Tag Archives: Raymond Smullyan

Gödel, Grammar, and Mathematics


This attempt at exposition of Gödel’s Incompleteness Theorem was inspired or provoked by somebody else’s attempt at the same thing, in a blog post that a friend directed me to (by means of a Twitter message on November 17, 2020). I wanted in response to set the theorem in the context of mathematics rather than computer science.

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On Plato’s Republic, 8

Index to this series

Plato is somehow quite challenging in the present reading, which is the first part (Stephanus 484–502d) of Book VI of the Republic. Socrates tries to work out the third wave from the previous reading. Significant features are several analogies or figures:

  • city as ship whose sailors neither know how to sail nor want to know;
  • people and sophist as beast and zoologist or zookeeper;
  • ruler as painter who compares a canvas with what the mind’s eye sees;
  • philosopher as seed that needs good soil, lest it become a noxious weed.

I concurrently discuss the Republic readings in a group formed through the Catherine Project, which now has the website just linked to. The same was true for Pascal in the winter and Chaucer in the summer.

Bookshelves in morning sun
Ayşecik Sokağı, Fulya, Şişli, İstanbul, October 14, 2021.
The order of the books on the shelves of the cases being like that of words on the lines of pages of an individual book, the ordering is chronological, by birth date of author, editor, or personal subject. The youngest author for now is Sally Rooney, and Zena Hitz is on the same shelf. Plato is on the opposite wall.

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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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