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December 14, 2010

Purity of Method

Posted by David Corfield

Last week I participated in the inaugural conference of the Association for the Philosophy of Mathematical Practice in Brussels. I decided to sketch some of my thinking on coalgebra, that Café people helped me formulate back here and here. This has turned into an article which will appear next year in Studies in History and Philosophy of Science.

Given twenty-five minutes to speak, there was only time to gesture at the connections with computer science, algebraic set theory, and analysis. It’s very tricky choosing a rich and interesting case study which is philosophically salient. To encourage the reader or listener to follow up the mathematics to understand what you’re saying, there must be a decent pay-off. An intricate twentieth century case study had better pack plenty of meta-mathematical punch. The trouble is that mathematics has become so enormously interconnected that when you pull on one strand, it’s easy to find yourself dragging along the whole edifice.

One of the talks I found most interesting was by Andy Arana, which managed to combine simple but deep mathematics, with a historical meta-mathematical debate of possibly current significance. He told us about work on projective geometry in the late nineteenth century, and in particular about reactions to the proof of Desargues’s theorem in real projective planar geometry. One reaction was to think it a theorem that ‘really’ belonged to projective space, rather than the plane. Certainly there’s a very straightforward three-dimensional proof. Others considered this an importation of foreign concepts into the proof – a statement about the plane should not use concepts from three-dimensional geometry.

One group who did not wish to keep planar and spatial geometry separate were the fusionists. Here’s Felix Klein from the Preface to Elementary Mathematics from an Advanced Standpoint

It has long been the custom in the schools as well as the university, first to study the plane and then, entirely separated from it, the geometry of space. On this account space perception, which we possess originally, is stunted. In contrast to this the ‘fusionists’ wish to treat the plane and space together, in order not to restrict our thinking artificially to two dimensions…

Andy has a strong interest in the notion of purity of proof, where a proof only uses concepts which belong to the theorem. This was a live issue for as important a mathematician as David Hilbert. In particular he discusses Desargues’s theorem in terms of the “Reinheit der Methode”, and shows by constructing a non-Desarguesian plane that an extra axiom is necessary.

One modern response, due to Michael Hallett, is to say that implicit within Desargues’s theorem, there are spatial notions. Therefore to invoke spatial concepts is not impure. Andy’s thought is that by the time we cash out this ‘implicit’ we’ll find so much of mathematics involved that just about any proof will count as pure, and the notion won’t do any work. This makes sense when we come to think of all the decategorified shadows of richer structure we like to discuss here.

Say we avoid the ‘implicit content’ approach, and stick with an idea of purity in terms of a tight reading of what is contained in the statement of the theorem. So, we have that no pure proof of Desargues’s theorem is possible, and presumably this is the case also for a whole raft of theorems. Then either we can lose interest in purity, or continue to think it has a mathematical or philosophical salience. I wonder how many readers would say that we need to follow wherever the mathematical structure leads us, whether pure in the strict sense or not.

Hilbert himself writes:

In many cases, our understanding is not satisfied when, in a proof of a proposition of arithmetic, we appeal to geometry, or in proving a geometrical truth we draw on function theory. Nevertheless, drawing on differently constituted means has frequently a deeper and justified ground, and this has uncovered beautiful and fruitful relations; e.g. the prime number problem and the ζ(x)\zeta(x) function, potential theory and analytic functions, etc. (p. 6 of Ivahn Smadja, Local Axioms in Disguise: Did Hilbert Really Dismiss Diagrams?)

By the way, John has written some fascinating material about Desarguesian planes. See especially week 145 and week 173, where we find out why the associativity of the coordinate algebra is necessary to extend a plane to a space, and so why there is an octonion plane (non-Desarguesian), but not an octonion space. It seems it comes down to the transitivity of the action of the group of collinearities on quadrangles within the plane.

Posted at December 14, 2010 5:11 PM UTC

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Re: Purity of Method

Interesting! Many examples of “impure proofs” that I can think of are now regarded as displaying a deep underlying connection between seemingly disparate fields of mathematics. If this were always the case, then I might be inclined to say that a desire for “purity” in proofs is misguided. Can anyone think of examples of “impure proofs” which you would not regard as displaying any deep underlying connection?

On the other hand, I can probably think of some cases where a pure proof gives more, or at least different, insight into “why” a theorem is true than an impure one does. But is it better to regard that through the lens of pure vs. impure, or is should it rather be subsumed under the general principle of “the more proofs we have, the better, since each one gives different insight”?

(Those are real questions, not rhetorical ones.)

Posted by: Mike Shulman on December 14, 2010 7:58 PM | Permalink | Reply to this

Re: Purity of Method

I can’t say that I’m persuaded by the value of purity. One of Arana’s arguments for it is to say that if we have a pure proof of a proposition, then if we withdraw our assent to any concept used in it, we have also withdrawn assent from the theorem statement itself. With an impure proof we may be left with an unproved theorem after such withdrawal. I suppose we might say this could happen when an unnecessary use of the axiom of choice is made. But perhaps there’s a better way of characterising the unnecessary use of axioms of a certain strength.

In the case at hand, Desargues’s theorem, it would be a little odd to withdraw assent from spatial projective geometry while maintaining it for planar projective geometry. In fact I’m struggling now to think what is involved in such assent.

Can anyone think of examples of “impure proofs” which you would not regard as displaying any deep underlying connection?

We can always add to a pure proof some unnecessary lines which use foreign concepts, but no doubt you meant where every line of the proof is necessary. Arana provides some candidates in On formally measuring and eliminating extraneous notions in proofs, including:

In a paper on a theorem of Tannaka and Krein in functional analysis, for instance, Salomon Bochner remarks that:

The proof of Krein is based on ideas of N. Wiener and I. Gelfand which are extraneous to the problem, and we are going to give a new proof which stays wholly within the technique of uniform approximation… (Bochner [1942], p. 56).

No less an expert than Robert Cameron, in a review of Bochner’s paper, confirms Bochner’s view that Krein’s proof involves “difficult concepts apparently extraneous to the problem”, while Bochner’s proof is “direct and relatively simple”.

The Bochner paper is ‘On a theorem of Tannaka and Krein’, Annals of Mathematics(2), 43, 56–58, 1942. But perhaps this is a case of ‘the more proofs the merrier’.

Posted by: David Corfield on December 15, 2010 9:16 AM | Permalink | Reply to this

Re: Purity of Method

Can anyone think of examples of “impure proofs” which you would not regard as displaying any deep underlying connection?

We can always add to a pure proof some unnecessary lines which use foreign concepts, but no doubt you meant where every line of the proof is necessary.

Certainly my own reaction to an “impure proof” in which I don’t see an underlying connection (except when I recognize that there are some really unnecessary steps) is to think that there must be an underlying connection that I just haven’t managed to see yet. (Whether the connection is “deep” is more of a question.) I may find this unsatisfying, but it says more about the limits of my own understanding than about some fault in the proof.

Posted by: Mark Meckes on December 15, 2010 3:15 PM | Permalink | Reply to this

Re: Purity of Method

What does it mean to “withdraw assent from a concept”?

Posted by: Mike Shulman on December 15, 2010 7:02 PM | Permalink | Reply to this

Re: Purity of Method

One test case is the Fundamental Theorem of Algebra.

There are those who say that it is a statement purely about the algebra of C\mathbf{C}, and should therefore be proved by purely algebraic methods. (I think the only purely algebraic method I’ve seen involves Sylow subgroups, which, while certainly algebraic, doesn’t seem that pure.)

Then there are analytical proofs, topological proofs, … And there’s a proof that Todd sent me a couple of weeks ago, to which I haven’t replied yet: sorry, Todd.

Posted by: Tom Leinster on December 14, 2010 8:44 PM | Permalink | Reply to this

Re: Purity of Method

If the complex numbers are defined (as they usually are) in terms of the real numbers, and the real numbers are defined (as they usually are) in terms of a topological/metric/order property (the Cauchy or Dedekind completion of the rationals) and not merely an algebraic one, then I don’t see how the Fundamental Theorem of Algebra could be regarded as “purely algebraic.”

Posted by: Mike Shulman on December 14, 2010 10:01 PM | Permalink | Reply to this

Re: Purity of Method

You could ask for a proof that the algebraic closure of a real closed field is given by adjoining a square root of -1. Real closed fields have a purely first-order definition. (For example, you can require that positive numbers have square roots, and all odd-degree polynomials have roots.)

Posted by: walt on December 14, 2010 10:37 PM | Permalink | Reply to this

Re: Purity of Method

Walt wrote:

You could ask for a proof that the algebraic closure of a real closed field is given by adjoining a square root of -1. Real closed fields have a purely first-order definition.

Right. But as you doubtless know, the proof that the real numbers are real closed still uses topology: for example, the intermediate value theorem.

So, the way I see it, by switching to this strategy you’re still basically using the fact that a continuous map f:D nD nf: D^n \to D^n whose restriction to the boundary has nonzero degree must be onto. You’re just using the case n=1n = 1 (where it’s the intermediate value theorem) instead of the case n=2n = 2 (where it’s my favorite topological proof).

I think it’s a very amusing joke that the Fundamental Theorem of Algebra is fundamentally about topology. So much for ‘purity of method’!

Maybe we should vote on some result to call the Fundamental Theorem of Topology — but with one condition, that its only known proof is algebraic.

An obvious candidate might be this result: “a continuous map f:D nD nf: D^n \to D^n whose restriction to the boundary has nonzero degree must be onto”. Doesn’t the easiest proof use algebraic topology? But does the only proof use algebraic topology? Not for n=1n = 1.

It’s somewhat painful, but extremely rewarding, to study Gauss’ first three attempts to prove the Fundamental Theorem of Algebra.

Posted by: John Baez on December 15, 2010 2:39 AM | Permalink | Reply to this

Re: Purity of Method

John said:

Maybe we should vote on some result to call the Fundamental Theorem of Topology — but with one condition, that its only known proof is algebraic.

Does the use of calculus for a topological result count as an impurity?

I’m thinking about the Riemann mapping theorem, you could try to prove the topological content of it using topological methods only. But I only know about proofs (both constructive and not constructive) using complex calculus to prove the stronger statement of the original theorem, therefore using the multiplicative structure of the complex numbers in a critical way, which is in turn irrelevant for the topological part.

Wikipedia states:

The Riemann mapping theorem is the easiest way to prove that any two simply connected domains in the plane are homeomorphic. Even though the class of continuous functions is vastly larger than that of conformal maps, it is not easy to construct a one-to-one function onto the disk knowing only that the domain is simply connected.

…which seems to indicate that there is a - constructive(?) - proof of the topological part of the theorem which does not use complex analysis, which I don’t know…

Posted by: Tim van Beek on December 15, 2010 10:09 AM | Permalink | Reply to this

Re: Purity of Method

Does the use of calculus for a topological result count as an impurity?

I’m thinking about the Riemann mapping theorem, you could try to prove the topological content of it using topological methods only.

For a more dramatic example, there’s Perelman’s proof of the Poincare conjecture via the complicated analytic machinery of Ricci flow. (And that’s not even mentioning geometrization in the middle somewhere.)

Posted by: Mark Meckes on December 15, 2010 3:06 PM | Permalink | Reply to this

Re: Purity of Method

Yes, but that’s also, ahem, far over my head…

But the situation with the Riemann mapping theorem seems simple enough to me:

Let 2\mathbb{R}^2 be the (unique up to isomorphism) real two dimensional topological vector space (or Banach space).

Puzzle: Prove that all simply connected open proper subsets of 2\mathbb{R}^2 are homeomorph.

Now, an executive summary of the usual approach: introduce a product on elements of 2\mathbb{R}^2 that turns it into a division algebra and define that complex differentiability of an endomorphism ff means that the differential dfd f is linear with respect to that product, which leads you to the Cauchy-Riemann differential equations, which in turn …among other things the Montel theorem, which can be used in the constructive proof of the Riemann mapping theorem to show that the constructed sequence of mappings converges to a biholomorphic mapping from a given simply connected proper subset to the (open) unit circle.

Now we could interrupt this line of reasoning by forbidding to use calculus (use pure topology, continuous functions, only), or by forbidding to use the complex product structure on 2\mathbb{R}^2 (use real calculus only).

Posted by: Tim van Beek on December 16, 2010 9:32 AM | Permalink | Reply to this

Re: Purity of Method

Tom wrote:

And there’s a proof that Todd sent me a couple of weeks ago, to which I haven’t replied yet: sorry, Todd.

No problem, Tom! I (well, it was mainly me) wrote up both the Artin-Schreier proof based on real closed fields, and the proof that I sent Tom a little while ago, in the nLab article fundamental theorem of algebra.

The second proof uses very primitive tools. It’s an argument one could almost imagine an 18th century mathematician cooking up.

Posted by: Todd Trimble on December 14, 2010 11:54 PM | Permalink | Reply to this

Re: Purity of Method

Interesting! I like the Artin-Schreier proof, although I agree that it’s questionably “pure” in the sense used here, since it drags in Sylow groups and Galois theory.

Posted by: Mike Shulman on December 15, 2010 1:14 AM | Permalink | Reply to this

Re: Purity of Method

There is something of the conflict between ‘process’ and ‘product’ in this. I have felt quite often that I relatively rarely explicitly use the exact statement of a result but more often glean something from its proof that allows me to push an analogous argument through in a new situation. That reaction may suggest that the really useful object of study is the proof and not the result that is proved. This is also apparent in the stimulus that a conjecture gives to a subject area. Lots of new theory is developed as people try to prove or disprove the conjecture, and it is not unknown for interest in an area to wane once a definitive answer to the conjecture is known. The process is at least as important as the product for the further development of the subject.

As an example of ‘pure’ v. ‘impure’, I like the homotopy theory of simplicial sets where weak equivalences are often defined using the geometric realisations, and therefore ‘impure’ topological spaces, although they can be also defined directly (with more work and probably more group theory).

Posted by: Tim Porter on December 14, 2010 9:25 PM | Permalink | Reply to this

Re: Purity of Method

Warning, shameless self-promotion:
How about a 4-dimensional proof of
Heron’s formula as sketched here? Does this violate purity of method?

We hope to have a preprint ready soon…

Posted by: Scott Carter on December 15, 2010 1:40 AM | Permalink | Reply to this

Re: Purity of Method

Would you be inclined to say that this is an explanatory proof? And perhaps that there’s something intrinsically 4D about Heron’s formula?

Posted by: David Corfield on December 15, 2010 3:03 PM | Permalink | Reply to this

Re: Purity of Method

David asks,
“Would you be inclined to say that this is an explanatory proof? And perhaps that there’s something intrinsically 4D about Heron’s formula?”

The answer to both questions is yes. Our central points are that the distributive law is a scissors congruence, and that if P and Q are planar polygons that are respectively congruent to P’ and Q’, then the cartesean products are scissors congruent.

The 4 dimensional nature is that one side of the equation can be written as 16 A x A where A is the area of the triangle. There is a beautiful 4-d hypersolid that is a triangle times triangle. 4 of these piece together to form a parallelogram times parallelogram — a type of 4-D parallelopiped. So this side of the equation is 4 such figures. These in turn are scissors congruent to hyper-rectangles.

On the other side is the hyper-volume of a hyper rectangle with edge lengths (a+b+c),(a-b+c),(a+b-c), and (-a+b+c). This chops into 81 pieces all except 9 of which cancel.
Those 9 pieces are reassembled via using a scissors congruence proof of the Pythagorean Theorem.

Posted by: Scott Carter on December 16, 2010 12:11 AM | Permalink | Reply to this

Re: Purity of Method

Looking to the Desargues case, would it be right to say that the non-desarguesian property is a measure of the obstruction to extending (embedding?) a projective plane to a projective space? If so, does it have a family resemblance to other kinds of obstruction, like the non-vanishing of the second Stiefel-Whitney class for spin structures.

Posted by: David Corfield on December 15, 2010 9:50 AM | Permalink | Reply to this

Re: Purity of Method

In some ways, it’s more than an analogy. In the setting of homotopy theory, the obstruction to an H-space having its multiplication being homotopy associative is the obstruction to creating a `projective 3-space’.

Posted by: jim stasheff on December 15, 2010 1:05 PM | Permalink | Reply to this

Re: Purity of Method

That’s sounds interesting. Is there anywhere I can read about that?

Posted by: David Corfield on December 15, 2010 3:01 PM | Permalink | Reply to this

Re: Purity of Method

I think it’s basically in Homotopy Associativity of H-spaces I. Each of the A nA_n conditions allows you to build another approximation to the ultimate delooping you would have if you had all the A A_\infty conditions (plus a technical condition like “grouplike”). The delooping is akin to an infinite-dimensional projective space, a kind of fancy version of the James construction, and the approximations are akin to finite-dimensional projective spaces.

Posted by: Todd Trimble on December 15, 2010 4:09 PM | Permalink | Reply to this

Re: Purity of Method

Thanks for blowing the trumpet for me.
But the James construction goes the other way -
builds up to an approximation of based loops on a suspension space.

A topological monoid M is in particular an A_\infty-space
and either of the constuctions of the classiying space is filtered by projective spaces, e.g. BS^1= ..\superset CPn…\superset CP2 \superset CP1

Posted by: jim stasheff on December 16, 2010 1:10 PM | Permalink | Reply to this

Re: Purity of Method

Oops. You’re right, I shouldn’t have said ‘James construction’.

Posted by: Todd Trimble on December 16, 2010 3:19 PM | Permalink | Reply to this

Re: Purity of Method

Maybe you should have said the Jim construction :)

Posted by: David Roberts on December 16, 2010 11:41 PM | Permalink | Reply to this

Re: Purity of Method

So there would then be an obstruction to extending something 3-dimensional to something 4-dimensional, and so on? But this doesn’t manifest itself in ordinary projective spaces, since once the Desarguesian obstruction is removed, all dimensions are possible?

Posted by: David Corfield on December 16, 2010 12:10 PM | Permalink | Reply to this

Re: Purity of Method

Exactly! Not at all sure what kind of `coordinates’ there are for such projective spaces.

Perhaps more accessible place to read ab out htis would be

PDF Clipboard Journal Article
MR0270372 (42 #5261) Stasheff, James $H$H-spaces from a homotopy point of view. Lecture Notes in Mathematics, Vol. 161 Springer-Verlag,

Posted by: jim stasheff on December 16, 2010 1:19 PM | Permalink | Reply to this

Re: Purity of Method

David wrote:

So there would then be an obstruction to extending something 3-dimensional to something 4-dimensional, and so on? But this doesn’t manifest itself in ordinary projective spaces, since once the Desarguesian obstruction is removed, all dimensions are possible?

I seem to remember the story goes like this. If GG is a topological group you can define its nerve BGB G as a topological space. When GG is the group of invertible elements in a topological field FF this is the infinite-dimensional projective space F F \mathbb{P}^\infty. But you can also try to build up F F \mathbb{P}^\infty stage by stage:

F 1F 2F 3 F\mathbb{P}^1 \hookrightarrow F\mathbb{P}^2 \hookrightarrow F\mathbb{P}^3 \hookrightarrow \cdots

The points of F 1F\mathbb{P}^1 are the 1-simplices in BGB G, the points of F 2F\mathbb{P}^2 are the 2-simplices in BGB G, and so on. Or something like that: I feel I have some of the details slightly wrong.

To do the first few stages of this construction, we don’t need FF to be a full-fledged topological field. But it needs to have better and better properties as we go up. It goes something like this:

If FF is a space with an element 00 you can define F 1F \mathbb{P}^1. If it has a product where every nonzero element has an inverse you can define F 2F \mathbb{P}^2. If the product is associative you can define F 3F \mathbb{P}^3and all the higher F nF \mathbb{P}^n’s!

That’s why there’s an octonionic projective plane but no higher octonionic projective spaces, and that’s why “once the Desarguesian obstruction is removed, all dimensions are possible.”

However, the story gets more involved if FF, instead of having a product that’s associative on the nose, has a product that’s associative only up to homotopy. This is what Jim Stasheff worked out in his famous book.

For example: if FF is associative up to homotopy, we can define 𝔽P 3\mathbb{F} P^3 but not necessarily any higher 𝔽P n\mathbb{F} P^n’s. If this homotopy obeys the pentagon identity, we can we can define 𝔽P 4\mathbb{F} P^4and all the higher 𝔽P n\mathbb{F} P^n’s!

I think you can guess the pattern from here on out.

Actually ‘fields’ are not required for most of this story to work. It’s only the multiplication we’re using here, not the addition. But it’s nice to think about fields (and their defective relatives, like the octonions) to make the connection between what Jim did and the old story of projective geometry. So, it might be nice to think about the projective geometry of ‘homotopy fields’, ‘A nA_n fields’ and the like. Maybe people already have. I know they’ve thought a lot about ‘field spectra’, which usually go by some other name. All this is part of what some people call ‘brave new algebra’. But I don’t know if anyone has thought about this high-powered stuff by trying to generalize the old-fashioned axiomatic approach to projective geometry!

Posted by: John Baez on December 17, 2010 4:17 AM | Permalink | Reply to this

Re: Purity of Method

John wrote (modified):

If this homotopy obeys the pentagon identity, we can we can define the projective 4-space and all the higher ones!

Of course he means obeys the pentagon identity on the nose, NOT just up to homotopy

and all the higher ones!

Is it obvious that the remaining 3 squares satisfy the next identity? some interchange property?

Posted by: jim stasheff on December 17, 2010 1:36 PM | Permalink | Reply to this

Re: Purity of Method

So what’s a familiar FF which does not obey the pentagon identity? Any famous A 3A_3-spaces which are not A 4A_4-spaces?

Posted by: David Corfield on December 17, 2010 2:45 PM | Permalink | Reply to this

Re: Purity of Method

An interesting point that does not seem to yet have been aired in this derives from the line:

Andy has a strong interest in the notion of purity of proof, where a proof only uses concepts which belong to the theorem.

Can someone say exactly what : “concepts which belong to the theorem” means?

The notion of purity is tricky because it to some extent depends on the subdivisions into which one divides mathematics. The ‘fusionists’ are perhaps interested in the unity of mathematics in some sense. The lesson of n-category theory may be to emphasise that unity, but also to suggest a revised version of purity which goes more with (another undefinable) term simplicity of proof.

Posted by: Tim Porter on December 15, 2010 10:11 AM | Permalink | Reply to this

Re: Purity of Method

Some motivation comes from proof theory and the idea of cut-free proofs. Andy quotes J. Alan Robinson saying of Erdos’s proof of Bertrand’s postulate

It is an example of how sometimes a proof draws on facts and ideas which seem wholly extraneous to the proposition being proved. (Gentzen’s Hauptsatz states essentially that if a proposition is [formally] provable at all, then it is formally provable using only notions which are present in the statement of the proposition itself.

Robinson continues though

It is hard to believe, is it not, that this assertion holds for informal “real” proofs?)

Unless a very tight line is kept on what is contained in the statement of a proof, the implicit will spill over. Relevant to the non-desarguesian nature of the octonion projective plane are cohomology, intransitivity of group action, and so on.

I like the idea of purity in your sense. Polanyi reckoned we should not let ‘rationality’ be replaced by ‘simplicity’, as if it were politer.

Posted by: David Corfield on December 15, 2010 12:27 PM | Permalink | Reply to this

Re: Purity of Method

In praise of PURITY in some sense.

The problem of proofs that use other areas of knowledge is perhaps that they are more ‘complex’ in some sense. Let me give an example:

Loday proved a result that extended a well known classification of homotopy 2-types (MacLane and Whitehead) in terms of crossed modules. He obtained one on homotopy n-types in terms of catn1^{n-1}-groups. In the process of the proof he went from the category of simplicial sets to that of spaces and back again several times. Very unpure! Both myself and separately Bullejos, Cegarra and Duskin, felt there was a ‘purer’ proof. I gave one that used very simple group theory with simplicial groups as well, they used an essentially parallel development and may once have used a spectral sequence argument(still algebraic). Both proofs were simpler and more constructive in an informal and probably almost a formal sense.

This last fact is relevant to the point about Gentzen’s Hauptsatz since if one looks at a theorem that is only known to have an impure proof, then possibly one should see if the notions which are present in the statement of the proposition are ‘impure’ themselves. The fundamental theorem of algebra relates to the complex numbers but how is that to be made precise without some recourse to completeness of some form or other and that is not purely algebraic. The formalised concepts in the theorem therefore cross ‘subject divisions’ as we usually think of them so we can expect them to require impure proofs.

Posted by: Tim Porter on December 15, 2010 6:00 PM | Permalink | Reply to this

Re: Purity of Method

The example of purity that springs to my mind is the first Polymath project, concerning the Density Hales-Jewitt Theorem. This is a basically combinatorial theorem but the only known proofs at the time used non-combinatorial techniques (ergodic theory). Tim Gowers, who started the project, was of the opinion that such a basically combinatorial theorem ought to have a combinatorial proof, and the project’s goal (successfully achieved) was to find one.

Two things occur to me about the philosophy behind this goal. One is what strikes me as the natural and legitimate motivation that if something seems as though it “ought to” exist, and is right in the middle of one’s area of interest, then it’s worth trying to find it!

The other is that various combinatorialists were complaining that they didn’t really understand the ergodic proofs properly because of lack of enough background in the underlying theory, while a combinatorial proof would hold out the possibility of yielding more insight. Of course, one could take this situation as an enticement to learn about another field—many people do—but life is short, the human brain is finite, and it seems to me a completely legitimate goal to try to gain greater insight into one’s own field, using the techniques of that field.

I don’t think there’s anything “wrong” with an impure proof. It’s just that … it’s impure; and purity seems like a very reasonable goal (one among other very reasonable goals, of course, some incompatible with purity).

Posted by: Tim Silverman on December 15, 2010 7:19 PM | Permalink | Reply to this

Re: Purity of Method

In re:

By the way, John has written some fascinating material about Desarguesian planes. See especially week 145 and week 173, where we find out why the associativity of the coordinate algebra is necessary to extend a plane to a space, and so why there is an octonion plane (non-Desarguesian), but not an octonion space. It seems it comes down to the transitivity of the action of the group of collinearities on quadrangles within the plane.

Sorry i somehow missed those posts of John - maybe before I joined the chorus. As an undergrad many years ago at U Michigan, I had the privilege and pleasure of a course in projective geometry from Yuri Rainich. It was from him that I discovered the optional nature of associativity, which has been the leitmotif of most of my research.

I was particularly struck by the fact that having a projective 3-space gives Desargues, cf the lack of such for the octonions. Some years later, I recalled it as I studied homotopy associativity which corresponds to just a ‘projective 3-space’, higher dim projective spaces require higher homotopies.

Posted by: jim stasheff on December 15, 2010 1:01 PM | Permalink | Reply to this

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