The discovery in 1846 of the planet Neptune was a dramatic and spectacular achievement of mathematical astronomy. The very existence of this new member of the solar system, and its exact location, were demonstrated with pencil and paper; there was left to observers only the routine task of pointing their telescopes at the spot the mathematicians had marked.

The most painful thing about mathematics is how far away you are from being able to use it after you have learned it.

It is unnecessary to understand electromagnetic theory before wiring a lamp or to study physics in order to repair a pump. We count on our fingers and give no heed to the proliferating implications of the act.

The Theory of Groups is a branch of mathematics in which one does something to something and then compares the result with the result obtained from doing the same thing to something else, or something else to the same thing.

To be sure, mathematics can be extended to any branch of knowledge, including economics, provided the concepts are so clearly defined as to permit accurate symbolic representation. That is only another way of saying that in some branches of discourse it is desirable to know what you are talking about.

There are... scientific works - star catalogues, for example - which are not art; but the theoretical structures of Gauss, Einstein, or Maxwell are original, individual, "very personal" responses and expressions of exactly the same kind as the creative works of Beethoven or Dostoievski.

The self-regulating mechanism of the market place cannot always be depended upon to produce adequate results in scientific research.