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.
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.