### Enigmas of Chance

I just finished reading Enigmas of Chance, the autobiography of mathematician Mark Kac (1914-1984). Not the most gripping book I’ve read, but it was sufficiently interesting to keep my attention for its quick 156 pages. Kac had some interaction with physics – he is most well-known for the Feynman-Kac formula and for other work in statistical physics, such as the “spherical model” for studying phase transitions (related to the Ising model).

One notable tidbit: Kac taught Weinberg and Glashow in the standard first-year graduate math methods course.

I learned that the “normal law” (Gaussian distribution) shows up mathematically in places that have nothing to do with “randomness.” Kac, along with Erdos, proved that the Gaussian shows up in the context of prime numbers. Using ${\nu (m)}$ to represent the number of prime factors of some integer ${m}$, they proved that the proportion of integers ${m}$ for which

$\displaystyle \log \log m + a \sqrt{2 \log \log m} < \nu (m) < \log \log m + b \sqrt{2 \log \log m}$

can be found by integrating the Gaussian

$\displaystyle \frac{1}{\sqrt{\pi}} e^{-x^2}$

from ${a}$ to ${b}$.

This quote, from pg. 111, could potentially be applied to some areas of physics:

“I have often claimed that if a subject is robust it should be insensitive to its foundations, and therefore too great an emphasis on the latter tends to produce a misleading slant. Concern about foundations should come, if at all, after one has a firm intuitive grasp of the subject.”

Having finished Kac’s autobiography, I started Halmos’s (I Want to be a Mathematician). For some reason I currently find autobiographies of mathematicians appealing.