Peirce and de Morgan
In Augustus de Morgan's Trigonometry and Double algebra from 1849, he replaces numbers by axiomatic algebraic symbols.
In the Single Algebra a symbol denotes a process either forwards or backwards, pictured as segments on a line. In order to deal with numbers of the form $a + ib$ de Morgan introduced the Double Algebra, where a line is given a direction in the plane. For a long time de Morgan tried to develop a Triple Algebra in the same spirit of pure logic, even after Hamilton showed that the Quadruple Algebra of quaternions was the next natural geometric step.
It was the originator of Category Theory, C. S. Peirce, who remained inspired by de Morgan's ideas and went on to develop the Theory of Signs and the triad philosophy. The under-appreciated Peirce is now recognised to have, amongst many other things, axiomatised arithmetic before Peano and to have discovered the ability of electrical circuits to do Boolean algebra.
In the Single Algebra a symbol denotes a process either forwards or backwards, pictured as segments on a line. In order to deal with numbers of the form $a + ib$ de Morgan introduced the Double Algebra, where a line is given a direction in the plane. For a long time de Morgan tried to develop a Triple Algebra in the same spirit of pure logic, even after Hamilton showed that the Quadruple Algebra of quaternions was the next natural geometric step.
It was the originator of Category Theory, C. S. Peirce, who remained inspired by de Morgan's ideas and went on to develop the Theory of Signs and the triad philosophy. The under-appreciated Peirce is now recognised to have, amongst many other things, axiomatised arithmetic before Peano and to have discovered the ability of electrical circuits to do Boolean algebra.
posted by Kea | 9:06 AM
2 Comments:
Thanks for the Peirce link. Fascinating bio. But how is his category theory as explained here related to what is understood as category theory today in mathematics?
I also liked the connection he made with logic and circuits. I did not realize that such a connection needed to be made at some point. It must have been as fundamental as realizing the distinction between counting and the concept of numbers.
Peirce is the giant of category theory and my master of philosophy ... but honestly Carl von Linné was the pragmatic developer in his ”Systema naturae” 300 years ago, it was revolutionary then and still is!!!
Post a Comment
<< Home