Tag Archives: Malcev

On Gödel’s Incompleteness Theorem

December 15, 2018 – 8:43 am

This is an appreciation of Gödel’s Incompleteness Theorem of 1931. I am provoked by a depreciation of the theorem.

I shall review the mathematics of the theorem, first in outline, later in more detail. The mathematics is difficult. I have trouble reproducing it at will and even just confirming what I have already written about it below (for I am adding these words a year after the original publication of this essay).

The difficulty of Gödel’s mathematics is part of the point of this essay. A person who thinks Gödel’s Theorem is unsurprising is probably a person who does not understand it.

Next to a slice from a tree trunk, a worn copy of the book Frege and Gödel: Two Fundamental Texts in Mathematical Logic, edited by Jean van Heijenoort

Continue reading →

By David Pierce | Posted in Archimedes, Categorical Thinking, Descartes, Euclid, Exposition, Hitchens, Mathematics, Philosophy, Philosophy of Mathematics, Science | Also tagged , , , , , , , , , , , , , , , , , , , , | Comments (16)

Boolean Arithmetic

May 5, 2018 – 7:38 am

Mathematics can be highly abstract, even when it remains applicable to daily life. I want to show this with the mathematics behind logic puzzles, such as how to derive a conclusion using all of the following premisses:

  1. Babies are illogical.
  2. Nobody is despised who can manage a crocodile.
  3. Illogical persons are despised.

The example, from Terence Tao’s blog, is attributed to Lewis Carroll. By the first and third premisses, babies are despised; by the second premiss then, babies cannot manage crocodiles.

George Boole, The Laws of Thought (1854), Open Court, 1940

Continue reading →
By David Pierce | Posted in Collingwood, Descartes, Exposition, G. H. Hardy, Language, Logic, Mathematics, Principles of Art | Also tagged , , , , , , , , , , , , , | Comments (7)