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’.)
Even in mathematics, you can find contrarian opinions that much of the field is meaningless. What we have is (or at least seems to be) logically proved from the basis of certain assumptions, but we could as easily have picked very different assumptions and proved different theorems instead. There is a prevailing opinion that certain assumptions (the mainstream foundations of mathematics) are correct or at least useful, but correctness ultimately reduces to an aesthetic judgement, and usefulness is known to be exaggerated.
How about accepting either empirical testability or a requirement that all claims be logically proven?
Even better demand that there be strict rules in the discipline, which the research must obey—be it logical provability, empirical testability or whatever else. It is still possible to make up unreasonable rules, but production of bullshit is a lot easier without rules. Which is the case of deconstructionism and related fields.
Even better demand that there be strict rules in the discipline, which the research must obey—be it logical provability, empirical testability or whatever else. It is still possible to make up unreasonable rules, but production of bullshit is a lot easier without rules.
Strict formal rules are a two-edged sword. If well designed, they indeed serve as a powerful barrier against nonsense. However, they can also be perverted, with extremely bad results.
In many disciplines that have been affected by the malaises discussed in this thread, what happens is that a perverse formal system develops, which then serves as a template for producing impressive-looking bullshit work. This sometimes leads to the very heights of ass-covering irresponsibility, since everyone involved—authors, editors, reviewers, grant committees… -- can hide behind the fact that the work satisfies all the highest professional expert standards if questioned about it. At worst, these perverse formal standards can also serve as a barrier against actual quality work that doesn’t conform to their template.
Just to be clear: by strict rules I don’t mean anything with significant subjective judgement involved, like peer reviews. I rather mean things like demanding testability, mathematical proofs, logical consistency and such. Also, not much rules governing the social life of the respective community, but rather rules applied to the hypotheses.
Also, I haven’t said that rules are sufficient. One can still publish trivial theories which nobody is interested to test, mathematical proofs of obscure unimportant theorems or logically consistent tautologies. But at least the rules remove arbitrariness and make it possible to objectively assess quality and to decide whether a hypothesis is good or bad, according to standards of the discipline.
The discipline’s standard of good hypothesis may not universally correspond to a true hypothesis, but I suspect that if the standards of the discipline are strict enough, either the correspondence is there, or it is easily visible that the discipline is based on wrong premises, because it endorses some easily identifiable falsehoods. (It would be too big a coincidence if a formal system regularly produced false statements, but no trivially false statements.)
On the other hand, when the rules aren’t enough formal, the discipline still makes complex false claims, but nobody can clearly demonstrate that their methods are unreliable because the methods (if there are any) can be always flexed to avoid producing embarrassingly trivial errors.
Just to be clear: by strict rules I don’t mean anything with significant subjective judgement involved, like peer reviews. I rather mean things like demanding testability, mathematical proofs, logical consistency and such. Also, not much rules governing the social life of the respective community, but rather rules applied to the hypotheses.
Trouble is, there are examples of fields where the standards satisfy all this, but the work is nevertheless misleading and remote from reality.
Take the example of computer science, which I’m most familiar with. In some of its subfields, the state of the art has reached a dead end, in that any obvious path for improving things hits against some sort of exponential-time or uncomputable problem, and the possible heuristics for getting around it have already been explored to death. Breaking a new path in this situation could be done only by an extraordinary stroke of genius, if it’s possible at all.
So what people do is to propose yet another complex and sophisticated but ultimately feeble heuristic wrapped into thick layers of abstruse math, and argue that it represents an improvement of some performance measure by a few percentage points. Now, if you look at a typical paper from such an area, you’ll see that the formalism is accurate mathematically and logically, the performance evaluation is carefully measured over a set of standard benchmarks according to established guidelines, and the relevant prior work is meticulously researched and cited. You have to satisfy these strict formal standards to publish.
Trouble is, nearly all this work is worthless, and quite obviously so. From a practical engineering perspective, implementing these complex algorithms in a practical system would be a Herculean task for a minuscule gain. The hypertrophied formalism often uses numerous pages of abstruse math to express ideas that could be explained intuitively and informally in a few simple sentences to someone knowledgeable in the field—and in turn would be immediately and correctly dismissed as impractical. Even the measured performance improvements are rarely evaluated truly ceteris paribus and in ways that reveal all the strengths and weaknesses of the approach. It’s simply impossible to devise a formal standard that wold ensure that reliably—these things are possible to figure out only with additional experimentation or with a practical engineering hunch.
Except perhaps in the purest mathematics, no formal standard can function well in practice if legions of extraordinarily smart people have the incentive to get around it. And if there are no easy paths to quality work, the “publish or perish” principle makes it impossible to compete and survive unless one exerts every effort to game the system.
That’s right, and I don’t disagree. Formal standards are not a panacea, never. But, do you suppose, in cases you describe, things would go better without those formal standards?
I am still not sure if we mean exactly the same thing, when talking about formal rules. Take the example of pure mathematics, which you have already mentioned. Surely, abstruse formalist descriptions of practically uninteresting and maybe trivial problems appear there too, now and then. And revolutionary breakthroughs perhaps more often result from intuitive insights of geniuses than from dilligent rigorous formal work. Much papers, in all fields, can be made more readable, accessible, and effective in dissemination of new results by shedding the lofty jargon of scientific publications. But mathematicians certainly wouldn’t do better if they got rid of mathematical proofs.
I do not suggest that all ideas in respectable fields of science should be propagated in form of publications checked against lists of formal requirements: citation index, proofs of all logical statements, p-values below 0.01, certificates of double-blindedness. Not in the slightest. Conjectures, analogies, illustrations, whatever enhances understanding is welcome. I only want a possibility to apply the formal criteria. If a conjecture is published, and it turns out interesting, there should be an ultimate method to test whether it is true. If there is an agreed method to test the results objectively, people aren’t free to publish whatever they want and expect to never be proven wrong.
If you compare the results of computer science to postmodern philosophy, you may see my point. In CS most results may be useless and incomprehensible. In postmodern philosophy, which is essentially without formal rules, all results are useless and incomprehensible, and as a bonus, meaningless or false.
I agree about the awful state of fields that don’t have any formal rules at all. However, I’m not concerned about these so much because, to put it bluntly, nobody important takes them seriously. What is in my opinion a much greater problem are fields that appear to have all the trappings of valid science and scholarship, but it’s in fact hard for an outsider to evaluate whether and to what extent they’re actually cargo-cult science. This especially because the results of some such fields (most notably economics) are used as basis for real-world decision-making with far-reaching consequences.
Regarding the role of formalism, mathematics is unique in that the internal correctness of the formalism is enough to establish the validity of the results. Sure, they may be more or less interesting, but if the formalism is valid, then it’s valid math, period.
In contrast, in areas that make claims about the real world, the important thing is not just the validity of the formalism, but also how well it corresponds to reality. Work based on a logically impeccable formalism can still be misleading garbage if the formalism is distant enough from reality. This is where the really hard problem is. The requirements about the validity of the formalism are easily enforced, since we know how to reduce those to a basically algorithmic procedure. What is really hard is ensuring that the formalism provides an accurate enough description of reality—and given an incentive to do so, smart people will inevitably figure out ways to stretch and evade this requirement, unless there is a sound common-sense judgment standing in the way.
Further, more rigorous formalism isn’t always better. It’s a trade-off. More effort put into greater formal rigor—including both the author’s effort to formulate it, and the reader’s effort to understand it—means less resources for other ways of improving the work. Physicists, for example, normally just assume that the functions are well-behaved enough in a way that would be unacceptable in mathematics, and they’re justified in doing so. In more practical technical fields like computer science, what matters is whether the results are useful in practice, and formal rigor is useful if it helps avoid confusion about complicated things, but worse than useless if applied to things where intuitive understanding is good enough to get the job done.
The crucial lesson, like in so many other things, is that whenever one deals with the real world, formalism cannot substitute for common sense. It may be tremendously helpful and enable otherwise impossible breakthroughs, but without an ultimate sanity check based on sheer common sense, any attempt at science is a house built on sand.
I don’t think we have a real disagreement. I haven’t said that more rigorous formalism is always better, quite the contrary. I was writing about objective methods of looking at the results. Physicists can ignore mathematical rigor because they have experimental tests which finally decide whether their theory is worth attention. Computer scientists can finally write down their algorithm and look whether it works. These are objective rules which validate the results.
Whether the rules are sensible or not is decided by common sense. My point is that it is easier to decide that about the rules of the whole field than about individual theories, and that’s why objective rules are useful.
Of course, saying “common sense” does in fact mean that we don’t know how did we decide, and doesn’t specify the judgement too much. One man’s common sense may be other man’s insanity.
Oh yes, I didn’t mean to imply that you disagreed with everything I wrote in the above comment. My intent was to give a self-contained summary of my position on the issue, and the specific points I raised were not necessarily in response to your claims.
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.
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.
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.
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.
Combined with
The vast majority of current mathematical theories … could be empirically tested by observing the behavior of a computing device programmed to do brute-force proofs within those systems.
is an intuition I agree with. As long as “empirically testable” includes “observing the behavior of a computing device”, that seems to be exactly where the “surprising utility” comes from.
Then you write
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
which is somewhat worrying, because the few papers I scrolled through seemed computationally implementable, assuming correctness (which is a mighty high standard for a preprint server in any case).
My experience has been that many mathematicians write articles with the assumption that if the reader can read the article, they also know what it’s “good for” in the engineering sense (which seems a somewhat delusional assumption to make). I think that if you read grant proposals, you’d get a better sense of the subfields that are “coupled with reality”—though again, only to a first approximation. Politics is the mind-killer almost everywhere.
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.
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.
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.
The danger I see is mathematicians endorsing mathematics research because it serves explicitly mathematical goals....I’d like us to decide to attack [the Rieman Hypothesis] 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.
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.
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.
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.
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.
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.
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.
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:
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.
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 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
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.
Again, your claim that I think the relationship between modern mathematics in the world is too indirect is simply putting words in my mouth.
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?
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?
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.
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
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.
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,
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.)
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.
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?)
(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.
No, the status quo is heavy subsidization. I have an essay on how there is too much art & fiction (http://www.gwern.net/Culture%20is%20not%20about%20esthetics.html) and one of my points is that the arts are heavily subsidized both directly and indirectly, which contributes to the over-supply.
I don’t buy the claim that copyright law amounts to a subsidy. Copyright law is an enforced monopoly, which is not the same thing.
Of course, you’re not focused on the specific works (which is what copyright grants a monopoly on) but on the industry as a whole. So perhaps monopoly on specifics amounts to a subsidy on generalities? But copyright law doesn’t have the same effects as a subsidy overall. A subsidy should lead to a higher quantity at a lower price, but copyright law surely leads to a higher price.
(Defenders of copyright usually argue that it also leads to a higher quantity, and I entirely agree with your scepticism that this would actually be a good thing. It’s obvious to me that copyright law is bad through and through, regardless of its effect on quantity. Still, anti-copyright activists have a valid point that it’s not obvious that copyright actually increases quantity either, since it makes distribution and derivative works harder.)
Nothing. If people wish to write as their recreation, that’s fine. I’m not arguing that gardens be banned either. The suggestions in my linked essay are that the subsidies be dropped and possibly a Pigovian tax imposed on commercial fiction.
(Am I being unreasonable in expecting people to read the essay which is all about how much fiction/art is produced, its value, and what we should do about it? It seems to me that much of the math discussion is isomorphic.)
I admit I read your essay very quickly, and skipped the footnotes.
I don’t think fiction is very heavily subsidized compared to the amount that’s produced. Copyright enforcement is the only thing you list that I think matters, and I believe we’d be drowning in fiction even without it.
The more I think about it, the harder I find justifying any subsidy.
No disagreement.
But higher production isn’t always good; production can be misguided or wasted
But if something is wasted what matters is not that there was too much of it, but that an opportunity to produce something else was lost. You’re focused on there being too much literature—when the relevant complaint is that there is too little of other things. An unread book does no harm. A year spent writing the book when something else could have been done with that year represents a lost opportunity. Maybe this is the focus of your concern, but it does not seem to be.
Suppose we all have 100 apples. Our lives do not revolve around apples, though we like them well enough. But still, 100 is too many
Indeed, most of these apples will be wasted if not sold, and this represents an opportunity lost to produce something else with the soil, but I think the analogy to novels is weak, as I will argue.
Now, can we apply this analogy? I don’t have 100 apples, but perhaps I have − 100 novels.
There are various ways in which 100 novels is not like 100 apples. For one thing, 100 novels is like 100 varieties of apple. You may prefer one variety of apple; your neighbor may prefer another. The novel Twilight, for example, appeals to many people and does not appeal to me. There are, meanwhile, novels that appeal to me but would probably not appeal to a typical fan of Twilight.
For another, creativity requires variation as well as selection. The vast majority of the variants are not selected, but that does not mean that they are wasted, because a reduction in variation reduces the raw material on which selection can act. In particular, in order that one brilliant writer be found, many must make the attempt. Reduce the number attempting, and you may well reduce the number of great writers found.
In short, if 1000 novels are written and only one is widely read and preserved, that does not necessarily mean the other 999 were wasted. They made up the variation that selection acted upon.
If the industry had imploded before Mistborn was published, I would have read Long Sun instead.
Sure, you can always manufacture hypothetical scenarios, and cherrypick real ones, in which the work of selection is already done, in which the superior variant and only the superior variant is produced in the first place. But that’s simply fantasy. In reality, variation is needed as raw material for selection.
The connection to other aspects of modern life and akrasia is apparent: there’s a Gresham’s Law whereby cheap yet unsatisfying works will push out more satisfying but more demanding entertainment. Humans suffer from hyperbolic discounting; we may know that in the long run, Mistborn will be forgotten when Long Sun is remembered, and that once we get started, we will enjoy it more—yet when the moment comes to choose, we prefer the choice of immediate pleasure.
I believe you have misapplied both Gresham’s law and hyperbolic discounting. For instance, there’s an important reason that Gresham’s law applies to money, and novels aren’t money.
Any field over a century old has built up a stock of masterpieces that could fill a lifetime.
This could be said at almost any point in history. You seem to be using it to imply that new works are unnecessary. But it would be equally good as an argument that Beethoven need not bother writing his masterpieces, since, after all, Bach had already written enough to fill a lifetime. But anyone who has listened to Beethoven knows that, even though Bach had already written enough to fill a lifetime, we are nevertheless enriched for having Beethoven, even though Beethoven necessarily displaces Bach to some extent.
Generalizing: even though we are already filled to capacity with art, literature, and music to spend all our lives on, we are nevertheless further enriched by new creation.
Society ought to discourage economically inefficient activities.
Yes, but efficiency is relative to what people want, which is difficult to discover except by observing their choices. And we see that they overwhelmingly choose contemporary fiction. My theory is that contemporary fiction really and truly does give the audience that chooses it greater satisfaction than most great old fiction, even though future generations will find most of it wanting. See for example that often Shakespeare will be updated in certain respects (such as setting—West Side Story, Ran, Forbidden Planet) for a new audience, and Shakespeare himself updated older stories for his own audience. For another example, the movie Clueless is an update of the Austen novel Emma. The novel Twilight takes place in contemporary America; in a hundred years it will be hopelessly out of date, but for much of its audience, Dracula by Bram Stoker, classic that it is, is not contemporary enough.
Because of this, there is a never-ending demand for contemporary fiction and for updates of old fiction, and this will keep writers in business indefinitely. You may judge this wrong by certain standards which you offer, but efficiency depends on what people want, and this is what they want. You don’t get to make the concept of efficiency mean something different.
Consumers of new art would be equally satisfied by existing art.
Evidently not. I see you argue against this, but I find your argument completely unpersuasive. What we have in front of us as evidence is consumer behavior. We see the choices people make. Against this you present hypotheticals and a couple of quotes from people. For example, someone whose grandson happens to be into old music at the moment.
Meanwhile we see that updates of classics, such as Clueless and West Side Story, do very well in the market. This validates the choice that the movie producers made, which choice is based in part on the assumption that there is a significant audience for an update—i.e., people who would in fact not be equally satisfied by the originals without update.
I wasn’t talking about subsidization, I was talking about taxation. The logic of the discussion was as follows: (1) Johnicolas said there should be an art tax; (2) I said “how would you do that?”; (3) wedrifid said “subject art to standard sales taxes”; (4) I pointed out that art already is subject to standard sales taxes—so far as I know it isn’t specifically exempt; hence wedrifid’s response doesn’t work as an answer.
The part of wedrifid’s comment that I quoted defined the scope of my remark, which you misunderstood.
Any meaningful discussion of taxation focuses on the net, not on arbitrary subdivisions and labels. If art were taxed at 50% sales tax but also came with a tax deduction of 100%, I would feel real physical pain to see someone argue ‘oh, but we are discouraging and taxing heavily artwork! Just look at that 50%!’
Which is why I bring up the subsidies. If art is being hugely subsidized, then just being taxed like everything else (in your impoverished sense) still leads to art being cheaper than it should.
That may or may not be a fair point to make, but in that case your comment should have begun with “Yes, but...” instead of “No...”.
On the merits, I disagree on every point: that there is too much art, that current art subsidies are “heavy”, and that art subsidies necessarily cancel out sales taxes for the purpose of interpreting government policy (which may simply be incoherent and non-uniform).
(I had let the parent be, not wanting to emphasise disagreement but the follow up prompts a reply.)
The proposal, as I understood it, was to have additional taxes specific to art.
I do not share your interpretation. The relevant quote is:
Similarly, I would tax visual art rather than subsidize it.
… A general sentiment regarding where he would place a slider on a simplistic one dimensional scale of financial incentive vs disincentive. It is definitely not a proposal for specific intervention in any particular jurisdiction.
Come to think of it your status quo claim is way off. The following is definitely not the status quo:
Don’t subsidise the field at all via grants or via university tax breaks.
Incidentally, investment in culture and education—even with respect to visual arts—is something I approve of. I just note that your questioning was rather disingenuous:
By paying inspectors to come to people’s houses to check whether they’ve drawn any pictures that day?
Taxation and subsidisation are well understood. This objection is silly (your other soldiers are better).
A general sentiment regarding where he would place a slider on a simplistic one dimensional scale of financial incentive vs disincentive.
Right; and he was wanting to place it to the right of zero (on the “disincentive” side) whereas you were talking about moving it from the left of zero to zero. This is the distinction I was pointing out.
Come to think of it your status quo claim is way off. The following is definitely not the status quo:
Don’t subsidise the field at all via grants or via university tax breaks.
See the very comment you linked, which contains a reminder that my “status quo” remark did not apply to that aspect.
By paying inspectors to come to people’s houses to check whether they’ve drawn any pictures that day?
Taxation and subsidisation are well understood. This objection is silly (your other soldiers are better).
The main point of that was to emphasize transaction costs of taxation. You will note that I immediately followed it by a more “reasonable” suggestion so as to forestall accusations of being overly rhetorical.
The obvious thing would be some sort of excise tax, like the “sin taxes” on alcohol and the like. That might extend to art supplies, but not necessarily; just charge it on the sale of the final product (if you sell it).
Not that I’m for this; otherwise I agree with your reaction to the proposal.
Allow me to clarify: Tax art rather than subsidize it, at a roughly comparable rate to other industries. I don’t think it matters much whether it’s exactly the same, slightly higher, or slightly lower.
One of the techniques of rational argumentation is called the “Principle of Charity”. When reading and interpreting what someone said, you should infer missing details in order to make their argument the strongest argument possible. In a lw-centric example, Eliezer’s idea of “The least convenient possible world” is the principle of charity, specialized to interpreting hypothetical situations.
I don’t understand the point of your paragraph explaining the principle of charity as if I might never have heard of it. If the implication is that I was being uncharitable to you by not interpreting “tax X” to mean “fail to exempt X from the default taxes”, I strongly disagree. When someone says, for example, that cigarettes should be taxed, they don’t just mean that the same sales taxes that apply to everything else should also apply to cigarettes (as if the default were to exempt cigarettes). Rather, they mean that there ought to be a specific tax on cigarettes in addition to whatever taxes would ordinarily apply, in order to discourage consumption of cigarettes. (This is known as a “sin tax”.)
In the context of the above discussion, the only reasonable interpretation of your remark was that you favored a sin tax on art, analogous to existing sin taxes (in some jurisdictions) on “harmful” products such as alcohol, cigarettes, and the like. If you hadn’t meant this, and simply meant that art should be treated like any other product, you would have simply said “I wouldn’t subsidize art”; as opposed to saying “I would tax art rather than subsidize it”, i.e. “not only would I not subsidize art, I would actually tax it”.
In case this needs still further clarification, the reason this is the only reasonable interpretation is that (so far as I know) art is not exempted from existing taxes. If it were, then the interpretation of “tax art” to mean “subject art to the same taxes as everything else” (i.e. “remove the exemption”) might make sense. As it is, however, “tax art” is highly misleading if what you mean is merely “remove subsidies” (where “subsidies” mean things like government grants, university salaries, etc, rather than tax-exemptions, which, again, don’t currently exist).
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”.
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.
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.
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.
Yes, one that fits in the class I described thusly:
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!”
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.
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’.)
Even in mathematics, you can find contrarian opinions that much of the field is meaningless. What we have is (or at least seems to be) logically proved from the basis of certain assumptions, but we could as easily have picked very different assumptions and proved different theorems instead. There is a prevailing opinion that certain assumptions (the mainstream foundations of mathematics) are correct or at least useful, but correctness ultimately reduces to an aesthetic judgement, and usefulness is known to be exaggerated.
Even better demand that there be strict rules in the discipline, which the research must obey—be it logical provability, empirical testability or whatever else. It is still possible to make up unreasonable rules, but production of bullshit is a lot easier without rules. Which is the case of deconstructionism and related fields.
prase:
Strict formal rules are a two-edged sword. If well designed, they indeed serve as a powerful barrier against nonsense. However, they can also be perverted, with extremely bad results.
In many disciplines that have been affected by the malaises discussed in this thread, what happens is that a perverse formal system develops, which then serves as a template for producing impressive-looking bullshit work. This sometimes leads to the very heights of ass-covering irresponsibility, since everyone involved—authors, editors, reviewers, grant committees… -- can hide behind the fact that the work satisfies all the highest professional expert standards if questioned about it. At worst, these perverse formal standards can also serve as a barrier against actual quality work that doesn’t conform to their template.
Just to be clear: by strict rules I don’t mean anything with significant subjective judgement involved, like peer reviews. I rather mean things like demanding testability, mathematical proofs, logical consistency and such. Also, not much rules governing the social life of the respective community, but rather rules applied to the hypotheses.
Also, I haven’t said that rules are sufficient. One can still publish trivial theories which nobody is interested to test, mathematical proofs of obscure unimportant theorems or logically consistent tautologies. But at least the rules remove arbitrariness and make it possible to objectively assess quality and to decide whether a hypothesis is good or bad, according to standards of the discipline.
The discipline’s standard of good hypothesis may not universally correspond to a true hypothesis, but I suspect that if the standards of the discipline are strict enough, either the correspondence is there, or it is easily visible that the discipline is based on wrong premises, because it endorses some easily identifiable falsehoods. (It would be too big a coincidence if a formal system regularly produced false statements, but no trivially false statements.)
On the other hand, when the rules aren’t enough formal, the discipline still makes complex false claims, but nobody can clearly demonstrate that their methods are unreliable because the methods (if there are any) can be always flexed to avoid producing embarrassingly trivial errors.
prase:
Trouble is, there are examples of fields where the standards satisfy all this, but the work is nevertheless misleading and remote from reality.
Take the example of computer science, which I’m most familiar with. In some of its subfields, the state of the art has reached a dead end, in that any obvious path for improving things hits against some sort of exponential-time or uncomputable problem, and the possible heuristics for getting around it have already been explored to death. Breaking a new path in this situation could be done only by an extraordinary stroke of genius, if it’s possible at all.
So what people do is to propose yet another complex and sophisticated but ultimately feeble heuristic wrapped into thick layers of abstruse math, and argue that it represents an improvement of some performance measure by a few percentage points. Now, if you look at a typical paper from such an area, you’ll see that the formalism is accurate mathematically and logically, the performance evaluation is carefully measured over a set of standard benchmarks according to established guidelines, and the relevant prior work is meticulously researched and cited. You have to satisfy these strict formal standards to publish.
Trouble is, nearly all this work is worthless, and quite obviously so. From a practical engineering perspective, implementing these complex algorithms in a practical system would be a Herculean task for a minuscule gain. The hypertrophied formalism often uses numerous pages of abstruse math to express ideas that could be explained intuitively and informally in a few simple sentences to someone knowledgeable in the field—and in turn would be immediately and correctly dismissed as impractical. Even the measured performance improvements are rarely evaluated truly ceteris paribus and in ways that reveal all the strengths and weaknesses of the approach. It’s simply impossible to devise a formal standard that wold ensure that reliably—these things are possible to figure out only with additional experimentation or with a practical engineering hunch.
Except perhaps in the purest mathematics, no formal standard can function well in practice if legions of extraordinarily smart people have the incentive to get around it. And if there are no easy paths to quality work, the “publish or perish” principle makes it impossible to compete and survive unless one exerts every effort to game the system.
That’s right, and I don’t disagree. Formal standards are not a panacea, never. But, do you suppose, in cases you describe, things would go better without those formal standards?
I am still not sure if we mean exactly the same thing, when talking about formal rules. Take the example of pure mathematics, which you have already mentioned. Surely, abstruse formalist descriptions of practically uninteresting and maybe trivial problems appear there too, now and then. And revolutionary breakthroughs perhaps more often result from intuitive insights of geniuses than from dilligent rigorous formal work. Much papers, in all fields, can be made more readable, accessible, and effective in dissemination of new results by shedding the lofty jargon of scientific publications. But mathematicians certainly wouldn’t do better if they got rid of mathematical proofs.
I do not suggest that all ideas in respectable fields of science should be propagated in form of publications checked against lists of formal requirements: citation index, proofs of all logical statements, p-values below 0.01, certificates of double-blindedness. Not in the slightest. Conjectures, analogies, illustrations, whatever enhances understanding is welcome. I only want a possibility to apply the formal criteria. If a conjecture is published, and it turns out interesting, there should be an ultimate method to test whether it is true. If there is an agreed method to test the results objectively, people aren’t free to publish whatever they want and expect to never be proven wrong.
If you compare the results of computer science to postmodern philosophy, you may see my point. In CS most results may be useless and incomprehensible. In postmodern philosophy, which is essentially without formal rules, all results are useless and incomprehensible, and as a bonus, meaningless or false.
I agree about the awful state of fields that don’t have any formal rules at all. However, I’m not concerned about these so much because, to put it bluntly, nobody important takes them seriously. What is in my opinion a much greater problem are fields that appear to have all the trappings of valid science and scholarship, but it’s in fact hard for an outsider to evaluate whether and to what extent they’re actually cargo-cult science. This especially because the results of some such fields (most notably economics) are used as basis for real-world decision-making with far-reaching consequences.
Regarding the role of formalism, mathematics is unique in that the internal correctness of the formalism is enough to establish the validity of the results. Sure, they may be more or less interesting, but if the formalism is valid, then it’s valid math, period.
In contrast, in areas that make claims about the real world, the important thing is not just the validity of the formalism, but also how well it corresponds to reality. Work based on a logically impeccable formalism can still be misleading garbage if the formalism is distant enough from reality. This is where the really hard problem is. The requirements about the validity of the formalism are easily enforced, since we know how to reduce those to a basically algorithmic procedure. What is really hard is ensuring that the formalism provides an accurate enough description of reality—and given an incentive to do so, smart people will inevitably figure out ways to stretch and evade this requirement, unless there is a sound common-sense judgment standing in the way.
Further, more rigorous formalism isn’t always better. It’s a trade-off. More effort put into greater formal rigor—including both the author’s effort to formulate it, and the reader’s effort to understand it—means less resources for other ways of improving the work. Physicists, for example, normally just assume that the functions are well-behaved enough in a way that would be unacceptable in mathematics, and they’re justified in doing so. In more practical technical fields like computer science, what matters is whether the results are useful in practice, and formal rigor is useful if it helps avoid confusion about complicated things, but worse than useless if applied to things where intuitive understanding is good enough to get the job done.
The crucial lesson, like in so many other things, is that whenever one deals with the real world, formalism cannot substitute for common sense. It may be tremendously helpful and enable otherwise impossible breakthroughs, but without an ultimate sanity check based on sheer common sense, any attempt at science is a house built on sand.
I don’t think we have a real disagreement. I haven’t said that more rigorous formalism is always better, quite the contrary. I was writing about objective methods of looking at the results. Physicists can ignore mathematical rigor because they have experimental tests which finally decide whether their theory is worth attention. Computer scientists can finally write down their algorithm and look whether it works. These are objective rules which validate the results.
Whether the rules are sensible or not is decided by common sense. My point is that it is easier to decide that about the rules of the whole field than about individual theories, and that’s why objective rules are useful.
Of course, saying “common sense” does in fact mean that we don’t know how did we decide, and doesn’t specify the judgement too much. One man’s common sense may be other man’s insanity.
Oh yes, I didn’t mean to imply that you disagreed with everything I wrote in the above comment. My intent was to give a self-contained summary of my position on the issue, and the specific points I raised were not necessarily in response to your claims.
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.
Combined with
is an intuition I agree with. As long as “empirically testable” includes “observing the behavior of a computing device”, that seems to be exactly where the “surprising utility” comes from.
Then you write
which is somewhat worrying, because the few papers I scrolled through seemed computationally implementable, assuming correctness (which is a mighty high standard for a preprint server in any case).
My experience has been that many mathematicians write articles with the assumption that if the reader can read the article, they also know what it’s “good for” in the engineering sense (which seems a somewhat delusional assumption to make). I think that if you read grant proposals, you’d get a better sense of the subfields that are “coupled with reality”—though again, only to a first approximation. Politics is the mind-killer almost everywhere.
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:
http://www-personal.umich.edu/~jlawler/von.neumann.html
http://www.math.rutgers.edu/~zeilberg/Opinion104.html
http://bentilly.blogspot.com/2009/11/why-i-left-math.html
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.
That’s the status quo. The proposal, as I understood it, was to have additional taxes specific to art.
No, the status quo is heavy subsidization. I have an essay on how there is too much art & fiction (http://www.gwern.net/Culture%20is%20not%20about%20esthetics.html) and one of my points is that the arts are heavily subsidized both directly and indirectly, which contributes to the over-supply.
I don’t buy the claim that copyright law amounts to a subsidy. Copyright law is an enforced monopoly, which is not the same thing.
Of course, you’re not focused on the specific works (which is what copyright grants a monopoly on) but on the industry as a whole. So perhaps monopoly on specifics amounts to a subsidy on generalities? But copyright law doesn’t have the same effects as a subsidy overall. A subsidy should lead to a higher quantity at a lower price, but copyright law surely leads to a higher price.
(Defenders of copyright usually argue that it also leads to a higher quantity, and I entirely agree with your scepticism that this would actually be a good thing. It’s obvious to me that copyright law is bad through and through, regardless of its effect on quantity. Still, anti-copyright activists have a valid point that it’s not obvious that copyright actually increases quantity either, since it makes distribution and derivative works harder.)
I think the major way fiction is subsidized is people producing fiction in spite of it not being at all lucrative for most of them.
What are you planning on doing about fan fiction?
Nothing. If people wish to write as their recreation, that’s fine. I’m not arguing that gardens be banned either. The suggestions in my linked essay are that the subsidies be dropped and possibly a Pigovian tax imposed on commercial fiction.
(Am I being unreasonable in expecting people to read the essay which is all about how much fiction/art is produced, its value, and what we should do about it? It seems to me that much of the math discussion is isomorphic.)
I admit I read your essay very quickly, and skipped the footnotes.
I don’t think fiction is very heavily subsidized compared to the amount that’s produced. Copyright enforcement is the only thing you list that I think matters, and I believe we’d be drowning in fiction even without it.
Some responses.
No disagreement.
But if something is wasted what matters is not that there was too much of it, but that an opportunity to produce something else was lost. You’re focused on there being too much literature—when the relevant complaint is that there is too little of other things. An unread book does no harm. A year spent writing the book when something else could have been done with that year represents a lost opportunity. Maybe this is the focus of your concern, but it does not seem to be.
Indeed, most of these apples will be wasted if not sold, and this represents an opportunity lost to produce something else with the soil, but I think the analogy to novels is weak, as I will argue.
There are various ways in which 100 novels is not like 100 apples. For one thing, 100 novels is like 100 varieties of apple. You may prefer one variety of apple; your neighbor may prefer another. The novel Twilight, for example, appeals to many people and does not appeal to me. There are, meanwhile, novels that appeal to me but would probably not appeal to a typical fan of Twilight.
For another, creativity requires variation as well as selection. The vast majority of the variants are not selected, but that does not mean that they are wasted, because a reduction in variation reduces the raw material on which selection can act. In particular, in order that one brilliant writer be found, many must make the attempt. Reduce the number attempting, and you may well reduce the number of great writers found.
In short, if 1000 novels are written and only one is widely read and preserved, that does not necessarily mean the other 999 were wasted. They made up the variation that selection acted upon.
Sure, you can always manufacture hypothetical scenarios, and cherrypick real ones, in which the work of selection is already done, in which the superior variant and only the superior variant is produced in the first place. But that’s simply fantasy. In reality, variation is needed as raw material for selection.
I believe you have misapplied both Gresham’s law and hyperbolic discounting. For instance, there’s an important reason that Gresham’s law applies to money, and novels aren’t money.
This could be said at almost any point in history. You seem to be using it to imply that new works are unnecessary. But it would be equally good as an argument that Beethoven need not bother writing his masterpieces, since, after all, Bach had already written enough to fill a lifetime. But anyone who has listened to Beethoven knows that, even though Bach had already written enough to fill a lifetime, we are nevertheless enriched for having Beethoven, even though Beethoven necessarily displaces Bach to some extent.
Generalizing: even though we are already filled to capacity with art, literature, and music to spend all our lives on, we are nevertheless further enriched by new creation.
Yes, but efficiency is relative to what people want, which is difficult to discover except by observing their choices. And we see that they overwhelmingly choose contemporary fiction. My theory is that contemporary fiction really and truly does give the audience that chooses it greater satisfaction than most great old fiction, even though future generations will find most of it wanting. See for example that often Shakespeare will be updated in certain respects (such as setting—West Side Story, Ran, Forbidden Planet) for a new audience, and Shakespeare himself updated older stories for his own audience. For another example, the movie Clueless is an update of the Austen novel Emma. The novel Twilight takes place in contemporary America; in a hundred years it will be hopelessly out of date, but for much of its audience, Dracula by Bram Stoker, classic that it is, is not contemporary enough.
Because of this, there is a never-ending demand for contemporary fiction and for updates of old fiction, and this will keep writers in business indefinitely. You may judge this wrong by certain standards which you offer, but efficiency depends on what people want, and this is what they want. You don’t get to make the concept of efficiency mean something different.
Evidently not. I see you argue against this, but I find your argument completely unpersuasive. What we have in front of us as evidence is consumer behavior. We see the choices people make. Against this you present hypotheticals and a couple of quotes from people. For example, someone whose grandson happens to be into old music at the moment.
Meanwhile we see that updates of classics, such as Clueless and West Side Story, do very well in the market. This validates the choice that the movie producers made, which choice is based in part on the assumption that there is a significant audience for an update—i.e., people who would in fact not be equally satisfied by the originals without update.
I liked the linked essay. I suspect an even stronger case could be made that there’s too much supply of news.
I wasn’t talking about subsidization, I was talking about taxation. The logic of the discussion was as follows: (1) Johnicolas said there should be an art tax; (2) I said “how would you do that?”; (3) wedrifid said “subject art to standard sales taxes”; (4) I pointed out that art already is subject to standard sales taxes—so far as I know it isn’t specifically exempt; hence wedrifid’s response doesn’t work as an answer.
The part of wedrifid’s comment that I quoted defined the scope of my remark, which you misunderstood.
Any meaningful discussion of taxation focuses on the net, not on arbitrary subdivisions and labels. If art were taxed at 50% sales tax but also came with a tax deduction of 100%, I would feel real physical pain to see someone argue ‘oh, but we are discouraging and taxing heavily artwork! Just look at that 50%!’
Which is why I bring up the subsidies. If art is being hugely subsidized, then just being taxed like everything else (in your impoverished sense) still leads to art being cheaper than it should.
That may or may not be a fair point to make, but in that case your comment should have begun with “Yes, but...” instead of “No...”.
On the merits, I disagree on every point: that there is too much art, that current art subsidies are “heavy”, and that art subsidies necessarily cancel out sales taxes for the purpose of interpreting government policy (which may simply be incoherent and non-uniform).
(I had let the parent be, not wanting to emphasise disagreement but the follow up prompts a reply.)
I do not share your interpretation. The relevant quote is:
… A general sentiment regarding where he would place a slider on a simplistic one dimensional scale of financial incentive vs disincentive. It is definitely not a proposal for specific intervention in any particular jurisdiction.
Come to think of it your status quo claim is way off. The following is definitely not the status quo:
Incidentally, investment in culture and education—even with respect to visual arts—is something I approve of. I just note that your questioning was rather disingenuous:
Taxation and subsidisation are well understood. This objection is silly (your other soldiers are better).
Right; and he was wanting to place it to the right of zero (on the “disincentive” side) whereas you were talking about moving it from the left of zero to zero. This is the distinction I was pointing out.
See the very comment you linked, which contains a reminder that my “status quo” remark did not apply to that aspect.
The main point of that was to emphasize transaction costs of taxation. You will note that I immediately followed it by a more “reasonable” suggestion so as to forestall accusations of being overly rhetorical.
The obvious thing would be some sort of excise tax, like the “sin taxes” on alcohol and the like. That might extend to art supplies, but not necessarily; just charge it on the sale of the final product (if you sell it).
Not that I’m for this; otherwise I agree with your reaction to the proposal.
Allow me to clarify: Tax art rather than subsidize it, at a roughly comparable rate to other industries. I don’t think it matters much whether it’s exactly the same, slightly higher, or slightly lower.
One of the techniques of rational argumentation is called the “Principle of Charity”. When reading and interpreting what someone said, you should infer missing details in order to make their argument the strongest argument possible. In a lw-centric example, Eliezer’s idea of “The least convenient possible world” is the principle of charity, specialized to interpreting hypothetical situations.
I don’t understand the point of your paragraph explaining the principle of charity as if I might never have heard of it. If the implication is that I was being uncharitable to you by not interpreting “tax X” to mean “fail to exempt X from the default taxes”, I strongly disagree. When someone says, for example, that cigarettes should be taxed, they don’t just mean that the same sales taxes that apply to everything else should also apply to cigarettes (as if the default were to exempt cigarettes). Rather, they mean that there ought to be a specific tax on cigarettes in addition to whatever taxes would ordinarily apply, in order to discourage consumption of cigarettes. (This is known as a “sin tax”.)
In the context of the above discussion, the only reasonable interpretation of your remark was that you favored a sin tax on art, analogous to existing sin taxes (in some jurisdictions) on “harmful” products such as alcohol, cigarettes, and the like. If you hadn’t meant this, and simply meant that art should be treated like any other product, you would have simply said “I wouldn’t subsidize art”; as opposed to saying “I would tax art rather than subsidize it”, i.e. “not only would I not subsidize art, I would actually tax it”.
In case this needs still further clarification, the reason this is the only reasonable interpretation is that (so far as I know) art is not exempted from existing taxes. If it were, then the interpretation of “tax art” to mean “subject art to the same taxes as everything else” (i.e. “remove the exemption”) might make sense. As it is, however, “tax art” is highly misleading if what you mean is merely “remove subsidies” (where “subsidies” mean things like government grants, university salaries, etc, rather than tax-exemptions, which, again, don’t currently exist).
Non-functional art quilting
Found while looking for the first link, and included for pretty
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.