## January 2, 2008

### Two Cultures in the Philosophy of Mathematics?

#### Posted by David Corfield

A friend of mine, Brendan Larvor, and I are wondering whether it would be a good idea to stage a conference which would bring together philosophers of mathematics from different camps.

Brendan is the author of Lakatos: An Introduction, and someone who believes as I do that one of our most important tasks is the Lakatosian one of attempting to understand the rationality of mathematics through the history of its practice.

By contrast, a much more orthodox philosophical approach to mathematics in the English-speaking world, well represented in the UK, is to address the question of whether mathematics is reducible to logic. To gain an idea of the current state of play here, you can take a look at What is Neologicism? by Linsky and Zalta. You can see from the final sentence of section 1 that organisational issues, such as whether category theory is a good language for mathematics, are irrelevant to them.

Now, it could be that the town is big enough for the both of us. Just as you may choose to work on trying to discover the real story behind elliptic cohomology, and have very little interest in random graph theory, so there might plausibly be Two Cultures of philosophy of mathematics. Still, I would like to see whether it’s possible to discover why we are led to ask such very different questions about mathematics.

One issue that will inevitable arise is the extent to which it matters whether you mathematicians prefer one of the two approaches. Minhyong Kim mentioned the misconceptions about each other’s fields that can afflict scientists and philosophers. If Brendan and I receive an appreciative nod from a mathematician, e.g., here and here, neologicists could well answer that this shows nothing. We’re not here to please mathematicians. Why should they have any good notion of how philosophy ought to conduct itself?

Nor, I take it, does it matter to them that wheras the original logicism of Frege and Russell took place within spitting distance of the mathematical work of central figures such as Dedekind and Hilbert, the current neo-logicism is not even on today’s mathematical radar.

Still, I think we ought to meet up. An external view on one’s work is never a bad thing, and I certainly learned from Alexander Paseau’s review of my book in Studies in History and Philosophy of Science.

Posted at January 2, 2008 1:40 PM UTC

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### Re: Two Cultures in the Philosophy of Mathematics?

It may also be beneficial to have a conference on pure and applied mathematics.

The latter seems to have done more for civiliztion.

Perhaps this over simplifies the relation of pure and applied mathematics:
Archimedes was a mathematician and mechanical engineer.
Newton was a mathematician and mechanical engineer.
Maxwell was a mathematician and electrical engineer.
Steinmetz was a mathematician and electrical engineer.

Posted by: Doug on January 2, 2008 6:14 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Doug wrote:

The latter seems to have done more for civiliztion.

There’s a simple explanation: when pure mathematics does something for civilization that nonmathematicians understand, it gets renamed ‘applied mathematics’.

Mathematics is like a tree: you can’t have fruits without roots.

Posted by: John Baez on January 2, 2008 9:53 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

This comment reminds me of a neat description of the difference between the disciplines of Artificial Intelligence and Computer Science: Researchers in AI study very challenging problems; whenever one of these difficult problems is solved, it ceases to be AI and becomes part of CS.

Posted by: Peter on January 6, 2008 4:16 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Hi John Baez,

I meant only to equate applied with pure mathematics rather than to suggest, as I did, that one branch is superior to the other.

For example, the Princeton, Stanford and RAND applied mathematician Richard Bellman is credited with inventing dynamic programming associated with optimal control theory.
Another Bellman biography states “In those days applied practitioners were regarded as distinctly second-class citizens of the mathematical fraternity.”

John von Neumann was a balanced mathematician with 60 each pure and applied mathematic papers.

Posted by: Doug on January 10, 2008 2:12 AM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Newton was a mathematician and mechanical engineer.

Hi Doug,

I am interested in Newton in the historical context. I would appreciate if you could elabarote on your justification for calling Newton a mechanical engineer.

Thank you.

Posted by: Pioneer1 on January 3, 2008 1:13 AM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Hi Pioneer1,

Recall that I am over simplifying or using the term engineer in the most liberal sense.

See Wiki Mechanical engineering first paragraph definition.

Newton built a machine, the Newtonian telescope [reflecting] from wiki listing of advantages and disadvantages.

Newton also experimented in alchemy, one might consider him a chemical engineer, in the most liberal sense.

Posted by: Doug on January 10, 2008 1:41 AM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

I think I misunderstood you. I thought you meant engineer in the sense we use today. But still I think that Galileo will fit in that list better than Newton. Galileo wrote a book used by engineers for a long time. He was an engineer in the tradition of Archimedes. Compared to Galileo Newton was not an engineer. Newton was a world builder, yes, but that’s not really engineering, it is theory.

Posted by: Pioneer1 on January 15, 2008 2:32 AM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

This would be a wonderful conference!

I’ve published lots of math, and have taught Philosophy (Epistemology, History of Scientific Revolution, The Frontiers of Ignorance, …). I find that this blog is near the center of, and in contact with, the best current work in the intersection.

David Corfield, the studies of and about Lakatos, and the oddities of neologicism are profound.

John Baez et al adds a great reality-testing set of insights as to how this all connects with Physics.

Great stuff!

I presume this idea will be floated next week at the big Math conference in San Diego?

Posted by: Jonathan Vos Post on January 2, 2008 7:29 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Jonathan vos Post wrote:

I presume this idea will be floated next week at the big Math conference in San Diego?

I doubt it. I never go to those big AMS conferences anymore — I’m not even a member — and David and Urs are too far away.

But, it makes sense to ask, just so café regulars and lurkers can meet up if they want to:

Does anybody reading this plan to go to the 2008 Joint Mathematics Meetings in San Diego from January 6th to 9th? If so, post a comment!

Posted by: John Baez on January 2, 2008 9:59 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

[…] too far away.

This reminds me: hadn’t you also been invited to QGT08 in Kolkata next week?

The poster carries your name. And mine for that matter. But I had to cancel, due to teaching duties, unfortunately. I am very much regretting that.

Is any $n$-Café reader attending this conference?

Posted by: Urs Schreiber on January 2, 2008 10:06 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Yes, I’d been planning to attend that conference, but I’ve been feeling sort of overwhelmed by overwork, and my wife had been travelling a lot during the fall, and flying to India during the first week of class would be pretty bad for me and the students, so I decided to cancel.

I’ve been having a very productive winter break, staying home and breaking through the logjam of half-written papers that had been making me miserable lately. So, my mood is different than when I cancelled that trip. But, I’m still very glad I don’t need to fly across the world in a few days from now. Instead, I can focus on finishing up that paper with Danny on the classifying space for 2-bundles!

Posted by: John Baez on January 2, 2008 10:52 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

I’ll be there. Furthermore - I’ll be talking. Twice! And both times, it’ll be on A-infinity stuff…

I would very much like to meet up with people around this blog, and bloggers in general, and interesting people in general. I’ll be using my swedish cell phone: +46706450283 for coordination; please drop me text messages there!

Posted by: Mikael Vejdemo Johansson on January 3, 2008 4:29 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

“I’ve … taught Philosophy (Epistemology, History of Scientific Revolution, The Frontiers of Ignorance, …).”

That makes me curious. If you have some texts online would you post the links?

Thanks,
Thomas

Posted by: Thomas Riepe on January 5, 2008 12:16 PM | Permalink | Reply to this

### If we had worlds enough and time; Re: Two Cultures in the Philosophy of Mathematics?

Dear Thomas, my degrees (Caltech and Umass/Amherst) are in Math (specialty was advanced mathematical logic), English Literature, and Computer Science. Plus PhD work (Thesis, yet All But Degree) in “Molecular Cybernetics”). Hence I have not taught Philosophy for college credit. This is because amateur passion does not equate with credentials in most USA universities (or even secondary schools). I am keenly interested in the Philosophy of Science, and Philosophy of Mathematics. But, rather, as follows:

CENTER FOR THE STUDY OF THE FUTURE, Ventura, CA 1995-Present
* The Center for the Study of the Future is a “supersite” of the Elderhostel organization, which has had over 2,000,000 adult students in the past 25 years
* I report directly to the co-founders and co-chairmen
* I have taught roughly 2,000 students of average age 65
* I have taught classes in Pasadena, Monrovia, San Diego, Ventura, Costa Mesa, Westwood, and Beverly Hills
* Courses which I developed and taught include:
* “The Search for Other Earths” – astronomy and planetary science
* “Time Machines” – Physics, Philosophy, and Fiction of Time Travel
* “New Paradigms” – the Structure of Scientific Revolution
* “How Do We Know What We Know” – Epistemology and Psychophysics
* “The Frontiers of Ignorance” – unsolved problems of science
* “Undersea Living” – the biology and history of undersea habitats
* “Human Evolution” – anthropology and archeaology

My formal teaching in colleges and universities and high schools is limited to Mathematics, and Astronomy.

I’d love to develoip my very extensive notes (many digitized in Wordperfect on an antique computer no longer functioning, but I think backed up to secondary storage, diskettes, even CD-ROM) of the philosophy classes that I taught to these motivated senior citizens, and their reactions to texts I worked from, with them, by Kuhn, Lakatos, and others.

But, without institutional support, grant, or book contract, I’m sad to say that it doesn’t rise high enough in my hierarchy of priorities. This comes from having an 11-room home, 3 cars, a son in law school, and other financial pressures.

But I’d love to discuss any of this offline with you.

Posted by: Jonathan Vos Post on January 5, 2008 7:56 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Having no feeling for the landscape of academic philosophy, there’s little I can say about the professional aspect of the conference. On the other hand, the general idea of such communication seems very nice. Therefore, if you think it appropriate, I can try to be helpful in various small ways:

-If the meeting is somewhere near the London area, I would gladly be a passive participant. Especially so if having some practicing mathematicians involved helps with obtaining funding, for example. (Then again, maybe it would count against you!)

-If you need it, I can provide help with recruiting other mathematicians, for example, Michael Harris. He’s normally quite occupied with family obligations, but if it’s just for a day or two, he should be able to come over from Paris. Of course I’m assuming again that having some mathematical participation with no specifically philosophical sophistication might at least be amusing for the professionals.

-One active participant I can recommend among mathematicians is Angus Macintyre. He’s of course among the most senior of mathematical logicians in the UK. But he’s also very communicative and well-cultured on a broad spectrum of mathematical issues. He might be quite willing to provide an overview lecture on the evolution of foundational mathematics, and its relevance today. I find this a fascinating development in foundations, that mainstream mathematical logicians think of logic as just being a proper branch of mathematics, and have seen some spectacular applications of logic to number theory and algebra. I myself would definitely like to know what philosophers think about developments of this sort. On the other hand, this is perhaps just a third direction not compatible with what you have in mind. In any case, I’ll be seeing him next week, so if you’d like me to sound him out, I’d be more than happy to.

-Finally, if there might be some symbolic meaning in having the meeting actually be hosted by a math department, I could probably arrange something at UCL.

Let me know.

MK

Posted by: Minhyong Kim on January 3, 2008 12:20 AM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Brendan and I were considering inviting Michael Harris. John Baez and I met him in Delphi this July at the ‘Mathematics and Narrative’ conference.

Interesting that you mention Angus Macintyre. Alexandre Borovik and I have some funding for a small workshop ‘New Directions in Philosophy of Mathematics’ and were considering inviting him. Alexandre works on the interface of model theory and group theory.

Macintyre has written some interesting papers, including the one here (see also the Lawvere paper there). I’d like to have heard this talk.

I find this a fascinating development in foundations, that mainstream mathematical logicians think of logic as just being a proper branch of mathematics, and have seen some spectacular applications of logic to number theory and algebra.

Right. One of my pathways into philosophy was via the category theoretic idea that logic was a facet of mathematics. I remember trying to work my way through Lambek and Scott’s Introduction to higher order categorical logic before starting my Masters.

Once started, however, I was told that there’s a difference between mathematical logic and philosophical logic, and even that it’s almost a pun they share the term ‘logic’. Further, I was told that it’s philosophical logic which does the philosophical work of telling us what our ontological commitments are (what we are committed to saying exists).

The paper on neologicism I linked to is engaged in this sort of quest, coming to the conclusion that mathematics is all expressible as some portion of third-order logic. This tells us then, supposedly, what sort of entities mathematics is about, and how we can come to know about them.

I can’t say I was ever so convinced by what I was told, so stuck to the task of elaborating Lakatos’s ideas on concept-stretching. You can read Russell and feel the excitement he conveys that at last philosophy has a wonderful new instrument for resolving age old problems. Somehow or other I just never got the reason why this ‘philosophical logic’ is so special.

At the IMA workshop on n-categories I met up with Steve Awodey, who although a professor in a philosophy department sees himself as a (category theoretic) mathematical logician, and has little time for, or dialogue with, philosophical logic.

Posted by: David Corfield on January 3, 2008 2:52 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Since you two (David Corfield and Minhyong Kim) are among my favorite people to talk to, you might have fun talking to each other. Maybe you could meet up at the January 9th conference on Categories, Logic and Physics at Imperial College? I think David said he was going there…

Posted by: John Baez on January 3, 2008 8:24 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Yes, it would be nice to meet up and I’ll try to create an occasion. But unfortunately, the meeting at Imperial is exactly the day Macintyre will be visiting me at UCL. Bad planning on my part.

When I found out, I scheduled Andreas Doering for a talk at the London Number Theory Seminar, in order to compensate.

MK

Posted by: Minhyong Kim on January 4, 2008 11:24 AM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Yes, I am going to the Imperial event. We’ll have to find another occasion to meet up.

Posted by: David Corfield on January 4, 2008 1:54 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Meanwhile, perhaps I can insert a few naive remarks/questions about logicism.

–To me, it has always seemed clear that the question

Can mathematics be reduced to logic?’

is entirely analogous to

Can properties of complex physical systems be reduced to classical or quantum mechanics?’

My impression is that this analogy is in fact implicit in most mathematicians’ attitude towards logicism. That is, for the second question, we all know that there is an obvious sense in which the answer is in the affirmative. Meanwhile, this fact is not terribly interesting or practical. And then, an increasing number of people seem to feel that the impracticality is even of conceptual importance.

Therefore, I had assumed that various subtle theorems notwithstanding, the claims of logicism should be essentially valid, but in a somewhat trivial sense. So to the extent that it’s worth anyone’s while to give an account of mathematical process, the focus should be on global principles whereby aggregate mathematical reasoning emerges (to use that awful word) out of the small steps formalized in logic. But it seems many clever people get bogged down instead in innumerable examples and counterexamples.

Is this a silly viewpoint that’s already been dispensed with?

–I stress that it’s not logic itself that’s asserted to be trivial, any more than classical mechanics is.

–It occurs to me as I write that even in the strong form, the claims of logicism are not as strong as those of mechanics. As far as I know, a logicist does not presume to have *predictive power*.

–I fully understand that many philosophers of mathematics might not care much about the opinions of mathematicians themselves. A famous quip says something to the effect that Art historians are to artists as ornithologists are to birds.’ There is a standard reading of this sentiment that’s popular among artists. But I’ve understood it to express in part the irrelevance of a bird’s *opinions* on ornithology.

MK

Posted by: Minhyong Kim on January 3, 2008 1:30 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

The curious thing is how little agreement there is amongst what I called above the ‘orthodox’ in English-language philosophy of mathematics.

This paper by two proponents of the scottish form of neo-logicism marks the difference from the Zalta-Linsky form.

Frege is quoted there as showing that their programme is closer to his:

The problem becomes, in fact, that of finding the proof of the sentence, and of following it up right back to the primitive truths. If in carrying out this process, one comes only to general logical laws and definitions, then the truth is an analytic one. [… ] [If the] proof can be derived exclusively from general laws, which themselves neither need nor admit proof, then the truth is a priori.

and

In virtue of the gaplessness of the chain of inferences it is achieved that each axiom, each presupposition, hypothesis, or however else one might want to call that which a proof rests upon, is brought to light; and thus one gains a foundation for the assessment of the epistemological nature of the proven law.

Personally, I find the following quotation from Frege much more interesting:

[Kant] seems to think of concepts as defined by giving a simple list of characteristics in no special order; but of all ways of forming concepts, that is one of the least fruitful. If we look through the definitions given in the course of this book, we shall scarcely find one that is of this description. The same is true of the really fruitful definitions in mathematics, such as that of the continuity of a function. What we find in these is not a simple list of characteristics; every element is intimately, I might almost say organically, connected with others.

Posted by: David Corfield on January 3, 2008 4:14 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

What confuses me about scottish neologicism is whether it matters to them whether their logical reconstructions of portions of mathematics get to the conceptual heart of those portions.

It might be the case that this is a purely ‘in principle’ exercise, where once it has been shown that portion X is expressible with an abstraction principle and second order logic, then the job is done, and knowledge about X is revealed to be a priori.

If this is all they’re doing I don’t think they’re being very Fregean. Jamie Tappenden has papers showing how Frege, the Riemannian, cared about carving out concepts correctly. Fruitfulness is key. If a neologicist doesn’t care about the fruitfulness of their abstraction principles then they’ve rejected an enormously important part of Fregean thinking.

Posted by: David Corfield on January 4, 2008 1:52 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

I’m probably not phrasing any of my questions properly because it’s been so long since I tried even vaguely to understand these issues. Nevertheless, if some concept of fruitfulness came up in Frege, maybe I can make another attempt.

It seems to be generally agreed upon that the intuitive statement of the (neo-)logicism is

Mathematics can be reduced to logic

and I would like to better understand the meaning of this. Approaching the matter more or less as a scientist, one needs to see what non-trivial questions become associated to this statement, and what would be regarded as significant progress on these questions. So let me propose another analogy by superficially comparing logicism with the theory of universal grammar. As I understand the latter, language is regarded as a natural phenomenon, and it would be considered a major achievement just to give a complete account of the principles for checking the correctness of all sentences. Would it similarly be agreed upon among logicists that showing logic to provide a good account of mathematical correctness is a major goal? I presume however, that this is not the only goal. How much further the scope of logicism extends seems to me a significant point in coming to grips with the difference between the views of philosophical logic and practicing mathematics.

To pursue the comparison a bit further, universal grammar would not claim to understand why some sentences are better’ than others, say Shakespeare (or John Baez) over Minhyong Kim. But perhaps logicism does make this kind of a claim about the reduction of mathematics? To relate this back to the Frege quote, is your impression that Frege was using `fruitfulness’ in the sense that a naive practicing mathematician would use the word in referring to a mathematical concept? (To start, I am avoiding an explanation of what that sense is.)

By the way, I apologize if I’m imposing the repetition of views that were thoroughly thrashed out in earlier discussions. I started following this blog in a systematic way rather recently.

MK

Posted by: Minhyong Kim on January 5, 2008 4:37 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

You wrote,

…is it your impression that Frege was using ‘fruitfulness’ in the sense that a naive practicing mathematician would use the word in referring to a mathematical concept?

I think Tappenden has done enough to establish this, bearing in mind that the term will have changed its sense a little over intervening decades, and there is variation between mathematicians. This is not so surprising in view of the fact that Frege was a mathematician, working in a mathematics department.

While one is told a story of the original logicists taking the next step after Weierstrass’ arithmetization of analysis, in fact Frege disliked his work. More had to be done to carve concepts correctly.

On p. 17 (443 in text) of that fruitfulness paper by Tappenden, he gives criteria for an example of the analysis of a piece of mathematics into principles which would be satisfying to Frege. Condition C says

The principle must have incorporated mathematically pregnant notions which made possible inferences of the sort which “increase order and regularity”, “reveal connections between matters apparently remote” etc.

Another lesser known fact about Frege was that he didn’t believe analysis into purely logic principles would be possible for geometry. He later gave up the thesis that it could even for arithmetic.

We can, of course, ask whether it matters what Frege thought. But should note the weight his name carries for analytic philosophers. It’s easy to forget that late nineteenth century German mathematicians were educated in a very philosophically sophisticated environment, dominated to a large extent by Kant. This makes it more plausible that Frege’s concerns are not the same as contemporary neo-logicists.

You also wrote,

Would it similarly be agreed upon among logicists that showing logic to provide a good account of mathematical correctness is a major goal?

I think the best way to understand the contemporary ‘orthodox’ scene is through a problem posed by Benacerraf. This is well enough described in Wikipedia:

In Mathematical Truth, he argues that no interpretation of mathematics (available at that time) offers a satisfactory package of epistemology and semantics; it is possible to explain mathematical truth in a way that is consistent with our syntactico-semantical treatment of truth in non-mathematical language, and it is possible to explain our knowledge of mathematics in terms consistent with a causal account of epistemology, but it is in general not possible to accomplish both of these objectives simultaneously. He argues for this on the grounds that an adequate account of truth in mathematics implies the existence of abstract mathematical objects, but that such objects are epistemologically inaccessible because they are causally inert and beyond the reach of sense perception. On the other hand, an adequate epistemology of mathematics, say one that ties truth-conditions to proof in some way, precludes understanding how and why the truth-conditions have any bearing on truth.

Mathematics is seen as a real thorn in the side. What are the concerns of fruitfulness, when we’re caught in the dilemma produced by our knowledge of 2 + 2 = 4? Either numbers exist (where? how to we come to know about them?), or we’d better be able to rewrite the sentence as a logical derivation from definitions and logical principles.

Posted by: David Corfield on January 7, 2008 10:52 AM | Permalink | Reply to this

### Tao, Fruitful or Truthful; Re: Two Cultures in the Philosophy of Mathematics?

“Fruitfulness is key. If a neologicist doesn’t care about the fruitfulness of their abstraction principles then they’ve rejected an enormously important part of Fregean thinking.”

An extremely interesting statement!

Did not Terry Tao list this as one of the (nonexlusive) markers of “good mathematics”?

Monday, April 23, 2007
Fruitful or Truthful
Reuben Hersh kindly allowed me to include in the discussion fragments of our e-mail dialogue on philosophy of mathematics. Here it goes:

Reuben: I understand your comment that the “social constructivist” philosophy may not be very “fruitful”. When I wrote a chapter in What is Math, Really? about criteria for a philosophy of math, I did not think of including fertility. Maybe I should have. I did stress truthfulness. The two do not seem to be identical. By no means do I mean to suggest that you are a Platonist, but I see an analogy. There is a general belief that Platonism can be helpful for problem solving, but that is not a very strong reason to believe it is true.

Alexandre: You have raised a very interesting point: I never thought about applying the concept of “truthfulness” to philosophy. Philosophy is not a natural science. Was existentialism truthful? Philosophy can be fruitful, however. Existentialism, for example, was fruitful because it generated a great literature; hence it touched something in human soul, which is a social practice proof of its fruitfulness.

I myself had strictly Vygotskian upbringing; however, Vygotskianism in mathematics appears to generate more paradoxes than give answers. You have mentioned one of these paradoxes: indeed, it is an established fact of centuries of social practice of mathematicians that Platonism is useful for problem solving. Moreover, Platonism was more fruitful than formalism (although perhaps not considerably more fruitful since formalism led to computers) and considerably more interesting than intuitionism and finitism.

TEN MATHEMATICAL ESSAYS ON APPROXIMATION IN ANALYSIS AND TOPOLOGY

Ten Mathematical Essays on Approximation in Analysis and TopologyTen Mathematical Essays

Edited By

Description
This book collects 10 mathematical essays on approximation in Analysis and Topology by some of the most influent mathematicians of the last third of the 20th Century. Besides the papers contain the very ultimate results in each of their respective fields, many of them also include a series of historical remarks about the state of mathematics at the time they found their most celebrated results, as well as some of their personal circumstances originating them, which makes particularly attractive the book for all scientist interested in these fields, from beginners to experts. These gem pieces of mathematical intra-history should delight to many forthcoming generations of mathematicians, who will enjoy some of the most fruitful mathematics of the last third of 20th century presented by their own authors.

cf.:
The Philosophy of Mathematics: An Introductory Essay - Google Books Result
by Stephan Körner - 1986 - Mathematics - 198 pages
Its failure suggests a modification of the original programme and is a source of much fruitful mathematics. But the logical status of the notion of an …

cf.

Not, of course, to be confused with:

ERIC #: EJ090184
Title: Fruitful Mathematics
Authors: Ranucci, Ernest R.
Descriptors: Algebra; Diagrams; Discovery Learning; Geometric Concepts; Instruction; Mathematics Education; Problem Solving; Secondary School Mathematics; Teaching Methods
Source: Mathematics Teacher, 67, 1, 5-14, Jan 74
Peer-Reviewed: N/A
Publisher: N/A
Publication Date: 1974-00-00
Pages: N/A
Pub Types: N/A
Abstract: To discover a generalization from a pattern of data, students need to know how to analyze the data. This is a description of how high school students can find a formula to predict the number of spherical fruits in a piling by using differences. (JP)

Posted by: Jonathan Vos Post on January 5, 2008 7:39 PM | Permalink | Reply to this

### Fruitful Fermat and Torricelli; Re: Tao, Fruitful or Truthful; Re: Two Cultures in the Philosophy of Mathematics?

Sorry. I left off the two specific Tao citations. The first, on what is Good mathematics, is probably familar to many here already.

The second is:

PCM article: Generalised solutions, where PCM = Princeton Companion to Mathematics.

The Companion also has a section on history of mathematics; for instance, here is Leo Corry’s PCM article “The development of the idea of proof“, covering the period from Euclid to Frege. We take for granted nowadays that we have precise, rigorous, and standard frameworks for proving things in set theory, number theory, geometry, analysis, probability, etc., but it is worth remembering that for the majority of the history of mathematics, this was not completely the case; even Euclid’s axiomatic approach to geometry contained some implicit assumptions about topology, order, and sets which were not fully formalised until the work of Hilbert in the modern era. (Even nowadays, there are still a few parts of mathematics, such as mathematical quantum field theory, which still do not have a completely satisfactory formalisation, though hopefully the situation will improve in the future.)

Following Tao’s link gets a wonderful PDF, which has passages on-topic here such as:

[p.5] “Examples of how the classical Greek conception of geometric proof was essentially followed but at the same time fruitfully modified and expanded are found in the works of Fermat, for example in his calculation of the area enclosed by a generalized hyperbola…”

[p.6] “The rules of Euclid-like geometric proof were completely contravened in proofs of this kind and this made them unacceptable in the eyes of many. On the other hand, their fruitfulness was highly appealing, especially in cases like this one in which an infinite body was shown to have a finite volume, a result which Torricelli himself found extremely surprising…”

What, I wonder, will 22nd century editions of Princeton Companion to Mathematics say about n-Category Theory? String Theory? QFT? Loop Quantum Gravity? Ed Witten? Greg Chaitin? Steve Wolfram? Richard Feynman? Experimental Mathematics? Kurt Godel?

Posted by: Jonathan Vos Post on January 5, 2008 9:27 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

A famous quip says something to the effect that ‘Art historians are to artists as ornithologists are to birds.’

The version I know is supposedly due to Feynman,

Philosophy of science is about as useful to scientists as ornithology is to birds.

Presumably this was meant to say not useful at all. If so, then for me philosophy or science (or both) has gone astray.

On the other hand, in view of habitat destruction, perhaps birds do need ornithologists.

Posted by: David Corfield on January 7, 2008 4:53 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

If I remember correctly, a consistency-proof of QFT exists only for uninteresting cases, e.g. when fields don’t interact and nothing happens?

Posted by: Thomas Riepe on January 5, 2008 12:30 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

You could try the papers on this archive.

Posted by: David Corfield on January 5, 2008 2:35 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Thanks! Exists there any investigation of the way mathematicians really think in philosophy? E.g. in case mathematical thinking makes use of something like aprioric ideas, how is that use distributed and how changes such a distribution (I found only accidentially that N. Hartmann suggested to investigate “categorial dynamics” ca. like the changes of opinions expressed here, but there may be others). An other question I wonder about is if Chaitins Number is considered by philosophers as relevant.

Posted by: Thomas Riepe on January 5, 2008 8:11 PM | Permalink | Reply to this

### Wolfram, Chaitin, Leibnitz; Re: Two Cultures in the Philosophy of Mathematics?

My recollection is that at last year’s 7th International Conference on Complex Systems, Steve Wolfram introduced Greg Chaitin, who traced his ideas back to both Godel and, he emphasized, Leibniz, his favorite philosopher.

Chaitin’s web site gives Leibnitz quotations to support Chaitin’s claim that Leibnitz was the father of Complexity Theory in a very modern sense, albeit for theological reasons intertwined with notions of “fecundity” and fruitfulness in Mathematical Physics.

I spoke at length with Wolfram, Chaitin, and James Gleick about this, and how it related to Feynman’s never-published critique of that specific Leibnitz notion (about a finite number of Physical Laws or an infinite number).

But I have no idea what “mainstream” Philsophy says about Chaitin. Anyone here know?

Posted by: Jonathan Vos Post on January 5, 2008 9:40 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

- “the way mathematicians really think”
I just found that David Corfields text on his book on that. But why your restriction to “what leading mathematicians of their day”? I would guess that it works better to investigate mathem.s in general and only later to define subgroups. E.g. such an approach on the development of musicians. To ask “how these “notions, conceptions, intuitions, and so on” are developed”, would fit into the mentioned idea of Hartmann and a funny idea how to accelerate the creation of sci. notions has been pursued by Gunkel . Apparently Gunkel inputs his creations into normal science, so one could perhaps see there like in a laboratory how new notions and concepts are processed in the sci. community. “Why do we persist in teaching certain ways of thinking” - when reading books about life in the middle ages nearly everything appears extremly strange, with exception of texts on medieval universities. Did science develop back to that? E.g. when John collects online books, I wonder if that makes sense because now as in the middle ages the reading skills seem to develop only at the end of university studies and therefore such collections are useless for the intended readership. “Connectivity of mathematics” - could this be just an illusion because we know only very little? Chaitins number seems to imply that.

Posted by: Thomas Riepe on January 6, 2008 7:52 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

David Corfields text links to an article about Cartier, who is quoted that Bourbaki-texts are “a disaster” as textbooks. But Deligne seems to have studied math when still in school with them. What do you think of EGA as textbook?
Because you mention greek philosophy and Platon, here the link to Gyburg Radke who wrote about Platonism and its concept of number some very interesting texts.

Posted by: Thomas Riepe on January 6, 2008 8:11 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Conc. how mathematicians (and philosophers?) really think, here a fascinating report about new ways to observe the semantic localisation of concepts in the living human brain. Apparently one could look how mathematicians brains proceed with mathematical terminology, relate this to the existing (acc. to the report very extensive) data and look for interesting individual differences and how they develop.

Posted by: Thomas Riepe on February 17, 2008 11:10 AM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Another fascinating insight into mathematical brains in Ioan James’
article in the Bulletin of the Royal Society of Medicine.

Ioan writes further:
Simon Baron-Cohen played a role, created quite a stir and led to my book. MF had observed that his Aspie patients were interested in mathematics and was writing a book on the subject He came to see me and proposed a collaboration. Meanwhile I had been lecturing on the subject to psychologists and mathematicians at various places, including Philadelphia, and have been in correspondence with various Aspie mathematicians. Also Simon B=C’s research group have investigated the connection between autism and mathematics. The Royal Society are running a conference in the autumn which should throw further light on the matter.

I’m looking forward tro the whole book.

Posted by: jim stasheff on February 17, 2008 1:34 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Here an article on musicians brains.

Posted by: Thomas Riepe on February 18, 2008 5:57 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Thomas Riepe wrote:

If I remember correctly, a consistency-proof of QFT exists only for uninteresting cases, e.g. when fields don’t interact and nothing happens?

No, the situation is not that bad! On my website you can find a free book describing a mathematically rigorous construction of interacting quantum fields in 2d spacetime. This is old 1970’s work of Irving Segal and his student Edward Nelson. A different approach to solving the same problem can be found in the famous book by Glimm and Jaffe: Quantum Physics: A Functional Integral Point of View.

Later, people constructed interacting quantum field theories in 3d spacetime — for a good review, try the book by Vincent Rivasseau, From Perturbative to Constructive Renormalization.

Things get really hard in 4 dimensions. If you can first show that $SU(2)$ Yang–Mills quantum field theory makes sense in 4d, and then show that the lightest particle has mass $\gt 0$, you will win a million dollars.

However, I’ve heard that people have recently constructed other interacting quantum fields in 4 dimensions. I’m waiting for more details before writing about this in This Week’s Finds — it would be a big deal.

Posted by: John Baez on January 10, 2008 2:50 AM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

If I remember correctly, a consistency-proof of QFT exists only for uninteresting cases, e.g. when fields don’t interact and nothing happens?

No, the situation is not that bad! On my website you can find a free book describing a mathematically rigorous construction of interacting quantum fields in 2d spacetime.

The question also depends on what exactly one counts as a QFT. There are different definitions floating around and surprisingly little work has been done on trying to relate them.

The approach called local- or algebraic quantum field theory, which adopts the definition:

A QFT is a certain cosheaf of algebras.

has been strongly motivated by the desire to understand the kind of 4-dimensional quantum field theory relevant for the real world.

A certain disdain among some of its leading practitioners can be felt (and heard in their talks) towards efforts to study QFTs in dimensions other than 4 and for fields not seen in nature.

Therefore it is both remarkable and a little ironic, that AQFT has to date – and that’s probably the statement Thomas Riepe had in mind – of all 4-dimensional QFTs been only able to handle the free field (at least I have never ever seen anything other in 4-dimensions discussed), while the axiomatics of AQFT has turned out to be a strikingly powerful tool for the analysis of two-dimensional QFT, in particular of 2-dimensional conformal quantum field theory.

In two dimensions, it turns out that local nets of von Neumann algebras are essentially an alternative to vertex operator algebras (even though here, too, there has been surprisingly little work (I know one single paper) concerned with understanding what exactly the relation is).

There are cool powerful classification theorems for 2-dimensional CFTs using AQFT, the kind of stuff you need to do things like deciding if Witten’s recent conjecture about CFTs of central charge 24 is correct.

While this is true, one must be careful: according to the definition of QFT which I think is the right one, the data provided by a solution to the AQFT axioms is not what is called a “full” QFT. Rather, in the CFT context at least, it is what is called a “chiral” QFT.

This says, and that’s not surprising if you look at the axioms, that if you regard a quantum field theory as a global thing which allows you to assign amplitudes (morphisms in some category, really), not just to local patches of your parameter space, but to arbitrary topologically nontrivial parts of parameter space (i.e. if you really regard QFT as a functor on cobordisms), then a solution to the AQFT axioms givew you only necessary, but not sufficient information, in general, to construct that assignment.

This is a problem that people haven’t even tried to address in more than two dimensions, as far as I can tell. But for 2-dimensional CFT, there is a powerful theory by Fjelstad, Fuchs, Runkel and Schweigert which entirely solves the problem of constructing a full 2D CFT from a solution to the AQFT axioms – but only in the special case that this solution to the AFTQ axioms happens to be what is called “rational”.

(This solution, by the way, it very beautiful: the statement is that the full CFT is uniquely fixed by the Morita class of a Frobenius algebra object internal to the representation category of the local net of algebras).

And even then, one has to be more careful: to really be able to turn the crank on the general FFRS solution, one needs to have first computed something called the “conformal blocks” of the local net of observables. Which has been done in some cases. But only in comparatively few.

Nothing is ever easy.

Posted by: Urs Schreiber on January 10, 2008 7:53 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Many thanks for the answers! An other question I’m curious about is what philosophy of physics says about the compatibility of general relativity and quantum theory. E.g. if I remember correctly, there exist situations where GR predicts “timelike loops”. Would that not conflict with the unpredictability of decay-events etc. for any observer?

Posted by: Thomas Riepe on January 18, 2008 4:14 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Here and here Fesenkos interesting thoughts on a possible use of model theory in physics and mathematics.

Posted by: Thomas Riepe on January 8, 2008 8:45 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

I realize this topic is a bit old, but I just discovered it, and thought that others might like to hear directly from one of the neo-logicists who are being discussed.

Anyway, a few points worth making:

In the original post David Corfield writes:

“By contrast, a much more orthodox philosophical approach to mathematics in the English-speaking world, well represented in the UK, is to address the question of whether mathematics is reducible to logic. To gain an idea of the current state of play here, you can take a look at What is Neologicism? by Linsky and Zalta.”

This is misleading at best. Since Russell’s paradox shattered Frege’s own hopes of reducing arithmetic (and real and complex analysis) to logic, very few philosophers have taken seriously the idea that mathematics is reducible to logic. The reason is simple: Logic (as understood today) has no ontological commitments, while mathematics has many.

This brings up a crucial issue: Neo-logicism (at least, the standard variant of it, as espoused by myself, Crispin Wright, Bob Hale, etc,) is not, strictly speaking, a version of logicism at all. There is no claim that mathematics is reducible to logic, since the abstraction principles used (e.g. Hume’s Principle) are not logical truths. Abstraction principles are viewed as being definitions of a certain kind. As a result, their consequences can be known a priori, etc. But the neo-logicist claim is not, and never has been, that mathematics can be reduced to logic alone (misleading nomenclature notwithstanding).

[It should be pointed out that there are alternative views which have also adopted the”logicist” or “neo-logicist” label, such as Neil Tennant’s project and the object-theory version proposed by Zalta and Linsky. In the philosophical literature, however, “neo-logicism” (without any qualification) is generally understood to refer to the views of Wright and Hale (and myself). Linsky and Zalta’s characterization of this version of neo-logicism is, unfortunately, misleading in this respect. Unfortunately, this confusion has persisted recently, necessitating the need to distinguish the ‘Scottish School’ from other variants of neo-logicism as in the Ebert/Rossberg paper.]

In a later post David also notes that he is unclear regarding “whether it matters to them whether their logical reconstructions of portions of mathematics get to the conceptual heart of those portions.” The reason for the unclarity is that there is substantial disagreement between neo-logicists (of the Scottish school) on exactly this issue. The principle in question is this (quoted from C. Wright’s “Neo-Fregean Foundations for Real Analysis”):

“Frege’s Constraint: That a satisfactory foundation for a mathematical theory must somehow build its applications, actual and potential, into its core – into the content it ascribes to the statements of the theory – rather than merely ‘patch them on from outside.’”

Wright rejects Frege’s Constraint as a general requirement on neo-logicist reconstructions of mathematical theories, arguing that whether it applies to a given theory depends on the nature of that theory (and, in particular, on the nature of its applications). In particular, he requires that our reconstruction of arithmetic satisfy it, but not our reconstruction of analysis. Hale, on the other hand, seems to require that Frege’s constraint be met across the board, and he demonstrates in some of his work how an account of analysis can be formulated within the neo-fregean framework that arguably meets the constraint. (I am also sympathetic to Frege’s Constraint, for what it is worth).

Finally, there is a recurring idea (both in David’s book, but also in discussion) that typical philosophers of mathematics are ignorant of of, or are too lazy to learn, ‘real’ mathematics and instead concentrate on three rather artificial areas: arithmetic, analysis, and set theory. The formulation of this idea is particularly offensive in the following quote from John Baez’s discussion of David’s book:

“Alas, too many philosophers seem to regard everything since Goedel’s theorem as a kind of footnote to mathematics, irrelevant to their loftier concerns (read: too difficult to learn).”

I think there are a number of misunderstandings regarding the way philosophy of mathematics has progressed that are at work here.

First off, it is clear that most philosophers of mathematics of any significance know a lot more mathematics than just these three areas. And there is definitely widespread agreement that the more math one knows (including recent mathematics) the better (and not just areas one might classify as ‘mathematical logic’). So why does discussion of these areas not pop up more within philosophy of mathematics? The underlying cause, as Baez noted, has to do with Gödel’s theorem, I think, but not for the reasons Baez suggests.

It is not that mathematics later than this is irrelevant, uninteresting, or too hard. Instead, the problem is that Gödel’s theorem (and various other limitative results) has shown the problem of accounting for the metaphysics and epistemology of mathematics to be much, much more difficult than it had previously appeared to be (and it appeared hard enough as it was).

As a result, philosophers of mathematics have restricted their attention to a small number of conceptually simple ‘test cases’ (arithmetic, analysis, and set theory) The underlying idea would seem to be that if we cannot adequately handle these cases, then there is little point in trying to tackle more complicated mathematical structures and theories. [Note that although ZFC is complicated, the intuitive notion of ‘set’ is rather simple.] In addition, some philosophers (but not all) think there is something particularly special and central about these specific theories (see the comments regarding Frege’s constraint above), and this might also be contributing to this emphasis on a few mathematical domains.

Additionally, I think that many philosophers of mathematics believe that if we can handle these three cases, then we should be able to deal with the rest of mathematics in a roughly similar manner. This might be optimistic (I suspect it is), but I certainly don’t think there is any knock-down argument to the effect that concentrating on these three domains is causing the thinkers in question to miss out on important aspects of the nature of mathematics (after all, pretty much all mainstream mathematics can be modeled in the powerset of the reals).

Posted by: Roy T Cook on February 16, 2008 12:29 AM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Roy, could you clarify a couple of points?

You quote David Corfield and then write:

Some people might find the “at best” provocative. Many things are worse than “misleading”, and some such things are accusations that aren’t conducive to productive discussion. It might help to keep the temperature down if you could clarify what you meant. (Or, of course, it might raise the temperature — but at least we’d know what you had in mind.)

At the end of the comment, you write:

pretty much all mainstream mathematics can be modeled in the powerset of the reals.

Can you explain what this means? I can’t see a way of understanding it that both makes it true and interprets the word “modeled” to mean anything similar to what I’d expect it to mean.

For example, suppose it means that pretty much all of the sets encountered in mainstream mathematics are in bijection with some subset of the reals. Suppose I accept this as true. Then it’s very far from what I would take “the modelling of mathematics” to mean. A model is presumably meant to be an accurate portrayal of reality — something that captures all of its essential aspects. Simply keeping track of which sets get used seems to me to be a million miles from the goal of capturing the essence of mathematics.

I’m not saying that this is what you mean. Probably it’s not. I’m just trying to explain what it is that I don’t understand about your statement. I guess you mean “modeled” in some technical or semi-technical sense; but what is it?

Posted by: Tom Leinster on February 17, 2008 1:36 AM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Yeah, I should have been more careful here (in two ways - this is of course the danger of typing out a quick rant on the internet, which causes me, like others, to perhaps be less careful than I should be).

Anyway, the “misleading at best” comment was meant in exactly the ‘provocative’ manner that you suggest. The problem, however, is not David’s in particular. Instead, I think that there is a widespread misunderstanding regarding what many philosophers of mathematics are up to - especially us neo-logicists. The terminology doesn’t help, of course, since neo-logicism is NOT a new variant of old-fashioned logicism. But the Zalta and Linsky paper doesn’t help either, since they characterize the debate between their own view and Wright/Hale style neo-logicism as a debate about what the most promising form of logicism is (i.e. which project has a better claim to having reduced mathematics to logic). Neo-logicists, however, never claimed that they were reducing mathematics to logic.

It is particularly telling that the titles of two of the best and most influential articles in the literature on this topic are both “Is Hume’s Principle Analytic” (one by Crispin Wright and the other by George Boolos). Notice that the title of both papers makes it clear that the issue is the analyticity of Hume’s Principle, not its status as a truth of logic!

The “misleading at best” comment was reflecting, not any anger or anything at particular authors posting here, or discussed here, but the more general fact that outside of the few specialists who work on this topic very few philosophers of mathematics seem to ‘get’ this point. All too often I tell people (philosophers of math) I work on neo-logicism and they say “well, that’s sort of pointless - after all, logicism is dead.”

Regarding the comment about the powerset of the reals - I did, in fact, mean the rather trivial embedding notion that you suggest (i.e. that the vast majority of mathematical structures studied in mainstream mathematics are isomorphic to some substructure of the powerset of the reals). While such embeddings are typically not all that illuminating, the point I was (perhaps none too clearly) trying to make is that, given such embeddings and playing Devil’s advocate for the moment, I don’t see any serious reason to suspect that limiting attention to arithmetic, analysis, and set theory will cause philosophers of mathematics to ‘miss’ crucial aspects of mathematics that ought to be incorporated into their accounts. The argument, in more detail, might go something like this: Assume that philosophers of mathematics successfully produce accounts that capture the ‘essence’ (your word!) of arithmetic, analysis, and set theory. Then, since the rest of mathematics can be embedded into the powerset of the reals, these theories are not, in some sense, all that different from the theories we have already accounted for, and thus it shouldn’t be too hard to generalize our account to other areas. Keep in mind that I also suggested I was a bit skeptical of moves like this. But I can understand the underlying persuasiveness of arguments along these lines.

Posted by: Roy T Cook on February 18, 2008 6:47 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

All too often I tell people (philosophers of math) I work on neo-logicism and they say “well, that’s sort of pointless - after all, logicism is dead.”

Well that is ignorance on their part if they say that.

But you might also be hearing some part of another complaint that bears on what Tom Leinster and you are beginning to discuss about modelling mathematics within the powerset of the reals.

What just about any mathematician (excluding some proof theorists) feels about projects like yours which aim to describe their discipline as the deduction of statements from a set of axioms/definitions/stipulations is that some part of the essence of mathematics has simply been overlooked.

This raises the question of what is not being captured, and the further question of whether that which is overlooked is the proper subject matter of philosophy.

To give an example, deciding which way to reformulate or extend a mathematical concept is taken by mathematicians to be a process which may be conducted rationally. Where then does the rationality reside? Following Lakatos, I take this to be a question philosophy can and should address.

Posted by: David Corfield on February 19, 2008 10:28 AM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Roy writes:

…there is a recurring idea (both in David’s book, but also in discussion) that typical philosophers of mathematics are ignorant of of, or are too lazy to learn, ‘real’ mathematics and instead concentrate on three rather artificial areas: arithmetic, analysis, and set theory.

I certain never meant to give any impression of laziness. Or am I to parse that ‘or’ in your sentence to allow the latter’s truth if in my book I only charged philosophers with ignorance?

But even then, ignorance was never the charge I was making. What I was attempting to say there was that I think philosophy has gone astray with regard to mathematics in that it largely fails to ask the right kinds of question of it. Given the questions you do choose to ask, I can see perfectly that arithmetic, analysis and set theory would be enough.

Meta-philosophical discussion is never an easy business. But Brendan and I genuinely hope that in the conference we plan we can learn to see what it is in the questions neo-logicists ask that forces them upon you. We also hope that anyone attending would come open to the proposal that philosophy might pose new questions.

Perhaps we might find a point of contact in

Frege’s Constraint: That a satisfactory foundation for a mathematical theory must somehow build its applications, actual and potential, into its core – into the content it ascribes to the statements of the theory – rather than merely ‘patch them on from outside’.

I’m not really sure what this means. I wonder what would constitute success for you in this project.

I’ve mentioned that I’m fond of Frege’s comment:

[Kant] seems to think of concepts as defined by giving a simple list of characteristics in no special order; but of all ways of forming concepts, that is one of the least fruitful. If we look through the definitions given in the course of this book, we shall scarcely find one that is of this description. The same is true of the really fruitful definitions in mathematics, such as that of the continuity of a function. What we find in these is not a simple list of characteristics; every element is intimately, I might almost say organically, connected with others.

Can you understand why a philosopher might be interested in this theme?

Posted by: David Corfield on February 18, 2008 10:30 AM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

David,

First off, re: the first passage you quoted. I do think that the comment is accurate when parsed in the charitable manner you suggest, but it is misleading at best (Ha! I used the loaded phrase aimed at myself!). Your own claim (if the reviews, etc. I have read are accurate) is merely that many philosophers of mathematics are ignoring ‘real’ mathematics to their detriment. You do, however, cite Baez’s discussion approvingly. And he does, quite clearly, make accusations of laziness or lack of ability.

At any rate, as you note, the real issue is the sort of question that should be asked by philosophers of mathematics. I don’t want to speak for any other neo-logicists (all 2 of them, at least of the Scottish School variety), but I personally would never claim that the questions asked by thinkers such as yourself were the wrong questions, or misguided. At the most, all I would claim is that we should answer simple questions about the ontological and epistemological status of mathematical objects BEFORE moving on to more subtle questions about particular mathematical disciplines. In other words, we need to know what it is we are talking about, and how we manage to talk about it, before we can start examining in detail how such talk plays a role in explanation, applications, etc. In addition, so long as we think that the epistemology and ontology of mathematics generally is relatively uniform, and that arithmetic, set theory, and analysis are relatively typical cases, then, with respect to these basic questions at least, restricting attention to these three areas is not harmful.

Now, I take it that the Frege quotation which you are fond of is, in fact, an expression of the general idea behind what we now call Frege’s constraint. The idea, applied to numbers, is that our account of number should, all in one go, explain ALL of the important aspects of number talk, that is, that “every element [of the concept of number] is intimately, I might almost say organically, connected with others”

Included amongst such aspects of (cardinal) number talk are:

(1) The idea that numbers are objects. (“Five is a number”)
(2) The idea that numbers can be ascribed adjectivally (“There are five apples.”)
(3) The natural applications of arithmetic.

Frege’s insight - the one retained by (Scottish School) neo-logicism, is that the idea underlying Hume’s Principle (that number is really a function from concepts to objects) can provide a natural, straightforward account of all of these uses.

Now, there are certainly other aspects of the notion of cardinal number that one might want to consider. And, for other mathematical concepts, the list of important features to be explained might be different. Nevertheless, if something like the Fregean project can be carried out for numbers (and it seems like it can), then this would seem to be a pretty strong argument in favor of exploring whether, and how, the account can be carried out for other domains.

Posted by: Roy T Cook on February 18, 2008 7:23 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

You say

…our account of number should, all in one go, explain ALL of the important aspects of number talk, that is, that “every element [of the concept of number] is intimately, I might almost say organically, connected with others”

You then mention three examples of number talk:

(1) The idea that numbers are objects. (“Five is a number”) (2) The idea that numbers can be ascribed adjectivally (“There are five apples.”) (3) The natural applications of arithmetic.

And then claim that Hume’s Principle can “provide a natural, straightforward account of all of these uses.” This principle says “for any concepts F and G, the number of F-things is equal to the number G-things if and only if there is a one-to-one correspondence between the F-things and the G-things.”

First then we need to see if there were other aspects of number talk which could prove more challenging to account for. For one thing it is noticeable just how many of mathematics’ most intricate theories are used in number theory. There’s a question then of whether we can extricate simple arithmetic from the rest of mathematics.

To take a simple example, it is the case that any prime number of the form $4 n + 1$ is expressible as the sum of the squares of two natural numbers. Do we need to check in what sense Hume’s principle explains this talk?

Second, while considering the application of numbers, we might wonder if to thoroughly explain their use to count fruit we should say something about the world and our being in it as cognitive agents. But perhaps all this work has been done by prior analysis of ‘concept’ and ‘object’ or ‘thing’.

Lastly, let’s note that Hume’s Principle sounds quite familiar to us here. It’s related to the process of decategorification from the category of (finite) sets to the set of cardinals. Perhaps then we can shed some more light on related uses of number theory.

Let’s consider the statements 3 + 2 = 4 + 1 and 2 + 3 = 3 + 2. John Baez gives us an account of what higher category theory has to say about associativity and commutativity, and it leads to deep waters. So the question arises of whether there is here some aspect of number talk which needs more than Hume’s Principle.

Posted by: David Corfield on February 20, 2008 1:58 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

I think that you are absolutely correct that the issue here (for the success of Scottish-school neologicists, at least) is whether there are “other aspects of number talk which could prove more challenging to account for.”

Now, the fact that we use a lot of intricate mathematics when doing number theory (Wiles’ proof is the example that is always trotted out here) is relatively simple to understand. Second-order arithmetic is categorical, so any truth stateable in the language of second-order arithmetic follows (semantically) from Hume’s Principle. Second-order logic is incomplete, however, so some of these consequences are not provable. Moreover, many of the theorems that are provable from these resources will have horribly complicated proofs. Thus, the utility of bringing in additional resources is obvious, since doing so may allow us to prove things we couldn’t prove before, or couldn’t prove at all. Nevertheless, there is no truth of arithmetic which is not ‘guaranteed’, in the appropriate sense, by Hume’s Principle.

Also, regarding the worries about how counting applies to the real world, it is striking that we can count apples but not ‘waters’. The reason for this, however, on the neo-logicist picture, has little to do with our role in the world, as cognitive agents (after all, the number of moons of mars would still be two, even if no one had ever been around to count them). Instead, the difference stems from the fact that APPLE, but not WATER, is a sortal concept. Scottish neo-logicists have worked hard on sorting out (pun intended) which concepts are sortal, and what criteria distinguish sortal from non-sortal concepts, since abstraction operators such as “number of” arguably only apply to sortal concepts (and it is this fact that explains, among other things, their applicability). I recommend the relevant chapters of Wright and Hale’s The Reason’s Proper Study (Oxford, 2001?) for details.

Regarding the idea that Hume’s Principle is just the “process of decategorification of the category of (finite) sets to the set of cardinals”, you note that Hume’s Principle sounds familiar. This brings up an interesting question. Mac Lane studied in Gottingenwith, among others, Bernays. So there is a good chance that he would have had at least a passing familiarity with Frege’s work, I would think. It would be interesting to know how much Fregean ideas influence the early (and perhaps later) development of category theory.

Posted by: Roy T Cook on February 25, 2008 6:59 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

One bone of contention I see concerns what it is to explain or account for an aspect of number talk.

In the case of number theoretic results with what are taken to be good ‘explanatory’ proofs which go via extraneous constructions (the vast majority of contemporary number theory), it is debatable whether merely being ‘guaranteed’ by Hume’s Principle suffices. As to how to settle this question, I’m not sure how to proceed. I believe it is important for philosophy to try to make sense of ‘explanatory’ in mathematics. And I don’t believe it is merely a psychological phenomenon. It’s something which points to the idea of carving concepts well, as revealed by what happens next in their usage and extension.

Seeing a number as a function on the set of primes is a good idea.

I can see a rationale for Wright rejecting the Frege Constraint for analysis. Where you can with some plausibility believe that applications of numbers to the world have largely exhausted their range with counting five apples, it seems unlikely that the story of the application of analysis is in any way near completion. Just look at discussions on this blog about smoothness.

On the other hand, as I suggested, it is possible to argue that Hume’s Principle doesn’t get to the heart of moving enumerated objects about in the world.

As for Fregean influence on Mac Lane, certainly the latter was very interested in philosophy. I don’t know how much the promoton of the idea of cardinal numbers as equivalence classes of sets owes to Frege. Frege himself seems to have learned from Riemann about this kind of technique, according to Jamie Tappenden.

Posted by: David Corfield on February 27, 2008 12:31 PM | Permalink | Reply to this

### Math, Chess, Music; Re: Two Cultures in the Philosophy of Mathematics?

I agree with David Corfield. What is it “to explain or account for an aspect of number talk”?

I go to colloquia and seminars at Caltech, in the Math, Applied Math, Physics, and CS departments, where “number talk” proliferates.

I even go for subdisciplines that are outside of my areas of understanding, in hopes of learning something.

I do usually learn something, and I seem to have acquired a reputation for asking superficially stupid questions which the speaker finds interesting. Sometimes I ask by explicitly channeling my mentor Richard Feynman, complete with thick Brooklyn accent. He had a metaphysics of Mathematics versus Physics which is well-explained in his writings and recorded lectures.

It is most interesting to observe talks which give impressionistic “sketches” of proofs, complete with weird whiteboard scribblings and/or digital projections of beautiful elaborate computer graphics and/or xeroxes of xeroxes of hand-written notes and/or literal hand-waving.

There is indeed a strong flavor of “extraneous constructions” and more. There’s sometimes a kind of pointing to the Platonic Ideal, a nod towards a reality external to the lecture hall, as if to say: “Look into the infinite directly, and see that I am right, or at least on the right track.” Formally, this does not bear on proof, and yet it has a social function parallel to formal proof.

It reminds me of informal gatherings of Chess masters, international masters, and grandmasters I have attended. I can’t play adequately myself, but I have an old friend, Ben Nethercott, who is a Tournament Director, even of the US Chess Open. I also socially am friends with the former U.S. Women’s Chess Champion.

They have conversations over a chessboard which are like bilingual discussions switching back and forth between languages. Here, one language is English (maybe with Russian or Chinese accent), and the other language is on the board. Example:

“Yes, but she was bluffing when she [speaker moves a rook on the board and grins]. Because of course he could simply [takes a pawn en passant, shrugs]. But, on the other hand, the fianchetto bishop [taps a bishop, makes a hand flick in a diagonal motion].”

Second Speaker: “No, no, you fall into the same trap. Because you see [wiggles the queen] and so [knocks over a knight], except of course [forefinger waves up and down, left and right, over a particular rank and file].”

I am not being glib in comparing Number Talk and Chess Talk. And I further make a commutative diagram by referencing Music Talk. As a child I had the wonderful experience of attending many Young People’s Concerts by Leonard Bernstein. This was music talk of genius. A small orchestra would play for a minute, interrupted by Lenny waving his hands, talking in apparent simple English, illustrated by his making little burst of sound on a piano. Children could follow the Music Talk, to some extent. That was genius-level lecture in the same class as Feynman, I believe.

Mathematics, Music, Chess. The three domains of human life where a child prodigy can be world-class. Why? I wave my hands towards a large literature on the subject.

Posted by: Jonathan Vos Post on February 28, 2008 3:04 PM | Permalink | Reply to this

### Re: Two Cultures in the Philosophy of Mathematics?

Regarding Mac Lane and Frege, I had the following from Colin McLarty:

Saunders’ fullest discussion of Frege is in “Mathematical Models: A Sketch for the Philosophy of Mathematics”, The American Mathematical Monthly, Vol. 88, No. 7 (Aug. - Sep., 1981), 462-472. This consists of the same not very detailed things Weyl says about Frege in Philosophy of Mathematics and Natural Science (1949). Saunders probably got them from Weyl in Göttingen though by 1981 they were also common knowledge. Two things specially:

Frege, Dedekind and later Russell founded arithmetic on set theory (Weyl pp. 11, 230).

Hilbert eventually axiomatized a formalist arithmetic (which merely concerns combinations of symbols) “unassailable even by the criticism directed against it by Frege” (Weyl p. 35).

These are the things I expect one heard in Göttingen in those days. But I am no Bernays scholar. And this is the view which survives in outline in Mac Lane’s Mathematics: Form and Function. There is another thing he likely heard but paid less attention to:

Pasch and Frege gave the first clear accounts of definition by abstraction (like Hume’s principle). (Weyl p. 12)

Actually Weyl finds the principle of abstraction early in Leibniz, and also credits Helmholtz, but I doubt that Saunders had any interest in either of them. And Saunders early and late was very interested in the practice of definitions by equivalence relations and I suspect that very interest drove all worry about the history of the idea out of his thinking.

I doubt Saunders looked at anything actually written by Frege, unless possibly Grundgesetze just enough to use it as a reference in his dissertation.

Posted by: David Corfield on February 28, 2008 8:10 PM | Permalink | Reply to this

### Post-modern web-centric doing Math; Re: Two Cultures in the Philosophy of Mathematics?

Over at John Armstrong’s fine Math blog, I gave an abstract (accidently omitting blogs) of how I actually do Mathematics, as opposed to how Philosophers contend that I should do Mathematics:

My default Math writing strategy is similar but slightly more post-modern:

On the average of more than once per day, over the past 5 years, find:

(a) simple but nontrivial solutions;

(b) to overlooked or shallowly-probed elementary problems;

(c) regardless of whether they need to be solved;

(d) deliver them as precisely and usefully formatted as possible;

(e) in an edited legitimate online venue such as the Online Encyclopedia of Integer Sequences, Prime Curios, or MathWorld;

(f) which links to dead-tree Math and science literature and online resources;

(g) which is date-stamped and has my email address so that people may contact me if interested, because the Killer App of the World Wide Web is Collaborationware;

(h) and consider that I have been starting with a very crude version 1.0; then

(i) iterating and deepening and collaborating optimally for the golden mean of my 2,500+ cardinality portfolio of Journal articles, Books, refereed International Conference papers, arXiv reprints, letters to the editor, newspaper columns, science fiction stories with math content, poems about Math or Science, screenplays or teleplays about Math/Science, and other stuff that someone will actually pay me to do (via salary, consulting fees, or grants).

Posted by: Jonathan Vos Post on February 19, 2008 7:27 PM | Permalink | Reply to this

### Re: Post-modern web-centric doing Math; Re: Two Cultures in the Philosophy of Mathematics?

Thanks for the hat-tip, JVP. I was actually hoping that a philosopher of mathematics would look at my adaptation of Paul Graham’s design philosophy to mathematics research.

Now, where would I find a philosopher of mathematics? Preferably one who’s just gotten tenure so he doesn’t have to worry about job security from looking at my silliness, while being young enough to find my silliness entertaining.

Posted by: John Armstrong on February 19, 2008 11:05 PM | Permalink | Reply to this

### Re: Post-modern web-centric doing Math; Re: Two Cultures in the Philosophy of Mathematics?

One interesting phenomenon that apparently occurs to some extent in most manufacturing, but particularly in computing where getting interfaces between “components” (of whatever sort) is so crucial, is “first mover advantage”. This is basically that reluctance to significantly change the “brand” of an installed system, partly just due to general “organisational inertia” but particularly because of the aforementioned complexity of interfaces, means that if you get something “workable” into the marketplace first, you’ll generally force any competition into being minor competitors. This applies even if your competitors are thinking slowly and trying to get things “right”.

This is partly why iterative refinement is emphasised in writings such as Paul Graham’s. The interesting question if you’re trying to apply this to mathematics: is it possible that in mathematical development a set of definitions/theory/notation/viewpoint/whatever that, whilst being completely “correct” might be suboptimal and yet, by virtue of being first, crowd out more carefully thought out, more optimal definitions/theory/notation/viewpoint/whatever?
Clearly publishing quickly has is advantageous to an individual researcher, but does the field as a whole have things that are significant “first mover artifacts”?

Note that this is different from the “paradigm shift” idea, in that those are supposed to happen when the weight of newly observed things unexplained by the old theory becomes too great. Here everything can be dealt with by both setups, but just much more “conveniently” in the one which came historically second (and thus lost out). You often see books referring to “using standard notation to avoid being confusing”, but anything deeper?

Posted by: bane on February 20, 2008 8:23 AM | Permalink | Reply to this

### Re: Post-modern web-centric doing Math; Re: Two Cultures in the Philosophy of Mathematics?

Firstly, I think that good mathematics will eventually push out “suboptimal”, or at least come to coexist. If it’s really a superior viewpoint, it will eventually prove itself.

On the other hand, I’m not in a position to deny the real-world consequences of research styles. If I publish fewer, but more polished or “complete” papers, my publication list looks thin. I’m less likely to make it past the first cut of any given hiring process.

What I’m thinking would be a good adjustment in strategy is not to rush in with a different idea, but for me to publish papers I’m currently viewing as “incomplete”. Get out a paper that works over an algebraically closed field, and later come back to do it over more general fields, and then commutative rings. Three papers instead of one, and all incremental refinements of the basic idea that’s there in the first.

Posted by: John Armstrong on February 20, 2008 3:00 PM | Permalink | Reply to this

### Re: Post-modern web-centric doing Math; Re: Two Cultures in the Philosophy of Mathematics?

Get out a paper that works over an algebraically closed field, and later come back to do it over more general fields, and then commutative rings. Three papers instead of one, and all incremental refinements of the basic idea that’s there in the first.

But preferably release all three papers at the same time, to prevent somebody reading the first one and beating you to the others!

Posted by: Jamie Vicary on February 20, 2008 4:26 PM | Permalink | Reply to this

### Re: Post-modern web-centric doing Math; Re: Two Cultures in the Philosophy of Mathematics?

Well, that’s exactly why I delay release in a lot of situations. But the upshot is that I can’t get hired because I haven’t published enough. But if I published partial results faster, I may get scooped.

Posted by: John Armstrong on February 20, 2008 5:15 PM | Permalink | Reply to this

### Re: Post-modern web-centric doing Math; Re: Two Cultures in the Philosophy of Mathematics?

If $p$ is someone’s ‘natural’ publication rate — the rate at which they’d publish if they had a permanent job — then by combining these two strategies, after a career of time $T$, they will have published $a p T - b$ papers, for their choices of positive constants $a$ and $b$. From what you’ve said, I guess you aim for $a \simeq b \simeq 3$. Time for a fun survey! Everybody else, what values do you use?

Posted by: Jamie Vicary on February 20, 2008 6:16 PM | Permalink | Reply to this

### Re: Post-modern web-centric doing Math; Re: Two Cultures in the Philosophy of Mathematics?

Jamie, you’re taking me way too literally here.

What I mean is that the first few points of Graham’s philosophy I’ve got. I work in an area not many people have taken explicit notice of, but which has things that need doing.

But left to my own devices I tend to build up papers that get as much generality as possible, even though I could get a useful special case down on paper more easily. What Graham’s method would entail is getting those partial results down and published rather than holding out until I have more thorough results.

Posted by: John Armstrong on February 20, 2008 9:42 PM | Permalink | Reply to this

### Re: Post-modern web-centric doing Math; Re: Two Cultures in the Philosophy of Mathematics?

“On the other hand, I’m not in a position to deny the real-world consequences of research styles.”

Note that I wasn’t putting any judgement on this. There’s a strong argument that there’d be much less advanced software around if we tried to “develop software properly” because the general pace of innovation would be less. (First mover’s tend to get displaced by applications that redefine what the target task is, eg, www essentially replacing gopher, etc.) I was more thinking that we’d like to believe that the better viewpoint will win in mathematics, but if we’re taking the computer analogy seriously it’d be interesting to look at the question empirically.

As trivial examples, I know that writing functions to the left of the their arguments is said to yield awkwardness because it’s natural to get “apply $h$ to $x$, then $g$ to result, then $f$ to that result” for left-right language readers as $(((x)h)g)f$ rather than $f(g(h(x)))$, particularly with some of the kind of category theory stuff. Do we stick with the current convention just because reprinting all the textbooks, and mentally convert “pre-change” papers is too much work for too little gain. Maybe, but there’s an argument that in the most common case of just $f(x)$ having the (more important) function before its argument makes pyschological sense. (Incidentally, I sat through one undergrad lecturer who periodically unconsciously changed that convention because that’s how he did research calculations; confused the heck out of me.)

But are there more signficant examples.

Posted by: bane on February 21, 2008 4:57 AM | Permalink | Reply to this

### “First Mover advantage” a myth; Re: Post-modern web-centric doing Math; Re: Two Cultures in the Philosophy of Mathematics?

Yes, I know what’s in the Econonomics/Game Theory textbooks. But my coauthor and I respectfully challenge the notion. The paper will explain all (cut and pasted from an email this weekend):

“Congratulations [Prof.] Philip [V.] Fellman [Southern New Hampshire University], Jonathan [Vos] Post:
We are pleased to announce that your submission to the 2006 International Conference on Complex Systems, Paper #089, ‘Complexity, competitive
intelligence and the “first mover” advantage’, has been accepted for the print proceedings of the conference.”

As to the protocol of paper-writing, Phil and I are still recovering from the 3 sessions we chaired at the 2007 International Conference on Complex Systems [I ran 2 tracks of Physics, he ran one on Consciousness] and we are knee-deep in preparing our papers for the 2009 International Conference on Complex Systems.

And I’m trying to boil down 1,000 pages of notes to a 4-page paper for “Nature” with my coauthor Thomas L. Vander Laan, M.D., F.A.C.S., on the mathematical Physics of the small intestine modeled with Fitzhugh-Nagumo equations and more, where I believe I have classified 8 distinct dynamics, including solitons. Solitary waves in peristalsis? That itself should be publishable. Am reading more deeply into soliton theory, thank you Terry Tao!

And teaching several days a week as substitute teacher (Math, Physics, Computers, English) in Pasadena Unified School District high schools and middle schools.

And trying to keep my average of one submission per day to OEIS and Prime Curios and the like.

And gave a short funeral oration for the great Applied Mathermatician, Parallel Computing Pioneer, and Bifurcation expert Herb Keller at Caltech yesterday. He was the equal to Feynman at Caltech for exuberence, eccentricity, chutzpah, and brilliance. Both taught me that ultimately there are no experts – if a problem is interesting, you should fling yourself into it regardless of your label, and become the expert as needed.

Posted by: Jonathan Vos Post on February 26, 2008 2:12 AM | Permalink | Reply to this

### Re: “First Mover advantage” a myth; Re: Post-modern web-centric doing Math; Re: Two Cultures in the Philosophy of Mathematics?

I’m assuming since google found it there’s no problem with a direct link to the paper. It looks interesting and I haven’t had time to do more than skim it. I do however notice that it’s primarily a theoretical economics analysis rather than looking at data gathered from real products/whatever. Certainly in anecdotal experience I’ve seen many examples of “psychological inertia” in choice of software: once you’ve made a choice you simply can’t summon the energy to change unless a staggeringly better product becomes available. So I view the issue of first mover advantage, particularly amongst real people, as still an open question.

But the paper looks very interesting and I’ll try to get around to it.

Posted by: bane on February 26, 2008 7:58 AM | Permalink | Reply to this
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