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Johnicholas comments on Some Heuristics for Evaluating the Soundness of the Academic Mainstream in Unfamiliar Fields - Less Wrong
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Yes.
However, imagine some abstruse mathematical theory that, in some "evaluate it on its own terms" sense, is true, but every correspondence that we attempt to make to the empirical world fails. I would claim that the failure to connect to an empirical result is actually a potent criticism of the theory - perhaps a criticism of irrelevance rather than falsehood, but a reason to prefer other fields within mathematics nevertheless.
I don't know of any such irrelevant mathematical theories, and to some extent, I believe there aren't any. The vast majority of current mathematical theories can be formalized within something like the Calculus of Constructions or ZF set theory, and so they could be empirically tested by observing the behavior of a computing device programmed to do brute-force proofs within those systems.
My guess is that mathematicians' intuitions are informed by a pervasive (yet mostly ignored in the casual philosophy of mathematics) habit of "calculating". Calculating means different things to different mathematicians, but computing with concrete numbers (e.g. factoring 1735) certainly counts, and some "mechanical" equation juggling counts. The "surprising utility" of pure mathematics derives directly from information about the real world injected via these intuitions about which results are powerful.
This suggests that fields within mathematics that do not do much calculating or other forms of empirical testing might become decoupled from reality and essentially become artistic disciplines, producing tautology after tautology without relevance or utility. I'm not deep enough into mathematical culture to guess how often that happens or to point out any subdisciplines in particular, but a scroll through arxiv makes it look pretty possible: http://arxiv.org/list/math/new
In my perfect world, all mathematical papers would start with pointers or gestures back to the engineering problems that motivated this problem, and end with pointers or gestures toward engineering efforts that might be forwarded by this result.
I don't have any good examples of actual irrelevant/artistic mathematics, but possibly:
"Unipotent Schottky bundles on Riemann surfaces and complex tori" http://arxiv.org/abs/1102.3006
would be an example of how opaque to outsiders (and therefore potentially irrelevant) pure mathematics can get. I'm confident (primarily based on surface features) that this paper in particular isn't self-referential, but I have no clue where it would be applied (cryptography? string theory? really awesome computer graphics?).
...
Why do mathematicians put up with this? I'll need to describe a mathematical culture a little first. These days mathematicians are divided into little cliques of perhaps a dozen people who work on the same stuff. All of the papers you write get peer reviewed by your clique. You then make a point of reading what your clique produces and writing papers that cite theirs. Nobody outside the clique is likely to pay much attention to, or be able to easily understand, work done within the clique. Over time people do move between cliques, but this social structure is ubiquitous. Anyone who can't accept it doesn't remain in mathematics.
Among other things, it sounds like you're expecting inferential distances to be short.
My intent was to demonstrate a particular possible threat to the peer review system. As the number of people who can see whether you're grounded in reality gets smaller, the chance of the group becoming an ungrounded mutual admiration society gets larger. I believe one way to improve the peer review system would be to explicitly claim that your work is motivated by some real-world problem and applicable to some real-world solution, and back those claims up with a citation trail for would-be groundedness-auditors to follow.
Actually, there's a vaguely similar preprint: http://arxiv.org/PS_cache/arxiv/pdf/1102/1102.3523v1.pdf
The danger I see is mathematicians endorsing mathematics research because it serves explicitly mathematical goals. It's possible, even moderately likely, that a proof of the Riemann Hypothesis (for example) would be relevant to something outside of mathematics. Still, I'd like us to decide to attack it because we expect it to be useful, not merely because it's difficult and therefore allows us to demonstrate skill.
Why such prejudice against "explicitly mathematical goals"? Why on Earth is this a danger? One way or another, people are going to amuse themselves -- via art, sports, sex, or drugs -- so it might as well be via mathematics, which even the most cynically "hard-headed" will concede is sometimes "useful".
But more fundamentally, the heuristic you're using here ("if I don't see how it's useful, it probably isn't") is wrong. You underestimate the correlation between what mathematicians find interesting and what is useful. Mathematicians are not interested in the Riemann Hypothesis because it may be useful, but the fact that they're interested is significant evidence that it will be.
What mathematics is, as a discipline, is the search for conceptual insights on the most abstract level possible. Its usefulness does not lie in specific ad-hoc "applications" of particular mathematical facts, but rather in the fact that the pursuit of mathematical research over a span of decades to centuries results in humans' possessing a more powerful conceptual vocabulary in terms of which to do science, engineering, philosophy, and everything else.
Mathematicians are the kind of people who would have invented negative numbers on their own because they're a "natural idea", without "needing" them for any "application", back in the day when other people (perhaps their childhood peers) would have seen the idea as nothing but intellectual masturbation. They are people, in other words, whose intuitions about what is "natural" and "interesting" are highly correlated with what later turns out to be useful, even when other people don't believe it and even when they themselves can't predict how.
This is what we see in grant proposals -- and far from changing the status quo, all it does is get the status quo funded by the government.
It's easier to concoct "real-world applications" of almost anything you please than it is to explain the real reason mathematics is useful to the kind of people who ask about "real-world applications".
From an assumption of wealth, that we humans have plenty of time and energy, I agree with you - the fact that someone is curious is sufficient reason to spend effort investigating. However, (and this is a matter of opinion) we're not in a position of wealth. Rather, we currently have important scarcities of many things (life), we have various ongoing crises, and most of our efforts to better ourselves in some way are also digging ourselves deeper in some other way, manufacturing new crises that will require human ingenuity to address.
Improvements to the practice of peer review would be valuable, to achieve more truth, more science, more technology.
You're putting words in my mouth by claiming I'm following a "inferential distances are short" heuristic. That would be like additionally requiring the groundedness-auditor ought to bottom out in the real world after a short sequence of citations. I never said anything like that.
Your claim that all mathematicians somehow have accurate intuitions about what will eventually turn out to be useful is dubious. Mathematicians are human, and information about the world has to ultimately come from the world.
Earlier I suggested "computations", that is, mechanical manipulations of relatively concrete mathematical entities, as the path for information from the world to inform mathematician's intuitions. However, mathematicians rarely publish the computations motivating their results, which is the whole point that I'm trying to make.
Adding the quantifier "all" is an unfair rhetorical move, of course; but anyway, here we come to the essence of it: you simply do not see the relationship between the thoughts of mathematicians and "the world". Sure, you'll concede the usefulness of negative numbers, calculus, and maybe (some parts of) number theory now, in retrospect, after existing technologies have already hit you over the head with it; but when it comes to today's mathematics, well, that's just too abstract to be useful.
Do you think you would have correctly predicted, as a peasant in the 1670s, the technological uses of calculus? I'm not even sure Newton or Leibniz would have.
Human brains are part of the world; information that comes from human thought is information about the world. Mathematicians, furthermore, are not just any humans; they are humans specifically selected for deriving pleasure from powerful insights.
Every proof in a mathematics paper is shorthand for a formal proof, which is nothing but a computation. The reason these computations aren't published is that they would be extremely long and very difficult to read.
I think we've both made our positions clear; harvesting links from earlier in this thread, I think my worry that mathematics might become too specialized is perennial:
Regarding the distinction between computation and proving, I was attempting to distinguish between mechanical computation (such as reducing an expression by applying a well-known set of reduction rules to it) and proving, which (for humans) is often creative and does not feel mechanical.
By "the computations motivating their results", I mean something like Experimental Mathematics: http://www.experimentalmath.info/
The issue here is about the "usefulness" of mathematical research, and its relationship to the physical world; not whether it is too "specialized". Far from adding clarity on the intellectual matter at hand, those links merely suggest that what's motivating your remarks here is an attitude of dissatisfaction with the mathematical profession that you've picked up from reading the writings of disgruntled contrarians. They may have good points to make on the sociology of mathematics, but that's not what's at issue here. Your complaint wasn't that mathematicians don't follow each other's work because they're too absorbed in their own (which is the phenomenon that Zeilberger and Tilly complain about); it was that the relationship between modern mathematics and "the world" is too tenuous or indirect for your liking. On that, only the Von Neumann quote (discussed here before) is relevant; and the position expressed therein strikes me as considerably more nuanced than yours (which seems to me to be obtainable from the Von Neumann quote by deleting everything between "l'art pour l'art" and "whenever this stage is reached").
As for computation: if your concern was the ultimate empirical "grounding" of mathematical results, the fact that all mathematical proofs can in principle be mechanically verified (and hence all mathematical claims are "about" the behavior of computational machines) answers that. Otherwise, you're talking about matters of taste regarding areas and styles of mathematics.
Indeed, people will always amuse themselves. But that doesn't mean they deserve an academic field devoted to amusing people within their own little clique. Should there be Monty Python Studies, stocked with academics who (somehow) get paid to do nothing but write commentary on the same Monty Python sketches and performances?
No, because that would be ****ing stupid. Their work would only be useful to the small clique of people who self-select into the field, and who aspire to do nohting but ... teach Monty Python studies. Yet the exact same thing is tolerated with classical music studies, whose advocates always find just the right excuse for why their field isn't refined enough to make itself applicable outside the ivory tower, or to anyone who isn't trying to say, "Look at me, plebes! I'm going to the opera!"
With that said, I agree that this criticism doens't apply to the field of mathematics for the reasons you gave -- that it is likely to find uses that are not obvious now (case in point: the anti-war prime number researcher whose "100% abstract and inapplicable" research later found use in military encryption). So I think you're right about math. But you wouldn't be able to give the same defense of academic art/music fields.
Well, um, thanks for bringing that up here, but of course I don't give the same defense of academic art/music fields; for those I would give a different defense.
There is.
Yes, one that fits in the class I described thusly:
And re: Monty Python Studies:
God help us all.
There's one funny quote I like about partially uniform k-quandles that comes to mind. Somewhat more relevantly, there's also Von Neumann on the danger of losing concrete applications.