The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method. Henri Poincare

+ (b + 1) = (a + b) + 1,

      which, apart from the difference of notation, is nothing but the equality (1), by which I have just defined addition.

      Supposing the theorem true for c = γ, I say it will be true for c = γ + 1.

      In fact, supposing

      (a + b) + γ = a + (b + γ),

      it follows that

      [(a + b) + γ] + 1 = [a + (b + γ)] + 1

      or by definition (1)

      (a + b) + (γ + 1) = a + (b + γ + 1) = a + [b + (γ + 1)],

      which shows, by a series of purely analytic deductions, that the theorem is true for γ + 1.

      Being true for c = 1, we thus see successively that so it is for c = 2, for c = 3, etc.

      Commutativity.—1º I say that

      a + 1 = 1 + a.

      The theorem is evidently true for a = 1; we can verify by purely analytic reasoning that if it is true for a = γ it will be true for a = γ + 1; for then

      (γ + 1) + 1 = (1 + γ) + 1 = 1 + (γ + 1);

      now it is true for a = 1, therefore it will be true for a = 2, for a = 3, etc., which is expressed by saying that the enunciated proposition is demonstrated by recurrence.

      2º I say that

      a + b = b + a.

      The theorem has just been demonstrated for b = 1; it can be verified analytically that if it is true for b = β, it will be true for b = β + 1.

      The proposition is therefore established by recurrence.

      Definition of Multiplication.—We shall define multiplication by the equalities.

      (1) a × 1 = a.

      (2) a × b = [a × (b − 1)] + a.

      Like equality (1), equality (2) contains an infinity of definitions; having defined a × 1, it enables us to define successively: a × 2, a × 3, etc.

      Properties of Multiplication.—Distributivity.—I say that

      (a + b) × c = (a × c) + (b × c).

      We verify analytically that the equality is true for c = 1; then that if the theorem is true for c = γ, it will be true for c = γ + 1.

      The proposition is, therefore, demonstrated by recurrence.

      Commutativity.—1º I say that

      a × 1 = 1 × a.

      The theorem is evident for a = 1.

      We verify analytically that if it is true for a = α, it will be true for a = α + 1.

      2º I say that

      a × b = b × a.

      The theorem has just been proven for b = 1. We could verify analytically that if it is true for b = β, it will be true for b = β + 1.

      IV

      Here I stop this monotonous series of reasonings. But this very monotony has the better brought out the procedure which is uniform and is met again at each step.

      This procedure is the demonstration by recurrence. We first establish a theorem for n = 1; then we show that if it is true of n − 1, it is true of n, and thence conclude that it is true for all the whole numbers.

      We have just seen how it may be used to demonstrate the rules of addition and multiplication, that is to say, the rules of the algebraic calculus; this calculus is an instrument of transformation, which lends itself to many more differing combinations than does the simple syllogism; but it is still an instrument purely analytic, and incapable of teaching us anything new. If mathematics had no other instrument, it would therefore be forthwith arrested in its development; but it has recourse anew to the same procedure, that is, to reasoning by recurrence, and it is able to continue its forward march.

      If we look closely, at every step we meet again this mode of reasoning, either in the simple form we have just given it, or under a form more or less modified.

      Here then we have the mathematical reasoning par excellence, and we must examine it more closely.

      V

      The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms.

      That this may the better be seen, I will state one after another these syllogisms which are, if you will allow me the expression, arranged in 'cascade.'

      These are of course hypothetical syllogisms.

      The theorem is true of the number 1.

      Now, if it is true of 1, it is true of 2.

      Therefore it is true of 2.

      Now, if it is true of 2, it is true of 3.

      Therefore it is true of 3, and so on.

      We see that the conclusion of each syllogism serves as minor to the following.

      Furthermore the majors of all our syllogisms can be reduced to a single formula.

      If the theorem is true of n − 1, so it is of n.

      We see, then, that in reasoning by recurrence we confine ourselves to stating the minor of the first syllogism, and the general formula which contains as particular cases all the majors.

      This never-ending series of syllogisms is thus reduced to a phrase of a few lines.

      It is now easy to comprehend why every particular consequence of a theorem can, as I have explained above, be verified by purely analytic procedures.

      If instead of showing that our theorem is true of all numbers, we only wish to show it true of the number 6, for example, it will suffice for us to establish the first 5 syllogisms of our cascade; 9 would be necessary if we wished to prove the theorem for the number 10; more would be needed for a larger number; but, however great this number might be, we should always end by reaching it, and the analytic verification would be possible.

      And yet, however far we thus might go, we could never rise to the general theorem, applicable to all numbers, which alone can be the object of science. To reach this, an infinity of syllogisms would be necessary; it would be necessary to overleap an abyss that the patience of the analyst, restricted to the resources of formal logic alone, never could fill up.

      I asked at the outset why one could not conceive of a mind sufficiently powerful to perceive at a glance the whole body of mathematical truths.

      The answer is now easy; a chess-player is able to combine four moves, five moves, in advance, but, however extraordinary he may be, he will never prepare more than a finite number of them; if he applies his faculties to arithmetic, he will not be able to perceive its general truths by a single direct intuition; to arrive at the smallest theorem he can not dispense with the aid of reasoning by recurrence, for this is an instrument which enables us to pass from the finite to the infinite.

      This instrument is always useful, for, allowing us to overleap at a bound as many stages as we wish, it spares us verifications, long, irksome and monotonous, which would quickly become impracticable. But it becomes indispensable as soon as we aim at the general theorem, to which analytic verification would bring us continually nearer without