Engine of Structuralism

The Gothic Alphabet

Euclides and his relation to equation

The illustration speaks for itself.

Cylinders and logic

The adulthood of our much loved Boole-algebra.

The Three Petals of Propositional Logic

With the category-theoretic notion of the adjunction somebody can easily define the basic propositional connectives in the preordered set of formulae and the consequence relation: For example ex falso quodlibet iff the very simple (constant) function which range is the singleton has a left adjoint. That was a half petal. Let's see the other two and a half...

Boolos, Gödel the Second, and Words of One Syllable

First of all, when i say "proved", what I will mean is "proved with the aid of the whole math." Now then: two plus two is four as you well know. And of course it can be proved that two plus two is four (proved, that is with the aid of the whole of math, as I said, though in the case of two plus two, of course we do not need the whole of math to prove that it is four). And, as may not be quiet so clear, it can be proved that it can be proved that two plus two is four, as well.
And it can be proved that it can be proved that it can be proved that two plus two is four. And so on. In fact, if a claim can be proved, then it can be proved that the claim can be proved. And that too can be proved.

Now: two plus two is not five. And it can be proved that two plus two is not five. And it can be proved that it can be proved that two plus two is not five, and so on.

Thus: it can be proved that two plus two is not five. Can it be proved as well that two plus two is five? It would be a real blow to math, to say the least, if it could. If it could be proved that two
plus two is five, then it could be proved that five is not five, and then there would be no
claim that could not be proved, and math would be a lot of bunk.

So, we now want to ask, can it be proved that it can't be proved that two plus two is five? Here's the shock: no, it can't. Or to hedge a bit: if it can be proved that it can't be proved that two plus two is five, then it can be proved as well that two plus two is five, and math is a lot of bunk. In fact, if math is not a lot of bunk, then no claim of the form "claim X can't be proved " can be proved.

So if math is not a lot of bunk, then, though it can't be proved that two plus two is five, it can't be proved that it can't be proved that two plus two is five.

By the way, in case you'd like to know: yes, it can be proved that
if it can be proved that it can't be proved that two plus two is five, then it can be proved that two plus two is five.

George Boolos: Gödel's Second Incompleteness Theorem Explained in Words of One Syllable
in: Logic, Logic, and Logic
Harvard University Press
Cambridge, Massachusetts
London, England
1998

Queen of calculi

Gerhard Gentzen’s sequent calculus. Here the falsum is the sequent ” ->”, which can be obtained by the usage of some cut-rules only (the rule in the box). But if we eliminate the cut-rule, the sequent calculus we gain in this way is deductively equivalent with the original one. So it is consistent. Voila.

The formulae are from Gerhard Gentzen: Recherches Sur La Déduction Logique