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July 25, 2007

Delphic Inspiration

Posted by David Corfield

I’ve returned from the sun of Delphi to the sogginess of England. John has already put up some pictures and a description of the event – Mathematics and Narrative – in his diary. I think the very best part of the meeting was the decision to have each participant be interviewed by another. The suggested length of two to three hours for this process seemed daunting, but it allowed a kind of conversation I’ve never known before. And to have two and a half hours of Barry Mazur’s undivided attention!

When it came to my turn to be interviewed, by my philosopher friend Colin McLarty, I began to see that Alasdair MacIntyre’s notion of a rational tradition of enquiry could be made to do some real work. We get rather used in the humanities to fairly loose schematic descriptions of phenomena, unlike in the hard sciences where predicted entities (such as categorified constructions) had better be found if we are not to lose faith. From the interview, we got the sense that this framework could point us easily to the difficulties other approaches face, and then explain them.

Perhaps we’ll see the Delphi meeting as one of those defining moments in getting a non-relativist practice-oriented philosophy of mathematics off the ground. Elsewhere, I interviewed a third member of this movement, Brendan Larvor, for the fourth edition of The Reasoner.

Posted at July 25, 2007 2:22 PM UTC

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Re: Delphic Inspiration

David, can you explain what you mean by a “non-relativist” philosophy of mathematics? (Or a “relativist” philosophy, if you prefer :-))

Posted by: Tom Leinster on July 25, 2007 4:00 PM | Permalink | Reply to this

Re: Delphic Inspiration

There’s a huge amount to say in answer to this. But here’s a first stab.

Relativism means many different things. I’m using it here with regard to mathematics to take the contrary position to mean that while any mathematician is historically situated, and so only has access to limited conceptual and material resources, they can still aim to contribute to a timeless formulation of their field.

We’re in a peculiar postion in mathematics in that the debate, if it merits the name, is almost exclusively confined to the truth of propositions, where we can then question, say, whether we might have used a different logic, or whether we had to choose to have ‘2 + 2 = 4’. A crude version might claim that had Brouwer managed to wield more power in the community, then perhaps the majority of you now would be working with intuitionistic principles.

I want to shift the debate to include the conceptual content of mathematics. That improvements take place in the understanding of whatever it is that mathematics is about, and not just according to us. We may express our best understanding of what it is about in terms of space, quantity, dimension, symmetry, etc., recognising that this understanding will itself likely be improved. So I want to maintain that there are good reasons for changes in mathematical theory, that some changes are made for good reasons, and that this ‘good’ is not just for us here now.

At the earlier Thales and Friends meeting, historians were keen to show us that mathematics could have gone in very different directions. Had not the political climate been such, French analysis would not have gone in a Cauchy-esque direction. Elsewhere, others have tried to argue that Hamilton’s work on quaternions was driven largely by his idiosyncratic metaphysical beliefs, without which they might never have seen the light of day.

Parallel theses have been argued about contingency in physics. There we have plausible accounts of how the construction of pieces of apparatus was only made possible due to fortuitous circumstances arising from, say, industry or war. The next step is to wonder whether certain theories could have been developed without the results recorded by this apparatus. A more daring form of relativism takes the recalcitrance of problems in Fluid Dynamics to arise from male dominance precluding the appearance of new forms of feminine language.

On the face of it, mathematicians have plenty of freedom as to which branches of mathematics to create. It’s hardly outrageous to argue that the field might have looked quite different, as does David Ruelle. Had our needs been very different, had we not been a warlike species requiring to know how to project missiles and to keep messages secure, which forms of mathematics might have prevailed?

I remain, though, inclined to adopt a realist position. I find it hard to conceive that we could have achieved a similar level of sophistication without devising, say, the concept of a Lie group. Consider I tell you that a thousand years from now number theorists will look back at this time and at the rise of the Langlands Program in particular, and see it as one big mistake. And if I tell you further that they had thrown everything away done after Gauss and had struck out in a different direction. When I put this to Barry Mazur, he could only think that something would have had to have gone wrong.

Posted by: David Corfield on July 26, 2007 11:10 AM | Permalink | Reply to this

Re: Delphic Inspiration

Thanks very much.

So let’s see if I understand this correctly. Everyone agrees that we have a choice as to what mathematics we do, and that our choices are influenced by the world around us. Wars may cause advances in mathematics related to ballistics, for example.

But a non-relativist philosophy of mathematics asserts something like: there is a core body of mathematics, independent of historical events, that must be discovered if mathematics as a whole is to advance. For instance, any civilization on any planet that hasn’t discovered Lie groups yet will be seriously held back in its mathematical progress until it does - there’s simply no way around. Is that right?

(I don’t mean ‘is that true?’; I mean ‘is that a correct understanding of the term non-relativist philosophy?’.)

‘There’s a divinity that shapes our ends, rough-hew them how we will.’

Posted by: Tom Leinster on July 26, 2007 4:51 PM | Permalink | Reply to this

Re: Delphic Inspiration

Looking back at this thread, I see that I put my response in the wrong place.

Posted by: David Corfield on August 27, 2007 8:16 AM | Permalink | Reply to this

Re: Delphic Inspiration

I’ve just been enjoying reading John’s ‘very rough draft’ of notes for his talk Why Mathematics is Boring at the Delphi Meeting. (See also the earlier blog post of the same name.) I look forward to provocatively-titled sequels such as Why nn-Categories Suck.

There’s a lot in John’s notes about abstracts and introductions of mathematics papers, and how they’re typically very forbidding to anyone outside the immediate speciality of the author. And this set me thinking.

Most people’s first reaction to this phenomenon is probably ‘that’s a bad thing. We should write papers to be widely accessible’. But recently I read a book called Communicating Science (Chicago University Press; I forget the author but I wouldn’t recommend it very much anyway), which sort of argued the opposite.

His point was that we should be honest about what we expect of the reader. In his opinion, it’s unfair to lure in a non-specialist reader with a widely accessible, technically-undemanding introduction if the substance of the paper is only comprehensible to specialists anyway. You’re simply wasting the reader’s time. Better to start with a sentence like ‘we prove the conjecture that quasidivisors on hypercompact ultraprimes are pseudo-Serre’, because that efficiently tells the reader that if they don’t understand those terms, it’s not worth their while continuing.

I’m probably exaggerating the author’s view slightly. But there’s a real dilemma here. Suppose that after much thought, you’ve decided to write the main body of your paper in such a way that it’s only comprehensible to specialists. (Whether you should do this is a separate debate - but let’s say for sake of argument that this is how it’ll be.) Suppose also that the broad ideas of your paper can be understood by a much larger crowd. What do you do?

One extreme: write the abstract and introduction so that they can be understood by as many people as possible. Use the introduction to describe in widely accessible terms as much of your paper as possible (which may make it quite long). Then as soon as you begin Section 1, switch to writing for specialists.

Opposite extreme: realize that only specialists are going to understand your paper all the way through, so begin as you mean to go on.

I have opinions on this, but I’d rather hear other people’s first. What do you think?

Posted by: Tom Leinster on July 25, 2007 11:02 PM | Permalink | Reply to this

Re: Delphic Inspiration

This may be only muddying the waters, but I think that to state a general answer to “how should one write such a paper”, one must handle the case that the theorem statement and its proof may be of different levels of accessibility.

Particularly as there is some level of “glory” associated to accessible statements with inaccessible proofs (FLT being only the most famous example).

Posted by: Allen Knutson on July 26, 2007 3:43 AM | Permalink | Reply to this

Re: Delphic Inspiration

I don’t think it’s muddying the waters. In fact, it’s more or less the same as the scenario I described.

The kind of work I do means that my papers are more often about exploring an idea than proving a theorem. In my scenario, I had in mind that the main thrust of the idea could be explained to a large audience, but to explore it in detail would require specialist knowledge. In your scenario, there is a theorem whose statement could be explained to a large audience, but whose proof requires specialist knowledge.

The key thing about this (common!) situation is that you can begin in a widely accessible way, but you can’t continue in a widely accessible way. The question is: in this situation, should you begin in a widely accessible way?

Posted by: Tom Leinster on July 26, 2007 5:01 PM | Permalink | Reply to this

Re: Delphic Inspiration

I shouldn’t want to give the impression that this non-relativist position is already well-defined. In fact this very exchange is a considerable addition to a sparse literature, certainly in the English language. The conceptual content of maths is rarely addressed. Generally, existence is the theme – do all mathematical entities exist or do none, with no way to descriminate between, say, E 8E_8 and the category of finite groups with the monster removed.

With a concept as recently devised as Lie groups (130 odd years ago) there is the risk that as time goes by humans themselves may learn to see them as of lesser importance. But at least we expect that our descendents should be able to see why we considered them so important, and that our theorising about them was an important step on the path to them.

As for other civilsations, as sci-fi writers know, there’s always the worry when dreaming up other intelligences of displaying a lack of imagination. Spock may have lacked emotion and have reasoned more accurately, but he approximates a certain kind of human.

So perhaps that’s too much to expect them to have discovered Lie groups per se. As a first, very minimal concession, perhaps they were clever enough to begin with groupoids, rather than their one object counterparts. But then why should we not be able to think up quite different paths we ourselves might have taken? Couldn’t the twists and turns of the development of the calculus and then analysis have happened quite differently?

I think the best solution is to think in terms of a form of convergence. Can we imagine a civilisation advanced enough to navigate through the galaxy, and capable of enormously better physical predictions at the very large and very small scale, who couldn’t make sense of Lie group theory?

Hmm, perhaps this invocation of aliens doesn’t take us too far. Maybe we have enough to be dealing with here on Earth. What if the Erdos-Gowers-Tao culture had received all professional mathematical attention. But even here Tao can speak of

nilsequences arising from flows on a quotient of a nilpotent Lie group. (p. 24)

and, of course, elsewhere when discussing differential equations.

Posted by: David Corfield on July 28, 2007 10:48 AM | Permalink | Reply to this

Re: Delphic Inspiration

I’m rather a lurker on here, but as someone who’s been getting back to reading math papers I may have a different perspective on this than you pros. I honestly like a clean Big Picture introduction more than anything else, but barring that I like papers dripping with references. Nothing is more disheartening than finding a paper that sounds very interesting that has than a half dozen references. When, not if, I get stuck it’s often much harder to find something to answer the questions I have.

Some of the best papers I’ve read have been review articles, because almost everything freakin’ sentence has a reference attached to it so I can get more details when something doesn’t make sense.

Admittedly, I think online resources such as Wikipedia have been making this a bit easier. If I can’t remember what a braided monoidal category is, I can look it up rather quickly.

Posted by: Creighton Hogg on July 26, 2007 4:37 AM | Permalink | Reply to this

Re: Delphic Inspiration

It would be madness to expect every paper to come with introductions accessible to those without much knowledge of the field. But imagine if there were a much greater range of online exposition of concepts and programmes, written for different levels of expertise, and that it was easy for an author to indicate where he or she locates her work in terms of a broader program of research.

Posted by: David Corfield on July 26, 2007 11:27 AM | Permalink | Reply to this

Re: Delphic Inspiration

Tom Leinster wrote:

His point was that we should be honest about what we expect of the reader. In his opinion, it’s unfair to lure in a non-specialist reader with a widely accessible, technically-undemanding introduction if the substance of the paper is only comprehensible to specialists anyway. You’re simply wasting the reader’s time. Better to start with a sentence like ‘we prove the conjecture that quasidivisors on hypercompact ultraprimes are pseudo-Serre’, because that efficiently tells the reader that if they don’t understand those terms, it’s not worth their while continuing.

I agree that if the sole goal of your paper is to prove that that quasidivisors on hypercompact ultraprimes are pseudo-Serre, and you aren’t planning to say anything that someone ignorant of these terms could find interesting, you might as well make your abstract reflect that, to keep them from wasting their time.

In fact, I think it’s a great idea for boring papers to have boring abstracts — and boring titles, too! A Journal of Boring Mathematics would be even better. devil

But, I argued in Delphi that math papers don’t need to be quite as inaccessible as they so often are. They can have introductions and abstracts that are accessible even to mathematicians who won’t want to read the whole paper. That way, more people can learn more things.

I suspect that about 90% percent of mathematicians who read a given paper’s title will stop there; about 90% who go on will stop after reading the abstract, and about 90% who go on further will stop after reading the introduction. So, I think authors should put a bit more work into pleasing and educating the 99.9% of their audience who do not read the whole paper.

Of course, there are lots of different kinds of papers, and lots of different kinds of authors, so I’m not looking for any sort of uniformity of approach.

But, I do want to make people think harder about whether they’re writing as well as they can. This requires a certain amount of moral pressure.

Posted by: John Baez on July 26, 2007 2:11 PM | Permalink | Reply to this

Re: Delphic Inspiration

My take on this is as follows. I enjoyed John’s boring talk pic. I am on the verge of becoming a passionate supporter of this ideology, but before I do so, I need to be assured about a certain foundational point.

Is it the primary goal of the mathematical world to advance mathematics?

I would love it to be so. Indeed, this seems to be an unwritten assumption in John’s talk. The last sentence of his abstract gives his ‘slamdunk reason’ for why mathematics shouldn’t be boring :

“This impedes the development of mathematics.”

Here is a thought experiment which shows, I think, that the goal of the mathematical community is not the advancement of mathematics.

Suppose an alien U.F.O. landed in Arizona, demanding to speak to the world leaders. They inform us that, unless the human race can prove the Riemann hypothesis by 2020, it will be regarded as unfit for preservation, and the earth will be demolished to make way for an interstellar highway.

Ask yourself : How would this affect the way mathematics is done?

Surely overnight we would see thousands of expository articles released on the internet, desperately explaining in the clearest and simplest way imaginable, how this or that research program might be able to contribute? Wouldn’t we see a plethora of online lectures given by the experts, frantically trying to teach their work in the most understandable way to those from other fields, in the frenzied hope that perhaps someone, someone out there could use those ideas. Wouldn’t the entire nature of mathematical journals be transformed overnight?

Since this is so far from the way the mathematical community operates at the moment, I am forced to conclude that the role of the mathematical community in world affairs is not to enhance human mathematics. Rather, it seems to be a sort of intellectual racket, an “old-boy’s club” where we get to travel around the world on taxpayer’s money, giving the occasional conference talk, and keeping up appearances by publishing the odd poor-quality paper in various overpriced journals.

Of course, I’m playing the devil’s advocate here. But can you refute this argument?

Posted by: Bruce Bartlett on July 26, 2007 3:02 PM | Permalink | Reply to this

Re: Delphic Inspiration

Alright, I just wanted to attack the notion of mathematics being an “old-boy’s club”. I don’t think it’s anything so malicious or exclusive on the part of the mathematicians, I think it’s just a variation on the tragedy of the commons.

I’ve never met a mathematician who would be opposed to more people knowing and understanding the mathematics that they love; however, it’s a lot of work to make papers more readable and it takes away time from doing the mathematics they love. The personal gain for an individual isn’t very obvious, since writing better papers doesn’t force anyone else to do the same, and they already understood the subject themselves so they won’t benefit from their own paper being clearer.

I think the whole thing is very similar to why professors who are good at research are sometimes very bad at teaching: while teaching the subject well has important long term benefits, in the short term it means they spent n-hours not doing their favorite research.

Of course, one subject on which I disagreed with some people in my old physics department is in the importance of teaching. I firmly believe that the forcing yourself to explain complicated ideas to other people makes you better understand your own work. There are plenty of professors I met who are not good teachers, and I’m sure some of them are far cleverer than I will ever be, but I’m going to say that they probably could have been even better had they taken more time to explain and teach.

Posted by: Creighton Hogg on July 26, 2007 3:39 PM | Permalink | Reply to this

Re: Delphic Inspiration

Interesting provocation here (that’s not meant as a criticism). I’d like to counter with a second gedanken: suppose that at the last minute in 2019 RH is proved, and it turns out the aliens were just having a laugh (a game of dare after too many Pangalactic Gargleblasters). Would “human mathematics” really be better off than if we had continued on the current path?

I think there’s also a massive assumption here that ability to communicate maths is positively correlated with the ability to do it, and I suspect that at the coal-face this might not be true. In which case, one has to ask whether it’s better to understand the maths “we” have already proved, or to crack on and try for the sharpest advances as soon as possible.

(Hope that wasn’t too incoherent…)

Posted by: Yemon Choi on July 26, 2007 6:27 PM | Permalink | Reply to this

Re: Delphic Inspiration

The question is what is it for human mathematics to be better off. This can’t be answered without a conception of the ‘good’ of mathematics (to use an Aristotelian term).

Presumably it can’t just be the extent of the furthest points of mathematical knowledge.

If 6 billion of the world’s population all thoroughly understood the contents of the arXiv up to yesterday, mathematics would be in a better state than if only one person understood the contents up to today’s arXiv and nobody else knew anything.

But once one addressed the question of the ‘good’ of mathematics, it seems difficult to avoid the larger question of the place of mathematics in the life of a political community. This is something the Greeks naturally did.

I think there’s also a massive assumption here that ability to communicate maths is positively correlated with the ability to do it, and I suspect that at the coal-face this might not be true.

Has that been assumed? I think what was assumed was that the good of mathematics would be better served by clearer communication.

But perhaps you are picking up a theme of this blog that mathematics got right is a lot simpler than it would appear to be, and that the kind of mathematician who can see through the tangle of complex formulations to pull out the essence of the situation should then be able to tell us of this clearer vision. I would agree that this involves some assumptions to which not all would adhere.

Posted by: David Corfield on July 27, 2007 8:25 AM | Permalink | Reply to this

Isaac Asimov; Re: Delphic Inspiration

Let me quote from my late friend, editor, and co-broadcaster:

“The rapid advance of science is exciting and exhilarating to anyone who is fascinated by the unconquerability of the human spirit and by the continuing efficacy of the scientific method as a tool for penetrating the complexities of the universe.” [Asimov’s New Guide to Science, 1984]

“The publications of scientists concerning their individual work have never been so copious - and so unreadable for anyone but their fellow specialists. This has been a great handicap to science itself, for basic advances in scientific knowledge often spring from the cross-fertilization of knowledge from different specialties. Even more ominous, science has increasingly lost touch with nonscientists. Under such circumstances, scientists come to be regarded almost as magicians - feared rather than admired. And the impression that science is incomprehensible magic, to be understood only by a chosen few who are suspiciously different from ordinary mankind, is bound to turn many youngsters away from science.”

[Asimov’s New Guide to Science, 1984]

“Physics was quickly eliminated, for it was far too mathematical. After years and years of finding mathematics easy, I finally reached integral calculus and came up against a barrier [he told me, JVP, that it was understanding WHY Integration By Parts worked; he could do it and get the right answer, but his intuition abandoned him, which caused ontological and/or epistemological panic].”

“I realized that that was as far as I could go, and to this day I have never successfully gone beyond it in any but the most superficial way…. Chemistry won by default….”

Posted by: Jonathan Vos Post on July 27, 2007 7:12 PM | Permalink | Reply to this

Re: Isaac Asimov; Re: Delphic Inspiration

Here is a short story about trying to make the mathematics more interesting and failing at that endeavor. Part of the failure is my own fault, but there is an interesting editorial prejudice that I haven’t quite sorted out. Jon Vos Post’s quotes from Asimov reminded me of the story. Please note, I am not particularly angry with the editorial process in the story, I am just miffed why the result is not more well known.

About a year or so ago Abhijit Champanerkar and I wrote the paper, A geometric method to compute some elementary integrals. The gist of the paper is, “If the integral from 0 to 1 of x^3 is equal to a quarter, what is itl a quarter of?” The answer is nice, it is a quarter of the volume of the 4-dimensional cube. More generally, when integrating x^n, from 0 to 1, one finds that the resulting area can be interpreted as the (n+1)-dimensional volume of a pyramid in the $(n+1)$-dimensional hypercube. A total of (n+1) congruent pyramids tile the hypercube. For lower dimensional examples try decomposing the cube into three pyramids each of volume 1/3.

The preceding paragraph is a punch line to a joke that, when it is told, everyone can find their favorite set up. When Abhijit and I wrote down the proof, we used linear algebra. I may the unique person in the universe who thinks that linear algebra is more simple than calculus, and one of my big soap boxes is that linear algebra, not calculus should be the entry level freshman course in state universities. So when we wrote the afore mentioned paper, we told the joke from a linear algebra point of view. We also gave an elegant one line formulation, and in the meantime we described a bunch of the combinatoric structure that one would see in a hypercube. One interesting aspect of the paper is that we could decompose the (n+1)-cube into (n+1)! congruent simplices.

What does one do with a paper like this? Send it to the dominant pedagogical journals in the field. Start at the top and see what the reaction is. Several editors rejected the paper. No one said explicitly, “Hey dudes, this is folklore.” The one reference that I received did not seem to me to address the issue directly. After the third rejection we found a journal that would, at least send it out for refereeing. The report was quite gracious and interesting. But the referee was one of those people who read the punch line and set up the joke from yet another point of view. That is what should be done! But the correspondence was quasi-private. The editor of the journal explicitly did not invite us to resubmit. The referee thought that she could write the paper better. Maybe so. If so, then she should write it, give us some credit, and get it into the literature.

OK, so here is an expository dilemma. I think that it is quite interesting that the (n+1)-dimensional cube can be decomposed into (n+1) congruent pyramids. A consequence of this geometric fact is that polynomial functions can be integrated (both definitely and indefinitely), and a limiting process is not needed. The whole situation is rigid, rectilinear, and elementary. Moreover, by examining the configuration, you can glean some higher dimensional intuition. Higher dimensional cubes, and higher dimensional simplices are important to know. Many don’t.

None of my friends had not heard the result.

For sport, I included the mathematica code that would let the reader have her very own hypercube on her desk top. My code is not that elegant, but that is not the point. It rotates your favorite higher dimensional object around in space. Since the TeX file is on the arXiv, you can cut and paste the code. Publishing code is not a bad thing. The code itself is replete with puns and zany variable names. At least it is not boring.

Sure, the paper might have been marketed better. I concede that. But now the paper is not among the literature in the published journals.

I, at least, think that higher dimensional configurations are not boring. When I show them to the kids in the neighborhood, they all go, “Wow!” Even though the ideas are nineteenth century ideas, they should be part of the general culture. So I have this paper, that has received more than its share of rejections from reputable pedagogical journals, does not have a home, and should be part of the general mythos of mathematics. Any ideas, beyond advertising on a lesser known blog?

Posted by: Scott Carter on July 28, 2007 2:36 AM | Permalink | Reply to this

Walter Fisher’s oncept of ‘story’; Re: Delphic Inspiration

Robert Sibley. No happy endings.

History records that there is nothing so powerful as a fantasy whose time has come.

– Historian Tony Judt, Reappraisals

“… concept of ‘story’ from Walter Fisher, a communications theorist who argues that humans are essentially storytellers, and that all communication – history, art, language, science, etc. – is a form of storytelling. That is to say, the world is a collection of ‘stories’ – or ‘narrative paradigms,’ to use Fisher’s terms – that we constantly examine for coherence and check against our experience as we attempt to create meaningful lives, individually and collectively.”

Posted by: Jonathan Vos Post on July 7, 2008 4:51 PM | Permalink | Reply to this

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