Great researchers in mathematics are certainly not ten times more numerous today than they were a century ago; indeed, it takes some audacity to argue that we have as many. (It's far from clear, for example, whether anyone alive today can bear close comparison with Poincaré.)
-
Show this thread
-
But if conditions today are so spectacularly more favorable to successful research in the mathematical sciences than a century ago, and the number of trained researchers has grown by at least an order of magnitude, why is there no corresponding growth in achievement?
1 reply 2 retweets 31 likesShow this thread -
Mathematics itself may be the most illuminating case to study, because a "depletion of low-hanging fruit" explanation of modern stagnation is least tenable there. All the fundamental laws of physics may already have been discovered, but nothing like this is true in mathematics.
2 replies 1 retweet 40 likesShow this thread -
Indeed, mathematics is demonstrably inexhaustible, and the exceedingly long history of the art records no fallow period during which its master practitioners believed they might be unable for fundamental reasons to discover deep new results of lasting interest.
1 reply 0 retweets 26 likesShow this thread -
Note that this contrasts strikingly with physics: in 1894, Michelson judged it likely that "most of the grand underlying principles have been firmly established," and that "the future truths of physical science are to be looked for in the sixth place of decimals."
1 reply 1 retweet 25 likesShow this thread -
No similarly eminent mathematician has mooted a similarly pessimistic view of the art's prospects. On the contrary: great mathematicians have tended to predict extraordinary things to result from the art's inevitable assimilation and refinement of recent breakthroughs.
1 reply 0 retweets 18 likesShow this thread -
Because mathematicians have the freedom to devise and pursue entirely new fields of research -- a freedom successfully exploited, repeatedly, by its greatest past masters -- the formidable intricacy of its current best-established fields is no bar to its further flourishing.
1 reply 1 retweet 22 likesShow this thread -
If at any particular epoch of mathematical history no low-hanging fruit remains on some particular mathematical tree, then mathematicians may choose to plant, cultivate, and harvest the fruit of entirely new trees. Indeed, when frustrated, they have often done exactly that.
2 replies 2 retweets 25 likesShow this thread -
So what is going on? Why is mathematical practice today not dramatically more successful than a century ago? Why is there no spectacular contemporary flourishing of the art, with entirely new fields opened up by ten times as many Poincarés, Hilberts, Cartans, and Noethers?
24 replies 4 retweets 62 likesShow this thread -
Replying to @MathPrinceps
Although there is an infinite amount of math to discover, as we progress new math becomes more difficult to discover, because one must first master all of the relevant math that is already known. In that sense, the low hanging fruit has been picked.
1 reply 0 retweets 7 likes
If it were necessary to master all the relevant math that is already known, no progress would ever result. Already in 1900, Hilbert maintained that this was an impossible undertaking.
-
-
Replying to @MathPrinceps @Singularitarian
One need not master all existing math. One does need to get to the edge in a given field. Even this simpler task gets more difficult as math becomes more developed. Moreover, the new math that gets developed does not tend to have the same impact since it isn't as fundamental.
0 replies 0 retweets 3 likesThanks. Twitter will use this to make your timeline better. UndoUndo
-
Loading seems to be taking a while.
Twitter may be over capacity or experiencing a momentary hiccup. Try again or visit Twitter Status for more information.