Attractors which lead to people being unable or afraid to attempt other activities are much less credible as values than those which leave one with many options.
This is strictly analogous to Aristotle’s claim that intellectual pursuits were best, as is clear because they are chosen by almost everyone who appreciates them and other sorts of activity. He’s right, but its broad personal development with a strong focus on intellectual pursuits of a polymathic nature which seems to actually be chosen by those who can choose it, reliably.
I expect people who are used to interacting with me to take it at my word that it is periodically falsified on details and revised. The closest that’s been written down is in Lakoff, but his model isn’t that far off.
Also—Terence Tao has pretty broad interests already within mathematics.
And I’m starting to suspect that the category “mathematics” is like the category “single-celled organisms.” It’s a convenient notation to lump together a class of objects that most humans ignore, but that are intrinsically much more variable than all the other objects that are more salient to most humans’ lives. Two single-celled organisms can be more different from each other than you are from an oak tree. Commutative algebra and non-commutative algebra can be more different, separated by a wider gulf of background knowledge, than genetics and economics. (That’s a little hyperbolic but I think some statements of that kind are actually true.)
So I think people with broad mathematical interests are really more “polymathic” than we tend to realize from the outside. Math is diverse. Terence Tao may in fact be more of a polymath than Tyler Cowen, if we’re talking about expertise in radically varied fields that are rarely studied together by one person.
The caveat is that lots of categories are like the category “single-celled organisms,” from what I can tell. I would guess that you are familiar with mathematics, so you can look at math and go “wow, there are so many sub-fields. Other fields aren’t like that!” Meanwhile someone familiar with biology might think “wow, there’s so much to study within biology; math doesn’t seem like that!”
Edit: on the other hand I could be wrong—if you’re not particularly familiar with mathematics research compared to biology research my story is inaccurate.
Attractors which lead to people being unable or afraid to attempt other activities are much less credible as values than those which leave one with many options.
I think I see what you’re saying here. Attractors that are irreversible dead ends are more likely to represent miscalculations. But still, that doesn’t rule out the possibility that some people really like working on abstract math, really value academic status, and/or have a large comparative advantage in doing math, and therefore for them going into math is a rational choice. Furthermore, it seems likely that such people would be more concentrated among the top mathematicians.
I think your other argument is that according to your model of human psychology, most people would find it more fun to be a polymath than to specialize in one area. That makes some sense to me, but obviously it’s hard to put a lot of credence in it without seeing some strong empirical evidence.
Based on these two arguments, it seems to me there’s a reasonable chance that Terrance Tao went into math by mistake, but your level of confidence still seems unjustified. Did I miss anything else?
I’m sure that there are many people who really like abstract math. Your right about my claim about polymaths. My best guess is that Tao was right to go into math but would be best served by staying in math but broadening his focus somewhat now. The fact that he doesn’t seem to be driven by any particular problem strengthens it, but I’m still not all that confident of it. I’m a lot more confident that he should put a bit of effort into becoming a multimillionaire in a small part of his spare time, simply by selling very expensive consulting services, and less effort into housekeeping, driving his own car, and whatever else he spends his time on other than math, but it’s possible to me that he has done so and possible that he has relatives and/or friends who would envy him and make it a net loss.
One can simply see if people produce less or different stuff after tenure as one measure.
So the claim is that anybody who could become a polymath, does? And that people who specialize narrowly do so because they lack the capacity to be broader?
I don’t know how you would look at someone who works in a specialized area and conclude that he could, or couldn’t, have been a polymath if he had tried. One imperfect proxy, I guess, would be if our specialist has well-developed skills outside his profession. Are mathematicians below average in outside skills, compared to the amount of focus needed for their work? I’ve known or encountered a sizable minority who could have been musicians or writers instead; I have the anecdotal impression that mathematicians don’t do too badly on that score. Of course, mathematicians do tend to be less casually conversant with non-math stuff than average.
Or maybe you’re talking about something different. Maybe the capacity to be a polymath is not just plenty of innate abilities at different things, but the will and desire to set up a self-designed, varied life instead of a fixed one. Some people with varied abilities choose to give up all their skills but one; some people refuse, and insist on blending or simultaneously using their different skills. In that case, it’s probably even harder to test the “capacity to be a polymath,” because it seems kind of tautological—but I would instinctively agree that very specialized people probably don’t have it.
Attractors which lead to people being unable or afraid to attempt other activities are much less credible as values than those which leave one with many options.
This is strictly analogous to Aristotle’s claim that intellectual pursuits were best, as is clear because they are chosen by almost everyone who appreciates them and other sorts of activity. He’s right, but its broad personal development with a strong focus on intellectual pursuits of a polymathic nature which seems to actually be chosen by those who can choose it, reliably.
I expect people who are used to interacting with me to take it at my word that it is periodically falsified on details and revised. The closest that’s been written down is in Lakoff, but his model isn’t that far off.
Also—Terence Tao has pretty broad interests already within mathematics.
And I’m starting to suspect that the category “mathematics” is like the category “single-celled organisms.” It’s a convenient notation to lump together a class of objects that most humans ignore, but that are intrinsically much more variable than all the other objects that are more salient to most humans’ lives. Two single-celled organisms can be more different from each other than you are from an oak tree. Commutative algebra and non-commutative algebra can be more different, separated by a wider gulf of background knowledge, than genetics and economics. (That’s a little hyperbolic but I think some statements of that kind are actually true.)
So I think people with broad mathematical interests are really more “polymathic” than we tend to realize from the outside. Math is diverse. Terence Tao may in fact be more of a polymath than Tyler Cowen, if we’re talking about expertise in radically varied fields that are rarely studied together by one person.
The caveat is that lots of categories are like the category “single-celled organisms,” from what I can tell. I would guess that you are familiar with mathematics, so you can look at math and go “wow, there are so many sub-fields. Other fields aren’t like that!” Meanwhile someone familiar with biology might think “wow, there’s so much to study within biology; math doesn’t seem like that!”
Edit: on the other hand I could be wrong—if you’re not particularly familiar with mathematics research compared to biology research my story is inaccurate.
I think I see what you’re saying here. Attractors that are irreversible dead ends are more likely to represent miscalculations. But still, that doesn’t rule out the possibility that some people really like working on abstract math, really value academic status, and/or have a large comparative advantage in doing math, and therefore for them going into math is a rational choice. Furthermore, it seems likely that such people would be more concentrated among the top mathematicians.
I think your other argument is that according to your model of human psychology, most people would find it more fun to be a polymath than to specialize in one area. That makes some sense to me, but obviously it’s hard to put a lot of credence in it without seeing some strong empirical evidence.
Based on these two arguments, it seems to me there’s a reasonable chance that Terrance Tao went into math by mistake, but your level of confidence still seems unjustified. Did I miss anything else?
I’m sure that there are many people who really like abstract math. Your right about my claim about polymaths. My best guess is that Tao was right to go into math but would be best served by staying in math but broadening his focus somewhat now. The fact that he doesn’t seem to be driven by any particular problem strengthens it, but I’m still not all that confident of it. I’m a lot more confident that he should put a bit of effort into becoming a multimillionaire in a small part of his spare time, simply by selling very expensive consulting services, and less effort into housekeeping, driving his own car, and whatever else he spends his time on other than math, but it’s possible to me that he has done so and possible that he has relatives and/or friends who would envy him and make it a net loss.
One can simply see if people produce less or different stuff after tenure as one measure.
So the claim is that anybody who could become a polymath, does? And that people who specialize narrowly do so because they lack the capacity to be broader?
I don’t know how you would look at someone who works in a specialized area and conclude that he could, or couldn’t, have been a polymath if he had tried. One imperfect proxy, I guess, would be if our specialist has well-developed skills outside his profession. Are mathematicians below average in outside skills, compared to the amount of focus needed for their work? I’ve known or encountered a sizable minority who could have been musicians or writers instead; I have the anecdotal impression that mathematicians don’t do too badly on that score. Of course, mathematicians do tend to be less casually conversant with non-math stuff than average.
Or maybe you’re talking about something different. Maybe the capacity to be a polymath is not just plenty of innate abilities at different things, but the will and desire to set up a self-designed, varied life instead of a fixed one. Some people with varied abilities choose to give up all their skills but one; some people refuse, and insist on blending or simultaneously using their different skills. In that case, it’s probably even harder to test the “capacity to be a polymath,” because it seems kind of tautological—but I would instinctively agree that very specialized people probably don’t have it.
By can I mean see themselves as having the option, that’s all.