Consider the sequence {1, 4, 9, 16, 25, ...} You recognize these as the square numbers, the sequence A_k = k². Suppose you did not recognize this sequence at a first glance. Is there any way you could predict the next item in the sequence? Yes: You could take the first differences, and end up with:

{4 - 1, 9 - 4, 16 - 9, 25 - 16, ...} = {3, 5, 7, 9, ...}

And if you don't recognize these as successive odd numbers, you are still not defeated; if you produce a table of the second differences, you will find:

{5 - 3, 7 - 5, 9 - 7, ...} = {2, 2, 2, ...}

If you cannot recognize this as the number 2 repeating, then you're hopeless.

But if you predict that the next second difference is also 2, then you can see the next first difference must be 11, and the next item in the original sequence must be 36 - which, you soon find out, is correct.

Dig down far enough, and you discover hidden order, underlying structure, stable relations beneath changing surfaces.

The original sequence was generated by squaring successive numbers - yet we predicted it using what seems like a wholly different method, one that we could in principle use without ever realizing we were generating the squares. Can you prove the two methods are always equivalent? - for thus far we have not proven this, but only ventured an induction. Can you simplify the proof so that you can you see it at a glance? - as Polya was fond of asking.

This is a very simple example by modern standards, but it is a very simple example of the sort of thing that mathematicians spend their whole lives looking for.

The joy of mathematics is inventing mathematical objects, and then noticing that the mathematical objects that you just created have all sorts of wonderful properties that you never intentionally built into them. It is like building a toaster and then realizing that your invention also, for some unexplained reason, acts as a rocket jetpack and MP3 player.

Numbers, according to our best guess at history, have been invented and reinvented over the course of time. (Apparently some artifacts from 30,000 BC have marks cut that look suspiciously like tally marks.) But I doubt that a single one of the human beings who invented counting visualized the employment they would provide to generations of mathematicians. Or the excitement that would someday surround Fermat's ...

Matt2

Matt2

Matt2

Matt2