Proof By Recurrence

This monotonous series of reasonings exposes a nature of uniformity at every step.

• The process is proof by recurrence.

We first show that a theorem is true for n = 1.

• We then show that if it is true for n−1 it is true for n.
• We conclude that it is true for all integers.

This proof by recurrence may be used for the proof of the rules of addition and multiplication – the rules of the algebraical calculus. This calculus:

• is an instrument of transformation.
• lends itself to many more different combinations than the simple syllogism.

But it is still:

• a purely analytical instrument
• incapable of teaching us anything new.

If mathematics had no algebraical calculus, no other instrument, then its development would immediately stop.

• But it uses the same process, as reasoning by recurrence, so it can continue its forward march.

5. Reasoning by recurrence

Reasoning by Recurrence is therefore mathematical reasoning par excellence.

• The essential characteristic of reasoning by recurrence is that it contains, condensed in a single formula, an infinite number of syllogisms.

They follow one another in a cascade*.

The following are the hypothetical syllogisms:

The theorem is true of the number 1.

• If it is true of 1, then it is true of 2
• Therefore it is true of 2.
• If it is true of 2, it is true of 3 and so on.

The conclusion of each syllogism serves as the minor of its successor.

Further, the majors of all our syllogisms may be reduced to a single form.

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

Thus in reasoning by recurrence, we confine ourselves to the enunciation of the minor of the first syllogism, and the general formula which contains as particular cases all the majors.

This unending series of syllogisms is thus reduced to a phrase of a few lines.

• This makes it easy to understand why every particular consequence of a theorem may be verified by purely analytical processes.

If, instead of proving that our theorem is true for all numbers, we only wish to show that it is true for the number 6 for instance.

• It will be enough to establish the first five syllogisms in our cascade.

We shall require 9 if we wish to prove it for the number 10.

• For a greater number we shall require more still
• But however great the number may be we shall always reach it and the analytical verification will always be possible.
• But however far we went we should never reach the general theorem applicable to all numbers, which alone is the object of science.

To reach it we should require an infinite number of syllogisms, and we should have to cross an abyss which the patience of the analyst, restricted to the resources of formal logic, will never succeed in crossing.

We cannot conceive of a mind powerful enough to see at a glance the whole body of mathematical truth.

The answer is now easy.

A chess-player can combine for 4-5 moves ahead. But he cannot prepare for more than a finite number of moves.

• If he applies his faculties to Arithmetic, he cannot conceive its general truths by direct intuition alone.
• To prove even the smallest theorem he must use reasoning by recurrence, for that is the only instrument which enables us to pass from the finite to the infinite.
• This instrument is always useful, for it enables us to leap over as many stages as we wish; it frees us from the necessity of long, tedious, and monotonous verifications which would rapidly become impracticable.

Then when we take in hand the general theorem it becomes indispensable, for otherwise we should ever be approaching the analytical verification without ever actually reaching it.

In this domain of Arithmetic, we may think ourselves very far from the infinitesimal analysis, but the idea of mathematical infinity is already playing a preponderating part, and without itnature of mathematical reasoning.

There would be no science at all, because there would be nothing general.