them any more than that a collection of beams and stones will make a cathedral. Mere haphazard construction does not produce the cathedral either. To reach the end it is necessary to have the end in view from the beginning. It is not only necessary to choose a route, but we must see that it is the route to be chosen. This implies a power of the mind which Poincaré calls intuition. It is that power which enables us to perceive the plan of the whole, to seize the unity in the matter at hand. This power is necessary not only to the investigator, but it is also necessary—in less degree, perhaps—to him who desires to follow the investigation. Why is it, he asks, that any one can ever fail to understand mathematics? Here is a subject constructed step by step with infallible logic, yet many do not comprehend it at all. Not on account of poor memory—that may lead to errors in calculation, but has little to do with comprehension of the subject. Sylvester, for example, was notorious for his inability to remember even what he himself had proved. It is not due to lack of the power of attention, for while concentration is necessary in the development of a demonstration, or in following a piece of logic, it does riot give this appreciation of mathematics. A mathematical demonstration is a series of inferences, but it is above all a series of inferences in a certain order. The important thing is the order, just as in chess the mere moving according to the rules is not enough, it is the plan of the game that counts. If one appreciates it, this order, this plan, this unity, this harmony, he need have no fear of a poor memory, nor need he weary his concentration. The student deficient in this power may learn demonstrations by heart, he may assent to each step as logically proved, yet he will know little of the theorem itself. Those who possess this kind of insight which reveals hidden relations, this divining power for the discovery of mines of gold, may hope to become investigators, creators. Those who do not have it must find it or give up the task. The great educational question of the day is the problem of the development of the intuition. If we learn to cultivate this spirituelle flower it will open all doors of invention and discovery of Jaws. It is an interesting problem for even the grade teacher. If it be true, as Boris Sidis and others have claimed, that there are superior methods of education (which seem really to lie along this line) then they must become the methods of future education. We will begin to educate for genius. One thing seems evident, that too prolonged adherence to the methods of rigid reasoning leads to sterility. In mathematics at least both logic and intuition are indispensable, one furnishes the architect's plan of the structure, the other bolts it and cements it together. Logic, says Poincaré, is the sole instrument of certitude, intuition of creation. Yet even the steps of a logical deduction are planned in their entirety by the intuition. In discussing the partial differential equations of physics[1]
- ↑ Amer. Jour. Math., Vol. 12.
