634 SCIENCE AND HYPOTHESIS 



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. 



I asked at the outset why 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 four or five 

 moves ahead; but, however extraordinary a player he may be, 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 our- 

 selves very far from the infinitesimal analysis, but the idea of math- 

 ematical infinity is already playing a preponderating part, and with- 

 out it there would be no science at all, because there would be nothing 

 general. 



The views upon which reasoning by recurrence is based may be 

 exhibited in other forms; we may say, for instance, that in any finite 

 collection of different integers there is always one which is smaller 

 than any other. "We may readily pass from one enunciation to an- 

 other, and thus give ourselves the illusion of having proved that 

 reasoning by recurrence is legitimate. But we shall always be brought 

 to a full stop we shall always come to an indemonstrable axiom, 

 which will at bottom be but the proposition we had to prove translated 

 into another language. We cannot therefore escape the conclusion 

 that the rule of reasoning by recurrence is irreducible to the principle 

 of contradiction. Nor can the rule come to us from experiment. Ex- 

 periment may teach us that the rule is true for the first ten or the first 

 hundred numbers, for instance; it will not bring us to the indefinite 

 series of numbers, but only to a more or less long, but always limited, 

 portion of the series. 



Now, if that were all that is in question, the principle of contradic- 

 tion would be sufficient, it would always enable us to develop as many 



