Nothing must be advanced in a positive manner. The mind of the pupil is to be the principal operator; it must instruct, convince, and confute itself; and when it arrives at some important truth or result, it must be through its own powers. It ought not even to perceive that it has been guided thither.
I think the universe is pure geometry - basically, a beautiful shape twisting around and dancing over space-time.
Men are liars. We'll lie about lying if we have to. I'm an algebra liar. I figure two good lies make a positive.
There is geometry in the humming of the strings, there is music in the spacing of the spheres.
God exists since mathematics is consistent, and the Devil exists since we cannot prove it.
Poetry is a form of mathematics, a highly rigorous relationship with words.
Mathematics is the most beautiful and most powerful creation of the human spirit.
Games give you a chance to excel, and if you're playing in good company you don't even mind if you lose because you had the enjoyment of the company during the course of the game.
I like mathematics because it is not human and has nothing particular to do with this planet or with the whole accidental universe - because, like Spinoza's God, it won't love us in return.
The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.
Drop a stone into water. It makes a sound, "glop" for a big stone, "splitch" for a small stone. Can you predict the pitch of the sound from the size of the stone? The usual teaching of mathematical theories is like a pyramid. Young people tend to become passive (if passionate) admirers of a structure built by old people, and problems they are taught to solve make them walk straight up to the peak. But what if we want to explore a natural mountain range, whose peaks are invisible among clouds, whose trails among trees are unknown? The problem of the sound of a stone falling into water is natural, so natural that every child knows the phenomenon and can wonder about it. The mathematics involved is extremely hard, so hard that it is not taught at any mathematics department in the world.
I am interested in mathematics only as a creative art.
There should be no such thing as boring mathematics.
Mathematics is written for mathematicians.
God made integers, all else is the work of man.
Music is the arithmetic of sounds as optics is the geometry of light.
There is no royal road to geometry.
There is no other science so admirably calculated to draw out the thinking faculties in children, and therefore none which forwards so effectually the purposes of elementary education. It is on this account that... all modern reformers of education, have devoted so much of their attention to arithmetic and geometry. These two studies, when pursued in the proper manner, go hand in hand, and form the very ground-work of an intellectual education. They are almost the only branches taught at schools, which call upon the pupil's judjement, prompt his mind to thought and reflection, and teach him to reason from given things to things unknown. They lead the pupil to institute comparisons, and to determine the relations which things bear to each other. Thus exercising every faculty of his mind, he acquires a habit of close attention, and strengthens and confirms his mental energies by concentrating them, and bringing all to bear upon the same point. With this regard it may be said that mathematics facilitate every other study. The mind which is once to a certain degree developed, and which has acquired the habit of thinking and reasoning, can easily apply the same powers to other branches of knowledge. Whatever study the pupil may now undertake, is entered upon more systematically. He is now accustomed to investigate for himself. He has not the same dependence on authority; nor does he lay so much stress on the conceptions of others, until they are made his own by a fair appeal to his understanding.