David Pierce: Induction and Recursion

D. Pierce, Induction and Recursion, The De Morgan Journal, 2 no. 1 (2012),  99-125.

From the Introduction:

In mathematics we use repeated activity in several ways:

  1. to define sets;
  2. to prove that all elements of those sets have certain properties;
  3. to define functions on those sets.

These three techniques are often confused, but they should not be. Clarity here can prevent mathematical mistakes; it can also highlight important concepts and results such as Fermat’s (Little) Theorem, freeness in a category, and Goedel’s Incompleteness Theorem.
The main purpose of the present article is to show this.

In the `Preface for the Teacher’ of his Foundations of Analysis of 1929, Landau discusses to the confusion just mentioned, but without full attention to the logic of the situation. The present article may be considered as a sketch of how Landau’s book might be updated.

Augustus De Morgan: Mathematical Induction

The first  paper of our Blog is the famous Penny Cyclopedia article of 1838 by Augustus De Morgan which contains a description of mathematical induction in the form it is used now in mathematical textbooks. PDF file of the paper.

Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician and a founder and the first President of the London Mathematical Society.  He formulated De Morgan’s laws and is seen as one of the creators of mathematical logic.

The paper was first published in The Penny Cyclopedia of the Society for the Diffusion of Useful Knowledge, vol. 12. London: Charles Knight and Co., 22, Ludgate Street, 1838. A scanned image of the original is available on Google Books, http://tinyurl.com/PennyCyclopedia

Please refer to this paper as A. De Morgan, Mathematical induction. The De Morgan Journal 1 no. 1 (2011), 1–2.