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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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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.
The inferential chain is: too specialized leads to small cliques of peers who can review your work, which allows mutual admiration societies to start up and survive, which leads to ungroundedness, which leads to irrelevance.
Again, your claim that I think the relationship between modern mathematics in the world is too indirect is simply putting words in my mouth. I have no difficulty with indirect or long chains of relevance; my problem is with "mathematics for mathematics sake", particularly if it is non-auditable by outsiders. Would you fund "quilting for quiltings sake", if the quilt designs were impractically large and never actually finished or used to warm or decorate?
Here is a way that I think our positions could be reconciled: If there were studies on the "spin offs" of funding mathematicians to pursue their intuitions (deciding who is a mathematician based on some criterion perhaps a degree in mathematics and/or a Putnam-like test), then citing those studies would be sufficient for my purposes. I believe this is far less restrictive than current grants, which (as you say) demand the grant-writer to confabulate very specific applications; graph theory funded by sifting social networks for terrorists, for example.
Non-functional art quilting
Found while looking for the first link, and included for pretty
I think the crucial thing is not so much demonstrating that there might be some use for some not-obviously useful math-- I doubt there's any way to do that usefully in the short run. An accurate answer can't be known for any but the most obvious cases, and just making up something that sounds vaguely plausible is all too easy, especially if money is riding on the answer.
Instead, I recommend working on understanding the process by which uses are found for pure math, and, if it makes sense, cultivating that process.
The non-sequitur occurs in the third step (or possibly the second, depending on what you've built into the meaning of "mutual admiration society"). The "mutual admiration" in question is based largely, even mostly, on the work that people do within the clique, and not simply on membership. Both within and between cliques, "relevance" is regulated by the mechanism of status: those mathematicians (and cliques) working on subjects that the smartest mathematicians find interesting (which, as I've argued, is the appropriate test for "relevance" in this context) will tend to rise in status, while areas where "important" problems are exhausted will likewise lose prestige. This doesn't work perfectly, and there is some random noise involved, of course, but in the aggregate statistical sense, this is basically how it works. Contrary to the conventional cynical wisdom, the prestige of mathematical topics does not drift randomly like clothing fashion (unless the latter has patterns that I don't know about), but is instead correlated with (ultimate) usefulness by means of interestingness.
It's already easy to trace the intellectual ancestry of any mathematics paper all the way back to counting: you simply identify the branch of mathematics that it's in, look up that branch in Wikipedia, and click a few times. So what else do you mean by "groundedness", if not that subjects which are fewer inferential steps away from counting are more "grounded" than subjects which are more steps away?
I still don't understand why you have a problem with "mathematics for mathematics' sake". Is interestingness not a value in itself? For me it certainly is, and this is the core of my argument for academic/high-IQ art -- an argument which also applies to mathematics, for all that mathematics also benefits from utilitarian arguments. "Quilting for quilting's sake" as you describe it just sounds like a form of visual art, and visual art is something I would indeed fund.
What would count as a successful "spin off" in your view?
In your first paragraph, you have excellently made my point; the social process of mathematics depends on between-clique evaluations. To the extent that those between-clique evaluations are impossible, the social process of mathematics becomes more like clothing fashion, and mathematical goals become decoupled from engineering or science applications.
As I said previously, my criticism of "mathematics for mathematics sake" is based on an attitude of scarcity - which I admit is an attitude rather than a fact. Similarly, I would tax visual art rather than subsidize it.
Successful spin offs of mathematics would be applications of mathematics to fields that have better arguments that their work is not idle amusement, status-seeking or fashion-following.
But they're never impossible, and of necessity they're always going on (since university positions, grant dollars, etc. are limited in number). The only question can be what criteria are being used. While it is conceivable that some fields could end up using criteria that are "arbitrary" (i.e. not ultimately correlated with fundamental values), my argument is that this is not the case in mathematics, due mainly to the strong IQ barrier to entry. (Generally, my view is that the higher someone's IQ, the more strongly impressing them is correlated with satisfying fundamental values.)
Mathematical cliques are not islands; in fact to the extent they become isolated, they lose prestige! There is a continuum of relatedness, with cliques clustering into "supercliques" of various levels. Mathematicians, particularly those with a taste for cynical humor, will joke about how it is supposedly impossible to understand the work of neighboring cliques; but the reality is that their ability to understand varies more or less continuously with distance, and more or less increases with one's rank within a clique or superclique.
To summarize, there shouldn't be much to worry about so long as status in mathematics remains correlated more strongly with IQ than with other variables such as social/political skills. (Given that they're still willing to (try to) award prizes to someone like Perelman, I'd say the field is in pretty good shape.)
This is most extraordinary. Just how prosperous would we have to get before you would allow people to have tax-free fun?
Assuming you meant it literally (and not just as a signal of something else), this scares the hell out of me. It sounds like we may have practically-incompatible utility functions.
(How would that even be implemented? By paying inspectors to come to people's houses to check whether they've drawn any pictures that day? Extra sales tax on art supplies?)
Allow the visual art industry to have all the usual taxes on goods sold, exhibition prices and education. Don't subsidise the field at all via grants or via university tax breaks. No commando raids on kindergartens to catch off-the-books, under-the-table finger painters required.