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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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I'm surprised that you don't mention the humanities as a really bad case where there is little low-hanging fruit and high ideological content. Take English literature for example. Barrels of ink have been spilled in writing about Hamlet, and genuinely new insights are quite rare. The methods are also about as unsound as you can imagine. Freud is still heavily cited and applied, and postmodern/poststructuralist/deconstructionist writing seems to be accorded higher status the more impossible to read it is.
Ideological interest is also a big problem. This seems almost inevitable, since the subject of the humanities is human culture, which is naturally bound up with human ideals, beliefs, and opinions. Academic disciplines are social groups, so they have a natural tendency to develop group norms and ideologies. It's unsurprising that this trend is reinforced in those disciplines that have ideologies as their subject matter. The result is that interpretations which do not support the dominant paradigm (often a variation on how certain sympathetic social groups are repressed, marginalized, or "otherized"), are themselves suppressed.
One theory of why the humanities are so bad is that there is no empirical test for whether an answer is right or not. Incorrect science leads to incorrect predictions, and even incorrect macroeconomics leads to suboptimal policy decisions. But it's hard to imagine what an "incorrect" interpretation of Hamlet even looks like, or what the impact of having an incorrect interpretation would be. Hence, there's no pressure towards correct answers that offsets the natural tendency for social communities to develop and enforce social norms.
I wonder if "empirical testability" is a should be included with the low-hanging fruit heuristic.
Sounds like a good idea until you realize that you are throwing out most math and philosophy with the bathwater.
How about accepting either empirical testability or a requirement that all claims be logically proven? (Much of microeconomics and game theory slides in under 'provable' rather than 'testable'. Quite a bit of philosophy fails under both criteria, but some of it approaches 'provable'.)
Pure mathematics per se may not be empirically testable, but once you establish certain correspondences - small integers correspond to pebbles in a bag, or increments to physical counting devices - then the combination of conclusion+correspondence often is testable, and often comes out to be true.
In some cases, combinations of correspondences+mathematically true conclusion gives a testably false conclusion about the real world, such as the Banach-Tarski paradox.
The problem here isn't the mathematics, but the correspondence. Physical balls are only measurable sets to a first approximation.
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".
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.