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@MathPrinceps | |||||
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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?
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Laurens Gunnarsen
@MathPrinceps
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26. stu |
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In particular, looking just at mathematics, we find the following names associated with profound innovations made in the first two decades of the last century: Frobenius, Burnside, Poincaré, Hilbert, Minkowski, Hadamard, Cartan, Takagi, Ramanujan, Weyl, Hecke, Noether, Banach.
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Laurens Gunnarsen
@MathPrinceps
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26. stu |
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It is impossible to argue that the first two decades of mathematical research in this century have produced any innovations as profound as group representation theory, functional analysis, dynamical systems theory, the geometry of fiber bundles, or class field theory.
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Laurens Gunnarsen
@MathPrinceps
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26. stu |
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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.)
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Laurens Gunnarsen
@MathPrinceps
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26. stu |
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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?
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Laurens Gunnarsen
@MathPrinceps
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26. stu |
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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.
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Laurens Gunnarsen
@MathPrinceps
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26. stu |
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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.
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Laurens Gunnarsen
@MathPrinceps
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26. stu |
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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."
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Laurens Gunnarsen
@MathPrinceps
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26. stu |
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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.
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Laurens Gunnarsen
@MathPrinceps
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26. stu |
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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.
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Laurens Gunnarsen
@MathPrinceps
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26. stu |
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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.
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Daniel O'Connor
@Singularitarian
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27. stu |
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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.
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Laurens Gunnarsen
@MathPrinceps
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27. stu |
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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.
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Elad kosloff
@eladkosloff
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27. stu |
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There are entirely new fields created in the last 60 years . To name a few: Computer science, geometric group theory, arithmetic geometry, category theory. Those are huge fields that may have earlier origins but are essentially new.
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Laurens Gunnarsen
@MathPrinceps
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27. stu |
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How many of these were created in the past twenty years?
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John Carlos Baez
@johncarlosbaez
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27. stu |
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My view of current-day mathematics is much more optimistic. So much amazing stuff is going on that the field isn't dominated by a few giants. When you get ten times as many Poincarés, Hilberts, Cartans, and Noethers they don't seem like such a big deal anymore. (1/2)
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John Carlos Baez
@johncarlosbaez
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27. stu |
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So many good new ideas are showing up that it's become much harder to keep up with them all. Voevodsky's homotopy type theory. Lurie's work on (infinity,1)-categories. Witten's work on string theory, M-theory and geometric Langlands (math, not physics). And much more. (2/2)
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Paul Johnson
@ptwiddle
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26. stu |
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I think you're overplaying the "no low-hanging fruit" - though we do plant new trees, we keep climbing the old ones as well, and it gets much harder, but huge breakthroughs do happen - classification of finite simple groups! The geometrization conjecture!
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Paul Johnson
@ptwiddle
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26. stu |
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Fields have been spectacularly rebuilt, sometimes multiple times - algebraic geometry, algebraic topology, logic. And whole new fields - graph theory, numerical methods using computers, category theory...
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Model Of Theory
@ModelOfTheory
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26. stu |
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Part of this must be that today's "superior" institutional support structures support production of acceptable work, which has indeed increased tremendously in volume, and neglect to support production of revolutionary work, support for which is much different in character.
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Timothy Gowers
@wtgowers
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27. stu |
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I think maths just grows so fast that important innovations will on average have an effect on a smaller proportion of the subject. So a fairer comparison might be between giants of mathematics 100 years ago and giants of some (largish) subdomain of mathematics now.
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Timothy Gowers
@wtgowers
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27. stu |
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One could argue (I'd be interested to know whether people think it would be correct though) that mathematicians like Gauss, Euler and Riemann had an even bigger impact than the ones you list from the early 20th-century.
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