How Mathematicians Think

Hadamard estimated that :

About 90% of mathematicians think visually, 10% think formally.

Usually, they think in steps:

  1. Get the right idea, often think vaguely about structural issues, leading to some kind of strategic vision;
  2. Tactics to implement it;
  3. Rewrite everything in formal terms to present a clean, logical story. (Gauss’s removal of ‘scaffolding’ – middle working steps)

Source: [NLB #510.922]

Ultimate Mathematician


A mathematician is a person who can find analogies between theorems;

A better mathematician is one who can see analogies between proofs


The best mathematician can notice analogies between theories.

One can imagine that the ultimate mathematician is one who can see analogies between analogies.

Father and son Mathematicians

Mathematics, unlike Medicine, is hardly inherited, as in “Like father like son” running in the family.

There are some exceptions in the history, the most outstanding Mathematician family is Bernoulli.

2. 祖冲之和儿子祖暅之
Zu Chongzhi (424 – 500 CE) and son computed the first
\pi = \frac{355}{113}
3.1415926 < \pi < 3.1415927

3. Emil Artin and son Michael Artin: both great algebraists.

Old Mathematicians Live Young

J.J. Sylvester, who coined the term Matrix, pointed out that Leibniz, Newton, Euler, Lagrange, Laplace, Gauss, Plato, Archimedes, and Pythagoras all were productive until their 70s or 80s.

“The mathematician lives long and lives young,” he wrote.

“The wings of the soul do not early drop off, nor do its pores become clogged with the earthly particles blown from the dusty highways of vulgar life.”

Sylvester himself was his 82nd year, in 1896, when he “found a new enthusiasm and blazed up again over the theory of compound partitions and Goldbach’s conjecture.”

Another mathematician Harold S. M. Coxeter (9/2/1907 – 31/3/2003) attributed his longevity to love of mathematics.