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It is the time of year for predictions: predictions for 2011, predictions for the next decade, predictions for the unspecified future. I searched a bit for predictions in mathematics, but it seems mathematicians are too wise to engage in this dubious activity. I found only two predictions:

(1) Two New Scientist writers, Samuel Arbesman and Rachel Courtland, predict that 2011 will not see the $P=NP$ problem resolved.

(2) Sir Michael Atiyah

suggested that the conjectured self-adjoint operator that could explain the Riemann hypothesis might be the Hamiltonian of quantum gravity

in a November talk at the Simons Center, as reported by Peter Woit.

Of course it is a stretch to call Atiyah's suggestion a "prediction." And every conjecture in mathematics is a prediction! Nevertheless, in the spirit of New Year's, I would be interested to hear any predictions on future developments in mathematics.

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closed as no longer relevant by Andrés E. Caicedo, Joseph O'Rourke, Harry Gindi, Charles Siegel, Felipe Voloch Jan 1 '11 at 0:20

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@Roy: I think that most of the opinions on that page are abhorrent where they pertain to pure mathematics and are irrelevant otherwise. – Harry Gindi Dec 31 '10 at 21:12
The Procrastinator's Club will soon release its predictions for 2010. They have a history of complete accuracy. – Michael Hardy Dec 31 '10 at 21:16
I predict this question is going to be closed. – KConrad Dec 31 '10 at 22:06
No longer relevant? It's only 8pm here in NY. :-) – sigoldberg1 Jan 1 '11 at 1:07

I've heard this type of prediction made:

In a few decades or so, the negation of the continuum hypothesis will be accepted as a consequence of some "obvious" axiom added to ZFC just as choice was added to ZF as an obvious axiom.

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How long was this ago? One must be pretty much blind to say things like that nowadays... – darij grinberg Dec 31 '10 at 20:49
@darij, would you care to elaborate? – Andrés E. Caicedo Dec 31 '10 at 21:00
@darij: This year; My use of obvious here is more of a technical nature. The argument was that we've accepted the axiom of replacement because it is strongly used in proofs. The proper forcing axiom, which implies $2^{\aleph_0} = \aleph_2$, is being advanced as an axiom to assume because of its many interesting and useful set-theoretical consequences including the existence of a Woodin cardinal in an inner model. The argument is that eventually we'll just wind up accepting such an axiom as natural. – Jason Dec 31 '10 at 22:02
Algebraists used to "accept" additional axioms some time ago due to their use in putting category theory on a solid fundament (Grothendieck universes and the likes). Nowadays this seems to be a thing of the past, and modern algebraists have switched to more constructivist and type-theoretical approaches (even finitism is on the rise now). This is just my personal feeling from various talks by algebraists. Note that I am from Germany rather than from France, so maybe this Grothendieck school is still alive but I just never met it. – darij grinberg Jan 1 '11 at 17:33

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