If I am given a formula and I am ignorant of its meaning, it cannot teach me anything, but if I already know it what does the formula teach me?
Students attend our lectures, not because the mathematics we teach ‘makes lots of fun’ for us, but because they believe they can learn some essential knowledge from us. And each of our young students has only one life to live. We should therefore be able to justify ourselves to our listeners with respect to what we teach them.
To appreciate the living spirit rather than the dry bones of mathematics, it is necessary to inspect the work of a master at first hand. Textbooks and treatises are an unavoidable evil…The very crudities of the first attack on a significant problem by a master are more illuminating than all the pretty elegance of the standard texts which has been won at the cost of perhaps centuries of finicky polishing.
In Samoa, when elementary schools were first established, the natives developed an absolute craze for arithmetical calculations. They laid aside their weapons and were to be seen going about armed with slate and pencil, setting sums and problems to one another and to European visitors. The Honourable Frederick Walpole declares that his visit to the beautiful island was positively embittered by ceaseless multiplication and division.
Thus metaphysics and mathematics are, among all the sciences that belong to reason, those in which imagination has the greatest role.
I suppose you are two fathoms deep in mathematics, and if you are, then God help you, for so am I, only with this difference, I stick fast in the mud at the bottom and there I shall remain.
An educated mind is, as it were, composed of all the minds of preceding ages.
Each problem that I solved became a rule which served afterwards to solve other problems.
This therefore is Mathematics:
She reminds you of the invisible forms of the soul;
She gives life to her own discoveries;
She awakens the mind and purifies the intellect;
She brings light to our intrinsic ideas;
She abolishes oblivion and ignorance which are ours by birth.
A good stack of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one.
The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver.
The art of doing mathematics consists in finding that special case which contains all the germs of generality.
The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, refuge from the goading urgency of contingent happenings, and the sort of beauty changeless mountains present to senses tried by the present-day kaleidoscope of events.
A child[’s]…first geometrical discoveries are topological…If you ask him to copy a square or a triangle, he draws a closed circle.
There is no more a math mind, than there is a history or an English mind.