Tag Archives: Timothy Gowers

Mathematics and Logic

Large parts of this post are taken up with two subjects:

  1. The notion (due to Collingwood) of criteriological sciences, logic being one of them.

  2. Gödel’s theorems of completeness and incompleteness, as examples of results in the science of logic.

Like the most recent in the current spate of mathematics posts, the present one has arisen from material originally drafted for the first post in this series.

In that post, I defined mathematics as the science whose findings are proved by deduction. This definition does not say what mathematics is about. We can say however what logic is about: it is about mathematics quâ deduction, and more generally about reasoning as such. This makes logic a criteriological science, because logic seeks, examines, clarifies and limits the criteria whereby we can make deductions. As examples of this activity, Gödel’s theorems are, in a crude sense to be refined below, that

  • everything true in all possible mathematical worlds can be deduced;

  • some things true in the world of numbers can never be deduced;

  • the latter theorem is one of those things.

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Logic of Elliptic Curves

In my 1997 doctoral dissertation, the main idea came as I was lying in bed one Sunday morning. Continue reading

Boolean Arithmetic

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

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Hypomnesis

When is a help a hindrance? The Muses have provoked this question. They did this through their agents, the cicadas, who sang around the European Cultural Center of Delphi, during the 11th Panhellenic Logic Symposium, July 12–5, 2017.

     Cicada, European Cultural Center of Delphi, 2017.07.15     

Cicada, European Cultural Center of Delphi, 2017.07.15

My question has two particular instances.

  1. At a mathematical conference, can theorems “speak for themselves,” or should their presenters be at pains to help the listener appreciate the results?

  2. When the conference is in Greece, even at one of the country’s greatest archeological sites, does this enhance the reading of ancient Greek texts, or is it only a distraction?

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