From a March 1940 letter to her brother, André, collected in A Life in Letters, which will be published next month by Harvard University Press. Translated from the French by Nicholas Elliott.
My dear brother—
The Greeks’ originality in terms of mathematics isn’t, as I see it, their refusal to accept approximation. There is no approximation in the Babylonian problems, and for a very simple reason: it’s because they are constructed from the solutions. Thus there are dozens (or hundreds, I don’t remember) of fourth-degree problems with two unknowns that all have the same solution. This shows that the Babylonians were interested only in the method and not in solving problems actually posed. They enjoyed supposing unknown what is given, and known what is not. It’s a game, obviously, that does the greatest honor to their conception of disinterested research (did they have scholarships and medals to stimulate them?). But it’s only a game.
This game must have seemed profane to the Greeks, or even impious. The Greeks did not see any value in a method of reasoning for its own sake. They valued it insofar as it allowed the effective study of concrete problems; not because the Greeks were avid for technical applications, but because their sole object was to conceive more and more clearly of the relationship between the human mind and the universe. Purity of soul was their only concern; imitating God was its secret; the study of mathematics helped to imitate God insofar as one saw the universe as subject to mathematical laws, which made the geometer an imitator of the supreme legislator. It’s clear that the Babylonians’ mathematical games, in which the solution was given before the data, were useless to this end. What was needed was data actually provided by the world or action on the world; so what was needed was to find ratios that did not require the problems to be artificially prepared to “come out right,” as is the case with whole numbers.
It’s for the Greeks that mathematics was truly an art. Its purpose was the same as their art, namely, to make perceptible a kinship between the human mind and the universe, to make the world appear as “the city of all rational beings.” And it was really made of solid matter, matter that existed, like that of all the arts without exception, in the physical sense of the word; this matter was space actually given, imposed as a de facto condition to all of man’s actions. Their geometry was a science of nature; their physics (I’m thinking of the Pythagoreans’ music, and especially of Archimedes’ mechanics and his study of floating bodies) was a geometry in which the hypotheses were presented as postulates.
I fear that today it is rather toward the Babylonian conception that we’re moving, playing games rather than making art. I wonder how many mathematicians today see mathematics as a process aimed at purifying the soul and imitating God. What’s more, it seems to me that the matter is lacking. There is a lot of axiomatics, which seems to be closer to the Greeks, but aren’t the axioms largely chosen at will? You speak of “solid matter,” but isn’t this matter essentially formed by the entirety of mathematical work accomplished to this day? In that case, current mathematics would be a screen between man and the universe (and consequently between man and God, as understood by the Greeks) instead of putting them in contact. But perhaps I’m disparaging it.
