Category Theory Illustrated - Logic

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2021-09-26 12:00:03

Now let’s talk about one more seemingly unrelated topic just so we can “surprise” ourselves when we realize it’s category theory. By the way, in this chapter there will be another surprise in addition to that, so don’t fall asleep.

Logic is the science of the possible. As such, it is at the root of all other sciences, all of which are sciences of the actual, i.e. that which really exists. For example, if science explains how our universe works then logic is the part of the description which is also applicable to any other universe that is possible to exist.

Logic studies the rules by which knowing one thing leads you to conclude (or prove) that some other thing is also true, regardless of the things’ domain (e.g. scientific discipline) and by only referring to their form.

Seeing this description, we might think that the subject of logic is quite similar to the subject of set theory and category theory, as we described it in the first chapter - instead of the word “formal” we used another similar word, namely “abstract”, and instead of “logical system” we said “theory”. This observation would be quite correct - today most people agree that every mathematical theory is actually logic plus some additional definitions added to it. For example, part of the reason why set theory is so popular as a theory for the foundations of mathematics is that it adds just one single primitive to the standard axioms of logic which we will see shortly - the binary relation that indicates set membership. Category theory is close to logic too, but in a quite different way.