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What is mathematics

January 12, 2017

Most 20th and 21st Century practitioners would say it is the study of structure. The idea that it has to do with numbers has historical relevance, but not much more. The 1,716 page book I wrote about my own research has a lot of numbers in it: every page, every chapter and section, every formal statement is numbered.   The text has quite a few (but fewer) numbers of which I bet the most common is 2π.

In the 20th Century there was a hugely successful concentration on the underlying logic of mathematics led first by Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (1872/05/18-1970/02/02).

Bertrand_Russell_transparent_bg.pngThis led in 1901 to his DEFINITION of mathematics;

The subject in which we never know what we are talking about, nor whether what we are saying is true.

The point is that when properly stated, all mathematical propositions are conditional. They say if something (“the hypothesis”) is true then something else (“the conclusion”) must also be true. It is beyond the scope of mathematics to investigate whether the hypothesis is true, so we never know or care whether the conclusion is true. This is in fact a leading reason for “the unreasonable effectiveness of mathematics” (1960, Eugene Wigner (1902/11/17-1995/01/01), my favorite 20th Century physicist.)


When ever the hypothesis is true then one can be confident that the conclusion is also true. Thus a single theorem may have application in MANY fields. That is why it is vitally important that a mathematician not think that he knows “what he is talking about”. Only the structure (the rules that the system follows) are important.

Godfrey Harold Hardy, FRS ((1877/02/07-1947/12/01)


friend and protector of both Bertrand Russell and Srinivasa Ramunajan (1887/12/22- 1920/04/26))


gave another interesting definition of mathematicians in 1940:

  • A mathematician, like a painter or poet, is a maker of patterns.
  • If his patterns are more permanent than theirs,
  • it is because they are made with ideas.

Mathematicians insist that their conditional statements be logically correct, but lots of things are logically correct. Only a keen sense of aesthetics leads one to formulate a theorem of lasting value.

Let me address what it feels like to do creative mathematics. To discover anything significant (unless you are an incredible genius, which I have never been) it takes months of struggling to understand a collection of ideas. Suddenly you do! (My best ideas all arrived when I was walking outdoors and most of the great mathematicians I have been privileged to know were hardy outdoor people.) After this flash of insight essentially everyone reports they feel as if they have understood something that existed already. However the only logically supportable position is that mathematics is a construction of human beings, so your insight did NOT pre-exist.

My hero Charles Darwin, FRS, RGS, FLS, FZS (1809/02/12-1882/10/19)


gave another humorous definition of a mathematician

  • A mathematician is a blind man in a dark room
  • looking for a black cat which isn’t there.

This is how it feels when you start thinking about a hard problem. The deeper point is that the “Theorem” is NOT there until you discover it.


I am proud to have been a creative mathematician. A few of my results have been judged to be important. Like the proof which Euclid published about 300 BCE proving the infinitude of primes, I feel assured that my best results will be remembered (perhaps by an exceedingly tiny cadre of scholars of the history of mathematics) as long as civilization endures. Most mathematical results were proved long before any application for them was found, so it is possible that my results might become really important. (They could conceivably be relevant to the unified theory of physics that would reconcile general relativity with quantum mechanics.)

Differential geometry was essentially a complete theory, before Albert Einstein ((1879/03/14-1955/04/18) Nobel laureate, Time Magazine Man of the Century)


realized that it was the language in which general relativity could be explained. This took him eight years from 1907 to 1915.

(As soon as I got to Harvard I got a carrel in Widener Library even though all the relevant mathematical books were at 2 Divinity Avenue with the faculty offices. I roamed the stacks finding thousands of fascinating books on many subjects. In particular I looked at the history of science. I found two interesting books relevant to this discussion. Both from about 1907. The first was a survey of current mathematics. It pointed to the beauty of number theory and differential geometry, but said neither could possibly have any practical relevance. General relativity was less than a decade in the future. While I was still a graduate student, number theory became the basis of all modern security codes. NSA stupidly tried to classify recent important research papers in the number theory. The other discussed Einstein as the greatest contemporary physicist, but mentioned that he had to be forgiven his crazy ideas about (special) relativity.  Similarly his 1922 Nobel Prize citation only mentions the photoelectric effect discovered in 1905 like special relativity.


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