Someone I respect and read every day, Seth Goden, came out today with a howler: “infinity is not a number“. This is a very 19th century way of thinking.
In fact if somebody says “There are no numbers that …”, well, maybe there are. Why don’t we assume there are, and see what happens? People have been doing this since Pythagoras.
Humanity has a long history of being dragged, kicking and screaming, into the realization they needed new kinds of numbers: fractions or fractured numbers or broken numbers; negative numbers, irrational numbers, imaginary numbers. People hated to extend the number system. But they did it anyway, because it helped them understand the world better, and build themselves a better life.
So now we come to infinite numbers. Up to the 19th century, infinity was not a number, it was the limit of a process. This became so ingrained that the first person to challenge it, Georg Cantor, was ruthlessly mocked and eventually confined to an insane asylum for the rest of his life. But today mathematicians accept Cantor’s “transfinite” numbers and have used them to infer that transcendental numbers must exist.
But Seth’s conclusion is not wrong here. Infinite numbers have not integrated themselves into our technology and our lives the way other kinds of numbers have. (Yet.) I agree completely with him that if you are chasing infinity, get used to disappointment. That is not the right way to deal with infinite numbers. We don’t know the right way. We’ve only had about a century and a half to deal with this new kind of number and history shows that other kinds of numbers have needed centuries to be accepted and used. But efforts like nonstandard analysis and surreal numbers show that both infinite and infinitesimal numbers (think 1/infinity) can be potentially useful ways of thinking about this. And that means they exist, just as much as the number 1 exists. (Does it? Can you point to it? No. Can you use it? Yes.)
This also relates to the question whether mathematics is discovered or invented. It’s invented, because it is human inventiveness that has created all these numbers. But you can’t just create willy-nilly. You have constraints, and those constraints must be discovered. So the answer is, it is both.
