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February 10, 2007

Why Do I Bother?

Posted by David Corfield

Terence Tao has placed on the ArXiv an article submitted to the Bulletin of the AMS, ‘What is good mathematics?’ After a substantial list of criteria of what constitutes a good piece of mathematics, there follows a lengthy case study of Szemerédi’s theorem to support a conjecture:

It may seem from the above discussion that the problem of evaluating mathematical quality, while important, is a hopelessly complicated one, especially since many good mathematical achievements may score highly on some of the qualities listed above but not on others. However, there is the remarkable phenomenon that good mathematics in one of the above senses tends to beget more good mathematics in many of the other senses as well, leading to the tentative conjecture that perhaps there is, after all, a universal notion of good quality mathematics, and all the specific metrics listed above represent different routes to uncover new mathematics, or difference stages or aspects of the evolution of a mathematical story.

As readers will know, I’ve been pushing for philosophers of mathematics to address the problems of values other than truth. In ‘Towards a Philosophy of Real Mathematics’, I moved beyond a Lakatosian conception of progress to tackle the arguments used for and against the extension of groups to groupoids. And in the past couple of years I’ve been advocating MacIntyre’s idea of a rational tradition of enquiry as governed by overarching dramatic narratives.

I’ve also been working on a conception of mathematical reality close to Michael Polanyi’s:

A new mathematical conception may be said to have reality if its assumption leads to a wide range of new interesting ideas. (Personal Knowledge: 116)

…while in the natural sciences the feeling of making contact with reality is an augury of as yet undreamed of future empirical confirmations of an immanent discovery, in mathematics it betokens an indeterminate range of future germinations within mathematics itself. (Personal Knowledge: 189)

But why need MacIntyre or Polanyi or I bother, if, without reading us, a mathematician can come to very similar conclusions?:

…the very best examples of good mathematics do not merely fulfil one or more of the criteria of mathematical quality listed at the beginning of the article, but are more importantly part of a greater mathematical story, which then unfurls to generate many further pieces of good mathematics of many different types. Indeed, one can view the history of entire fields of mathematics as being primarily generated by a handful of these great stories, their evolution through time, and their interaction with each other. I would thus conclude that good mathematics is not merely measured by one or more of the “local” qualities listed previously (though these are certainly important, and worth pursuing and debating), but also depends on the more “global” question of how it fits with other pieces of good mathematics, either by building upon earlier achievements or encouraging the development of future breakthroughs.

Well, for one, further to what Tao says, I would like to insist that if the mathematical community does not act to promote the telling of these ‘great stories’, then it is failing to be fully rational. Second, there’s the question of how these stories are to be told. This is perhaps not as straightforward as one might think (see here). Lastly, thinking about MacIntyre’s original motivation, mathematical rationality involving value judgements as it does, this ought to tell us something about the way other rational communities should be organised. Try running through Tao’s quotations, substituting ‘political-ethical’ for ‘mathematical’ and you have something approaching MacIntyre’s moral realism.

Posted at February 10, 2007 9:15 AM UTC

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Re: Why Do I Bother?

At first, from reading your title, I feared this would be a lament about the hopelessness of getting analytic philosophers of mathematics to take seriously the questions you’re raising. I’m glad it’s not!

Philosophy departments should take these questions very seriously. They should also hire you!

Posted by: John Baez on February 10, 2007 5:46 PM | Permalink | Reply to this

Re: Why Do I Bother?

I wonder if the barrier is psychological (or maybe even just a practical reality)? The parts of mathematics that are popular in analytic philosophy – foundations, modal logic, interpretations of probability – are areas where philosophers can be genuinely more expert than mathematicians. The question of what makes good mathematics is one that it would be difficult for philosophers to be better at talking about than mathematicians would be, so they shy away from it.

Posted by: Walt on February 11, 2007 5:43 AM | Permalink | Reply to this

Re: Why Do I Bother?

I hope not. It would seem overly restrictive if philosophy could only discuss subjects at a non-expert level. It seems to leave quite thin line to walk.

On one side, philosophers shy away from active areas of mathematical research. Then what they say has to do with mathematical practice only insofar as it is superficial, and it is substantial only insofar as it has nothing to do with mathematical practice.

On the other side, philosophers could start making normative prescriptions for how mathematics “should” be from outside the group who are actually doing the mathematics and (as a rule) couldn’t care less what those philosophers think.

The way could be widened, though, if those studying the philosophy of mathematics would at least try to be conversant with how mathematics is actually done by being conversant with an actual active field of mathematical research. Like, if only there was a philosopher out there who could read and speak the language of n-categories along with practicing mathematicians and physicists who actively work on them.

Hey wait…

Posted by: John Armstrong on February 11, 2007 6:26 AM | Permalink | Reply to this

Re: Why Do I Bother?

The solitary philosopher is like the proverbial small town with one lawyer. You need two lawyers to do any real business.

Posted by: Walt on February 12, 2007 12:40 AM | Permalink | Reply to this

Re: Why Do I Bother?

One other aspect to add to Tao’s account concerns the rivalry that may exist between different stories. What is there to say about the rationality of establishing one account over another? Cartier may tell us a story about how Connes’s and Grothedieck’s versions of space get fused. Connes may then wish to retell aspects of this narrative (pp. 20-21).

Posted by: David Corfield on February 12, 2007 2:56 PM | Permalink | Reply to this

Re: Why Do I Bother?

It seems to me we can try grafting in older ideas from the philosophy of history. Would it be accurate to couch the Connes/Grothendieck story in terms of a dialectic? Two different ideas conflicting, then merging into the new idea that proceeds forwards.

Posted by: John Armstrong on February 12, 2007 3:07 PM | Permalink | Reply to this

Re: Why Do I Bother?

Yes, dialectic is a good concept to have in mind here. Lakatos certainly saw ‘Proofs and Refutations’ as an illustration of dialectical reasoning in modern mathematics. As one might expect, though, the term has different shades of meaning. For our purposes we might start out from Aristotle’s conception, loosely a means of scutinising someone’s assumptions.

This snippet from Connes’ paper raises an objection to Cartier’s merger:

There are some similarities between noncommutative geometry and the theory of topoi as suggested by the diagram proposed by Cartier in [3], in the case of the cross product of a space by a group action or of a foliation which can be “treated” in the two ways… It is crucial to understand that the algebra associated to a topos does not in general allow one to recover the topos itself in the general noncommutative case…It then becomes clear that the invariants that are defined directly in terms of the algebra possess remarkable “stability” properties which would not be apparent in the topos side.

If the dialogue were to continue, we would expect to see either Cartier responding that this supposed defect is not a defect in his view, or else conceding that Connes has a good point.

Posted by: David Corfield on February 12, 2007 3:38 PM | Permalink | Reply to this

Re: Why Do I Bother?

I’m afraid that if one wants to understand that particular story, a psychoanalytic approach might reveal more than a dialectic…

Posted by: lieven on February 12, 2007 3:39 PM | Permalink | Reply to this

Re: Why Do I Bother?

Was this a coincidence, or did you know of Cartier’s Grothendieck et les motifs? This is

le texte développé d’une conférence donnée à Cerisy-la-Salle en septembre 1999, dans le cadre du Colloque “Mathématiques et psychanalyse”, organisé par P. Cartier et N. Charraud.

Posted by: David Corfield on February 12, 2007 4:04 PM | Permalink | Reply to this

Story is a Force of Nature; Re: Why Do I Bother?

“… the very best examples of good mathematics … are more importantly part of a greater mathematical story… the history of entire fields of mathematics as being primarily generated by a handful of these great stories…”

This suggests that the aesthetics of Mathematics involves the narrative. Narrative is thus, not just important to writing a Math paper well, but to fitting into a meta-narrative.

That is, there are 1-narratives (a properly motivated definition, theory, proof); 2-narratives (a Math paper which is a good read); 3-narratives (the evolution of a branch of Mathematics; 4-narratives (the evolution of the entire History of Mathematics); and 5-narratives (the interaction of the History of Mathematics with other 4-narratives, i.e. with the History of Physics, the History of Philosophy). That looks like n-Category Theory in a way…

Stories about mathematical objects, stories about stories, stories about stories about stories. Transormations between stories. Cutting and pasting stories. Inverting a story. Twist endings. Subplots braided into a novel. Novel made into Film made into Novel. Problems of trasnlation between languages.

Interactions between narratives and evolution: that suggests the evolution by natural selection of pairs of mathematical ideas sexually reproducing, as it were. That is, Mimetics.

The human brain has evolved to be very good at narratives. There have been conferences asd books on that notion. Psychoanalysis is only one approach to human narrative.

Mathematical Biographies, as vastly stimulated by Eric Temple Bell’s “Men of Mathematics” are also narratives. Bell’s double (triple?) life shows that some narratives may indeed be subject to psychoanalysis.

“Never let a fact get in the way of a good story” is a problem for Math. There is also a duality between Story and Plot, which I will not go into here.

Posted by: Jonathan Vos Post on February 12, 2007 5:27 PM | Permalink | Reply to this

Re: Story is a Force of Nature; Re: Why Do I Bother?

“Never let a fact get in the way of a good story” is a problem for Math.

On the other hand, there’s often a lot to be gained from this view.

“A polynomial of degree nn has exactly nn roots” is a great story. Unfortunately x 2+1x^2+1 is a fact that gets in the way when dealing with polynomials over the reals. What we need is a new viewpoint that exends the old one in which the story is true, and moving to complex numbers (and an understanding of “multiplicities” of roots) does nicely.

As I understand it, that’s basically how complex numbers were first introduced – trying to find a place the facts and the story can both be true.

Posted by: John Armstrong on February 12, 2007 5:49 PM | Permalink | Reply to this

Re: Story is a Force of Nature; Re: Why Do I Bother?

“A polynomial of degree n has exactly n roots” is a great story.

As I understand it, that’s basically how complex numbers were first introduced – trying to find a place the facts and the story can both be true.

Well, that’s a common story (2story?) about how complex numbers were introduced, but the facts can get in the way. (On the other hand, we never let such historical facts get in the way of motivation of new concepts for students!)

As I understand it, complex numbers were first seriously used by Cardan [Cardano] to describe real solutions to cubic equations. After all, nobody minded much that x2 + 1 had no (real) roots (the story about n roots came later), but Cardan’s formula for the cubic used complex numbers even when a cubic had 3 (real) roots!

Now we need a 3story, explaining how my 2story and your 2story relate.

Posted by: Toby Bartels on February 12, 2007 7:37 PM | Permalink | Reply to this

Negative root; Re: Story is a Force of Nature; Re: Why Do I Bother?

“… the Greek geometer Diophantus (first or third century AD) rejected negative solutions to equations, and the Indian mathematician Bhaskara (1114-ca. 1185) comments on the negative root of the quadratic equation, ‘The second value is in this case not to be taken, for it is inadequate; people do not approve of negative roots’ (Wells 1986, p. 20)”

Weisstein, Eric W. “Negative.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Negative.html

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 20-21, 1986.

So now, another 3story, or a 4-story needed?

I would say that both Diophantus and Bhaskara, tremenousl geniuses, ultimately suffered from what Sir Arthur C. Clarke calls “Failure of Nerve” [dual to Failure of Imagination]. Writers and editors, for narrative reasons, sometimes back away from the story ending that they originally accepted. Victorian drama rewrote Romeo and Juliet to have a happy ending. “Podkayne of Mars” was edited to save the life of the heroine, even though Heinlein felt it crucial to the story that she die because of the consequences of her action. One of the most-discussed story in all Science Fition is “The Cold Equations.” It puts Mathematical narrative fatally against human interest narrative.

Posted by: Jonathan Vos Post on February 12, 2007 8:01 PM | Permalink | Reply to this

Re: Story is a Force of Nature; Re: Why Do I Bother?

Let us not forget Rafael Bombelli.

we never let such historical facts get in the way of motivation of new concepts for students!

But don’t obscure the fact that there was not just one motivation for the complex numbers. Besides Bombelli’s use of imaginary numbers in Cardano’s formula, there was the desire to understand the factorisation of polynomials, one reason being to be able to integrate via partial fractions.

Encourage students to read John Stillwell’s excellent Mathematics and Its History. For the general public there’s Mazur’s Imagining Numbers: (particularly the square root of minus fifteen).

Posted by: David Corfield on February 12, 2007 8:04 PM | Permalink | Reply to this

Re: Story is a Force of Nature; Re: Why Do I Bother?

But don’t obscure the fact that there was not just one motivation for the complex numbers. Besides Bombelli’s use of imaginary numbers in Cardano’s formula, there was the desire to understand the factorisation of polynomials, one reason being to be able to integrate via partial fractions.

I did forget about Bombelli, but I didn’t even know about the motivation by partial fractions!

Although now that you mention it, I recall that when I first took calculus (gosh, 18 years ago now?; I’ve known calculus for most of my life!), I noticed that we needed both arctangents and logarithms to integrate rational functions (once we got our hands on their roots, one way or another), but I only needed logarithms if I used complex numbers. (Follow that far enough, with the description of arctangents in terms of logarithms of complex numbers, and you end up with Euler’s formula.)

Posted by: Toby Bartels on February 13, 2007 12:24 AM | Permalink | Reply to this

Re: Story is a Force of Nature; Re: Why Do I Bother?

Now we need a 3story, explaining how my 2story and your 2story relate.

I propose “omission”. As in, “I left that part out”.

Cardano wanted to find other roots of some cubics. He knew they were there (looking at the graphs) but needed to figure how to calculate them. At first, the complex numbers were weird intermediary step justified by the fact that they gave the right answers, and they gave three of them.

At any rate, surely they knew that there couldn’t be more than three roots to a cubic, and it’s a natural problem to want to find all that there are (hopefully 3).

Posted by: John Armstrong on February 12, 2007 8:40 PM | Permalink | Reply to this

GA Operators; Re: Story is a Force of Nature; Re: Why Do I Bother?

(1) Yes, Unapologetic John Armstrong, “Omission” is a story-transforming operator, analogous to gene deletions.

Other operators include:

(2) Point Mutation (We like your screenplay very much, but could you make Romeo a girl, and play up the lesbian angle?);

(3) Recombination (Okay, we like the beginning chapters, but we want to throw out the last 3 chapters and replace them with the first three chapters of your submitted sequel, and release that as one book);

(4) Inversion (good plot, but the story is better if told in reverse, got me? We open with Saddam Hussein on the gallows, and then work back to his childhood in Tikrit);

(5) Chromosome duplication (okay, we have two copies of the whole story, but the second one is told from the point of view of the villain from the first).

Given a fitness associated with each story (as defined formally by a multidimensional fitness landscape), where the fitness is a real in [0,1] and so formally a probability, we apply the genetic algorithm.

Good stories drive out bad stories, in the long run, per the theorems proved by John Holland [Adaptation on Natural and Artificial Systems, 1976; MIT Press edition, 1992] where we consider a story as represented by a string of alphanumeric characters. Note how different this is from a film, which is a time-parameterized sequence of 2-D images, with different edit operators and rules of interpretation, as first explicated by Eisenstein.

Posted by: Jonathan Vos Post on February 12, 2007 9:04 PM | Permalink | Reply to this

Re: GA Operators; Re: Story is a Force of Nature; Re: Why Do I Bother?

as first explicated by Eisenstein

Sergei or Ferdinand?

Posted by: John Armstrong on February 13, 2007 7:12 PM | Permalink | Reply to this

Re: Story is a Force of Nature; Re: Why Do I Bother?

John Armstrong wrote (about Cardano)

looking at the graphs

But graphs hadn’t been invented yet!

Posted by: Tim Silverman on February 13, 2007 6:56 PM | Permalink | Reply to this

Re: Story is a Force of Nature; Re: Why Do I Bother?

Okay, fine, I give. My knowledge of history is spotty here, and I was evidently misinformed from the get-go about motivations.

There are other stories that illustrate the same point, but I’m sure they’ll be blown apart as well. Grothendieck seeking to exploit the analogy between field extensions and covering spaces leading to topoi (as related in MacLane-Moerdijk) comes to mind.

And even if the explicit examples I recall are shot down, does everyone here go so far as to deny that progress is sometimes made in part in an effort to find a new viewpoint from which the “good story” and the facts both hold?

Posted by: John Armstrong on February 13, 2007 7:10 PM | Permalink | Reply to this

Re: Story is a Force of Nature; Re: Why Do I Bother?

progress is sometimes made in part in an effort to find a new viewpoint from which the “good story” and the facts both hold

Sure, and we can still use the same example (assuming that I have my facts right! ^_^).

The original story was that Cardan’s algorithm found all positive real roots of any real cubic polynomial. But Bombelli (or whoever it was) noticed that sometimes an intermediate step called for the square root of a negative number! That would be all right if, in such circumstances, the root being sought didn’t exist (in modern terms, was negative or imaginary), but the fact was that the algorithm sometimes contained such a step even when a perfectly good (positive real) root existed. So he used complex numbers in carrying out Cardan’s algorithm, getting the correct result. From this humble beginning, the modern understanding of complex numbers developed, and the (new) facts fit the (old) story. And then for good measure, they allowed the story to be enriched as well! (So now Cardan’s algorithm gives, with multiplicity, 3 complex roots to any cubic complex polynomial.)

Here is a nice link, found by a search for ‘Bombelli’ and ‘Cardano’: http://www.clarku.edu/~djoyce/complex/cubic.html

Posted by: Toby Bartels on February 13, 2007 10:31 PM | Permalink | Reply to this

4 of your and Polanyi’s quotes; Re: Why Do I Bother?

David, you say:

(1) “in the past couple of years I’ve been advocating MacIntyre’s idea of a rational tradition of enquiry as governed by overarching dramatic narratives.”

(2) “I’ve also been working on a conception of mathematical reality close to Michael Polanyi’s”

(3) [Polanyi] “A new mathematical conception may be said to have reality if its assumption leads to a wide range of new interesting ideas. (Personal Knowledge: 116)”

(4) [Polanyi]: “while in the natural sciences the feeling of making contact with reality is an augury of as yet undreamed of future empirical confirmations of an immanent discovery, in mathematics it betokens an indeterminate range of future germinations within mathematics itself. (Personal Knowledge: 189)”

I am fascinated by your phrasing in (1) “governed by overarching dramatic narratives” as it suggests that the governing [kybernetes as source of term] implies lawful interaction of narratives, and overarching suggests multiple levels, as I’ve tried to address.

I am fascinated by your saying in (2): “conception of mathematical reality” as I believe this is at the core of the Philosophy, History, and Sociology of Mathematics, and of its interaction with natural sciences.

I am fascinated by Polanyi saying in (3): “… if its assumption leads to a wide range of new interesting ideas…” because we can’t get our hands on “ideas” but can study the ideas projected to the lower dimensionality of character strings in papers, books, transcripts of talks, snailmails, emails, and blogs. The Genetic Algorithm gives us one was to formally handle evolution of character strings interacting with an interpreter evaluating them and returning a scalar “fitness.”

Claude Shannon’s Communications Theory also gives us a way to deal with ensembles of character strings, in terms of source, transmitter, channel, noise, receiver, and destination, via entropy and other mathjematical constructs, and using both probability theory and finite field theory and Galois theory.

In (4) Polanyi writes: “… indeterminate range of future germinations within mathematics itself…” This is wonderful. “Indeterminate range” is well-put, and relates to what Nature called (was it in their 125th anniversary issue) “The Fronteiers of Ignorance.”

“Future Germinations” strongly suggests a biological metaphor ro me, with sexual reproduction, which is why I phrased things in a Genetic Algorithm sense.

I have my own theories of “dramatic narrative” as a much-published poet, fiction author, playwright, and academic writer. I am deeply interested in how this relates to Mathematics as practiced, as formalized, as published.

I am eager to learn more from you and your colleagues.

Thank you!

Posted by: Jonathan Vos Post on February 16, 2007 8:09 PM | Permalink | Reply to this

Re: Why Do I Bother?

Regarding narrative, Noah Snyder writes advice for graduate students

Mathematicians are people and people love stories. People don’t want to hear your theorems, they want to hear the story you’re telling with them. Similarly if you want to understand a field you need to distill it into stories.

So do this at every opportunity. When you’re learning about other people’s work guess what the narrative is that it fits into and see if you can make it work. Give a talk on that story. It doesn’t have to be what the author actually had in mind, but it needs to be a good interesting story.

Posted by: David Corfield on May 18, 2009 11:27 AM | Permalink | Reply to this

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