Tag Archives: Raymond Smullyan

Mathematics and Logic

I continue with the mathematics posts, taking up, as I did in the last, material originally drafted for the first.

Designated for its own post, material can grow, as has the material of this post in the drafting. Large parts of it are taken up with

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

  2. Gödel’s logical theorems of completeness and incompleteness.

I have 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. This makes logic a criteriological science, since it seeks, examines, clarifies and limits the criteria whereby we can make deductions. As examples of this activity, Gödel’s theorems are 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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