Consider the sequence {1, 4, 9, 16, 25, ...} You recognize these as the square numbers, the sequence Ak = 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...
"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."
No, there's a conflation of two things here.
Have you ever really looked at a penny? I'm looking at a 1990 penny now. I know that if you look at the front and you see the bas-relief of Lincoln, and the date 1990, and it's a penny, then you can be sure that the back side will have a picture of the Lincoln memorial. It works! And you can find all sorts of connections. Like, there's a single "O" on the front, in the name GOD in the phrase IN GOD WE TRUST. And there's a single "O" on the back, in the phrase ONE CENT. One O on the front, one O on the back. A connection! You could make lots and lots of these interconnections between the front and the back of the penny, and draw conclusions about what it means. You could invent a discipline of Pennyology if only somebody would fund it.
Is it true that Pennyology is implicit in pennies? In a way. Certainly the pennies should exist before the Pennyology. But the pennies are only whatever they are. The existence of pennies doesn't tell us much about what the practitioners of the discipline of Pennyology will actually notice. They might never pay attention to the pair of O's. There could be a fold in Lincoln's coat that after the proper analysis provides a solution to the whole world crisis, and they may never pick up on it. While it's predictable that different independent attempts at Pennyology would have a whole lot in common since after all they all need to be compatible with the same pennies, still they might be very different in some respects. You can't necessarily predict the Pennyology from looking at the penny. And you can't predict what mathematics people will invent from observing reality.
You can predict some things. A mathematics that invents the same 2D plane we use and that proves a 3-color theorem has something wrong with it. But you can't predict which things will be found first or, to some extent, which things will be found at all.
If there's a reality that mathematics must conform to, still each individual version of human mathematics is invented by humans.
Similarly with physics. Our physics is invented. The reality the physics describes is real. We can imagine a platonic-ideal physics that fit the reality completely, but we don't have an example of that to point at. So for example before Townsend invented the laser, a number of great physicists claimed it was impossible. Townsend got the idea because lasers could be described using Maxwell's equations. But people thought that quantum mechanics provided no way to get that result. it turned out they were wrong.
Actual physics is invented. Certainly incorrect physics must be invented. There's nothing in reality that shows you how to do physics wrong.