One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed. This gives students the idea that there is a book somewhere with all the right answers to all of the interesting questions, and that teachers know those answers. And if one could get hold of the book, one would have everything settled. That's so unlike the true nature of mathematics.
The student is best taught who is told the least
Newton's health, and confusion to mathematics.
Nature imitates mathematics.
Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality.
I Once wrote: "In mathematics process and result are equivalent. "
Mathematics is distinguished from all other sciences except only ethics, in standing in no need of ethics.
Without mathematics there is no art.
My basic mathematics is rather weak, so when some of the theories are broken into equations, I get rather lost.
Mathematics is the science of patterns, and nature exploits just about every pattern that there is.
The practice of first developing a clear and precise definition of a process without regard for efficiency, and then using it as a guide and a test in exploring equivalent processes possessing other characteristics, such as greater efficiency, is very common in mathematics. It is a very fruitful practice which should not be blighted by premature emphasis on efficiency in computer execution.
A mathematics teacher is a midwife to ideas.
In mathematics, if I find a new approach to a problem, another mathematician might claim that he has a better, more elegant solution. In chess, if anybody claims he is better than I, I can checkmate him.
. . . nature seems very conversant with the rules of pure mathematics, as our own mathematicians have formulated them in their studies, out of their own inner consciousness and without drawing to any appreciable extent on their experience of the outer world.
No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game
The instruction of children should aim gradually to combine knowing and doing. Among all sciences mathematics seems to be the only one of a kind to satisfy this aim most completely.
Mathematics in general is fundamentally the science of self-evident things.
It is here [in mathematics] that the artist has the fullest scope of his imagination.
It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment.
A line is length without breadth.