"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.
"Secondly, if Y is a prime, it immediately follows that n cannot be the greatest prime, for Y is greater than n. Hence, however great may be any prime that we may assume, there will always be one that is greater, and even if we do not succeed in expressing it in figures, we see that it must certainly exist. Thus by studying carefully a particular case—the prime n, which was assumed to be the greatest possible one—we have arrived at a general theorem which states that there is no limit to the number of primes. Is not that, too, a triumph of intuition?"
"Certainly," said Einstein. "But you must not overlook the fact that a theorem of this kind cannot be ranked with a theorem of a fundamentally axiomatic character. The one you have discussed has been derived by a clever process of reasoning, but it does not exhibit the characteristic of a momentous discovery. This theorem of Euclid can be imagined absent from science without the content of truth in science being essentially effected. Compare with it a theorem of axiomatic significance, such as Galilei's Law of Inertia, or Newton's Law of Gravitation. Theorems such as the latter are characterized by being starting-points of knowledge that are inexhaustible in the consequences that may be deduced from them. Your question, earlier, as to whether I consider the deductive method superior to the inductive, was not formulated in correct terms. To this I answered above that the inductive method as a means of discovering general truths usually appears over-estimated. The proper form of the question is: Which truths are of the higher order, those that are found inductively, or those that lead to further deduction? There can scarcely be doubt about the answer."
"No, that is certainly true. If I understand your meaning rightly, the answer may be expressed by an allegory. Intuition of the highest order creates treasure-mines, those of lesser degree individual articles of value that are significant in themselves, although they cannot be compared with the inestimable value of the mines. The fact that the highest intuition is found in minds with a mathematical trend makes it appear possible that Kant's remark may gain more and more credence in the future. It already applies in a measure to subjects to which it seemed inapplicable during Kant's lifetime, for example, in Psychology, in which the relations between stimulus and response have been established mathematically only since the Weber-Fechner Law was set up; and also, since the time of Quetelet, in Moral Science and Sociology, we learn from mathematical methods of statistics and probability that even Man as an active being is subjected to mechanical causality. At any rate it seems manifest that Kant's remark, that in every science there is just as much truth as there is mathematics, has received additional support in recent times."
"That may be admitted," concluded Einstein, "without recognizing his remark as an axiom. It is still far removed from making possible unassailable deductions, and will never quite succeed in doing so; yet it may claim equal significance as a beautifully expressed idea with that of Pythagoras, which asserts number to be the nature of all things."
III
"The lines of demarcation between 'conceptual knowledge' (Erkennen) and 'perceptual knowledge' (Kennen) are being drawn more and more closely nowadays. The former is regarded as being the exclusive possession of the highly developed human mind, and the latter as being characteristic of the lower intelligence of other living creatures. Is this not a pronounced case of anthropomorphism, and does it not mislead us to form opinions that we should at once disown if we succeed in stepping out of our human frames even for a moment?"