Metamathematics become important only when we deal with the choice of axioms and the legitimacy of derivations in a theory. But, fortunately for us, most of the mathematics we use today is not concerned with these issues, and hence the part that is not metamathematics I will call pure mathematics or just mathematics.

 

The joy of pure mathematics is difficult to communicate. Here are some concepts which have deeply impressed me: Cauchy's integral theorem says that the values of an analytic function on a closed curve decide its value anywhere inside the contour. The  determinant of a compound matrix can be written as a power of the determinant of the original matrix. Three celebrated theorems, by Laplace, Cauchy, and Jacobi, about matrices can be stated elegantly, in terms of compound matrices. Riemann hypothesizes that the complex zeros of his simple analytic function lie on a vertical line in the complex plane. A philanthropist offers a million dollar prize for anybody who proves the hypothesis.

 

This blog is about pure mathematics.

 

 

      CAUCHY AND THE FORT

Cauchy was keen on impressing his emperor with his analytic abilities, like his fellow mathematician Fourier. Cauchy walked around the enemy fort and just listening to the soldiers talking outside, he was able to give a detailed account of enemy's plan of attack being hatched inside the fort. Some say, this episode is the basis for Cauchy's famous integral theorem and is as significant as Newton’s inference from the falling apple.

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