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 Last Theorem, or the factoring problem in RSA cryptography... and yet these are as implicit in the definition of the natural numbers, as are the first and second difference tables implicit in the sequence of squares.

This is what creates the impression of a mathematical universe that is "out there" in Platonia, a universe which humans are exploring rather than creating. Our definitions teleport us to various locations in Platonia, but we don't create the surrounding environment. It seems this way, at least, because we don't remember creating all the wonderful things we find. The inventors of the natural numbers teleported to Countingland, but did not create it, and later mathematicians spent centuries exploring Countingland and discovering all sorts of things no one in 30,000 BC could begin to imagine.

To say that human beings "invented numbers" - or invented the structure implicit in numbers - seems like claiming that Neil Armstrong hand-crafted the Moon. The universe existed before there were any sentient beings to observe it, which implies that physics preceded physicists. This is a puzzle, I know; but if you claim the physicists came first, it is even more confusing because instantiating a physicist takes quite a lot of physics. Physics involves math, so math - or at least that portion of math which is contained in physics - must have preceded mathematicians. Otherwise, there would have no structured universe running long enough for innumerate organisms to evolve for the billions of years required to produce mathematicians.

The amazing thing is that math is a game without a designer, and yet it is eminently playable.

Oh, and to prove that the pattern in the difference tables always holds:

(k + 1)² = k² + (2k + 1)

As for seeing it at a glance:

Think the square problem is too trivial to be worth your attention? Think there's nothing amazing about the tables of first and second differences? Think it's so obviously implicit in the squares as to not count as a separate discovery? Then consider the cubes:

1, 8, 27, 64...

Now, without calculating it directly, and without doing any algebra, can you see at a glance what the cubes' third differences must be?

And of course, when you know what the cubes' third difference is, you will realize that it could not possibly have been anything else...