The n-Category Café

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.

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).

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.

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.

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.

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.

My take on this is as follows. I enjoyed John’s boring talk . 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?

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?