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December 16, 2008

What to Make of Mathematical Difficulties

Posted by David Corfield

T. R. drew to our attention a conference dedicated to Grothendieck. One of the papers there is Mathematics and Creativity (presumably written by Leila Schneps), which contains the passage:

Pierre Cartier observed that when Grothendieck took interest in some mathematical domain that he had not considered up till then, finding a whole collection of theorems, results and concepts already developed by others, he would continue building on this work ‘by turning it upside down’. Michel Demazure described his approach as ‘turning the problem into its own solution’. In fact, Grothendieck’s spontaneous reaction to whatever appeared to be causing a difficulty - nilpotent elements when taking spectra or rings, curve automorphisms for construction of moduli spaces - was to adopt and embrace the very phenomenon that was problematic, weaving it in as an integral feature of the structure he was studying, and thus transforming it from a difficulty into a clarifying feature of the situation. (p. 8)

This brings to mind the idea of Lakatos in Proofs and Refutations that one should be open to difficult cases, and not just exclude them by monster-barring.

In his long case study concerning the Euler formula,

VE+F=2, V - E + F = 2,

he advises that if someone presents you with a ‘polyhedron’ which doesn’t fit, such as the picture frame, don’t limit your theorem to apply only to ‘proper’ polyhedra, that is, ones without holes. Rather, find out what it is about picture frames, and ultimately tori, which leads to their not satisfying the formula. You will arrive at a more comprehensive theory.

Lakatos presents a rather limited methodology for dealing with such ‘counterexamples’ via proof analysis, but he was pointing us in a useful direction. Also rather important is the ability to sense when a difficulty is likely to be a fruitful one.

Posted at December 16, 2008 2:33 PM UTC

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Re: What to Make of Mathematical Difficulties

When Grothendieck was `pursuing stacks’ in his letters to Ronnie Brown and myself, one of the main points that came across was his search for `concepts’ to handle the mathematical `difficulties’ that arose. Getting the definition right for a concept that was to encompass new difficult cases and examples as well as extending and relating to known ones is central to a lot of that sort of Methodology of Mathematics.

‘The proof of the pudding is in the eating’and the concepts arising in this blog show some good eating! but even here we can see people inching towards concepts as, perhaps, incomplete ‘categorification’ processes turn out to be not quite good enough to do what seems to be needed. (I feel we have not yet a thorough idea of what that process of categorification really involves, although there are insights on several different types of process involved.)

Perhaps Lakatos’ description is not adequate to describe all the processes involved but no really in depth study of methodology seems to have been undertaken by mathematicians and mathematically inspired philosophers (David, do you detect a hint!?)

Posted by: Tim Porter on December 17, 2008 12:45 PM | Permalink | Reply to this

Re: What to Make of Mathematical Difficulties

…but no really in depth study of methodology seems to have been undertaken…

Aside, of course, from your work (and here) and mine.

Posted by: David Corfield on December 17, 2008 5:24 PM | Permalink | Reply to this

Genus and genius; Re: What to Make of Mathematical Difficulties

“when a difficulty is likely to be a fruitful one” is the key meta-concept.

Lakatos was very clever in his choice of exemplifying the need to properly generalize Euler’s formula. Generalizing in dimension is essential also, besides nonconvex polyhedra with nonzero genus, resulting in simplicial complexes and flag manifolds.

I like Terry Tao’s style of anthropomorphizing obstructions to a proof. To make up a quote: “This number wants to be a prime, but is vigorously opposed by this structure, which we now look at more carefully…”

Posted by: Jonathan Vos Post on December 20, 2008 5:11 PM | Permalink | Reply to this

Re: Genus and genius; Re: What to Make of Mathematical Difficulties

“when a difficulty is likely to be a fruitful one” is the key meta-concept.

And I suspect one which would be quite resistant to analysis, although some descriptions of allusive examples of this phenomenon could well prove illuminating.

Does anyone have candidates for case studies of when what appeared to be an obstacle to a theory proved to contain the key to a large theory?

Posted by: David Corfield on December 23, 2008 12:11 PM | Permalink | Reply to this

Re: Genus and genius; Re: What to Make of Mathematical Difficulties

Not sure what you mean by a `case study’
but the fact that H-spaces and h-maps do NOT form a category might be relevant

Posted by: jim stasheff on December 23, 2008 2:01 PM | Permalink | Reply to this

Re: Genus and genius; Re: What to Make of Mathematical Difficulties

The idea of a ‘case study’ in philosophy of science is to illustrate a point of philosophical interest by relating the history of a particular episode. They are also known as ‘case histories’, as in Conant’s Harvard Case Histories in Experimental Science.

Posted by: David Corfield on December 24, 2008 1:48 PM | Permalink | Reply to this

Algebraic Number Theory is such a case analysis; Re: Genus and genius; Re: What to Make of Mathematical Difficulties

As I recalled my Math professors telling me in the 1960s, and as detailed in [Stewart, I. and Tall, D., Algebraic Number Theory and Fermat’s Last Theorem, 3rd ed. Wellesley, MA: A K Peters, 2000] one can argue that the entire discipline of Algebraic Number Theory evolved over centuries from fruitful difficulties in cracking Diophantine equations, most particularly Fermat’s Last Theorem.

A proof of Fermat’s Last Theorem would be published, someone would see a flaw in the proof – whoops, you assumed unique prime factorization here and it isn’t so – and this would focus attention of a fruitful difficulty.

As reminded by MathWorld (This entry contributed by David Terr):

Algebraic number theory is the branch of number theory that deals with algebraic numbers. Historically, algebraic number theory developed as a set of tools for solving problems in elementary number theory, namely Diophantine equations (i.e., equations whose solutions are integers or rational numbers). Using algebraic number theory, some of these equations can be solved by “lifting” from the field Q of rational numbers to an algebraic extension K of Q.

More recently, algebraic number theory has developed into the abstract study of algebraic numbers and number fields themselves, as well as their properties.

Posted by: Jonathan Vos Post on December 23, 2008 4:33 PM | Permalink | Reply to this

Re: Algebraic Number Theory is such a case analysis; Re: Genus and genius; Re: What to Make of Mathematical Difficulties

But is this the same type of thing as what David was talking about? Fermat’s last theorem stood as a monumental challenge, of course, but I got the sense that “difficult cases” in David’s sense pointed more toward cases that received opinion had considered, if not quite pathological, then sources of mathematical nastiness (like non-reduced schemes). So one problem would be: how can one sense which instances of so-thought ‘nastiness’ are potentially worth embracing and building a theory around [like creating a pearl]?

Contrast the cases of non-reduced schemes and “points with internal spin” in the construction of moduli spaces, which Grothendieck embraced, against his plea for a “tame topology” in Sketch of a Program, which would seem to be in more of a “monster-barring” direction. Apparently there were some pathologies that even Grothendieck didn’t think would lead in fruitful directions.

Posted by: Todd Trimble on December 24, 2008 1:23 PM | Permalink | Reply to this

Re: Algebraic Number Theory is such a case analysis; Re: Genus and genius; Re: What to Make of Mathematical Difficulties

And I guess there’s not much more to be said about when to bar and when to embrace than to relate examples of master-mathematicians in action.

Posted by: David Corfield on December 24, 2008 1:50 PM | Permalink | Reply to this

Godel made a pearl of Whitehead/Russell’s irritants; Re: Algebraic Number Theory is such a case analysis; Re: Genus and genius; Re: What to Make of Mathematical Difficulties

One could make the case that Kurt Godel built a theory around the difficulties that Whitehead and Russell tried to avoid, where difficulties include what the “Principia Mathematica” authors (I worked through all 3 volumes of the paperback in my last summer before college) considered pathological paradoxes, and Godel simply incorporated through Godel numbering to prove Undecidability.

Posted by: Jonathan Vos Post on December 24, 2008 4:16 PM | Permalink | Reply to this

Re: What to Make of Mathematical Difficulties

Here is the link to an other very interesting conference.

Posted by: T.R. on December 22, 2008 4:26 PM | Permalink | Reply to this

Re: What to Make of Mathematical Difficulties

Andreas Holmstrom posts on his blog texts of the talks on the recent Grothendieck conference.

Posted by: Thomas on January 21, 2009 8:30 PM | Permalink | Reply to this

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