Wednesday, April 20, 2016

"... finally I'm becoming stupider no more."-- Paul Edős

Erdos in the mentorship role (the constant coffee cup in foreground) with ten-year-old Terence Tao, a math genius (photo taken in 1985)


Tuesday, April 19, 2016

Hilary Putnam RIP


The great philosopher Hilary Putnam died on March 16.

With professors of the caliber of Quine, Reichenbach and Carnap, one would end up either a fusty analytic philosopher or smart as a whip. Putnam was the latter. I'd like to just talk here about Putnam's direct contribution to a subject dear to me: Mathematics.

Putman's point?

Math is as real and essential as physical entities. 

His argument opens with a simple fact: math remains the indispensable language of science (judging by its growth, this indispensability of math will not stop any time soon).

If mathematical entities are indispensable for some of our best scientific theories, we should have ontological commitment to mathematical entities.

Putnam is not alone here. Quine, another important logician and math inclined philosopher, referred to this general dependence of science to math as math's "ontological rights" to science. 

Why is π real? 

π qua number is both transcendent and irrational (a badge of honor in number theory). π 's randomness is actually useful for mathematical analysis and computational theory. π is a good friend of number series, circles, spheres, ellipses, tori, waves (in nature). The pervasiveness of π is no accident and pretending otherwise is unfair to math's foundationality.

Take a look at this marvel of simplicity and richness:

e^{i \pi} + 1 = 0.

Euler's equation connects fundamental numbers like i π, e, 1, and 0.  Follow the sinewy development of the formula here, which takes steps of mathematical audacity on behalf of Euler (end of page 4).

Are we to unproblematically accept that 
Euler's identity doesn't express something "out there" in the world?