Jean-Yves Girard - Lectures at IHP thematic trimester : Semantics of proofs and certified mathematics

Lectures given in April 2014. You can really tell he was at a French University in 1968!

Qu'est-ce qu'une réponse ? (l'analytique)

47:53 On Gentzen's distinction between implication and cut and Lewis Carroll's What the Tortoise said to Achilles. Around the time he was presenting these lectures in Paris I was homeless in Caranavi, Bolivia writing to people in Cambridge about exactly this subject.

He doesn't seem to have thought of the possibility that Lewis Carroll, who was a geometer, might have been talking about something more than symbolic logic. The form of the five common notions in the Elements is that of formal rules of inference, and that is how they are used implicitly in proofs:

(A) Things that are equal to the same are equal to each other.
(B) The two sides of this Triangle are things that are equal to the same.
(Z) The two sides of this Triangle are equal to each other.

Consider whether the character of (A) is that of a statement about a geometric fact or a statement about logical inference. In the former sense it is stating a geometric fact and in the latter it is stating a valid inference. So When (A) is applied to (B) it is operating in the former sense, and when (A) is applied to (Z) it is affirming it as a valid inference. Hence (A) is here acting as a function, (B) as its argument and (Z) as the implied consequence.

Readers of Euclid will grant, I suppose, that Z follows logically from A and B, so that any one who accepts A and B as true, must accept Z as true?"

"Undoubtedly! The youngest child in a High School -- as soon as High Schools are invented, which will not be till some two thousand years later -- will grant that."

What starts the infinite regress is this (a Hypothetical reader!):

And might there not also be some reader who would say 'I accept A and B as true, but I don't accept the Hypothetical '?"

The essence of the argument I am trying to make is that Euclid's Elements is not only about geometry, it also about logic, arithmetic and measure in general.

It is this hypothetical reader for whom the Elements was written:

"And if some reader had not yet accepted A and B as true, he might still accept the sequence as a valid one, I suppose?"

"No doubt such a reader might exist. He might say 'I accept as true the Hypothetical Proposition that, if A and B be true, Z must be true; but, I don't accept A and B as true.' Such a reader would do wisely in abandoning Euclid, and taking to football."

What greets this person at the end of Proposition 18 of Book XIII is the constructed space measured by perfect uniform spheres from which the hypotheses of Proposition 1 of Book I were drawn. Then they see that in fact the circles must intersect, because that is by construction: the circles are cross-sections of intersecting spheres centred on a line and each passing through the centre of the other. From this we see then that the planar geometry emerges as an arbitrary section through the whole space in which the equilateral triangle happens to be, and that gives meaning to the stereometry when we consider other distinct sections and relate them together. This is the substance of Lewis Carroll's (1867) An elementary treatise on determinants : with their application to simultaneous linear equations and algebraical geometry.

The first four Common Notions allow the constructions of the Elements to be composed together, so that one may apply them in any context which meets the conditions of the premisses. For example, having constructed an icosahedron in a sphere, one can construct a dodecahedron in a different sphere which shares the length of its sides with those of the icosahedron. The fifth Common Notion says that you can continue this process in either direction. The postulates too are described in the original Greek in a way which suggests that this how one ought to read Euclid. As Fitzpatrick notes. The Greek present perfect tense indicates a past action with present significance. Hence, the 3rd-person present perfect imperative could be translated as “let it be postulated”, in the sense “let it stand as postulated”, but not “let the postulate be now brought forward”. The literal translation “let it have been postulated” sounds awkward in English, but more accurately captures the meaning of the Greek.

Note also the structural similarity of the five common notions with the five postulates: