I had thought, long and intently, about these theses to discover as nearly as possible what distinguished their content from the usual view. The fundamental differences suggest an abundance of ideas whose importance grows in value as we apply them to various cases as illustrations. And I feel convinced that we shall yet have to occupy ourselves with these words of Einstein, which present themselves as a confession, as with the famous "hypotheses non fingo" that Newton set up as the idea underlying his work.
The latter as well as the former implies something negative: it denies something. In Einstein's words there is apparently a repudiation of the really creative act in discovery; he lays stress on the gradual, methodical constructive factors, not omitting to emphasize intuition. There is no other course open to us but to seek indirectly a synthesis of these conceptions, and to eliminate what is apparently contradictory in them.
I consider this possible if we decide to subdivide the discovery into a series of individual acts in which succession takes the place of instantaneous suddenness The creative factor may then remain intact; indeed, it attains a still higher degree of importance if we imagine to ourselves that a series of creative ideas must be finked together to make possible a single important discovery.
The original idea never springs fully equipped and armed like Minerva out of the head of its creator. And it is wise to bear in mind that even Jupiter had to suffer in his head a period of pregnancy accompanied with great pain. It is only in the after-picture that Pallas Athene appears with the attribute of suddenness. It is the nature of our myth-building imagination to leap over the actual act of birth so as to give a more brilliant form to the finished creation.
We feel great satisfaction when we learn that Gauss, the Prince of Mathematicians, declared in one of his valuable flashes of insight: "I have the result, only I do not yet know how to get to it." For in this utterance we see above all that he emphasizes a lightning-like intuition. He has possession of a thing, which is, however, not yet his own, and which can only become his own when he has found the way to it. Is this contradictory? From the point of view of elementary logic, certainly; but methodologically, by no means. Here it is a question of: Erwirb es um es zu besitzen! This makes necessary a series of further intuitions along the road of invention, and of construction.
This is, then, where that phase commences, which Einstein denotes by the word "gradual," or "by steps." The first intuition must be present; its presence as a rule usually guarantees that further intuition will follow in logical sequence.
This does not always happen. In passing, we discussed several special cases from which particular inferences may be drawn. The powerful mathematician Pierre Fermat has presented the world with a theorem of extremely simple form which he discovered, a proof of which is being sought even nowadays, two and a half centuries after he stated it. In easy language, it is this: the sum of two squares may again be a square, for example, 52 + 122 = 132, since 25 + 144 = 169; but the sum of two cubes can never be a cube, and, more generally, as soon as the exponent, the power index n, is greater than 2, the equation xn + yn = zn can never be satisfied by whole number values for x, y, and z; it is impossible to find three whole numbers for x, y, and z, which, when substituted in the equation, give a correct result.
This is certainly true; it is an intuitive discovery. But Fermat's assertion that he possessed a "wonderful proof," is for very good reasons open to contradiction. No one doubts the absolute truth of the theorem. But the later inspiration, the next step after the intuition, has occurred neither to Fermat nor to anyone else. It cannot be established whether his remark about the proof was due to a subjective error, or was baseless. In any case it seems probable that Fermat had arrived at the result per intuitionem without knowing the way to it. His creative act stopped short; it was only a first flare of a conflagration, and did not fulfil the condition that Einstein associates with the conception of a logically complete method.
We may, indeed, pursue this case of Fermat still further. He had enunciated another theorem, again per intuitionem, namely, that it was possible to construct prime numbers of any magnitude by a formula he gave. Euler later showed by a definite example that the theorem was false. It was stated in a letter to Pascal written in 1654 in the words: the result of squaring 2 continuously and then adding 1 must in each case be a prime number, that is, 22k + 1 must always be a prime no matter what value k may have. Fermat added: "This is a property for the truth of which I answer." Euler chanced to try k = 5, and found that 232 + 1 = 4,294,967,297, which may be represented as the product of 641 and 6,700,417, and hence is not a prime.
It is conceivable that no Euler might have lived, and that no one else might have discovered this contradiction. What would then have been the position of this "discovery" of Fermat?