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September 9, 2006

Mathematics and Virtual Reality

Posted by David Corfield

If we consider the obstacles which stand in the way of certain actions, by and large they fall into two classes. Physical laws prevent us from tossing up a ton weight boulder into the air unaided, while social laws prevent us from going around murdering people. In the latter case, we may be able to perform the act physically, but are prevented by our obedience to the law which proscribes it. One tradition, whose history is far too complex to begin to describe here, has sought to make of these two classes one, arguing that some social laws, such as that forbidding murder, are natural laws, that is, are drafted in accordance with the nature of man, and, for many in that tradition, in accordance with the nature of his Creator.

When we come to experience mathematics, we can’t help but feel the force of the obstacles put in our way to prevent us from doing what we like. At first, these may appear to be arbitrarily imposed on us by our teachers. We wanted to add 23\frac{2}{3} to 34\frac{3}{4} and make 57\frac{5}{7}, but the spoilsports put a big red cross by it. If we are fortunate, we will come to understand why rules, such as those for the addition of fractions, are being imposed. Inspired by the beautiful coherence of these rules, we may eventually turn our hand to research. Now, instead of forever butting up against the disapproval of our teachers, we encounter some freedom at last. Our supervisor suggests we look at this paper and that to see if we can generalise their results in a certain sort of direction. Freedom of a sort, then, but clearly we can’t just define things how we like. Not only must we reason logically, but we feel ourselves severely restricted in how we define our concepts. When we get it right, unthought of consequences should ensue.

Indeed, very commonly a researcher’s explorations in one area will lead them, apparently inexorably, to the concepts of a very different field. When we come to describe the experience of this reality we are butting up against, options are rather slim. As a first choice, we might compare it to physical reality. But the differences here seem too great. We might then liken working in mathematics to playing according to the rules of the game of chess. This takes us beyond the realm of physical possibility. So we could move our bishop for our first move, but then we simply wouldn’t be playing chess. Also, like mathematics, we can play chess in our heads without a physical chess set. However, the analogy doesn’t take us very far. The rules of chess seem far too arbitrary. Certainly, they’re nicely crafted to make for a game that has absorbed people for their whole lives. But engaging in mathematics is more like being able to change some of the rules of the game. If we took all such card and board games together the analogy would be closer. But where then is the analogue to the discovery of surprising links? It would be as though someone could discover a new strategy in bridge, and a chess player then recognise how it could help her game.

Where can we find a new point of comparison? Alexandre Borovik has located one in the field of virtual reality.

Mathematics is an interactive multiplayer game. Its virtual reality is constantly affected by your actions and by the actions of other players.

  • Why is it one single game for the entire world and thousands, if not millions, of players?
  • What makes the game stable?
  • Why does not it crash?
  • What is the nature of the shared game space for all players?

Mathematics is then compared by Borovik to a “Massively-Multiplayer Online Role-Playing Game” (MMORPG). And he goes on to pose the questions:

What is the nature of intrinsic and unintended laws of MMORPGs? Why do virtual world economies of MMORPGs obey the same laws as the real world economies? In particular, why do many virtual worlds suffer from inflation?

This brings us back to social laws, and the question of their arbitrariness. For some the appearance of the same economic laws in the virtual world will reflect the inevitably of the truths of economics. But another response would be to argue that one should expect common phenomena to appear in the virtual world and our world as these MMORPGs’ economies are modelled so closely on our economy, and this certainly isn’t the only possible economy. A further debate might then ensue about whether there is an ideal economy. Someone in the Natural Law tradition, for example, would invoke the concept of a ‘fair price’, that is, one which does justice to all parties, understanding the virtue ‘justice’ in a specific way.

It may appear that I am being led away from mathematics, but let me end by recalling that Grigory Perelman, a player of the Massively-Multiplayer Online Role-Playing Game that is mathematics, has turned down both a prestigious award within that game, as well as $1 million of the money we use in the ‘real world’ game. Many have speculated about his reasons. Perhaps they are ethical. According to The New Yorker:

Mikhail Gromov, the Russian geometer, said that he understood Perelman’s logic: “To do great work, you have to have a pure mind. You can think only about the mathematics. Everything else is human weakness. Accepting prizes is showing weakness.” Others might view Perelman’s refusal to accept a Fields as arrogant, Gromov said, but his principles are admirable. “The ideal scientist does science and cares about nothing else,” he said. “He wants to live this ideal. Now, I don’t think he really lives on this ideal plane. But he wants to.”

Posted at September 9, 2006 9:12 AM UTC

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Re: Mathematics and Virtual Reality

The Field’s medal only comes with $15K I think. Maybe his mother will convince him to take (his share of) the million if its offered by the Clay Institute. I think the wording says the prize is for the first complete and correct statement of a proof. The Chinese claim that some bits of Perleman’s proof could not be understood, and they had to invent stuff to fill the gaps. This was an ambit claim for the prize, but I reckon they’ll be delighted with a minor share.

Posted by: Robert on September 9, 2006 12:07 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

[…] has turned down […]

I was wondering about this. Something strikes me as strange. Consider this.

A (to B): “Many thanks for the interesting blog entry you wrote! I will put a prominent link to it on my own website to make a broader public aware of your very noteworthy work.”

B: “I will not accept this gratitude of yours. I turn it down.”

A: “But I just want to express my respect for your work.”

B: “I do not accept your respect. In order to be able to write truly deep blog entries, I must remain undisturbed by the effect they have on other people.”

Why can’t they just award a person whatever prize? Why does a prize have to be accepted?

Maybe the reason is this: there is really unofficially a second prize being awarded. This time by the chosen laureate. The chosen laureate decides if he awards the jury the prize of accepting their prize. In this case, maybe, he didn’t deem the jury worthy of that prize he had to offer.

Or else, I am not sure how rejecting a prize - with all the media attention that brings with it - is more helpful in concentrating on research than just saying: “thanks”.

Posted by: u on September 9, 2006 12:41 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

Perhaps he felt that if he’d accepted he’d have been obliged to receive it in Madrid and give a plenary talk. But then surely most people would have been happier had he accepted but made his excuses for Madrid. Perhaps any intrusion from outside his world, even recognition from afar, was to be blocked. I wonder how common it is for mathematicians to enter the mathematical world as a form of escape from the everyday world.

Altogether not very Platonic. After having the eye of your soul opened to the forms by studying mathematics for ten years, and a period studying dialectic, the philosopher king of The Republic takes on administrative duties, returning to the cave to deal with those still shackled there.

Posted by: David Corfield on September 9, 2006 1:13 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

I love the MMORG analogy precisely because it makes clearly the arbitrary, socially constructed nature of rules like ‘success’, ‘worthy of a chair’, ‘worthy of publication’ and so on.

Could one imagine a different practice of mathematics where the relative social positions of, say, category theory and analysis are inverted? Of course. But not many people play that game.

What would be lovely would be to see a similarly nuanced analysis of ‘accepted proof’. I don’t of course mean proof in the sense of proof theory - I mean it in the sense of ‘what is published as a proof in a reputable journal’. Like most working mathematicians, I have published ‘proofs’ which turned out not to work. I noticed first, thank goodness, and fixed it. But that’s not the point: I don’t honestly believe the edifice of mathematics would be very different if a large number of published proofs turned out to be, in detail, false. But I do believe it would be very different if the rule that determines what we accept as a proof were different.

Posted by: d on September 9, 2006 8:20 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

d wrote:

I love the MMORG analogy precisely because it makes clearly the arbitrary, socially constructed nature of rules like ‘success’, ‘worthy of a chair’, ‘worthy of publication’ and so on.

You might mean two things by this.

  1. However the mathematical community organised itself, it would be arbitrary and socially constructed which pieces of research got recognised as important, and which people got elected to chairs.
  2. The way the mathematical community is organised now is such that it is arbitrary and socially constructed which pieces of research get recognised as important, and which people get elected to chairs. But things need not be like this.

These are very different proposals. The first is more in accord with what I, after MacIntyre, have termed a ‘genealogical’ theory of enquiry. The second with what I have termed ‘traditional’.

Posted by: David Corfield on September 10, 2006 10:32 AM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

David Corfield wrote:

You might mean two things by this.

  1. However the mathematical community organised itself, it would be arbitrary and socially constructed which pieces of research got recognised as important, and which people got elected to chairs.
  2. The way the mathematical community is organised now is such that it is arbitrary and socially constructed which pieces of research get recognised as important, and which people get elected to chairs. But things need not be like this.

I mean the first: and thank you for the link.

Concerning my riff on proof, I suppose I am proposing a counterfactual: what would mathematical practice be like if we had a different notion of ‘proof’ to the ‘usual’ one. After all, the current Perelman/Yau controversy suggests that everyone does not necessarily agree about what a proof is. (I’m not defending Yau, simply pointing out that there is an issue here.) What if we valued intuitive but wooly ideas more than we do today? Or less? What if we demanded a full machine checked proof before we accepted something?

Posted by: d on September 10, 2006 5:03 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

If I could press you a little harder on this. Are you implying that:

  1. No piece of mathematics is of any greater intrinsic value than any other piece.
  2. Some pieces are of greater value, but however the mathematical community organises itself, it will not be able recognise this value, at greater than chance levels, ever.
  3. Some pieces are of greater value, but however the mathematical community organises itself, it will not be able recognise this value, at greater than chance levels, in time to reward the relevant researchers, although it has a good idea of the intrinsic value of older mathematics.

1 or 2 would be depressing. Aren’t modular forms more worthy of our attention than neutrosophic rings? 3 is less depressing.

Concerning my riff on proof

I’m slightly wary of following you down your riff. It’s not that I don’t find it interesting, but the topography of the philosophy of mathematics is such that within minutes of your uttering the word ‘proof’, you tumble headlong into a deep canyon, where the only sound you’ll hear are echoed voices chanting about the rational warrant for our belief in propositions of arithmetic. For a mathematician on why truth isn’t what’s philosophically interesting about mathematics, see Michael Harris’s “Why Mathematics?” you might ask.

I’m enormously more interested in the research directions mathematicians take.

Posted by: David Corfield on September 10, 2006 5:54 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

David Corfield wrote:

If I could press you a little harder on this. Are you implying that:

  1. No piece of mathematics is of any greater intrinsic value than any other piece.
  2. Some pieces are of greater value, but however the mathematical community organises itself, it will not be able recognise this value, at greater than chance levels, ever.
  3. Some pieces are of greater value, but however the mathematical community organises itself, it will not be able recognise this value, at greater than chance levels, in time to reward the relevant researchers, although it has a good idea of the intrinsic value of older mathematics.

Sorry, none of the above. I am implying that value is time and reader dependent, and that the tenure-weighted average across the community moves, too. I am not saying this is a good thing or a bad thing, just that like any semiotics, ‘mathematical value’ is context dependent. Just as relativising ‘proof’ to ‘proof within the internal language of the topos I happen to be working in today’ is liberating and useful, so relativising ‘value’ to ‘value within the society of mathematicians I respect’ is liberating too.

We might think that we can agree on general principles of value, and perhaps you and I can, but they would be hardly recognisable by mathematicans of 100 years ago, let alone 200. Not only do we not have a Platonic ideal of ‘value’, we don’t need one.

I’m slightly wary of following you down your riff. It’s not that I don’t find it interesting, but the topography of the philosophy of mathematics is such that within minutes of your uttering the word ‘proof’, you tumble headlong into a deep canyon, where the only sound you’ll hear are echoed voices chanting about the rational warrant for our belief in propositions of arithmetic.

Agreed, but don’t abandon your good empiricist tendencies. ‘Proof’, as Alice might say, having learnt her lesson, means what I want it to mean today. Or rather what the community wants it to mean. Just (in the bath, if need be - you needn’t write it down) speculate on what mathematical praxis would be like if ‘published proof’ meant something slightly different.

Posted by: d on September 11, 2006 12:11 AM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

From virtual reality to neutrosophic rings in only 8 comments - we’re sinking fast!

I’d be willing to make a devil’s-advocate case that almost any mathematical entity, studied with sufficient persistence and cleverness, could rival modular forms in interest (though I don’t actually believe this).

But when I tried to learn what a neutrosophic ring is, so I could take a stab at this devil’s-advocacy, here’s the first thing I saw:

… a neutrosophic group in general does not have group structure. We also define yet another notion called pseudo neutrosophic groups which have group structure.

The authors then go on, not to define neutrosophic groups, but to define a specific example, the “neutrosophic group associated to a group”.

We see here then not a competing mathematical concept so much as a competing style of doing mathematics, which begins by breaking three currently widespread rules:

  1. Unless there’s a darn good reason, an adjective should limit the scope of a term.
  2. Exception to rule 1: the adjective “pseudo” can extend the scope of a term.
  3. When introducing a new term, that term should be defined.

Most mathematicians (e.g. myself) will classify math that starts by breaks too many of such rules as “bad”, and lose interest in it and stop reading.

One can imagine a variant of d’s riff where we study how these rules were developed - and what math might be like if some of these rules were dropped, or others were added.

In particular, math does not always increase in rigor: when the Roman Empire took over Greece, we saw a massive decrease in rigor as people wrote simplified textbooks based on simplified textbooks based on the original Greek stuff. This may have led to a massive loss of mathematical knowledge. This could happen again.

I don’t think we’re on the brink of a severe decline in mathematical knowledge, but it’s worth noting that we always have to work to pass on good rules of thumb, a few of which I listed above. Making these rules more explicit and explaining their value might help.

As for David’s fear that talking about “proof” will make us

… tumble headlong into a deep canyon, where the only sound you’ll hear are echoed voices chanting about the rational warrant for our belief in propositions of arithmetic.

I don’t think we need to worry about that here! We’re not analytic philosophers, after all.

d was proposing a much more interesting exercise, namely to study the effects of different standards of proof on the growth of mathematics:

What if we valued intuitive but wooly ideas more than we do today? Or less?

For example; how should we respond to the wave of wooly but fascinating ideas entering mathematics from physics, like Feynman path integrals? Should we take a “buccaneering” attitude as Atiyah suggests, or a more cautious attitude, as Mac Lane would have us do? Are “physicist’s proofs” useful, or even a substitute for mathematician’s proofs - or are they just dangerous? These are questions real mathematicians are grappling with now, and the philosophers of real mathematics should be interested too.


Posted by: John Baez on September 11, 2006 4:08 AM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

how should we respond to the wave of wooly but fascinating ideas entering mathematics from physics, like Feynman path integrals?

Maybe before getting into math motivated by quantum field theory, it might be worth emphasizing that lots of math we have today was motivated by physics, or was shaped in its interaction with physics.

This might maybe be interpreted as a hint that the reality described by mathematics is not all that “virtual” at all.

The work of a couple of Fields medalist’s points in this direction: Witten of course, Kontsevich certainly. Maybe even the application of Ricci flow in the Poincaré conjecture stuff, for instance.

Following this argument to its extreme end would possibly amount to postulating that the “laws of nature” are in the end nothing but the laws of mathematics. I don’t want to go that far here, but I could come up with supportive evidence of at least a weak form of this statement:

Consider the struggle that physicists went through to find the basic mechanism behind the best theories of the fundamental constituents of our world we have: gravity and the standard model.

Looking at it from a modern perspective, an alien mathematician might shrug and wonder why it took so long to realize that all that is necessary is writing down the first few scalar invariants of a connection.

For gravity

(1) XRdvol \int_X R dvol

for the other forces

(2) XF 2. \int_X F^2 \,.

These expressions are so very elementary from the mathematical point of view, and yet have taken physicists such an enormous amount of man-years to distill.

So maybe, many years into the future when physicists have an even deeper understanding of the “laws of nature”, these will turn out to be even more basic “laws of mathematics”.

Posted by: urs on September 11, 2006 4:02 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

the “laws of nature” are in the end nothing but the laws of mathematics.

That’s precisely Roland Omnes’ position. He calls it physism. I discussed his book Converging Realities in a post and review.

Even if physics is a wonderful source for maths, it could be argued that mathematicians need to have their wits about them to sustain levels of rigour.

Posted by: David Corfield on September 11, 2006 4:25 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

Urs wrote:

Following this argument to its extreme end would possibly amount to postulating that the “laws of nature” are in the end nothing but the laws of mathematics. I don’t want to go that far here…

This isn’t the right place to talk about that extreme view, but interested readers should see:

Posted by: John Baez on September 11, 2006 4:28 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

This isn’t the right place to talk about that extreme view

But this extreme view is one possible reply to David’s question:

“Why is it that we tend to feel like encountering so many laws of nature while doing pure mathematics?”

Max Tegmark, Is “the theory of everything” merely the ultimate ensemble theory?

I haven’t read it in total. But from what I have seen it looks like I am expecting more unity within the mathematical world than Tegmark seems to imagine. I guess that’s where category theory enters the game.

Posted by: urs on September 11, 2006 4:50 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

David’s question:

“Why is it that we tend to feel like encountering so many laws of nature while doing pure mathematics?”

Eek, did I really pen such an ill-phrased question?

I’ve been meaning to save this up for the dark days of December when we run out things to chat about here at the Café, but so that you can let you minds wander over it in an idle moment, I wonder to what extent the mathematics used in a scientific theory is capturing not so much the structure of the world, but rather the structure of our interaction with the world. For an extreme view of this see Caticha’s The Information Geometry of Space and Time.

Posted by: David Corfield on September 11, 2006 5:15 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

Eek, did I really pen such an ill-phrased question?

Sorry, I was trying to paraphrase what I thought the underlying question was.

Posted by: urs on September 11, 2006 5:33 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

In the Harris paper you mention, we read:

“The case is exceptional, however; “certifying knowledge,” in Rosental’s sense, is as such relatively unimportant to mathematicians, and I suspect Perelman’s close readers would describe what they are doing as attempting to understand his proof rather than “certifying” it as knowledge (for the sake of the community, or a generous benefactor, or philosophers or sociologists).”

This is a useful comment. For myself, when I read mathematics, I find myself trying to understand. A good deal of learning and of practising mathematics is involved in such experiences, it would seem. It would also seem that the process of understanding at least implicitly involves an understanding that a piece of mathematics is correct, or incorrect. After all, in the case of a new result, one needs to know whether it can be used.

So I don’t see that here there is rift between the experience of understanding and the experience of certifying knowledge, although the latter might be more restricted than the former.

Posted by: Dennis on September 11, 2006 8:43 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

Dennis wrote:

So I don’t see that here there is rift between the experience of understanding and the experience of certifying knowledge, although the latter might be more restricted than the former.

Yes, no rift. But the idea of Harris, Atiyah, Thurston,… is that where you have ‘might be’ you should have ‘is’, or even ‘is considerably’, unless, that is, you expand what you mean by certifying knowledge far beyond what is usually meant.

The key question then becomes: What is it to understand? Or, in Thomistic terms, What is it for mind to be adequate to its objects? MacIntyre claims:

“…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.

This would need to be tailored for mathematical understanding.

Posted by: David Corfield on September 12, 2006 8:10 AM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

“Yes, no rift. But the idea of Harris, Atiyah, Thurston,… is that where you have ‘might be’ you should have ‘is’, or even ‘is considerably’, unless, that is, you expand what you mean by certifying knowledge far beyond what is usually meant.”

I suppose that is what I was suggesting above. One cannot even read a paragraph of a math text without, in a way, certifying knowledge (that is, experiencing that the line of reasoning is certain). Without this, the reading soon comes to a halt, understanding ceases, and the book is put back on the shelf. This is quite unlike reading other material, like the daily newspaper. The term “certifying knowledge” may have a narrow connotation, that of an activity consisting of figuratively stamping a mathematics paper with the word “truth”. So if something like this is what the expression means, mathematicians do very little of it. But then, it may be that a strawman is being set up.

A similar situation obtains in other descriptions of ascertaining the correctness of mathematics. It is common to read that ascertaining correctness consists of establishing syntactic correctness in a logical calculus. Again, this does not capture the mathematical experience.

In mathematics, theoretical discussion of establishing truth, certainty, etc., does not have much play (from what I can tell), largely, I suspect, because there is little dispute. Practical certainty regarding accepted results has managed to obtain. (There is some theoretical discussion of long proofs, diagrams-use in mathematics, and computationalism with regard the issue of certainty. But disputes are being worked out in practice, it seems.)

Philosophical discussion of certainty seems to get stalled, then, in two ways: narrow construals of certainty and lack of interest in the philosophical dimension of the issue in mathematics. But isn’t the practice of mathematics emerged in attempts to achieve certainty, to nail proofs down, make the reasoning transparent, substitute the awkward for the elegant? (The certainty I have in mind is closely associated with perfection, beauty, elegance, economy, cleverness, etc., in mathematics.) Some aspects of some activities do not directly involve achieving certainty. Mathematicians, e.g., can disagree about whether an axiomatic system is appropriate. But the goal of the system is to axiomatize. I can’t think of any activity in mathematics which does not entail the end of certainty in some way, even if the road to that end is long and winding. Certainty exerts a pull. Modern mathematics is an activity, after all, in which axiomatization is central.

Talking about mathematics in this way does not seem at odds with the content of the MacIntyre quotation which you give above. A story along his lines might be told about mathematics in which certainty is seen as a distinctive feature of the whole enterprise. (Another philosophical quest – my primary interest – is: how does practical certainty arise, does it come in varieties, in degrees, etc.?)

Posted by: Dennis Lomas on September 16, 2006 4:26 AM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

Talking about mathematics in this way does not seem at odds with the content of the MacIntyre quotation which you give above.

I agree. A part of the dramatic narrative of mathematics is the story of successive usages, and understandings of those usages, of axioms. Though the quest is for something timeless, the quest itself takes place in time.

Posted by: David Corfield on September 18, 2006 2:24 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

d wrote:

relativising ‘value’ to ‘value within the society of mathematicians I respect’ is liberating…

We might think that we can agree on general principles of value, and perhaps you and I can, but they would be hardly recognisable by mathematicans of 100 years ago, let alone 200. Not only do we not have a Platonic ideal of ‘value’, we don’t need one.

You’re very close here to the MacIntyrean philosophy I’ve been espousing. Let’s see if I can get you totally onboard.

You can believe that what has been taken to be of central importance in mathematics has changed greatly over the centuries and will continue to do so, while still believing there is a something towards which mathematics is moving. For Aristotle, this something is a perfected understanding. We and our descendents may never achieve this, or may achieve it but not know we have achieved it, but still a notion of rational enquiry presupposes its possibility. Part of what it means to have progressed towards that end is that you can explain what was partial about the viewpoints of your predecessors. Both this explanation and your current viewpoint may in turn be partial, as your descendents would be able to explain.

Realising that what are held to be the values of mathematics is context dependent is ‘liberating’, you say, but in a way which requires you to be more responsible, if you accept the notion of an end or telos. As a responsible mathematician you should always put your current conceptions of value into question. One important way of doing this is through getting to know the history of your tradition. Not the sanitized ‘royal-road-to-me’ type history, but one which serious seeks to put the present into question by examining the past.

Here’s Robert Langlands:

Despite strictures about the flaws of Whig history, the principal purpose for which a mathematician pursues the history of his subject is inevitably to acquire a fresh perception of the basic themes, as direct and immediate as possible, freed of the overlay of succeeding elaborations, of the original insights as well as an understanding of the source of the original difficulties. His notion of basic will certainly reflect his own, and therefore contemporary, concerns.(The Practice of Mathematics: 5)

A perhaps more readily available way to put your views to the test is to interact with other mathematicians. In a Clay Mathematics Institute interview, Terence Tao speaks of the importance of “being exposed to other philosophies of research, of exposition, and so forth”. Tao also claims that “a subfield of mathematics has a better chance of staying dynamic, fruitful, and exciting if people in the area do make an effort to make good surveys and expository articles that try to reach out to other people in neighboring disciplines and invite them to lend their own insights and expertise to attack the problems in the area.”

One of the virtues required of the MacIntyrean practitioner is a readiness to acknowledge the current weaknesses or apparent resourcelessness of their mathematical position. You will sooner find whether you are on the right path this way.

One good way of stepping in the Aristotelian direction would be to make available research proposals and referees comments. They may not all be of the quality of Grothendieck’s Esquisse, but their publication would be of tremendous value.

There is, of course, much more to say, but as I have tried to say it in my How Mathematicians paper, I’ll stop here.

Posted by: David Corfield on September 11, 2006 8:37 AM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

David Corfield wrote:

We and our descendents may never achieve this, or may achieve it but not know we have achieved it, but still a notion of rational enquiry presupposes its possibility.

I’m not sure that particular diagram possesses a colimit, let alone an ω-filtered one. Or, to be less irritatingly arch, I’m not sure there is one telos. It is purely taste, but I find Deleuze’s* mathematics as a history of problematizations a much more satisfying ontology than thinking of it as a discourse refining some essential truths.

(* He wrote ‘knowledge’ rather than ‘mathematics’, but the sense is clear.)

One of the virtues required of the MacIntyrean practitioner is a readiness to acknowledge the current weaknesses or apparent resourcelessness of their mathematical position. You will sooner find whether you are on the right path this way.

Of course: this is clearly a good thing.

One good way of stepping in the Aristotelian direction would be to make available research proposals and referees comments. They may not all be of the quality of Grothendieck’s Esquisse, but their publication would be of tremendous value.

That’s a good idea. Or how about making all referee’s reports public, after the fact. Those shabby little supervisor reviewing student paper arrangements one sometimes comes across should be exposed…

Posted by: d on September 11, 2006 4:22 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

d wrote:

It is purely taste, but I find Deleuze’s mathematics as a history of problematizations a much more satisfying ontology than thinking of it as a discourse refining some essential truths.

“It’s purely taste…” marks one of the problems for those that have followed Nietzsche. If truth is

“A mobile army of metaphors, metonymies, anthropomorphisms, a sum, in short, of human relationships which, rhetorically and poetically intensified, ornamented and transformed, come to be thought of, after long usage by a people, as fixed, bunding, and canonical. Truths are illusions which we have forgotten are illusions, worn-out metaphors now impotent to stir the senses, coins which have lost their faces and are considered now as metal rather than currency?”,

then what of their own pronouncements? Won’t they be condemned to spend a life escaping structures imposed upon them, their academic position (Nietzsche and Foucault), and more importantly that imposed by their own writing? Either they are left forever searching new ways of writing, or they are happy for their writings to become a new mobile army. But why is one army better than any other?

Having spent rather a lot of my early years trying, and I think partially succeeding, to understand one difficult French thinker - Jacques Lacan, I have been reluctant to take on another like Deleuze. If you’d like the café to discuss Deleuze, we could. Either in this thread, or you could e-mail something for a new post.

Posted by: David Corfield on September 12, 2006 9:05 AM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

David Corfield wrote:

Won’t they be condemned to spend a life escaping structures imposed upon them, their academic position (Nietzsche and Foucault), and more importantly that imposed by their own writing? Either they are left forever searching new ways of writing, or they are happy for their writings to become a new mobile army. But why is one army better than any other?

Undoubtedly there is a problem with theories of value in this setting: and certainly many of the French school(s) would recognise this as an issue. And yes, there is a fundamental relativism there. But personally it doesn’t bother me nearly as much as the problems of the single telos.

If you’d like the café to discuss Deleuze, we could. Either in this thread, or you could e-mail something for a new post.

Thank you for the kind suggestion. There are perhaps a few loose analogies here, from my rather limited understanding of him. The idea of micro-id seems rather close to the logic locally around a point for instance. But I don’t understand Deleuze well enough to make the points well without devoting a month to rereading him. Would anyone else with a better understanding care to kick off?

Posted by: d on September 12, 2006 12:22 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

David writes:

You can believe that what has been taken to be of central importance in mathematics has changed greatly over the centuries and will continue to do so, while still believing there is a something towards which mathematics is moving. For Aristotle, this something is a perfected understanding. We and our descendents may never achieve this, or may achieve it but not know we have achieved it, but still a notion of rational enquiry presupposes its possibility.

I don’t think the notion of rational understanding presupposes some concept of “perfected understanding”. On the contrary, I think the concept of a “perfected” understanding is a red herring that confuses discussions about rational enquiry. One can make perfect sense of the word “better” without there being a “best”. Or, in math jargon: you can have a partially ordered set without a maximal element.

So far, it seems to me, we’re able to continually improve our understanding of even the simplest issues without ever running into some final “perfect” understanding. I think it’s an empirical question, and an open one, whether we’ll ever decide we understand some things “perfectly”.

I think this “perfection” idea in Aristotle had to do with some distinctively Greek distaste for the apeiron - the unlimited. They didn’t seem to get the idea of partially ordered sets with infinite chains. If they had, Zeno’s paradoxes wouldn’t have seemed paradoxical. And the whole First Cause Argument, or “Cosmological Argument”, seems to be based on a distaste for infinite chains. As summarized at Wikipedia, it goes like this:

  1. Every effect has a cause(s).
  2. Nothing can cause itself.
  3. A causal chain cannot be of infinite length.
  4. Therefore, there must be a first cause; or, there must be something which is not an effect.

Which assumption seems the least self-evident? To me it’s number 3. But maybe Aristotle found it convincing.

Mathematical digression: you can easily turn the above argument into a proof that a partial ordered set with no infinite chains has a maximal element. But, there could be lots of maximal elements if the partially ordered set fails to be linearly ordered - as indeed causal relations typically do. So, this argument for god doesn’t rule out polytheism!

Similarly, as d points out, it doesn’t rule out the possibility of lots of distinct perfected understandings.

Another funny thing: in a linearly ordered cosmology, the “first cause” comes before all the rest, while the “perfected understanding” comes after all the rest. There’s some sort of duality between “efficient cause” and “final cause” (telos) going on here.

But now I’m getting a bit slap-happy. I don’t actually think we can get far in philosophy by reducing everything to partially ordered sets, though I think they’re a handy way to make precise certain naive assumptions. My serious point is just that we can - and often should - talk about improvement without the concept of perfection.


Posted by: John Baez on September 12, 2006 5:40 AM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

John writes:

I think it’s an empirical question, and an open one, whether we’ll ever decide we understand some things “perfectly”.

I don’t think it’s an empirical question. How could we ever know we had a perfected understanding? How could we preclude the possibility that some intelligent mind could show our understanding to be limited?

A point to note here is then when returning over 700 years to resume a line of philosophical thought, along with it come some very foreign ideas. We’ve be banding about ‘we’ and ‘our’ happily enough, but what are we including and excluding with such pronouns? In the 13th century there were many other postulated intelligences. But even in the 21st we might want to consider our evolutionary descendents.

Anyway, as I say, I don’t think it’s an empirical question. The notion of ‘perfected understanding’ can only be argued for on the grounds that it is presupposed by the very nature of rational enquiry. You raise a very interesting objection to this in terms of partially ordered sets. Rather than there being one maximal element, there might be none, or there might be more than one. Let’s consider the latter first.

What could it mean to say that there are two, or more, perfected understandings? Perfected understandings of what? To say they are different means that there must be some way in which one shows understanding of its object which is unavailable to the other. But doesn’t this simply imply that the understanding of that second intelligence is not perfected? Perhaps one could draw an analogy with maximal states of information about a quantum system. Such maximal states are not complete. There may be many distinct such states. But could this structure be found in an enquiry such as mathematics, something like: if you know all there is to know about 3-manifolds the Thurston way, you can’t know all there is to know the quantum groups-knot theory way.

The other possibility is for there to be no perfected understanding. This seems to be a stronger challenge. I’ll have to think about it.

This is a new experience for me, having my MacIntyrean Thomism put to the test. I must say, as a secular sort of person, I’m intrigued to find out how much necessarily goes along with it. Why is it that MacIntyre, Polanyi, and Collingwood, three philosophers of the twentieth century who think along these lines as regards science, all are led to a form of theological metaphysics?

Posted by: David Corfield on September 12, 2006 8:49 AM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

John wrote:

I think it’s an empirical question, and an open one, whether we’ll ever decide we understand some things “perfectly”.

David wrote:

I don’t think it’s an empirical question. How could we ever know we had a perfected understanding? How could we preclude the possibility that some intelligent mind could show our understanding to be limited?

We can’t - and that’s precisely because it’s an empirical question. It’s like the question of whether a certain runner has reached perfection in running. If some guy beats everybody else hands down, and the record stands for a few centuries, I bet most people would be willing to say this guy reached perfection in running. And, I think it would be fine to say that. But, tomorrow someone could come along and beat him. Then we’d know we’d been wrong.

Similarly, if our progress in understanding the equation 2+2=4 reaches a plateau at some point, and stays like that for a few centuries, I’d feel happy saying we’ve reached a perfect understanding of this equation. But, tomorrow someone could come along and prove me wrong, by coming up with a better way to understand it.

I don’t mind the fact that empirical statements are never certain; this is part of their nature, and it shouldn’t stop us from making them.

The notion of ‘perfected understanding’ can only be argued for on the grounds that it is presupposed by the very nature of rational enquiry.

In that case I think it’s a dead duck, for reasons I began to explain before.Why must we invoke “perfection” to understand the meaning of “improvement”? Perfection is just some weird ideal limiting case of improvement.

It’s possible, of course, that you’re using “perfected understanding” in some special Aristotelian sense that I don’t perfectly understand. I’m just using it to mean an understanding that cannot be bettered.

(If we assume a linear ordering on “understandings”, this is the same as an understanding that’s better than all others. But in general, I’m suspicious of assuming that any sort of human quality like “perfection of understanding” is linearly ordered. This is something that has to be demonstrated, not assumed! And, usually the fuzziness of concepts kicks in before one can prove something so demanding as trichotomy.)

Why is it that MacIntyre, Polanyi, and Collingwood, three philosophers of the twentieth century who think along these lines as regards science, all are led to a form of theological metaphysics?

Good question. Maybe because, following Aristotle, they don’t believe in posets without suprema?

More seriously, I think it’s important to avoid the pernicious effects of relativism without falling into the arms of Catholic doctrine. So, I hope you manage it.


Posted by: John Baez on September 12, 2006 10:04 AM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

This is all very well, but it doesn’t get the baby bathed, or Kleinian 2-geometry furthered. On the other hand, if you’re trying to save my immortal soul from embracing catholic doctrine, perhaps we should proceed.

John wrote:

David wrote:

I don’t think it’s an empirical question. How could we ever know we had a perfected understanding? How could we preclude the possibility that some intelligent mind could show our understanding to be limited?

We can’t - and that’s precisely because it’s an empirical question.

Here, wait a minute! If the proposition is “There is a time at which some intelligence will achieve perfected understanding of some object.”, that’s neither falsifiable nor confirmable. It’s evidently not falsifiable like Popper’s classic example of a proposition “There is some Latin utterance such that if spoken at some place and at some time, a horned devil will appear.” But nor is it confirmable being a nasty mixture of ‘there exists’ and ‘for all’. We can disconfirm a potential confirming candidate, but not confirm it.

I’m not sure what your example of the runner shows. It appears to me to be saying that we can never conclusively confirm a proposed instance of perfection, which is what I was saying. I suppose you’re also suggesting that that we can get by using ‘perfection’ in a loosish sense, and don’t need it for any other purpose. Let’s see. Part of the trouble with the analogy is that we think we have a clear idea of the goals of running. But what if, like at my nieces’ Steiner school, in the future winners are chosen by how fast they can run or how far they can throw, but by how gracefully they perform. Athletics is hardly a tradition of rational enquiry, but if we’re buying into the parallel we have to recognise the possibility of the goals of the tradition changing radically.

And this brings us to the problem of the infinite chains. If you conceive of us as living in a infinitely shelled ‘Matrix’, where at each emergence into the next shell you find good arguments to convince yourself that what you understood in the previous shell was wrong, including what you understood of previous shells, so that now you might believe that 2 shells ago, you had greater understanding than in your last shell, would you care to leave any particular shell you find yourself in?

Perhaps it’s because your fields are maths and physics where it’s plausible to think that at each stage of history that the representation of the poset of greater understanding justs gets augmented by the latest elements, or at worst a slight reshuffling of the order in a spot of less importance, that you’re happy to get by with ‘improvement’.

“I’m suspicious of assuming that any sort of human quality like “perfection of understanding” is linearly ordered.

We only wanted a partial ordering.

Posted by: David Corfield on September 12, 2006 11:35 AM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

How’s this for an argument for why a theory of rational enquiry precludes the possibility that there be no perfected understanding. John illustrated this idea with the image of a partially order set with an infinite increasing chain possessing no maximal element. He also remarked “I don’t actually think we can get far in philosophy by reducing everything to partially ordered sets”. Let’s see why he’s right.

I’m going to suggest that a tradition of enquiry presupposes considerably more strucure than a poset. Take the enquiry at two moments in time A and B. What is it for people at time B to say that their understanding is better that at time A? Something along the lines of: all the best reasoning we consider now tells us that everything that the A people could do well, we can also know how to do, and we can do more. Some of the frustrations they experienced, we can understand now why they occurred. But the B people should also recognise that they may be wrong in their understanding. People at a later time C may be able to explain B’s frustrations, etc., and at the same time show that some things the A understood had been overlooked by B, or that B’s explanations of A’s failures were partial.

The trouble with the poset image is that we forget that we’re looking on it from a God’s eye viewpoint. What is really the case though is that through the history of an enquiry the representation of the poset of improved understanding changes. Previous representations are overturned. Where the B thought they had improved on A, we may now say their research had degenerated.

Now, presumably the Thomist would want to say that to partake in a rational enquiry, which as we have seen involves the making of judgements as to what is an improvement over what else, there is presupposed a perfected understanding. We know our judgements may be wrong or partial, but we do what we do on the basis that we are doing our best to coincide with a perfected understanding which not only knows how things are, but also understands the limitations of other understandings. Without this notion, what sense does it make to say we are doing our best?

Posted by: David Corfield on September 12, 2006 9:54 AM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

David wrote:

Now, presumably the Thomist would want to say that to partake in a rational enquiry, which as we have seen involves the making of judgements as to what is an improvement over what else, there is presupposed a perfected understanding. We know our judgements may be wrong or partial, but we do what we do on the basis that we are doing our best to coincide with a perfected understanding which not only knows how things are, but also understands the limitations of other understandings. Without this notion, what sense does it make to say we are doing our best?

First, I don’t think we need to know what “doing our best” means to understand rational enquiry. We just need to say when we’re doing a bit better: making progress, improving our understanding.

For this, it helps to have some concept of when a given understanding is better than another.

But why in heaven’s name do we need to define better as most resembling the perfect? That seems to get it all backwards. I’d prefer to define “perfect” as “better than everything else”, and let someone with more spare time worry about whether perfection is ever achieved.

Of course there’s an old tradition, going back at least to Plato, of basing ones theory of knowledge on The Good. From the Republic:

Then what gives the objects of knowledge their truth and the mind the power of knowing is the Form of the Good.

If we further equate The Good with “perfection”, than one might say X is better than Y if it partakes more of perfection. Is your Thomist following some tack like this?

(Amusing side note:

Once Plato even promised to give a lecture on The Good. But when the day came, all he did was do a geometry proof.)

Posted by: John Baez on September 12, 2006 12:02 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

If we further […] then one might […]

May I expect that when the two of you think about this long enough you will come up with a good formalizaton that makes these considerations precise? I am thinking of the way formerly vague concepets like “theory”, “computation” and “model” have been made precise.

In fact, could it be that what is needed here is a formalization of the notion of “model” as used in physics? In physics, a “model” is some kind of approximation to a complex situation.

We improve our understanding by finding more refined models. A model gives a better understanding than another model if it contains the other in some way as a special case (often as some kind of limit).

So maybe one should consider some entity “R” (a reality) and let models be forgetful morphisms

(1)RM. R \to M \,.

A model M 1M_1 is better than a model M 2M_2 if

(2)RM 2 R \to M_2

factors through

(3)RM 1. R \to M_1 \,.

Or something like that.

Depending on how this is set up in detail, one could then analyze precisely what structure there is on the collection of models (a partial order, a total order, or something more complex).

Posted by: urs on September 12, 2006 12:20 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

Surely the best chance to use a formalization to achieve anything concerning progress is in a field like physics. There is surely progress in a discipline such as history, yet I can’t see it being formalized. (Strangely it would seem harder to do this in maths than physics, but then as the formal tools would be part of maths itself perhaps it’s not so surprising.) Now as you may imagine many people have tried to do what you say: formulate physical theories or models formally, and look to see what happens to successive theories. Unfortunately, some rather unpleasant pieces of formal apparatus have been used, and I don’t think category theory has been given a proper go.

One objection you’d face concerns what precisely to include in the model. Bob brought up the issue of including measurement within a theory. But there are many philosophers who think there’s a whole lot more to physics than looking for maps between reality and a mathematical model. Morgan and Morrisson’s ‘Models as Mediators’ (CUP) comes to mind.

Still I’d like to see the history of the physics of the electron set out in category theoretic form.

Posted by: David Corfield on September 12, 2006 2:38 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

Surely the best chance to use a formalization to achieve anything concerning progress is in a field like physics.

That might be. On the other hand, I got the impression that what you were talking about involved generally the attempt to incrementally understand some reality RR. Be it the reality of mathematical propositions or that of computational structures - or the reality of physics.

So there is some approximation process involved in any case. “Model building.”

As John wrote #

For this, it helps to have some concept of when a given understanding is better than another.

Now just replace “understanding” by “model” (which is quite justified, I think).

Posted by: urs on September 12, 2006 3:41 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

All I can say is that I’ve seen a few attempts trying to do something along the lines you suggest, and they’ve all struck most people as pretty worthless. Particularly famous were a group of German structuralists, who every few years augmented their model of science with some new feature. The last I saw of their work they were saying things like “A scientific theory is an 11-tuple…”.

I’m happy to be proved wrong, but I’d sooner go along with a view that

It is more rational to accept one theory or paradigm and to reject its predecessor when the later theory or paradigm provides a stand-point from which the acceptance, the lifestory, and the rejection of the previous theory or paradigm can be recounted in more intelligible historical narrative than previously. An understanding of the concept of the superiority of one physical theory to another requires a prior understanding of the concept of the superiority of one historical narrative to another. The theory of scientific rationality has to be embedded in a philosophy of history.

Posted by: David Corfield on September 12, 2006 4:11 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

a few attempts trying to something along the lines you suggest, and they’ve all struck most people as pretty worthless.

I can imagine that.

So let’s do better!

I am thinking of something like this:

Suppose for the moment that “reality” RR is a category of morphisms

(1)R=[C,Hilb] R = [C,\mathrm{Hilb}]

from some SMCC CC to Hilb\mathrm{Hilb}, within a suitable doctrine.

But maybe this category is hard to understand in its entirety. So instead of trying to describe it in its totality (maybe we don’t even know the precise nature of CC, for instance), we set out and build models approximating it.

As a concrete example: maybe CC equals 2SCob conf2S\mathrm{Cob}_\mathrm{conf}, the category of super-conformal 2-dimensional cobordisms. That makes RR pretty involved.

But in fact there is a much more tractable model of RR in this case, namely

(2)M=[C,Hilb], M = [C',\mathrm{Hilb}] \,,

where CC' contains just one-dimensional cobordisms with branching and joining of cobordisms allowed.

Assume we can make precise the notion of taking a 2-cobordisms and shrinking its diameter until it looks like a 1-cobordisms. Assume this provides us with a functor

(3)CC C \to C'

which sort of “forgets” the deviation of a 2-cobordisms from its “center-of-mass”.

This would induce a functor

(4)RM R \to M

from reality RR to a model of reality MM.

The model does not capture the full nature of RR, but it does capture some crucial aspects of it.

What this means in detail should be expressible rigorously in terms of the various functors involved.

Of course, there are even simpler models of RR, which still capture some important aspects of it. For instance if we let CC'' be the category of 1-corbordisms but without any splitting or joining allowed. Then me may consider a model

(5)M=[C,Hilb]. M' = [C'',\mathrm{Hilb}] \,.

This would forget a little more than MM did, and we’d have a chain of forgetful functors

(6)RMM. R \to M \to M' \,.

Historically, people went from MM' (interactionless single quantum particles) to MM (interactions in Feynman diagrams), while increasing their knowledge of physical reality.

Some speculate that physical reality is the above RR (or, more precisely, that the above RR is a model of physical reality, namely of a world where particles are really strings shrunk to a point).

Part of of the motivation behind this speculation is precisely the fact that we do have a nice functor

(7)RM. R \to M \,.

So now it gets interesting: it might be that there is another reality RR', for which MM is also a good model

(8)RM, R' \to M \,,

but which is incompatible with RR.

This difficulty of identifying RR in the present example is known as the difficulty of identifying the “physics beyond the standard model”.

So this highlights another interesting aspect that the functorial model concept could have.

In general, we just have a chain of models

(9)MMM, M'' \to M' \to M \,,

usually evolved (in the opposite direction) in the course of history. The big question is if there is a sensible “limit” to such a chain of models, namely a true “reality” to which these models are converging.

Maybe we can have chains of such models that continue indefinitely without “converging”. Maybe not.

If suitably formulated, such questions might be formalizable along the above lines.

I don’t know. Maybe not. But that’s what I have in mind here.

The above example, however, is pretty realistic, I think. (Heh, “realistic”…)

Posted by: urs on September 12, 2006 4:41 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

Interesting. I’ll have to think about this. Right now I’m preparing for this workshop in Berlin. With John in Hangzhou, you’ll be left to handle the café and all its awkward customers alone through to Sunday.

Posted by: David Corfield on September 13, 2006 1:18 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

I like this train of thought, but does it really help you make progress toward understanding RR? It almost seems like formalized common sense :)

Posted by: Eric on September 16, 2006 7:15 AM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

but does it really help you make progress toward understanding RR?

Maybe.

What I wrote reminds me a little of what is actually being done in general relativity. There, a central question, which attracts quite some attention, is which model Newtonian gravity is an approximation to.

Clearly, Einstein gravity (“general relativity”) is a model more refined than Newtonian gravity (since it contains the latter as a limiting case) and does match all observations so far very nicely (at least if you accept evidence for dark matter#).

But Einstein gravity might itself be just an approximation to something even more refined, or it might be indistinguishable at current experimental level from similr but different theories.

Since lots of people like to dream up lots of alternative theories to Einstein gravity, there is a certain pressure to find some framework in which to decide which model fits reality to what extent, and how these models interdepend.

There is in fact an attempt to formalize this question, known as the parameterized post Newtonian-formalism.

And this entire question is indirectly related to the situation which I talked about above #, since the decision between RR and RR' in my example usually tends to affect the corrections to Einstein gravity.

So I think in as far as one is willing to discuss what it means to improve our understanding of the world #, attempts to formalize this question as in PPN formalism or along the lines I tried to sketch could be of help. Maybe.

It almost seems like formalized common sense :)

True. My example was inspired by how other pieces of “common sense” have been formalized (better maybe: made precise) by category theory.

For instance the question of what one means by saying that a concept in mathematics is “natural” is what got category theory started, in the first place.

Category theory also tells you what it means for two concepts to be “equivalent”, and so on.

By itself, all these “formalizations” may seem trivial. But once these concepts are clearly identified they tend to give rise to interesting spin-offs.

Posted by: urs on September 17, 2006 4:36 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

Someone else blogging about formalizing concepts like ‘natural’ is Alexandre Borovik. Here he takes on ‘explicit’.

Posted by: David Corfield on September 17, 2006 9:07 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

Interesting.

I should maybe say that I feel that what I tried to talk about above should, in its spirit, not so much be compared with formalizing all those common adjectives like “natural”, or “explicit”, or “equivalent”, but rather with the way to talk about “syntax”, “semantics”, “theories” and “computation” that we discussed #.

In fact, the way I suggested to look at the issue it might be entirely internal to the existing field of categorical (quantum) logic.

I don’t know if maybe Mike is still reading this. But he might know (certainly John does) if there is some established concept describing for instance situations where a model

(1)f:Csomething f : C \to \text{something}

of some SMCC CC factors through the model of a “simpler” SMCC CC'

(2)C something C = C, \array{ C &\to & \text{something} \\ \downarrow && \uparrow \\ C' &=& C' } \,,

where “simpler” probably means that CCC \to C' is forgetful in some sense.

Posted by: urs on September 19, 2006 12:51 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

In fact, the way I suggested to look at the issue it might be entirely internal to the existing field of categorical (quantum) logic.

The key point I’d like to make is that any such formal endeavour is in some sense ‘internal’ to a field. On top of this there must be an act of assent to the adequacy of the field to the task set it, and to the appropriateness of the task. Such an act of assent takes place within a history of enquiry. The philosophical quest for a formal field which will liberate us from history is vain.

Posted by: David Corfield on September 19, 2006 2:05 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

The key point I’d like to make is that any such formal endeavour is in some sense ‘internal’ to a field.

I agree with that. But isn’t the same true when you don’t formalize?

Only that then everyboody may have a different personal internalization implicitly assumed. A situation that is almost bound to lead to misunderstandings.

Posted by: urs on September 19, 2006 2:22 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

Of course! There are many, many benefits to formalisation. It’s just worth reminding people from time to time, and this is probably less necessary amongst mathematicians and physicists, that your commitment to a formalism doesn’t come for free. Gian-Carlo Rota wrote a stinging such reminder to analytic philosophers, The Pernicious Influence of Mathematics Upon Philosophy, asserting that watching them use formalism was like watching someone pay for groceries with Monopoly money.

Posted by: David Corfield on September 19, 2006 2:33 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

asserting that watching them use formalism was like watching someone pay for groceries with Monopoly money.

That’s a nice analogy. I guess I see what he means. The problem is: many (most) subjects are so hideously complex (and/or vague) that every attempt to formalize them is bound to be hopeless.

You will correct me if I am wrong, since what I say next is just a conclusion based on cursory impressions, not based on a systematic analysis:

isn’t it true that, over the centuries, lots of subfields started out as subfields of philosophy, only to become commonly regarded instead as subfields of “science” as soon as people managed to formalize them sufficiently?

The step from “natural philosophy” to modern physics might be just the most prominent one.

Posted by: urs on September 19, 2006 2:52 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

isn’t it true that, over the centuries, lots of subfields started out as subfields of philosophy, only to become commonly regarded instead as subfields of “science” as soon as people managed to formalize them sufficiently?

as soon as people thought they had managed to formalize or operationalize them.

Yes, economics and psychology spun off from philosophy. There are then a curious range of relationships philosophy can have with its children, which range from gazing on them from afar (mathematics) to wanting to play an important role in their lives (physics).

Posted by: David Corfield on September 20, 2006 10:59 AM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

Alexandre Borovik gives the link to this very fascinating essay by Gromov, which discusses many issues occuring in this blog too.

Posted by: Thomas on December 27, 2009 8:55 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

Posted by: Thomas on December 28, 2009 9:29 AM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

I think you’re surreptiously relying on an outside the flow of things perspective. If so, for the Thomist, this would be understandable as such a perspective is a presupposition of being engaged in the flow. Not that we could know we’d ever managed to think in accord with this perspective, just that such a perspective is presupposed. Without such a regulative presupposition, there is no point to enquiry.

For this, it helps to have some concept of when a given understanding is better than another.

This we agree upon. Now do you also agree that this understanding of when a given understanding is better than another is again something to be judged against other such meta-understandings? So here we are with our rival understandings at time C. You understand earlier theory B to be an advance over A, while I don’t. You think your meta-understanding, which is really just a part of your understanding, is better than mine. And so we go on, being as virtuous as we can in trying recognise our prejudices. Our successors will make their own minds up about comparisons between our understandings, and may well disagree with each other. If so, their accounts of the history of the tradition will be very different in the place they accord to us.

Now what do I mean when I say “My understanding, including all that meta-understanding, is better than yours”. Do I really just mean, “From my perspective, my understanding is better than yours.”? Don’t I have one eye on the past, and another on the future (and if I had a third, one on the present too)? “My account of the history of the tradition leading up to today is better than yours and future generations will judge that my understanding was better than yours.”

But is this enough? Besides a clause to the effect that the future generations be rational (more on this later), I don’t just want my ideas to be thought to be right 10 generations ahead, only for this judgement to be overturned 20 generations ahead and ever after. I’d also surely be depressed if I ever came to believe that every ten generations opinion would oscillate between thinking my understanding far superior to yours, followed by a regime which made the reverse judgement. (By this I don’t mean I care about my understanding as mine but about what that understanding is about.) I’d much rather it be found that your understanding was superior to mine ever after, that what we argued about found some resolution in the future. If we knew no such issue in our field ever found resolution would we proceed?

So I’m hoping there’s a chain of improvements in understanding with a certain stability to it. Where successive members of the chain can make good sense of the earlier stages, realise their partialities, etc. And I’m also hoping there aren’t a whole bunch of other such chains making very different judgements about issues in my field.

But is this enough? If an angel (ah, the doctrine) whispered in my ear that there is something built into the human brain which means that in our field of study, however much it seems like we’re getting at the truth, we will always be led astray, and if I believed that voice, would I continue with my work? In other words, I seem to want any future resolution to be arrived at for good reasons, which may not be accessible to me now, but which relate to our descendents minds’ becoming, through new theories, equipment, etc., more adequate to the objects of the field. I don’t just want our descendents for all time to judge my understanding better than yours, I want them to be right about it.

Your comment on Plato is to the point, as Aquinas’ feat was to reconcile Aristotle with an Augustinian Neo-Platonism where God has much to do with The Good.

Posted by: David Corfield on September 12, 2006 1:41 PM | Permalink | Reply to this

Re: Mathematics and Virtual Reality

Even the devil would have the devil of a job playing devil’s advocate to neutrosophic rings. But it would be excellent training for young mathematicians to be asked to defend various constructions. Imagine how much a primary school kid would learn by building a devil’s advocate case for a return to Roman numerals, or a shift to binary for everyday life. This would be an exercise in the mathematics criticism called for by Lakatos and Ronnie Brown referred to in footnote 1 of my ‘How Mathematicians’ paper. Or perhaps, for the more advanced student, why Baez and Dolan are hopelessly wrong wanting to think of the spectrum of a ring as a groupoid rather than as a set.

Should we take a “buccaneering” attitude as Atiyah suggests, or a more cautious attitude, as Mac Lane would have us do?

That Jaffe-Quinn debate was a wonderful eruption through the placid front presented by mathematics to the world. But as we never had a second round of responses we never learned whether apparent differences of opinion were really so. The volcanic model of debate can’t be the best way.

Posted by: David Corfield on September 11, 2006 9:11 AM | Permalink | Reply to this

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