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October 8, 2008

New Directions in the Philosophy of Mathematics

Posted by David Corfield

To celebrate the founding of MIMS, the mathematics department of the recently unified Manchester University, it was proposed that various workshops named ‘New Directions in…’ be run. They kindly agreed to allow Alexandre Borovik and me to organise one of these workshops on the Philosophy of Mathematics.

So, on Saturday 4 October, we began with Mary Leng, a philosopher at Liverpool, talking about whether the creation of mathematical theories, e.g., Hamilton’s quaternions, gives us any more reason to think mathematical entities exist than does the discovery of new consequences within existing theories. She concluded that it does not – both concern the drawing of consequences from suppositions, e.g., “Were there to be a 3 or 4-dimensional number system sharing specified properties with the complex numbers, then…”.

George Joseph, author of The Crest of the Peacock, then told us about sophisticated work in 16th century Kerala concerning expansions of trigonometric functions, and better proved equivalents to the proto-calculus of Wallis. He then moved on to China where in search of the equally tempered scale, the 16th century Chinese mathematician Zhu calculated the value of 2 1/122^{1/12} to an extraordinary number of decimal places. Discussion centred around the question of why we continue to ignore non-Western roots of modern mathematics.

Marcus Giaquinto, a philosopher from University College London, talked about visual intuition and proof. He explained how a great deal of care is needed in assuring ourselves of the validity of a geometric demonstration. This follows up on the work of Ken Manders (Pittsburgh) to understand when Euclidean diagrammatic reasoning is valid.

Angus MacIntyre, a model theorist at Queen Mary College London, told us about the limited interest for mainstream mathematics of incompleteness results. He argued that the kind of Diophantine equations appearing in incompleteness results are not of the kind number theorists deal with, and he explained to us how he is trying to show that Wiles’ proof of Fermat’s Last Theorem can be written in first-order Peano arithmetic.

I finished off the talks by speaking about my paper Lautman and the Reality of Mathematics. This argues that philosophy should study mathematics less as (potentially) concerning abstract entities, but rather as concerning the development of certain ideas. While Lautman pointed us to some excellent examples of this phenomenon, the Galoisian idea and the idea of duality, it seems less clear to me that they are situated somehow superior to mathematics as he believed.

We ended with a brief general discussion, which included a deliberately provocative comment from the philosopher John Kennedy that the majority of mathematicians don’t think hard enough about basic concepts of identity and relation, and that this marked an impoverishment of mathematics since the days of Hilbert.

It occurred to me just before I began my talk that I should have said more about Michael Harris’s comments, discussed at my old blog:

Perhaps we might find richer pickings in answering a challenge posed by Michael Harris in “Why Mathematics?” You Might Ask that philosophers “…have a duty, it seems to me, to account for terms like “idea” and “intuition” – and “conceptual” for that matter – used by human mathematicians (at least) to express their value judgments.” (p. 17). Take the term idea:

Nothing in the life of mathematics has more of the attributes of materiality than (lowercase) ideas. They have “features” (Gowers), they can be “tried out” (Singer), they can be “passed from hand to hand” (Corfield), they sometimes “originate in the real world” (Atiyah) or are promoted from the status of calculations by becoming “an integral part of the theory” (Godement). (p. 14)

Harris makes the very useful point that my own use of the term is liable to a certain slippage:

Corfield uses the same word to designate what I am calling “ideas” (“the ideas in Hopf’s 1942 paper”, p. 200) as well as “Ideas” (“the idea of groups”, p. 212) and something halfway between the two (the “idea” of decomposing representations into their irreducible components for a variety of purposes, p. 206). Elsewhere the word crops up in connection with what mathematicians often call “philosophy,” as in the “Langlands philosophy” (“Kronecker’s ideas” about divisibility, p. 202), and in many completely unrelated conections as well. Corfield proposes to resolve what he sees as an anomaly in Lakatos’ “methodology of scientific research programmes” as applied to mathematics by

a shift of perspective from seeing a mathematical theory as a collection of statements making truth claims, to seeing it as the clarification and elaboration of certain central ideas… (p. 181)

He sees “a kind of creative vagueness to the central idea” in each of the four examples he offers to represent this shift of perspective; but on my count the ideas he chooses include two “philosophies,” one “Idea,” and one which is neither of these. (p.16)

Point taken. I’ll see what I can do.

Posted at October 8, 2008 1:57 PM UTC

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Re: New Directions in the Philosophy of Mathematics

The talk of Marcus Giaquinto got me thinking a little about Euclid’s first proposition. This says that the you can construct an equilateral triangle ‘on’ any line segment, by which I mean you can draw an equilateral triangle which has that line segment as one of its sides. The proof is apparently notorious because it requires you to deduce something from a diagram.

Marcus Giaquinto was arguing that you can get round the philosophical problems associated with arguing from the diagram, but as far as I could see there was still a mathematical problem.

The proof of the proposition goes as follows (and I deliberately won’t give you a picture to go with it). For each of the two endpoints of the interval, construct the circle with that point as its centre and with the the interval as its radius. This gives you two circles. These intersect at two points. Pick one of the intersection points and take that to be the third vertex of your equilateral triangle. QED

There is at least one dodgy step in the above proof. The one I’m thinking of is the assertion “These intersect at two points.” Where does that come from? Well you draw a picture and see that it’s true. Hmmm. I thought Euclid was all about deducing things from axioms. So let’s step back a little.

We could try to do the same construction in spherical geometry rather than Euclidean geometry. This is the geometry of the surface of the Earth, so “straight line” means part of a great circle, that is part of a biggest possible circle such as the equator or any circle passing through both the north and south poles. This kind of geometry satisfies Euclid’s axioms except the so-called parallel postulate.

However, Euclid’s first proposition fails for spherical geometry. If you take your interval to be greater than a third of the circumference of the sphere then there is no equilateral triangle on that interval. If you construct the two circles on that interval then they do not intersect.

I’ve tried to illustrate this with a little animation on YouTube.

So we could play at being Euclid and assert that just because we can draw an interval on the ground (think of doing it in the sand on the beach with a stick), then construct the two circles as above and observe that they intersect, we can therefore conclude that it will be true for any interval we draw on the ground. But that’s not true. If we draw a line from the South Pole to Sheffield then it won’t work.

This seems to mean that any proof of the first proposition must require the parallel postulate. Euclid certainly doesn’t mention it.

So Euclid’s first proof is decidedly iffy. Does anyone know any way around this? I mean I guess this is a very well known problem (for instance, this webpage mentions that the proof doesn’t work for other geometries), so there must be better proofs.

Posted by: Simon Willerton on October 8, 2008 6:55 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

Well, if you believe Wikipedia (and it’s never clear that you should), you need to go beyond Euclid’s axioms for the proof.

Posted by: Aaron Bergman on October 8, 2008 7:11 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

Ah, I guess you mean the Wikipedia article on Euclidean geometry. About a third of the way down it says:

His axioms, however, do not guarantee that the circles actually intersect, because they are consistent with discrete, rather than continuous, space. Starting with Moritz Pasch in 1882, many improved axiomatic systems for geometry have been proposed, the best known being those of Hilbert, George Birkhoff, and Tarski.

The issue here is a slightly different one, being that the model could be discrete and have the parallel postulate. I wasn’t thinking of the discrete case because Marcus explicitly excluded discrete models for some reason which I can’t remember.

I have to say that this has left me a little surprised. How could I be a professional mathematician – professional, ahem, geometer, even – and not know that that Euclid’s Elements, upheld as the quintessence of the axiomatic method, was a sham from Proposition 1?

Posted by: Simon Willerton on October 8, 2008 9:26 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

It’s not obvious to me (at least not after 5 minutes’ thought and 5 minutes’ Googling) what discrete geometry meets all 5 of Euclid’s postulates. 2\mathbb{Z}^2 as the set of points, with lines defined in the obvious way, doesn’t meet the fifth postulate, because many lines that would intersect in 2\mathbb{R}^2 do not do so in 2\mathbb{Z}^2.

Can anyone point to an example of a simple discrete model of the full set of axioms?

Posted by: Greg Egan on October 9, 2008 7:27 AM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

From ‘Mathematical Logic and the Foundations of Mathematics’ by G. T. Kneebone, Hilbert took

a certain number field Ω\Omega, consisting of all the algebraic numbers that can be obtained by ‘beginning with the number 1 and applying a finite number of times the four rational operations: addition, subtraction, multiplication, division, and the fifth operation |1+ω 2|| \sqrt{1 + \omega^2}|, where ω\omega denoted any number already obtained by means of the five operations’. (p. 198)

Powers of this field act as models for all of his axioms except linear completeness, which wasn’t one of Euclid’s.

Posted by: David Corfield on October 9, 2008 8:54 AM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

Thanks, David. That’s quite complicated!

Unless I’m confused, which is always possible, I realised shortly after my last comment that 𝔽 2\mathbb{F}^2 ought to satisfy Euclid’s axioms for any field 𝔽\mathbb{F}, because the linear algebra that allows you to prove the axioms for 2\mathbb{R}^2 should go through identically. And not only would that work for finite fields, it would work for the rationals. So wouldn’t 2\mathbb{Q}^2 satisfy Euclid’s 5 axioms, while also lacking equilateral triangles in most cases?

Posted by: Greg Egan on October 9, 2008 11:48 AM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

The trouble is that Euclid’s axioms are a bit vague. E.g., what does

To describe a circle with any center and radius

mean?

It’s easier to think about Hilbert’s axioms. Then you’ll see that 2\mathbb{Q}^2 fails for HC1. Given a line segment whose endpoints both have rational coordinates, and given a point with rational coordinates with a ray heading off through another such point, we can’t necessarily mark off a line segment of the same length.

Posted by: David Corfield on October 9, 2008 12:19 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

This is a very, very minor and pedantic point, and doesn’t affect the overall point of your comment, but in Euclid “To describe a circle with any center and radius” isn’t an axiom but a postulate. From what I understand, Euclid’s own view of the distinction between them is a matter of considerable controversy; if it’s analogous to Aristotle’s, as has traditionally been assumed, then axioms would be (supposed to be) general architectonic principles that are convincing in themselves, whereas postulates restrict the subject matter to a particular field of study (and so simply have to be permitted in order to talk about the subject at all). But there are plenty of ancient Greek views of the distinction that don’t match up with Aristotle’s (e.g., one view is that axioms are statements and postulates classes of permitted constructions). One of the difficulties with reading Euclid himself is that we really can only guess what he thought the differences between definitions, axioms, and postulates were (and, a fortiori how they would line up with modern notions), and yet it’s almost certain that he thought there was some important difference.

Posted by: Brandon Watson on October 9, 2008 3:33 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

Oh yes, good point. I once used to know something about this. Heath has an interesting discussion in section 3, chapter IX of his Introduction to Books I and II. I seem to recall a bewildering variability of use of the terms – axiom, common notion, hypothesis, definition and postulate.

Posted by: David Corfield on October 9, 2008 4:57 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

Well Euclid would require you to be able to draw circles of any radius at a point, but that can’t happen on a sphere.

Maybe you should try to derive it from Hilbert’s axioms. Your spherical geometry not only fails on the parallelism axiom, but also on

HB3. If AA, BB, and CC are three distinct points lying on the same line, then one and only one of the points is between the other two.

Posted by: David Corfield on October 9, 2008 9:15 AM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

Sheesh that is a great animation. I’m a bit confused by what happens at 7 seconds into the video. Howcome the red circle is no longer “attached” to the end point of the black interval?

Posted by: Bruce Bartlett on October 9, 2008 12:01 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

I think that’s an optical illusion caused by the shading. Would it be possible, Simon, to slow the animation down about 3-fold?

Posted by: David Corfield on October 9, 2008 12:05 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

Would it be possible, Simon, to slow the animation down about 3-fold?

Sure, I will also put in some actual circles so that you can see the boundary of the discs better. Unfortunately the code is on my laptop at home, so I can’t do it now.

[I’ve deleted my rant here about maple 11’s poor performance on my mac due to its over-jazzy, over-corpulent front-end.]

Posted by: Simon Willerton on October 9, 2008 1:43 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

Here’s a new version of the animation: it’s slower and it has the circles bounding the discs explicitly drawn.

[I don’t think I can replace videos on YouTube – I have to put up a new one.]

Posted by: Simon Willerton on October 10, 2008 10:21 AM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

David C. said:

Well Euclid would require you to be able to draw circles of any radius at a point, but that can’t happen on a sphere.

It does depend on quite how you define a circle. I could say that a radius rr circle is the locus of the points obtained by travelling in a straight line a distance rr away from the centre. That will give a circle of any radius, the circumference of that circle will be 2πsin(r)2\pi sin(r) on a radius 1 sphere.

Euclid gives a definition, according to one translation, as follows

A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another.

So on a sphere you certainly have for any ordered pair of points a circle which passes through the first point and has the other point as its centre.

Posted by: Simon Willerton on October 9, 2008 12:15 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

Now, you’ve given up on there being a unique distance between two points.

Also Euclid means by line what we might call line segment, e.g., definition 3 ‘The ends of a line are points.’ So you now have that for any two points there are infinitely many lines with those points as ends.

I don’t think Euclid’s axioms can stand the strain of all this. You need Hilbert.

Posted by: David Corfield on October 9, 2008 12:28 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

It seems to me that to follow the spirit of Euclid’s “proof”, one has to argue that doing it for one diagram is “generic”.

This means essentially a limited affine-ness, where both isotropic scaling and translation change nothing (but skews and non-isotropic scalings do). Spherical geometry indeed does not obey this invariant, while plane geometry does.

Can this be shown to be equivalent to the parallel postulate?

Posted by: Aaron on October 10, 2008 12:36 AM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

David, I completely agree with you that Hilbert’s axioms are to be preferred over Euclid’s. I guess I just remain interested in seeing what can be salvaged from Euclid with the bare minimum of tinkering.

Simon, I think I can put my finger on the point where the parallel lines postulate can be used to distinguish between the planar and spherical geometries in the proof of Euclid’s first proposition. It’s not sufficient, as I think the case of 2\mathbb{Q}^2 shows, but if you take the view that Euclid was tacitly assuming that he was working on what we’d now call a real manifold, the parallel lines postulate helps us see why the two circles must intersect.

We have a line segment on which we wish to build an equilateral triangle. Call the endpoints A and B. Draw a line L through A that meets AB at a right angle. Then draw a line M through B that is parallel to line L.

Draw circles centred at A passing through B, and vice versa; these meet the lines L and M at points C and D respectively.

A C B D Line L Line M

Now, it’s the fact that lines L and M never meet that (along with the assumption that we’re working with real distances) that forces the arcs BC and AD to intersect.

But if the lines L and M can cross, and the points C and D lie past the cross-over point, we lose the rock-solid guarantee that the arcs BC and AD will have to intersect. Looking down from the north pole on a sphere where A and B lie on the equator, for a large enough separation of A and B we get something like this:

A C B D Line L Line M
Posted by: Greg Egan on October 10, 2008 6:24 AM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

I just wanted to add to this that I do realise that it would take quite a bit more work to formulate precisely a sixth postulate that would require the arcs AD and BC to intersect (especially if we wanted this sixth postulate to hold true in spherical geometry as well as planar geometry). In the above, I’m merely trying to back up Simon’s intuition that the parallel lines postulate – despite getting no mention in Euclid’s own “proof” of his first proposition – should play a crucial role.

Posted by: Greg Egan on October 10, 2008 7:44 AM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

Michael Harris points out to me by e-mail how odd it is that I discuss a philosopher making the accusation that “the majority of mathematicians don’t think hard enough about basic concepts of identity and relation” on a blog entitled the n-Category Café. I must say I was surprised that the many mathematicians there didn’t defend themselves, or lynch their accuser.

But then his accusation was no worse than Plato’s in Book VII of The Republic. After describing the mathematics needed for the future philosopher king, he writes:

Socrates: Do you not know that all this is but the prelude to the actual strain which we have to learn? For you surely would not regard the skilled mathematician as a dialectician?

Glaucon: Assuredly not, he said; I have hardly ever known a mathematician who was capable of reasoning.

Soc: But do you imagine that men who are unable to give and take a reason will have the knowledge which we require of them?

Soc: No one will argue that there is any other method of comprehending by any regular process all true existence or of ascertaining what each thing is in its own nature; for the arts in general are concerned with the desires or opinions of men, or are cultivated with a view to production and construction, or for the preservation of such productions and constructions; and as to the mathematical sciences which, as we were saying, have some apprehension of true being – geometry and the like – they only dream about being, but never can they behold the waking reality so long as they leave the hypotheses which they use unexamined, and are unable to give an account of them. For when a man knows not his own first principle, and when the conclusion and intermediate steps are also constructed out of he knows not what, how can he imagine that such a fabric of convention can ever become science?

Glauc: Impossible, he said.

Soc: Then dialectic, and dialectic alone, goes directly to the first principle and is the only science which does away with hypotheses in order to make her ground secure; the eye of the soul, which is literally buried in an outlandish slough, is by her gentle aid lifted upwards; and she uses as handmaids and helpers in the work of conversion, the sciences which we have been discussing.

Posted by: David Corfield on October 9, 2008 2:54 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

I will take the bait. Why would I want to think
“about basic concepts of identity and relation”?

Posted by: Eugene Lerman on October 10, 2008 3:32 AM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

Because you want a good way to say when two constructions are the ‘same’? Sameness for structured sets seems to involve isomorphism, while for categories equivalence. I’ll leave the sermon about evil to John and also refer you to his Week 209 to hear that

the logician Michael Makkai presented his astounding project of redoing logic in a way that completely eliminates the concept of “equality”.

By the way does anyone know how I can read Makkai’s First Order Logic with Dependent Sorts, with Applications to Category Theory? When I open a chapter my reader just flicks through the pages very quickly.

Posted by: David Corfield on October 10, 2008 8:17 AM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

David wrote:

When I open a chapter my reader just flicks through the pages very quickly.

That’s bizarre. The chapters are gzipped postscript files. Maybe your reader isn’t smart enough to gunzip them before trying to read them? I don’t know why that would have this particular strange effect… but, maybe you need to use the UNIX ‘gunzip’ command or some equivalent to decompress them before reading them. When on Windows, I use WinZip, which is free.

In category theory, dependent types are a nice trick to make it impossible to ask whether or not morphisms ff and gg are equal when they don’t have the same source and target. This is a great way to avoid getting caught up in nonsense questions. Makkai takes this much further in his work on ω\omega-categories.

Posted by: John Baez on October 11, 2008 2:51 AM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

Normally I just open a ps.gz file with GSview and all is fine. Here I get a DSC error message. Anyway, Makkai tells me he will make a single pdf file of the whole piece soon.

Posted by: David Corfield on October 14, 2008 10:42 AM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

And there (1.7 MB) it is.

Posted by: David Corfield on October 14, 2008 3:48 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

Eugene wrote:

Why would I want to think “about basic concepts of identity and relation”?

You are thinking about basic concepts of identity and relation, and you know perfectly well why it’s important. After all, you wrote a paper whose abstract starts with the sentence:

The goal of this survey paper is to argue that if orbifolds are groupoids, then the collection of orbifolds and their maps need to be thought of as a 2-category.

So: you’re saying that instead of equality being the appropriate notion of identity for maps between orbifolds, we should use isomorphism.

David mentions my ‘sermons’ about the evil of using equations rather than specified isomorphisms between objects in a category. You don’t need my sermons — you’re preaching them yourself!

Posted by: John Baez on October 11, 2008 2:40 AM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

John, I disagree. What I was thinking about was: what’s a good definition of an orbifold and of a map between two orbifolds? It then happened that I had to deal with 2-categories, but that’s another story…

To get back to the original “taunt.” Mathematicians are not thinking about “basic concepts of identity and relation” because they are not philosophers.

Posted by: Eugene Lerman on October 13, 2008 5:18 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

I believe that there is some support for Eugene Lerman’s statement:

“Mathematicians are not thinking about ‘basic concepts of identity and relation’ because they are not philosophers.”

In Kuhn’s “Nature of Scientific Revolution” he says that, when a paradigm is working for most of the community, nobody argues Philosophy. That is “Normal Science.”

But when there are so many anomalies and contradictions and failure to predict, then the community begins to argue Philosophy, especially foundational and metaphysical. That’s a sign of “Revolutionary Science.”

Posted by: Jonathan Vos Post on October 15, 2008 9:30 AM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

But, so Cartier says in 1997:

When I began in mathematics the main task of a mathematician was to bring order and make a synthesis of existing material, to create what Thomas Kuhn called normal science. Mathematics, in the forties and fifties, was undergoing what Kuhn calls a solidification period. In a given science there are times when you have to take all the existing material and create a unified terminology, unified standards, and train people in a unified style. The purpose of mathematics, in the fifties and sixties, was that, to create a new era of normal science. Now we are again at the beginning of a new revolution. Mathematics is undergoing major changes. We don’t know exactly where it will go. It is not yet time to make a synthesis of all these things—maybe in twenty or thirty years it will be time for a new Bourbaki. I consider myself very fortunate to have had two lives, a life of normal science and a life of scientific revolution.

There is talk on the categories list of setting up a new Bourbaki.

Posted by: David Corfield on October 15, 2008 9:44 AM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

Eugene wrote:

Mathematicians are not thinking about “basic concepts of identity and relation” because they are not philosophers.

Maybe we shouldn’t generalize too much about the motives of mathematicians. I am thinking about basic concepts of identity and relation! That’s why I’m interested in nn-categories and new approaches to logic that avoid ‘equality’. And that’s why I’m always looking for situations where people are using equations when they should be using isomorphisms — or nn-categories when they should be using (n+1)(n+1)-categories. That’s why I like differential geometry where ‘smooth categories’ replace ‘smooth spaces’. And that’s why I found it interesting that people work with a category of orbifolds when they obviously should be using a 2-category. And that’s why I was happy when you wrote a paper about this.

Only later did I actually start giving a damn about orbifolds for their own sake. That happened when I discovered that you can use ‘orbifold Euler characteristic’ to classify wallpaper groups. Three of my favorite themes meet here: categorification, Euler characteristic, and symmetry! So, I decided orbifolds must be cool.

But apparently you got interested in them first, and only later became dissatisfied with how people usually formalize them.

Posted by: John Baez on October 13, 2008 8:48 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

Of course, Eugene’s interest in orbifolds was strong enough for him to write a paper about them. Mine wasn’t. So, one could still argue that a ‘merely philosophical’ interest in some topic isn’t enough to motivate mathematicians to do serious research on it. I don’t know.

Posted by: John Baez on October 13, 2008 9:57 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

It could be argued that changes in practice regarding ideas such as identity and relation will occur naturally as mathematicians go about their daily business, and that explicit discussion is unnecessary. Most of us at the Café seem to think otherwise, however. For one thing, explicit formulation may help the spread of these changes to other branches and even to other disciplines.

Another constituency, one close to John’s heart, is the next generation of mathematicians. It seems to me that he sees conceptual clarification in cutting-edge research to be of a piece with the explanatory organisation of existing material to teach students.

Lawvere makes similar points in the passage quoted in the conclusion of my paper:

It is my belief that in the next decade and in the next century the technical advances forged by category theorists will be of value to dialectical philosophy, lending precise form with disputable mathematical models to ancient philosophical distinctions such as general vs. particular, objective vs. subjective, being vs. becoming, space vs. quantity, equality vs. difference, quantitative vs. qualitative etc. In turn the explicit attention by mathematicians to such philosophical questions is necessary to achieve the goal of making mathematics (and hence other sciences) more widely learnable and useable. Of course this will require that philosophers learn mathematics and that mathematicians learn philosophy.

Posted by: David Corfield on October 14, 2008 9:40 AM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

I hope my previous comments caused no offence, since none was intended. But we clearly have different approaches to mathematics.

To set the record straight: I wrote the survey paper on orbifolds as stacks because when I tried to explain my work with Malkin on prequantization of orbifolds, (see also this ) nobody knew what I was talking about.

Posted by: Eugene Lerman on October 14, 2008 4:05 PM | Permalink | Reply to this

Re: New Directions in the Philosophy of Mathematics

No offence, of course.

The speculation is whether ‘conceptual/philosophical’ work helps the individual and/or community.

Interesting to note that you write something which involves what we saw as ‘conceptual/philosophical’ when you wanted to explain something else.

Posted by: David Corfield on October 14, 2008 6:18 PM | Permalink | Reply to this

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