"It would be pedantic to seek to establish a fundamental difference, even if we may regard the non-mathematical intuition of Goethe as a very striking case. Moreover, as I have often emphasized, all great achievements of science start from intuitive knowledge, namely, in axioms, from which deductions are then made. It is possible to arrive at such axioms only if we gain a true survey of thought-complexes that are not yet logically ordered; so that, in general, intuition is the necessary condition for the discovery of such axioms. And it cannot be denied that, in the great majority of minds with a mathematical tendency, this intuition exhibits itself as a characteristic of their creative power." "From these remarks it would appear that you value deduction considerably higher than induction. Perhaps in using these catchwords I am expressing myself a little vaguely; it seems to me that great things have been achieved, too, by using inductive processes." "Let us first define what each of these terms means. Deduction is the derivation of the particular from the general, whereas induction is the process of deriving the general from the particular case. Now, quote any example of a brilliant achievement, which you feel illustrates the power of the inductive method. Of whatever kind your example may be, you will soon become aware of the difference in the significance of the two processes." "For me the most perfect example of induction is given by certain reasoning of Euclid. The question was whether there is a finite or an infinite number of primes (that is, numbers that cannot be divided without leaving a remainder except by unity). Euclid found an elegant proof that the total number is infinite by the following strictly inductive reasoning. If the total number were finite there would have to be a _greatest_ prime. Let us call it _n_, and then form the product of all primes up to _n_ and including it, finally adding one, thus: 2 x 3 x 5 x 7 x 11 x 13 ... _n_, plus 1. This new number, say Y, is certainly greater than _n_, and now there are two possibilities, either _n_ is prime or it is not prime. "If it is not prime, it must be divisible by some existing prime. But the primes up to and including _n_ cannot divide exactly into Y, as there is always a remainder, namely, 1. Hence Y must be divisible by an existing prime X greater than _n_. This contradicts the assumption that _n_ is the greatest prime, for X is shown to be greater than _n_.