The methods of mathematics are the main topic of the course, not a long list of finished mathematical results with such highly polished proofs that t… - Richard Hamming

Mathematics, being very different from the natural languages, has its corresponding patterns of thought. Learning these patterns is much more important than any particular result... They are learned by the constant use of the language and cannot be taught in any other fashion.

The calculus is probably the most useful single branch of mathematics. ...I have found the ability to do simple calculus, easily and reliably, was the most valuable part of mathematics I ever learned.

To summarize, from simple counting using the God-given integers, we made various extensions of the ideas of numbers to include more things. Sometimes the extensions were made for what amounted to aesthetic reasons, and often we gave up some property of the earlier number system. Thus we came to a number system that is unreasonably effective even in mathematics itself; witness the way we have solved many number theory problems of the original highly discrete counting system by using a complex variable.