation whatever to how certain people may be about it. I have no idea whether mercury boils above 1000 degrees centigrade, but that doesn’t stop me believing that there is a fact of the matter about whether it does or doesn’t.

Posted by: Tim Silverman on November 16, 2009 6:25 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

You probably know the examples I am about to trot out: Appel & Haken proof of the four color theorem, Hales’ proof of Kepler conjecture, the classification of finite simple groups.

Are we certain these theorems are true? How certain?

As difficult theorems go, I'm quite certain about the four-colour theorem: it's been verified by computer!

Posted by: Toby Bartels on November 17, 2009 5:34 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Also, you seem to want not to have “true facts” that nobody knows. But since I don’t have any problem with this in the material world (I don’t know, offhand, the boiling point of mercury, but that doesn’t mean I think it doesn’t have a boiling point), I don’t see why it should be a problem in the mathematical world.

The trouble is, you’ve opened the door to “true facts” no one can know, which have weird consequences. For example, consider “Julius Caesar would have liked spicy tomato salsa.” Now, this is a counterfactual (tomatos being a New World plant, and Caesar being Roman) that cannot ever be tested (since Caesar is dead). Do we really want to defend the position that this proposition has a truth value?

Let’s suppose we do – i.e., it’s a fact one way or the other, but nobody can know it. Observe that this possibility poses a fatal objection to verificationist or instrumentalist readings of propositions, since it means there are propositions with no way of giving evidence for or against them, even in principle.

Now, giving up on logical positivism and the Vienna Circle is no great tragedy, but Einstein’s key conceptual breakthrough in discovering relativity was precisely to view time and space in instrumental fashion. Since the purpose of logic is to formalize effective reasoning, it seems rather to miss the point to rule one of the most spectacularly effective examples of reasoning out of bounds!

Posted by: Neel Krishnaswami on November 16, 2009 3:15 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

But we make statements like this all the time!

“If we were to put Julius Cæsar into a catapult with thus-and-so tension on the arm, when launched his body would fly in a parabola described by the curve…”

How is the fact that Julius Cæsar isn’t around to test this different from the fact that high school physics students don’t *actually* fire off cannon? The problem in your example is tied up with matters of personal opinion and taste, which are all-but-impossible to generalize and predict under any curcumstances.

Without counterfactuals prediction is useless and science is absurd. I’m willing to grant that the notion of “truth” may have to be revised to handle counterfactuals, but throwing it out altogether is nonsense.

Posted by: John Armstrong on November 16, 2009 3:57 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

John Armstrong said:

How is the fact that Julius Cæsar isn’t around to test this different from the fact that high school physics students don’t *actually* fire off cannon?

There’s that, too. You’re right that this has to be dealt with when we are trying to understand truth in its full complexity.

Posted by: Tim Silverman on November 16, 2009 6:52 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Neel said:

For example, consider “Julius Caesar would have liked spicy tomato salsa.” Now, this is a counterfactual (tomatos being a New World plant, and Caesar being Roman) that cannot ever be tested (since Caesar is dead). Do we really want to defend the position that this proposition has a truth value?

I have no intention whatever of making my treatment of facts depend on my treatment of counterfactuals, and indeed I don’t think counterfactuals have objective truth values, except relative to some general theory that crosses multiple instances. (I.e. within the context of a scientific theory, it makes sense to talk about what its counterfactual models predict, but only its actual real-world predictions can have truth-values. However, with actual scientific theories, you also have to deal with the fact that they are approximations, which is a much more complicated and tricky issue.)

Also:

Einstein’s key conceptual breakthrough in discovering relativity was precisely to view time and space in instrumental fashion.

I strongly deny this. The instrumental approach was simply a way to bootstrap up to a new theory of spacetime. The distinction between an instrumental and a realist epistemology (say) is a metaphysical one, and so can’t depend on the actual form of the scientific theory.

Posted by: Tim Silverman on November 16, 2009 6:42 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

The distinction between an instrumental and a realist epistemology (say) […] can’t depend on the actual form of the scientific theory.

I don't know whether Neel's history is correct, but he didn't say this.

Posted by: Toby Bartels on November 17, 2009 5:35 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Neel said:

Einstein’s key conceptual breakthrough in discovering relativity was precisely to view time and space in instrumental fashion.

In my reply, I said

The distinction between an instrumental and a realist epistemology (say) […] can’t depend on the actual form of the scientific theory.

about which Toby Bartels said

I don’t know whether Neel’s history is correct, but he didn’t say this.

Not in so many words, no. I took him to be implying it (or perhaps providing evidence for it), but maybe that was an inference too far. In the context of the rest of his post, I see now that that can’t have been the main point. But I’m not entirely sure how it does relate to his main point, since instrumentalism isn’t the negation of naïve realism.

Posted by: Tim Silverman on November 17, 2009 9:44 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I think that most forms of realism and ideas like verificationism (and its cousin instrumentalism) are not compatible; that is, put together they entail contradictions. I don’t think this is a particularly controversial point.

Furthermore, I think that the *process of reasoning* that Einstein used to reach special relativity involved aggressive use of the instrumentalist perspective. This means that if you want to argue that Einstein’s reasoning process (and not just his final theory) was a good one, you cannot endorse realism. Since I do want to argue this, I don’t endorse realism. (You apparently regard it as am ad-hoc dispensable trick, which is also a consistent position, though one that honestly leaves me a little incredulous.)

However, I don’t want to argue globally for instrumentalism, either – for example, I think it has had a catastrophic effect on research in social psychology and psychometrics. I just don’t want to rule it out of bounds before the game starts.


Posted by: Neel Krishnaswami on November 17, 2009 2:03 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Neel said:

Furthermore, I think that the *process of reasoning* that Einstein used to reach special relativity involved aggressive use of the instrumentalist perspective. This means that if you want to argue that Einstein’s reasoning process (and not just his final theory) was a good one, you cannot endorse realism.

I very strongly disagree with this on multiple levels. I could point to both the difference between the “context of discovery” and the “context of justification” and the fact that, in the course of making valid discoveries, people make all sorts of mistakes, and believe all sorts of erroneous things, which can ultimately end up leading them to a correct view.

If one is mistaken in ones belief in the existence of some entity, for instance, a reasonable path from belief in its existence to belief in its non-existence passes via belief in some sort of qualified or partial existence. (This is an over-simplification of my views on what was actually going on, but I think it will do here.)

Since I do want to argue this, I don’t endorse realism. (You apparently regard it as am ad-hoc dispensable trick, which is also a consistent position, though one that honestly leaves me a little incredulous.)

It seems entirely reasonable to me. If you look through the various bits of philosophical fluff that scientists have insisted were essential to their work over the ages, you’d end up having to believe in any amount of nonsense.

For instance, you’d have to endorse the various kinds of religious Platonism invoked by scientists as varied as Johannes Kepler and Richard Owen to explain aspects of their work; you’d need to endorse the mystical rays by which everything influenced everything else in various medieval philosophical systems, which indirectly led to (for instance) Alhazen’s unification of the “intromission” and “visual ray” theories of vision and to Fracastoro’s (admittedly vague and confused) discovery of contagion in disease.

And so forth. I place very little weight on the actual metaphysics behind these ideas, however useful they may be as source of imagery and models or pointers to fruitful directions of research.

Posted by: Tim Silverman on November 17, 2009 7:33 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

For example, consider “Julius Caesar would have liked spicy tomato salsa.” Now, this is a counterfactual (tomatos being a New World plant, and Caesar being Roman) that cannot ever be tested (since Caesar is dead). Do we really want to defend the position that this proposition has a truth value?

Let’s suppose we do – i.e., it’s a fact one way or the other, but nobody can know it. Observe that this possibility poses a fatal objection to verificationist or instrumentalist readings of propositions

I don't find this example particularly problematic. Keeping in mind that there is very slim chance that I'll have a convincing argument for it, I can still give it a hand-waving operational interpretation, to wit:

Although extremely unlikely, there is a small chance that, through some strange circumstances, Julius Caesar came upon some spicy tomato salsa. Suppose that he did, and suppose further that he came across it in more or less normal circumstances (however irregular the circumstances that brought the salsa to him might have been), since we don't want his enjoyment to be influenced by great hunger etc. So, if (in the sense of conditional probability) this unlikely event ever happened, did he like it?

Believers in absolute truth (even about reality) may not like this interpretation. Very probably, Julius Caesar never came across spicy tomato salsa, so the proposition that he liked it when he did is meaningless (or defaults to ‘false’ if you insist on using words in that way). But my pronouncements about whether he liked or would have liked something are based on my knowledge, and I don't (and never will) know with 100% certainty that he didn't ever try any, so this conditional probability is meaningful to me.

By the way, I'm not arguing that all counterfactuals can be understand with conditional probabilities. Even in this case, I don't think that my operational interpretation captures precisely the meaning of the original; it's just a reasonable first approximation.

Posted by: Toby Bartels on November 17, 2009 5:34 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Hi Toby,

I basically like this story, except that I’d be reluctant tell it in terms of probabilities, since probability theory does all sorts of fancy things I don’t fully understand.

Instead, I’d say that the verification process you described (finding Caesar and feeding him tomato salsa, and seeing that he liked it) is what constitutes a proof of Caesar liking tomato salsa. Then, this example just happens to be a proposition whose proof we cannot produce, due to practical considerations. So we cannot ascribe it a definite truth value, even though it’s a meaningful proposition. In other words, I think a certain form of verificationism is compatible with constructive reasoning.

I should add I don’t want to simply replace classical logic with constructive logic as the One True Logic. There are good mathematical ideas that intuitionism can’t naturally accomodate (e.g., the view that natural numbers are finite but without a definite upper bound) in the same way that there are ideas that classical logic stumbles over (e.g., the indivisibility of the continuum).

Posted by: Neel Krishnaswami on November 17, 2009 2:02 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I’m OK with some vagueness in our concepts of “knowledge” or “understanding”, which seem inherently quite psychological and complicated (as well as logical, physical and epistemological), but I don’t want that much vagueness to adhere to something as basic as “truth”.

It's clear from what you've written that you don't want that much vagueness in truth, but I don't see how truth could possibly be less vague, in practice, than knowledge.

Posted by: Toby Bartels on November 17, 2009 5:32 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Toby said:

It’s clear from what you’ve written that you don’t want that much vagueness in truth, but I don’t see how truth could possibly be less vague, in practice, than knowledge.

But I don’t see how truth could be more vague than knowledge, since the concept of truth is prior to that of knowledge. (Knowledge being “true justified belief” or some approximation.)

Posted by: Tim Silverman on November 17, 2009 9:48 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

since the concept of truth is prior to that of knowledge. (Knowledge being “true justified belief” or some approximation.)

Here it is!
Where your (arbitrary) premisses explain the rest of your stance.
I for one (and I expect not to be alone) see knowledge as prior to truth.
Truth is a fairly elaborate concept compared to knowledge.
I see rats and jellyfish and even bacteria holding some form of “knowledge” without having anything of a concept of truth or even any “concept” at all.

Posted by: J-L Delatre on November 17, 2009 10:23 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Knowledge being “true justified belief” or some approximation.

Well, that's what Plato said. I don't think that I've read his argument, so maybe he has a good one, but the idea that truth is more basic than knowledge doesn't match my thinking.

Posted by: Toby Bartels on November 17, 2009 7:58 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

If I remember correctly, “alethe” is not “truth”, but “untwisted” and “unskewed”. It is more an esthetic concept.

Posted by: Thomas on November 17, 2009 11:07 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Re: Alethe

I always thought it was something like undying, immortal, eternal, enduring, abiding.

Posted by: Jon Awbrey on November 18, 2009 4:36 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Heidegger translated it as “un-hiddenness (Unverborgenheit)”, but I guess that is wrong, perhaps intentionally (H. Arendt wrote her tale of Heidegger as fox about the dishonesty issue in his philosphy). “Unwithdrawn” would fit better. That connects with the religious roots and what W. Benjamin called “Aura” (induced by his Hoederlin studies, Hoelderlin had it from his try to reconstruct ancient greek mentality). Initially, the “aura” showed in temple slaves, whose blood attracted spirits who then could be eavesdropped by the priests, and the priests then interpreted what they heard with “logos”. In philosophers, both roles were united in one person, whose performances therefore contained paradoxa as seal of quality.

Posted by: Thomas on November 18, 2009 11:35 AM | Permalink | Reply to this

Re: The Deadly Hallows

Yes, the closest thing to a literal meaning that Perseus gives is a privative of λήθω = λανθάνω, “not escaping notice”, thus “unconcealed”.

Then again, for the Greek, to be taken in battle, to die, is a form of concealment in the underworld of Hades. Cf. Hell.

Posted by: Jon Awbrey on November 18, 2009 3:45 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

John said:

I can imagine a “mentalist” philosophy that believes primarily in the existence of minds, and secondarily in anything that any mind perceives.

But then you hit the other problem I raised: what reason do I have to believe in the existence of other minds? My reason to believe in John Baez’s mind comes largely from his writings on the internet, which I obtain by staring at patterns on computer screens—that is, by sensory experience of material objects. (Since I’ve also met him in person, I also have independent evidence—but again from sensory experience of material objects (shaking his hand and so forth)—that his is specifically a human mind, though that’s something I already strongly suspected on the basis of other evidence and my general knowledge of how the material universe works in this neck of the woods.)

I don’t see how my belief in other minds can justifiably be stronger than my belief in the (sensorally observed, material, inanimate) evidence for them.

And if I believe that my computer exists only because I’m observing it (or that truths about it only hold because I’m observing them), then, since the evidence for other minds goes through my observation of my computer screen, my experience of shaking people’s hands, etc, I should believe the same—or something even weaker—about other minds. So we’re back to solipsism again.

Posted by: Tim Silverman on November 16, 2009 11:07 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

So we’re back to solipsism again.

I am afraid that, willy nilly, we cannot hope for more than solipsism, except just like Salvador Dali opting for Critical Paranoia we can probably fare quite healthily with “critical solipsism”.
Our inner experience has no other witness than ourself and out of it we build everything else, but we can reasonably assume that the similarities we see to other people are based on more deeply grounded causes.
We hypothesize some “reality” but we cannot gather any better evidence of this reality than the “coincidences” we see about persistence, consistency and reproducibility of perceived phenomenons.
My own hypotheses are even beyond dualism into some kind of “trialism” (?), there is 1) some independent reality “out there” from which 2) our minds are an outgrowth as well as many other “things” and with our minds we build 3) images of all this stuff, including (some) parts of our minds too.

Posted by: J-L Delatre on November 16, 2009 2:24 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

J-L Delatre fascinated me with his belief that:

I am afraid that, willy nilly, we cannot hope for more than solipsism

You are entitled to this point of view, which I cannot fault on grounds of incoherence, but personally I always feel, if you’re going to be a solipsist, why not go the whole hog and accept Humean scepticism.

I’ll need to think about this some more, though. I’ve never argued with anyone who thought I didn’t exist before.

Posted by: Tim Silverman on November 16, 2009 6:50 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I’ve never argued with anyone who thought I didn’t exist before.

I guess it can be disconcerting at first. I can't speak for J-L Delatre, but if it makes you feel any better, I believe that you exist in reality. It's just that I don't believe that reality (the real world) is ontologically fundamental.

Posted by: Toby Bartels on November 17, 2009 5:35 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Toby said:

I believe that you exist in reality.

Er … thanks, I guess.

It’s just that I don’t believe that reality (the real world) is ontologically fundamental.

That strikes me as, if anything, even odder. Remind me again, what do you believe is ontologically fundamental?

Posted by: Tim Silverman on November 17, 2009 9:52 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

It’s just that I don’t believe that reality (the real world) is ontologically fundamental.

That strikes me as, if anything, even odder. Remind me again, what do you believe is ontologically fundamental?

I guess that I haven't said so on this thread, sorry.

When discussing ontology with philosophers, I am a hard solipsist. What is fundamental is my own perceptions (including memories, dreams, etc). Reality, like mathematics, is a story that I tell to tie these together into what seems like a coherent narrative (to ‘explain’ them). Tim Silverman and Toby Bartels are both features of this story, so they are both ‘real’. (The latter, as it happens, is particularly directly connected to what is fundamental; one might say that Toby ‘has’ these perceptions.)

Mathematics is a different story, so mathematical objects are not real, although (as Mike Shulman pointed out to me a while ago) it is also important to look that the morphisms between the stories, so it's not like mathematics is irrelevant to reality either.

Posted by: Toby Bartels on November 17, 2009 8:07 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

When discussing ontology with philosophers, I am a hard solipsist. What is fundamental is my own perceptions (including memories, dreams, etc). Reality, like mathematics, is a story that I tell to tie these together into what seems like a coherent narrative (to “explain” them).

This pretty much summarize my own position as well.
Interesting that there is such a close match in spite of (I guess) wildly different life histories.

Posted by: J-L Delatre on November 18, 2009 4:46 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

For those of you who followed my Platonic link to the EMS Newsletter, you may have noticed Davies response to Gardner. It usefully provides mention of several other approaches to the issue of Platonism, but Davies himself reminds me of the currently popular ‘new atheists’. He even accuses Mazur of being a ‘theist’ and seems to imply some deadly consequence of believing in Platonic realism. But gives no examples. All I can think of is such a belief leading to claiming to know what is and what is not ‘mathematics’. Cf. my first year put down by a more analytic fellow student - ‘Topology? That’s not mathematics’, but see also the article about Weierstrass in the same issue.

Are we so sure that we know what is ‘real’? Without that, what are we discussing?

Posted by: jim stasheff on November 18, 2009 2:23 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Are we so sure that we know what is ‘real’? Without that, what are we discussing?

I have absolutely no worries about what is real (in what appears to be the meaning you want to use) because I will never know nor will you, thus I just keep updating my “reality” and this seems good enough.

Posted by: J-L Delatre on November 18, 2009 5:47 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

[…] seems to imply some deadly consequence of believing in Platonic realism. But gives no examples. All I can think of is such a belief leading to claiming to know what is and what is not ‘mathematics’.

Yes, there is a certain danger there, although John Baez's version of platonism, at least, is free of it.

Constructive mathematics seems a little weird if you believe that the law of excluded middle (and even more for the axiom of choice) is simply true. Why not allow yourself to use known facts? It can be understood now as part of logic, topos theory, or the like, but people might still argue that, say, constructive analysis is really logic and not analysis at all.

Even stranger is mathematics that accepts classically false axioms, such Brouwer's principle of continuous choice¹ or Markov's version of the Church–Turing thesis². Taken literally, a mainstream mathematical realist will consider these axioms simply false and the theorems in these field unproved (and often also false). Yet such mathematics can be very beautiful, as much so as the beautiful results (such as they are) of the axiom of choice.

Simple constructive mathematics is getting respect now, but classically false mathematics is still very undeveloped. (The main exception that I know of is Synthetic Differential Geometry; although this can be understood as topos theory, people aren't afraid to write about spaces and maps as such, without always thinking of them as objects and morphisms. But it is not general mathematics as Brouwer and Markov had in mind.) Someone should write a book of examples to put it in forwards in people's minds. I have a title for them: ‘Non-Cantorian paradises’.

  1. Every entire relation from \mathbb{N}^{\mathbb{N}} to \mathbb{N} contains a continuous function (using the discrete topology on \mathbb{N} and then the product topology on \mathbb{N}^{\mathbb{N}}).

  2. Every partial function from \mathbb{N} to \mathbb{N} is computable in the Church–Turing sense.

Posted by: Toby Bartels on November 18, 2009 9:12 PM | Permalink | Reply to this

geometry and logic

Simple constructive mathematics is getting respect now, but classically false mathematics is still very undeveloped. (The main exception that I know of is Synthetic Differential Geometry; although this can be understood as topos theory, people aren’t afraid to write about spaces and maps as such, without always thinking of them as objects and morphisms.

I see what you mean and have nothing really to add to it, but will take this as an opportunity to make the following remark:

I believe that the idea of synthetic differential geometry would be way more wide-spread if the logic-perspective on it were de-emphasized and the relation to standard constructions in geometry more emphasized.

Just recently I spoke to somebody well familiar with algebraic geometry, who said he’d never wanted to look much into SDG as that would involve accepting weird logic.

In other words: here is somebody well familiar with one of the central models of the axioms of SDG, uses them all day, but when asked thinks that SDG is something arcane that he can’t possibly find the leisure to try to understand.

And of course the reason is that every second mathematician is well familiar with thinking about any particular topos. They do it all the time. What is less familiar is the change of perspective that makes one think inside the topos. But if you don’t want to do that, the loss of not doing it is moderate (unless you have really good reasons for doing it in the first place, in which case you’re the kind of person who won’t be mixed up about anything here anyway).

And this way maybe I can come back to Toby’s statement after all: generally speaking, I think that all this non-classical math, constructive math, is easily sold to everybody in the world when you start by presenting it externally and only later, when people are familiar with this, point out that there is the internal poinr of view, too.

Everyone doubting that doubting the axiom of choice is sensible will readily believe that not every epimorphism of sheaves has a global section. And that’s really all there is to it, largely. Everything else is just change of perspective. Useful as that may be to somebody at some point of his personal development.

Posted by: Urs Schreiber on November 19, 2009 1:08 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

OK, I think I see the point now. It reminds me of the question of where the visual perception takes place. Is it in the retina? No, the retina is to dumb for that. Is it in the primary visual cortex? Well, no, since no light actually reaches it and in any case, there is further processing of the information from the primary visual cortex, so it’s not the end of the line etc.

Is this is what’s being meant by John’s rejoinder?

But it still doesn’t detract from Todd’s point that

” An unobserved “truth,” truth in a vacuum if you will, is no truth at all.”

We are just arguing about what one could reasonably mean by a truth being observed. I don’t think it has to lead to a recursion or to imaginary elves.

If you press me hard enough, I would probably say that truth is a social construct, and not something absolute or eternal or existing outside spacetime etc.

Posted by: Eugene Lerman on November 13, 2009 10:03 PM | Permalink | Reply to this

GR as a foundation for Spatiotemporal Logic? Re: The Wand Chooses The Wizard

I agree that “there is a problem with saying ‘truth of assertions becomes manifest at points in time.’ There are no ‘points in time’: in our universe the closest approximation is a point in spacetime.”

So, is GR a foundation for replacement of Temporal Logic with Spatiotemporal Logic? ‘Temporal Logic’ is used to describe any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. It is sometimes also used to refer to tense logic, a particular modal logic-based system of temporal logic introduced by Arthur Prior in the 1960s. Subsequently it has been developed further by computer scientists, notably Amir Pnueli, Yde Venema [formal description of syntax and semantics, questions of axiomatization. Treating also Kamp’s dyadic temporal operators (since, until)] et al.

The logical language of Tense Logic [which lives in a Newtonian Time] contains, in addition to the usual truth-functional operators, four modal operators with intended meanings as follows:

P “It has at some time been the case that …”
F “It will at some time be the case that …”
H “It has always been the case that …”
G “It will always be the case that …”

How should we combine that with worldlines in Special Relativity, let alone with more complicated trajectories in GR?

Has anyone effectively made a athematical Physics Logic in which one can discuss Frame Dragging of Truth Values?

Posted by: Jonathan Vos Post on November 14, 2009 5:49 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Well, as John advised, I reflected a bit on what I remember of Plato. Then I realized that Todd was right. The position I formulated was a softcore version of Platonism. After all Plato does indicate in various places that the world of forms is *more* real than the sensible world. But still, I don’t suppose he would have insisted that every mathematical concept belongs to that realm.

Posted by: Minhyong Kim on November 10, 2009 11:27 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Minhyong wrote:

But still, I don’t suppose he would have insisted that every mathematical concept belongs to that realm.

There’s an amusing passage in the Parmenides where Parmenides asks Socrates if there are Forms — or ‘ideas’ — corresponding to hair, mud, or dirt:

While Socrates was speaking, Pythodorus thought that Parmenides and Zeno were not altogether pleased at the successive steps of the argument; but still they gave the closest attention and often looked at one another, and smiled as if in admiration of him. When he had finished, Parmenides expressed their feelings in the following words:-

Socrates, he said, I admire the bent of your mind towards philosophy; tell me now, was this your own distinction between ideas in themselves and the things which partake of them? and do you think that there is an idea of likeness apart from the likeness which we possess, and of the one and many, and of the other things which Zeno mentioned?

I think that there are such ideas, said Socrates.

Parmenides proceeded: And would you also make absolute ideas of the just and the beautiful and the good, and of all that class?

Yes, he said, I should.

And would you make an idea of man apart from us and from all other human creatures, or of fire and water?

I am often undecided, Parmenides, as to whether I ought to include them or not.

And would you feel equally undecided, Socrates, about things of which the mention may provoke a smile? — I mean such things as hair, mud, dirt, or anything else which is vile and paltry; would you suppose that each of these has an idea distinct from the actual objects with which we come into contact, or not?

Certainly not, said Socrates; visible things like these are such as they appear to us, and I am afraid that there would be an absurdity in assuming any idea of them, although I sometimes get disturbed, and begin to think that there is nothing without an idea; but then again, when I have taken up this position, I run away, because I am afraid that I may fall into a bottomless pit of nonsense, and perish; and so I return to the ideas of which I was just now speaking, and occupy myself with them.

Yes, Socrates, said Parmenides; that is because you are still young; the time will come, if I am not mistaken, when philosophy will have a firmer grasp of you, and then you will not despise even the meanest things; at your age, you are too much disposed to regard opinions of men.

Posted by: John Baez on November 11, 2009 1:46 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Yes, I sometimes feel doubtful about log[log[sin(log(log(cos(v)+tan 2(v))e e v+2 99+arcos(3.65995v 2)dv]].\log[\log[\int \frac{\sin (\log(\log (\cos(v)+\tan^2(v))}{e^{e^{v+2^{-99}}}+\arcos(3.65995v^2)} dv]]. Perhaps I’ll become more tolerant with age.

Posted by: Minhyong Kim on November 11, 2009 2:06 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Fo some reason, that passage also reminds me of Pablo Neruda and his generically romantic selections of ‘common things’.

Posted by: Minhyong Kim on November 11, 2009 2:17 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

“But the brand of Platonism I find most interesting is that espoused by Plato.”

I too find that more interesting. E.g. I have no clear idea about their concept of numbers. And if such theoretical issues were really so important for the antique Platon scholars and schools. Interesting too is how the contact to iranian/other eastern philosophy in late antiquity modified late antique Platonism and if/how that was relevant for the development of science mentality in renaissance. A friend recommended to look at McEvilley’s “The Shape of Ancient Thought”, but that is not in the library here.

Posted by: Thomas on November 11, 2009 12:35 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

John said:

combining Platonism with a materialistic concept of existence creates a parody of Platonism that’s much too easy to refute

and approvingly quoted Wikipedia:

A Form is aspatial (outside the world) and atemporal (outside time).

That’s certainly what Plato ought to believe, from a modern standpoint. But it’s far from clear to me that it’s what he does believe. For instance, his theory of recollection requires the souls of the dead (or at least the right sort of dead) to perceive the Ideas “directly” in some sort of way, but this appears to be some sort of physical process or act, that has to occur at some sort of location in the universe (viz. the abode of the dead—Hades or the moon or wherever that might be) at some sort of time (viz. while they are dead). If souls could directly implement or reflect Ideas in some non-physical way while dead (or as Plato might say, alive), then they should also be able to do so while alive (or as Plato might put it, dead). But they don’t.

Likewise the Demiurge is supposed to be copying the Ideals in some sort of physical process which requires observing them.

This basically unresolved relation between material and immaterial entities pervades, as far as I can see, the whole Platonic tradition. Even in Plotinus, who is reasonably austere, there is this horrible vacillation just under the surface over whether hypostases are subtypes or physical productions; what appears to me to be for all practical purposes the actual materiality of the soul and its source (and destination) appears in much more overt form in writers who have been swimming in the pool of Platonist ideas without necessarily being Platonists, such as pseuso-Dionysius—or in parts of the Hermetic corpus, where the journey of the soul through states of perfection is quite overtly a journey through actual celestial spheres, at real, identifiable (albeit distant) locations within the universe. All this seems to be me to be quite in the spirit of Plato.

It’s all very well to say this is

a parody of Platonism that’s much too easy to refute

but this is the collection of ideas that actually persisted over many centuries, so a great many people obviously did not think it refuted: in fact, for those whose philosophy takes a religious turn (or whose religion takes a philosophical turn) it would seem to me that this (admittedly confused) relation between material and immaterial is precisely a particularly appealing feature of the philosophy.

Posted by: Tim Silverman on November 11, 2009 12:16 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Actually, I find it very incongruous that you should bring up Plato in this discussion of mathematical Platonism. Like Todd, I understand the latter as a strictly modern philosophy, named after Plato but not attributed to him. And as a philosophy developed by mathematicians, it deserves my serious consideration. And since I disagree with it, it's naturally what I would attack.

Plato himself is another matter. For one thing, he's not a mathematician, so that's sufficient explanation if his ideas of mathematics don't resonate with me. (In contrast, David Corfield, while not exactly a mathematician, understands and discusses the philosophical issues brought up by contemporary mathematics, which for obvious reasons Plato does not.) Indeed, whenever I read anything by Plato about any subject, it doesn't take long before it becomes clear that, however interesting and important he may be historically, he has nothing to teach me. I would not bother to attack his positions.

Posted by: Toby Bartels on November 11, 2009 5:51 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Toby wrote:

Actually, I find it very incongruous that you should bring up Plato in this discussion of mathematical Platonism. Like Todd, I understand the latter as a strictly modern philosophy, named after Plato but not attributed to him. And as a philosophy developed by mathematicians, it deserves my serious consideration.

So who are these mathematicians, and who among them most ably argues in favor of Platonism?

I’m sorry that Plato has nothing to teach you. I find a lot of ancient philosophy more interesting than modern philosophy, because thinking hadn’t congealed quite so much, or gotten locked into its current patterns — like the separation of thought into ‘disciplines’ corresponding to academic departments, and all that. With the Greeks I feel like they were thinking about everything for the first time — fresh. A lot of modern stuff smells stale by comparison.

For example, I find it very refreshing that Plato is interested in mathematics as a tool for the cultivation of the soul — an aspect rather neglected in modern philosophy of mathematics!

Posted by: John Baez on November 11, 2009 8:40 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

So who are these mathematicians, and who among them most ably argues in favor of Platonism?

David cited Bernays. I don't know enough to confirm or deny that Bernays argued it most ably.

Of course, we could also take your philosophy as the one on the table. J-L Delatre is already asking the questions that I would ask to clarify it.

To clarify: I do find Plato interesting, just historically rather than philosophically. And there are ancient philosophers whom I find interesting philosophically (Laozi, the Cynics, Qoheleth, …), but not Plato.

Posted by: Toby Bartels on November 11, 2009 4:56 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Toby wrote:

David cited Bernays. I don’t know enough to confirm or deny that Bernays argued it most ably.

You’re actually confirming my impression that of all the people arguing against platonism in modern mathematics, few if any have read anything by anyone who supports this position.

Indeed, if you read the article by Bernays that David cited, you’ll see that Bernays scarcely argues for platonism – and skims very quickly over what platonism even means! Maybe he wrote more elsewhere. But he doesn’t sound like much of a platonist here.

He writes:

If we compare Hilbert’s axiom system to Euclid’s, ignoring the fact that the Greek geometer fails to include certain [necessary] postulates, we notice that Euclid speaks of figures to be constructed whereas, for Hilbert, system of points, straight lines, and planes exist from the outset. Euclid postulates: One can join two points by a straight line; Hilbert states the axiom: Given any two points, there exists a straight line on which both are situated. “Exists” refers here to existence in the system of straight lines.

This example shows already that the tendency of which we are speaking consists in viewing the objects as cut off from all links with the reflecting subject.

Since this tendency asserted itself especially in the philosophy of Plato, allow me to call it “platonism.” The value of platonistically inspired mathematical conceptions is that they furnish models of abstract imagination. These stand out by their simplicity and logical strength. They form representations which extrapolate from certain regions of experience and intuition.

And that’s about it, as far as explaining ‘platonism’.

Now, first of all, I think it’s strange to hold up Hilbert’s use of the word ‘exists’ in axiomatic geometry as an example of ‘platonism’ — except in a very diluted sense of ‘platonism’.

I have trouble imagining that Hilbert was asserting lines “exist in the system of lines, cut off from all links with the reflecting subject”. I would instead imagine Hilbert using the word ‘exists’ in what Todd called a ‘formal’ sense, governed by certain rules of logic like

ϕ(a)xϕ(x) \phi(a) \implies \exists x \phi(x)

In short, I imagine Hilbert as a formalist rather than a platonist.

Second, I don’t see a worked-out philosophical theory here — certainly no arguments that mathematical objects ‘really exist’, or analysis of what it means to ‘really exist’.

Instead, I get the feeling that Bernays regards platonism as a helpful attitude to take: “the value of platonistically inspired mathematical conceptions is that they furnish models of abstract imagination.” This is not far from what you folks have been calling ‘fictionalism’.

And it’s amusing how he says that platonism treats mathematical objects as “cut off from all links with the reflecting subject”, but says their value is that they “furnish models of abstract imagination”.

In short: the arguments against mathematical platonism feel to me like shadowboxing — a one-sided fight against an imaginary opponent. When I ask people to dig up an actual platonist, David obliges, but this platonist doesn’t seem much like a platonist, and nobody seems to have read what he wrote.

Perhaps some anti-platonist mathematicians are actually arguing against their own ‘emotional platonism’. We feel mathematical objects exist independently of us, but we ‘know’ that they can’t. This is what Manin seems to be saying.

Posted by: John Baez on November 11, 2009 6:13 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Well, I'm afraid that I don't find Bernays's arguments very coherent to begin with. First he states the canard (or maybe not a canard, if we're not sure what he means by ‘platonism’) that platonism implies the law of excluded middle; by page 7, he moves on to the converse of this, saying that one step (a necessary step, it seems from context, but maybe he meant sufficient instead) in rejecting platonism is to become a constructivist. I don't see how this makes any sense using his explanation of ‘platonism’, although I can somewhat understand it if ‘platonism’ is vaguely defined to simply mean orthodoxy.

But if that's the first description of the concept, then it's unsurprising that it might not be very well worked out. If you ever see me arguing against ‘mathematical Platonism’, I'm probably thinking of something like this (Stanford Encylopedia).

Posted by: Toby Bartels on November 11, 2009 6:49 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

“mathematics as a tool for the cultivation of the soul”: Does that work? E.g. “… up to 80 percent of high-achieving high school students and 75 percent of college students admit to cheating …”, how does that correlate with and how is that influenced by math education?

Posted by: Thomas on November 11, 2009 5:28 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Certainly mathematics as a tool for the cultivation for the soul won’t work if we never mention this idea and put all our emphasis on getting the right answer — or still worse, getting good scores on exams.

Posted by: John Baez on November 11, 2009 6:18 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

This is rather fascinating in the light of a book I’m reading, Masters of Theory by Andrew Warwick, about the teaching of mathematics at Cambridge in the 19th century, and in particular, the complete takeover by written exams, which came to drive every aspect of teaching. In the middle of the century, conservatives such as Whewell complained bitterly that examinations were destroying the traditional goal of Cambridge mathematics which was to cultivate the soul (and, perhaps more importantly, the character). By cultivating the soul Whewell and his allies meant, in truly Platonic fashion, the inculcation of aristocratic virtues and attitudes, particularly ideals of public service, into the next generation of judges, parliamentarians, landowners and clergymen, etc. In the eighteenth century, degrees were still granted on the basis of a verbal disputation, as had been true since the universities were created in the middle ages, and this made public speaking a key virtue. With written exams, the ability to speak fluently and plausibly in public was replaced by the ability to solve technical problems rapidly under pressure, an ability derided as too middle-class and practically-oriented.

In the last quarter of the nineteenth century, as the British aristocracy entered its terminal decline, these middle-class values became triumphant, and Cambridge virtuosity in mathematical physics became widely admired abroad and emulated throughout the Empire. In the early years of the twentieth century, pure mathematicians insisted on separating themselves off from mathematical physics, the exam was reformed, and the problem-solving tradition went into a slow decline.

At its height, from about 1820–1900, the mathematics examination was so central to Cambridge mathematics that it was quite normal for Cambridge researchers to publish their results in the form of exam questions. I mean literally that the first appearance of a new research result would be as a question on that year’s Tripos. Likewise, several mathematical physicists from the Cambridge tradition described research as being essentially a process of converting natural phenomena into exam questions.

Another peculiar feature of this period was that the official teaching infrastructure was so slow to catch up with the changes brought about by the growing importance and difficulty of the writen examination that students of any ambition derived all their knowledge from private coaches (this sense of the word “coach” derives from this period, though it probably originated in Oxford), and didn’t attend official college and university lectures since these were useless.

Posted by: Tim Silverman on November 11, 2009 8:28 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

And these coaches would prescribe a very strict program for their charges, down to details of diet and exercise. This was necessary to cope with the extraordinarily gruelling nature of the exams. Wikipedia tells us

In 1854, the Tripos consisted of 16 papers spread over 8 days, totalling 44.5 hours.

Posted by: David Corfield on November 12, 2009 11:50 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Indeed, many of them became quite accomplished amateur athletes; on the other hand, having a nervous breakdown was recognised as a common danger of studying without adequate exercise. The whole system is kind of amazing to behold, though not one I would want to be part of; it puts one in mind of the top-level performers of extreme sports, except in pursuit of a productive (or “productive”) goal, a sort of industrial problem-solving process in mathematics. It’s all very Victorian.

Posted by: Tim Silverman on November 12, 2009 1:43 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

And Edward Routh, the greatest of all the coaches, was, during busy terms, teaching 48 hours a week! Plus 2 hours a day of exercise to keep his strength up. Sometimes, when I contemplate Victorian institutions, I feel like a Visigoth looking at the Colosseum.

Posted by: Tim Silverman on November 12, 2009 1:55 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Sometimes, when I contemplate Victorian institutions, I feel like a Visigoth looking at the Colosseum.

Yes, I too am filled with an overwhelming urge to destroy. (^_^)

Posted by: Toby Bartels on November 12, 2009 5:39 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

<Splork!>

Posted by: Tim Silverman on November 12, 2009 6:41 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

When a thread gets huge, like this one, I hit that little button that says ‘view chronologically’. But that doesn’t always work.

We were having a big philosophical argument, and then Tim Silverman asserted:

Splork!

Hmm. Only later did I realize he was imitating a Goth throwing a rotten tomato at the Colosseum.

Posted by: John Baez on November 12, 2009 7:32 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

JB chided:

We were having a big philosophical argument, and then Tim Silverman asserted:

Splork!

Apologies. I will try to remember to quote what I’m replying to.

Posted by: Tim Silverman on November 12, 2009 7:50 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

cultivating the soul or character?
when I lived in college (Brasenose) my first year as a grad student at Oxford
one Don was designated officially as my `Morals Tutor’

Posted by: jim stasheff on November 12, 2009 2:50 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Jim Stasheff said:

when I lived in college (Brasenose) my first year as a grad student at Oxford one Don was designated officially as my ‘Morals Tutor’

I seem to remember something like this when I was at Cambridge, too, though the years have eroded the details from my memory.

Posted by: Tim Silverman on November 13, 2009 10:34 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Since the Platonist’s “World of ideas” potentially contains everything and anything (re the Rado Graph) could not it be said that rather than being “discovered” or “invented” the math structures are choosen over less interesting contraptions.
It seems to me that Manin emphasize that with his distinction between programs and (bad) problems.

Posted by: J-L Delatre on November 8, 2009 11:34 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

TT: . So for example, the precise conditions on the category of sets which enable the construction of a bicategory of relations is revealed, after some analysis, to be axioms as ensconced in the notion of regular category (the newly minted concept). Do we say that we ‘discovered’ regular categories?

Apparently, yes - we now “see” what was unseen before.

At least you said in posing this very example that the construction “is revealed”. This means that it existed beforehand in veiled form, and the veil is now lifted.

For a non-platonist, nothing can be revealed since it doesn’t exist before construction.

Posted by: Arnold Neumaier on November 9, 2009 1:41 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Yes, Arnold, I already said in the part that you snipped that one could say there was a process of logical discovery at work. The words “is revealed” were selected by me in full consciousness of the echoes they would have for those of a Platonist mindset, but you see, in this case I gave them a slightly different meaning: they refer more to the results of a logical analysis which turns up consequences and deductions which were not considered before. That’s hardly the same as subscribing to the ontological commitments of a Platonist.

I think you’d have to be a straw-man non-platonist to seriously maintain that mathematical practice is all about invention and never about discovery, especially when speaking at the level of human emotions, which is what we were talking about. The specific type of discovery I was referring to in my example is on a linguistic and logical level, about unsuspected or half-suspected consequences of some set of conditions being considered, consequences whose disclosure may indicate that this is a set of assumptions well worth putting forward. The act of putting it forward is however an act of human creativity and inventiveness at a specific time and situation and preparation within culture; it is not something which suddenly appears as if handed down from God.

Posted by: Todd Trimble on November 9, 2009 2:46 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

Arnold: while I’m on the subject of selective quotation: it would be nice and courteous if, when selectively quoting from comments which are back a ways in an ever-growing thread, you would link back to them, so that you have readers just a click away from deciding whether you are quoting from and replying to comments fairly, instead of making them hunt the comments down.

If you right-click on ‘Permalink’ at the bottom of the comment you are replying to, you can go to the menu option “Copy link location”. Then, to create a link in your post, you type

<a href=“[link location]”>[linking text]</a>

where you just paste in the copied link location between the quotes using your right mouse button.

Posted by: Todd Trimble on November 9, 2009 3:34 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

and for those of us without a mouse much less a right mouse button?

Posted by: jim stasheff on November 10, 2009 1:10 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

I don’t have a mouse per se myself, so ‘mouse’ is being used in a generalized sense here. Whatever you use to click on things, there ought to be a left-click button/control and right-click one. I have these things right by my touchpad that I use to move the cursor.

Posted by: Todd Trimble on November 10, 2009 1:31 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

For a non-platonist, nothing can be revealed since it doesn’t exist before construction.

In this statement you are (implicitly) trying to enforce your meaning of the word “exist” onto non-platonists because you cannot fathom that others can have a different construance than yours on such a “basic” concept.
If you could you would not be a platonist.
This is why such discussions are spinning in circles forever.
To a non-platonist existence is only about checking if some fancy idea he came up with stands to consistency within the realm of his current discourse.
It is only an analytical truth not an absolute ontological one, ontology is only a technical scaffolding subservient to the “current theory”.
In the end this amounts to… definedness!
And yet, to the non-platonist existence is “revealed” too because before he checked he had ZERO EVIDENCE pro or con.

Posted by: J-L Delatre on November 10, 2009 10:45 AM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

For example, if a student asks for an example of two noncommuting matrices, we may just toss something off quickly: invent a counterexample.

Assuming we're not pressed for time, I would do this differently. I would call on 8 students in turn to each ‘randomly’ name a number, arrange them in order into 2-by-2 matrices, and then check to see if these matrices commute. (My point would be that most pairs of matching square matrices don't commute, and so I can be reasonably confident that the result of this process will be noncommuting matrices.)

It doesn't feel like such matrices are either invented or discovered, indeed not chosen in any way (except in the sense that humans' picking of random numbers is probably really only pseudo-random). They are simply pulled out of a hat!

Posted by: Toby Bartels on November 9, 2009 7:05 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

The question is now posted on mathoverflow too.

Conc. Platonism: Giordano Bruno apparently had an interesting variant of it, described in his dialogue ‘On Heroic Frenzies’. But I don’t think I understand really what he had in mind.

Posted by: Thomas on November 9, 2009 3:56 PM | Permalink | Reply to this

Re: The Wand Chooses The Wizard

MO shows that the issue is a bit very unpopular.

Posted by: Thomas on November 9, 2009 10:57 PM | Permalink | Reply to this

Re: Interview with Manin

A short but sort of sweetly written essay by Barry Mazur, Mathematical Platonism and its Opposites. Has someone already mentioned this?

Posted by: Todd Trimble on November 12, 2009 2:53 AM | Permalink | Reply to this

Re: Interview with Manin

I don’t think so. It’s nice. However, I disagree with his claim that:

If we adopt the Platonic view that mathematics is discovered, we are suddenly in suprising territory, for this is a full-fledged theistic position. Not that it necessarily posits a god, but rather that its stance is such that the only way one can adequately express one’s faith on it, the only way one can hope to persuade others of its truth, is by abandoning the arsenal of rationality, and relying on the resources of the poets.

If this were true, then I think it would also be true of materialism: the idea that material objects are real, and that truths about them are discovered.

In both cases, people repeatedly encounter certain objects — mathematical objects or material objects — and find that they have persistent properties: properties that don’t change just because one wants them to. In both cases, one may conclude that these objects are real — that they have an existence independent of us, and that when we first encounter them we are discovering them rather than ‘making them up’.

In both cases it would be very hard to convince a determined skeptic of this. Does a physical object or mathematical object exist before anyone has witnessed it? I see no way to prove this.

However, I don’t think we need to ‘abandon the arsenal of rationality’ when trying to convince people that it makes good sense to attribute an independent existence to material or mathematical objects. Mainly what we need is a careful analysis of our concepts.

For example: what we do we mean when we say we’re ‘attributing independent existence’ to something? It’s a very tricky, problematic concept. But I think if we start to understand what it means, we’ll see why people want to do it, and why they’re actually justified in doing it, both for physical and mathematical objects.

So, in trying to convince someone of Platonism, the last thing I’d want to do is ‘abandon the arsenal of rationality’. I’d want to expand it!

Posted by: John Baez on November 13, 2009 3:03 AM | Permalink | Reply to this

Re: Interview with Manin

“the last thing I’d want to do is ‘abandon the arsenal of rationality’. I’d want to expand it!”

An analogy: Classical greek sculptures were about visualizing ‘abstract’ concepts; 19th century scholars concluded that this forces them to have been colourfree because ‘abstractionated’=’reduced’, ancient greeks made them probably for the same reason very colourfull because for them ‘abstractionated’=’enriched’.

Posted by: Thomas on November 13, 2009 10:26 AM | Permalink | Reply to this

Re: Interview with Manin

I did allude to Barry Mazur’s article back here.

Something that bothers me about commonly adopted philosophical treatments of forms of enquiry, such as mathematics, is the overwhelming attention given to truth as a property of linguistic assertions. One might expect that this predominance would only be justified if the ends of enquiry were primarily the compilations of true assertions. I certainly don’t take mathematics to have this as its primary end.

I prefer to think of enquiry aiming toward the ‘adequacy of mind to its objects’, something described here by Alasdair MacIntyre

…it is important to remember that the presupposed conception of mind is not Cartesian. It is rather of mind as activity, of mind as engaging with the natural and social world in such activities as identification, reidentification, collecting, separating, classifying, and naming and all this by touching, grasping, pointing, breaking down, building up, calling to, answering to, and so on. The mind is adequate to its objects insofar as the expectations which it frames on the basis of these activities are not liable to disappointment and the remembering which it engages in enables it to return to and recover what it had encountered previously, whether the objects themselves are still present or not. (Whose Justice? Which Rationality?, p 56)

Making this account appropriate for mathematics there is still some work to do, but I don’t take the intangibility of mathematical objects to be a major problem. At the very least we can all agree that we know what it means for one person’s mind to be more adequate to mathematics than another’s, and this doesn’t consist in the ability to spout a superset of truths.

Posted by: David Corfield on November 13, 2009 9:24 AM | Permalink | Reply to this

Re: Interview with Manin

Something that bothers me about commonly adopted philosophical treatments of forms of enquiry, such as mathematics, is the overwhelming attention given to truth as a property of linguistic assertions.

Yeah, I have opinions about all of this stuff, and it's fun to waste my time on it, but it's kind of a shame that professional philosophers should be distracted by it.

Posted by: Toby Bartels on November 13, 2009 6:12 PM | Permalink | Reply to this

Re: Interview with Manin

I wasn’t intending to suggest it was a waste of time. In fact the right kind of thinking about such matters often leads to mathematical insights. Somewhere (nLab - when discussing its scope) we touched on the idea that constructive considerations tie in often with category theoretic ones.

It’s interesting how a certain kind of philosophical consideration does this at times, like Finsler’s coalgebraic maximalizing axiomatisation arising from a kind of realism.

Posted by: David Corfield on November 13, 2009 7:32 PM | Permalink | Reply to this

Re: Interview with Manin

I wasn’t intending to suggest it was a waste of time.

But is it useful for philosophers to argue about whether mathematical objects really exist? To study how mathematicians' opinions influence their mathematics, yes, but that's not the same thing.

Posted by: Toby Bartels on November 13, 2009 8:35 PM | Permalink | Reply to this

Re: Interview with Manin

There’s certainly proper work to be done in understanding how mathematical activity fits with other parts of our mental activity. We must piece together sensation, imagination, idea, thought, language, object, and our account had better be such as to make good sense of mathematical thinking. Whether we end by taking ‘object’ to be properly applied to whatever it is mathematics is about is not to be settled beforehand.

I would warn philosophers off from trying to say what mathematics must be about. Collingwood does the same for pronouncements about physics. We shouldn’t say things of the kind ‘physics is the study of matter’, as though there’s some preexisting conception of matter, when really it is just a (often out of date) conception which has filtered through to the philosopher.

Physics goes where physics must. To do useful work about the fundamental presuppositions of a field you must be thoroughly immersed in the discipline. If someone does not feel they can manage this, historical studies of the changes in these presuppositions is very useful work.

Posted by: David Corfield on November 14, 2009 9:55 AM | Permalink | Reply to this

instead of Interview with Manin

I can’t keep up with @20 comments a day headed Interview with Manin
how about heading further comments with some other title
so I might guess which ones to look at?

GOM = grumpy old man

Posted by: jim stasheff on November 14, 2009 6:32 PM | Permalink | Reply to this

Re: Interview with Manin

Toby wrote:

But is it useful for philosophers to argue about whether mathematical objects really exist?

I think it’s much more useful for them to sort out what different people mean when they say mathematical objects do or do not exist. And I think the smart ones do this.

David wrote:

I would warn philosophers off from trying to say what mathematics must be about. Collingwood does the same for pronouncements about physics. We shouldn’t say things of the kind ‘physics is the study of matter’, as though there’s some preexisting conception of matter…

Feynman: “Philosophers say a great deal about what is absolutely necessary for science, and it is always, so far as one can see, rather naive, and probably wrong.”

Posted by: John Baez on November 14, 2009 10:49 PM | Permalink | Reply to this

Re: Interview with Manin

“Philosophers say a great deal about what is absolutely necessary for science” - Such pseudo-science programs apparently included philosophers of science.

Posted by: Thomas on November 15, 2009 9:49 AM | Permalink | Reply to this

Re: Platonism

A recent issue of the EMS Newsletter has Martin Gardner’s take on the issue with a response by EB Davies - apparently part of an ongoing stream of discussion of Platonism:

www.ems-ph.org/journals/journal.php?jrn=news

Posted by: jim stasheff on November 14, 2009 2:56 PM | Permalink | Reply to this

Re: Platonism

The Gromov interview in the current issue is very interesting too, thanks for the link!

Posted by: Thomas on November 14, 2009 8:56 PM | Permalink | Reply to this

Re: Interview with Manin

Egad! I’m not as regular here as some of you, but isn’t it some kind of record to have built up over two hundred entries in two weeks? Clearly, truth is a matter of pressing concern…

Posted by: Minhyong Kim on November 16, 2009 10:53 PM | Permalink | Reply to this

Re: Interview with Manin

A fascinating talk by Voevodsky recently. It is about new ideas on how to deal with inconsistency issues in mathematics. The final part on his (and his coworkers’) program sounds like a formalization of the idea of mathematics as cultural activity sketched by Manin. It reminds to me a bit to how Manin and other russian mathematicians (e.g. Kontsevich, Drinfeld) took up concepts from th. physics like Feynman integrals, independent of consistency and well-definedness associated with them.

Posted by: Thomas on October 6, 2010 3:15 PM | Permalink | Reply to this

Re: Interview with Manin

You may be curious about the coming film portrait of Yuri Manin by Agnes Handwerk and Harrie Williams, known by their Doeblin docu.

Posted by: Thomas on June 1, 2011 9:13 AM | Permalink | Reply to this

New Interviews with Manin

A great collection of 28 very short interviews, covering parts of his mathematics (unfortunately not his work on p-adic modular forms), teaching students, history of mathematics, his psychology seminar, and more.

“The twentieth century return to Middle Age scholastics taught us a lot about formalisms. Probably it is time to look outside again. Meaning is what really matters.”

Posted by: Thomas on February 16, 2012 9:31 AM | Permalink | Reply to this

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