Is it your contention that modern musicians write Clasical minuets and Baroque fugues which are in some cases better than the best of the older works that are still listened to, but that no-one cares because much of the value of those works is in their role in a canon?
I myself would guess that none of the works produced over the past hundred years would be judged by the majority of an impartial audience to be significantly more compelling than (for example) Bach’s Chaconne.
It should really be pretty easy to do a controlled experiment with a naive population to see how common it is for modern artists to be able to impress an audience as much as their 18th and 19th century precursors did.
I’m seriously interested in someone performing some experiments on this subject.
Same here.
I guess that my major reason for holding the contrary position was largely because modern musicians and composers, more than painters and authors, are the results of university education and I fairly strongly suspect university education of destorying artistic ability and distracting artists with intellectual games that simply lack the merits of the fields that the academic subjects are derived from. I suspect this in math as much as in music, and I think Von Neumann agreed with me, as this quote suggests.
I’m impressed that you’re familiar with the Von Neumann quote (which is sadly little known in the mathematical community but which my friend Laurens is fond of); but on the face of things it doesn’t seem to directly support your paragraph above. Explain further if you’d like?
Several points here:
My impression is that there are issues of bad social/cultural institutions destroying artistic ability outside of academia. I have some friends artistically genuine who have spent some time as painters and become disillusioned with the signaling games and hypocrisy present within the communities of painters that they’ve come across. Note that fledging painters and authors face greater financial pressures than academics and that this can lead to perverse incentives (to appeal to the lowest common denominator or to current fashions for greater marketability). See Minhyong Kim’s comment here.
The absence of new empirical sources for mathematics seems to me more a consequence of the stagnation of theoretical physics than the social structure of the mathematical community.
In my own view insufficient emphasis on exposition has played a significant role in whatever stagnation has occurred within the mathematical community since, e.g. the 1800′s. The barrier to entry has gotten progressively higher as mathematics has developed and in such a setting, in absence of strong efforts to to cast background material in an accessible and readily digestible form, the pressures toward specialization and fragmentation get progressively stronger. There aren’t career-based incentives for expository work so mathematicians who are interested in exposition either conform to the research-based publish or perish norms or leave.
I’d be happy to compile an annotated list of relatively accessible survey papers if you’d be interested and find it useful for getting a sense for the state of some of contemporary mathematical research.
The absence of new empirical sources for mathematics seems to me more a consequence of the stagnation of theoretical physics than the social structure of the mathematical community.
From my perspective in applied math things are exactly opposite. We’re practically drowning in empirical sources begging for better numerics. There is hardly a scientific or engineering field today that couldn’t be revolutionized by a breakthrough in a relevant area currently under study in the applied math community. The problem isn’t lack of necessity, inspiration, or motivation, the problem is the problems are damn hard, and I’m far from convinced that we attract the best minds.
Even with very clear, objective goals, progress can still be painfully slow!
In my own view insufficient emphasis on exposition has played a significant role in whatever stagnation has occurred within the mathematical community since, e.g. the 1800′s.
I very much agree. Incentives are out of line for most areas of science, but I think the fallout from this is especially poignant for math. I view part of our discipline’s responsibility as curating the common mathematical knowledge used by all other disciplines.
Like mathematicians, musicians who only write for experts are unmoored from empirical feedback and are thus dependent on unusually good taste if they are to do something valuable. It’s not fair to expect people who can’t evaluate their work to conclude that they have such good taste even if they are acknowledge to be very smart.
Fledging painters can paint for both the lowest common denominator AND for themselves if they want to. Academics can’t do popular work without that counting against them with other academics.
There’s lots of need for math in complexity theory and other domains. Quantum computing for instance. Really all over the place. Crypto is very popular. Probably lots of engineering examples.
I’d be happy if someone else we both know who shares this concern would review that list.
Like mathematicians, musicians who only write for experts are unmoored from empirical feedback and are thus dependent on unusually good taste if they are to do something valuable. It’s not fair to expect people who can’t evaluate their work to conclude that they have such good taste even if they are acknowledge to be very smart.
I agree. I guess the reason why I took pause is because Von Neumann’s quote does not immediately suggest that he concurs with
I fairly strongly suspect university education of destorying artistic ability and distracting artists with intellectual games that simply lack the merits of the fields that the academic subjects are derived from.
his quotation more just obliquely raises the possibility that you suggest.
Fledging painters can paint for both the lowest common denominator AND for themselves if they want to.
It’s not clear to me that this is true. Doesn’t it depend on what the lowest common denominator is? Van Gogh painted for himself and is reported to have been unable to support himself by selling his paintings. Are you suggesting that things have changed since his time? If so, how and under what evidence?
Academics can’t do popular work without that counting against them with other academics.
My observation has been that this is mostly true.
There’s lots of need for math in complexity theory and other domains. Quantum computing for instance. Really all over the place. Crypto is very popular. Probably lots of engineering examples.
(1) It should be noted that Von Neumann was in large measure an applied mathematician and so it’s natural to expect him of being biased in favor of applied topics.
(2) There is widespread agreement among most sophisticated contemporary mathematicians as to the high aesthetic value of some of the pure mathematical achievements over the past few decades. I agree that it’s difficult for an outsider to quickly ascertain that there’s something substantive going on here but I give you my word for whatever it’s worth :-).
(3) I’ve had a less pronounced positive aesthetic response to most applied math topics than to some of my favorite pure mathematical topics. I’ve found that applied topics are more ad hoc and lacking in internal coherence and a large part of what I find compelling about pure math is the high degree of internal coherence.
As I’ve said elsewhere in this thread, the danger of generalizing from one example here is very serious and I do not question to sincerity of those who are passionate about applied topics (or pure topics like graph theory and elementary analytic number theory which I personally find disappointingly ad hoc and lacking in internal coherence but which some people eagerly devote their lives to).
(4) My own interest in applied math topics comes more from the applications than from the math involved.
(5) I would differentiate between “early” applied math (e.g. of the type that Newton and Maxwell did) which was closer to pure math (on account of the fact that so little of either was developed and the fact that it was necessary to develop pure math further in order to get to the point of being able to do something useful) and modern applied math; the former is not necessarily representative of the latter.
(6) Some theoretical and mathematical physicists draw a sharp distinction between physics and applied math. I don’t have subject matter knowledge here; but have the rough impression that theoretical physics has more internal coherence than most applied mathematical topics. This is why I referenced theoretical physics in particular as a source of inspiration that seems to have dried up on account of lack of empirical feedback.
(7) The utilitarian value of the pure mathematical achievements alluded to above is questionable in light of the fact that (a) very few people are able to appreciate them and (b) they don’t have foreseeable technological applications in the near future. The second point has heightened significance in light of the fact that it seems very possible that an intelligence explosion is not far off.
I’m interested in popularizing some of the pure mathematical achievements that are regarded among elite pure mathematicians as being of great aesthetic value. Aside from the immediate enjoyment attached to enriching people’s lives; my interest in doing this is with a view toward giving people opportunities to develop heightened aesthetic sense and spreading humanistic values, in particular making a case that there are things in the world beautiful enough so that it’s worth working toward the long-term survival of the human race. I feel like I’ve seen a “promised land” of the sorts of intellectual experiences that lie beyond the boundaries of current human mind space.
I’d be happy if someone else we both know who shares this concern would review that list.
Sure, I would too. I’ll probably get around to it sometime over the next few weeks.
My one caveat with your claims here is that Van Gogh was severely insane, which probably impaired his ability to support himself quite a bit.
Also, how likely does it seem to you that applied math lacks pure math’s aesthetic value because its done by less aesthetically sensitive people (positive feedback loop) rather than because it couldn’t be like classical applied/pure math?
My one caveat with your claims here is that Van Gogh was severely insane, which probably impaired his ability to support himself quite a bit.
This is a fair point. I would guess / vaguely remember that there are plenty of examples of psychologically sound great painters who had trouble making a living but don’t have a list off hand. Laurens suggested Cezanne as an example but I have not independently verified that he qualifies. I don’t know very much about visual arts.
Note that in general there’s a selection effect where artists/scientists who have dim prospects for making a living doing what they do are disproportionately likely to leave relative to other artists/scientists of their quality. It’s hard to know how significant this selection effect is in a given domain.
Also, how likely does it seem to you that applied math lacks pure math’s aesthetic value because its done by less aesthetically sensitive people (positive feedback loop) rather than because it couldn’t be like classical applied/pure math?
There’s almost certainly some effect of this type; I’m uncertain as to how large the effect. Two relevant points:
To the extent that applied math involves features specific to how humans interact with the world (i.e. taking into account contingent constraints specific to human needs) arbitrariness creeps in on account of the fact that humans were generated by a random process.
There seem to be arbitrage opportunities for mathematical expertise to bring clarity to areas that were previously somewhat obscure. For example, SarahC has suggested that principle component analysis might be utilized to better understand what’s referred to as autism (a fact that I haven’t previously seen discussed explicitly). The existence of such apparent arbitrage opportunities suggests that there may be quite fertile unexplored ground within applied math.
I’m interested in popularizing some of the pure mathematical achievements that are regarded among elite pure mathematicians as being of great aesthetic value.
Sweet! Do you know of any existing works that attempt this (perhaps at a higher level of sophistication)? Also, what are the mathematical achievements you would focus on?
One good compilation of “pure mathematical achievements that are regarded among elite pure mathematicians as being of great aesthetic value” is Proofs from THE BOOK.
I myself would guess that none of the works produced over the past hundred years would be judged by the majority of an impartial audience to be significantly more compelling than (for example) Bach’s Chaconne.
Same here.
I’m impressed that you’re familiar with the Von Neumann quote (which is sadly little known in the mathematical community but which my friend Laurens is fond of); but on the face of things it doesn’t seem to directly support your paragraph above. Explain further if you’d like?
Several points here:
My impression is that there are issues of bad social/cultural institutions destroying artistic ability outside of academia. I have some friends artistically genuine who have spent some time as painters and become disillusioned with the signaling games and hypocrisy present within the communities of painters that they’ve come across. Note that fledging painters and authors face greater financial pressures than academics and that this can lead to perverse incentives (to appeal to the lowest common denominator or to current fashions for greater marketability). See Minhyong Kim’s comment here.
The absence of new empirical sources for mathematics seems to me more a consequence of the stagnation of theoretical physics than the social structure of the mathematical community.
In my own view insufficient emphasis on exposition has played a significant role in whatever stagnation has occurred within the mathematical community since, e.g. the 1800′s. The barrier to entry has gotten progressively higher as mathematics has developed and in such a setting, in absence of strong efforts to to cast background material in an accessible and readily digestible form, the pressures toward specialization and fragmentation get progressively stronger. There aren’t career-based incentives for expository work so mathematicians who are interested in exposition either conform to the research-based publish or perish norms or leave.
I’d be happy to compile an annotated list of relatively accessible survey papers if you’d be interested and find it useful for getting a sense for the state of some of contemporary mathematical research.
From my perspective in applied math things are exactly opposite. We’re practically drowning in empirical sources begging for better numerics. There is hardly a scientific or engineering field today that couldn’t be revolutionized by a breakthrough in a relevant area currently under study in the applied math community. The problem isn’t lack of necessity, inspiration, or motivation, the problem is the problems are damn hard, and I’m far from convinced that we attract the best minds.
Even with very clear, objective goals, progress can still be painfully slow!
I very much agree. Incentives are out of line for most areas of science, but I think the fallout from this is especially poignant for math. I view part of our discipline’s responsibility as curating the common mathematical knowledge used by all other disciplines.
Like mathematicians, musicians who only write for experts are unmoored from empirical feedback and are thus dependent on unusually good taste if they are to do something valuable. It’s not fair to expect people who can’t evaluate their work to conclude that they have such good taste even if they are acknowledge to be very smart.
Fledging painters can paint for both the lowest common denominator AND for themselves if they want to. Academics can’t do popular work without that counting against them with other academics.
There’s lots of need for math in complexity theory and other domains. Quantum computing for instance. Really all over the place. Crypto is very popular. Probably lots of engineering examples.
I’d be happy if someone else we both know who shares this concern would review that list.
I agree. I guess the reason why I took pause is because Von Neumann’s quote does not immediately suggest that he concurs with
his quotation more just obliquely raises the possibility that you suggest.
It’s not clear to me that this is true. Doesn’t it depend on what the lowest common denominator is? Van Gogh painted for himself and is reported to have been unable to support himself by selling his paintings. Are you suggesting that things have changed since his time? If so, how and under what evidence?
My observation has been that this is mostly true.
(1) It should be noted that Von Neumann was in large measure an applied mathematician and so it’s natural to expect him of being biased in favor of applied topics.
(2) There is widespread agreement among most sophisticated contemporary mathematicians as to the high aesthetic value of some of the pure mathematical achievements over the past few decades. I agree that it’s difficult for an outsider to quickly ascertain that there’s something substantive going on here but I give you my word for whatever it’s worth :-).
(3) I’ve had a less pronounced positive aesthetic response to most applied math topics than to some of my favorite pure mathematical topics. I’ve found that applied topics are more ad hoc and lacking in internal coherence and a large part of what I find compelling about pure math is the high degree of internal coherence.
As I’ve said elsewhere in this thread, the danger of generalizing from one example here is very serious and I do not question to sincerity of those who are passionate about applied topics (or pure topics like graph theory and elementary analytic number theory which I personally find disappointingly ad hoc and lacking in internal coherence but which some people eagerly devote their lives to).
(4) My own interest in applied math topics comes more from the applications than from the math involved.
(5) I would differentiate between “early” applied math (e.g. of the type that Newton and Maxwell did) which was closer to pure math (on account of the fact that so little of either was developed and the fact that it was necessary to develop pure math further in order to get to the point of being able to do something useful) and modern applied math; the former is not necessarily representative of the latter.
(6) Some theoretical and mathematical physicists draw a sharp distinction between physics and applied math. I don’t have subject matter knowledge here; but have the rough impression that theoretical physics has more internal coherence than most applied mathematical topics. This is why I referenced theoretical physics in particular as a source of inspiration that seems to have dried up on account of lack of empirical feedback.
(7) The utilitarian value of the pure mathematical achievements alluded to above is questionable in light of the fact that (a) very few people are able to appreciate them and (b) they don’t have foreseeable technological applications in the near future. The second point has heightened significance in light of the fact that it seems very possible that an intelligence explosion is not far off.
I’m interested in popularizing some of the pure mathematical achievements that are regarded among elite pure mathematicians as being of great aesthetic value. Aside from the immediate enjoyment attached to enriching people’s lives; my interest in doing this is with a view toward giving people opportunities to develop heightened aesthetic sense and spreading humanistic values, in particular making a case that there are things in the world beautiful enough so that it’s worth working toward the long-term survival of the human race. I feel like I’ve seen a “promised land” of the sorts of intellectual experiences that lie beyond the boundaries of current human mind space.
Sure, I would too. I’ll probably get around to it sometime over the next few weeks.
Thanks for the very thoughtful and clear post.
My one caveat with your claims here is that Van Gogh was severely insane, which probably impaired his ability to support himself quite a bit.
Also, how likely does it seem to you that applied math lacks pure math’s aesthetic value because its done by less aesthetically sensitive people (positive feedback loop) rather than because it couldn’t be like classical applied/pure math?
This is a fair point. I would guess / vaguely remember that there are plenty of examples of psychologically sound great painters who had trouble making a living but don’t have a list off hand. Laurens suggested Cezanne as an example but I have not independently verified that he qualifies. I don’t know very much about visual arts.
Note that in general there’s a selection effect where artists/scientists who have dim prospects for making a living doing what they do are disproportionately likely to leave relative to other artists/scientists of their quality. It’s hard to know how significant this selection effect is in a given domain.
There’s almost certainly some effect of this type; I’m uncertain as to how large the effect. Two relevant points:
To the extent that applied math involves features specific to how humans interact with the world (i.e. taking into account contingent constraints specific to human needs) arbitrariness creeps in on account of the fact that humans were generated by a random process.
There seem to be arbitrage opportunities for mathematical expertise to bring clarity to areas that were previously somewhat obscure. For example, SarahC has suggested that principle component analysis might be utilized to better understand what’s referred to as autism (a fact that I haven’t previously seen discussed explicitly). The existence of such apparent arbitrage opportunities suggests that there may be quite fertile unexplored ground within applied math.
I have always been intrigued by Kenneth Rexroth’s take.
Thanks for the link. That was a delightful essay.
Sweet! Do you know of any existing works that attempt this (perhaps at a higher level of sophistication)? Also, what are the mathematical achievements you would focus on?
One good compilation of “pure mathematical achievements that are regarded among elite pure mathematicians as being of great aesthetic value” is Proofs from THE BOOK.