It's getting late (at least where I'm at). I'm recovering from a cold & hoping the Benadryl is going to give me a nice buzz & help me sleep. So I thought I would pass along this nice little mind-bender that's been around for a while, and for which the debate has recently been rekindled. It basically comes down to this: Is mathematics an invention of humanity, or a discovery of humanity?
It seems like a simple question, but it touches on philosophy (metaphysics), Plato, and the fundamental nature of mathematics. Is math a discovery of fundamental truths or is it a human construct to express fundamental truths? Some of the greatest minds in human history have taken a crack at it, here's your chance.
Let's start with a "simple" truth.....
Now, on one level, it's a fundamental truth of existence. It should be true within this universe, whether humanity is here to know it or not. It should be as true today as it was 10,000 years ago. And it should be as true here on Earth as it would be if you were floating somewhere in the Eagle Nebula, 7,000 light years away. And so far no one has "invented" a way to make 2+2=5 & still have it make any kind of sense.
On the other hand, what are "2" & "4"? Mathematics has sometimes been described as the language of science & as such could be described as a tool. Mathematical concepts are not observable. A pulsar, Earth's atmosphere, a certain species of frog all exist in nature as observable things to study within an empirical framework. The number "1" only exists as a human construct on a piece of paper, classroom board, or computer screen. It's a symbolic representation of an idea used to express other ideas, just like +, -, $, %, and even the period I'm going to end this sentence with right now.
And if that wasn't enough, all of this leads to Plato.
Those who espouse discovery note that mathematical statements are true or false regardless of personal beliefs, suggesting that they have some external reality. But this leads to some odd notions. Where, exactly, do these mathematical truths exist? Can a mathematical truth really exist before anyone has ever imagined it?
[...]Plato is the standard-bearer for the believers in discovery. The Platonic notion is that mathematics is the imperturbable structure that underlies the very architecture of the universe. By following the internal logic of mathematics, a mathematician discovers timeless truths independent of human observation and free of the transient nature of physical reality. "The abstract realm in which a mathematician works is by dint of prolonged intimacy more concrete to him than the chair he happens to sit on," says Ulf Persson of Chalmers University of Technology in Sweden, a self-described Platonist.
However, Plato has his detractors.
If the mathematical ideas are out there, waiting to be found, then somehow a purely abstract notion has to have existence even when no human being has ever conceived of it. Because of this, [Barry] Mazur, [a mathematician at Harvard University] describes the Platonic view as "a full-fledged theistic position." It doesn’t require a God in any traditional sense, but it does require "structures of pure idea and pure being," he says. Defending such a position requires "abandoning the arsenal of rationality and relying on the resources of the prophets."
Indeed, Brian Davies, a mathematician at King's College London, writes that Platonism "has more in common with mystical religions than with modern science." And modern science, he believes, provides evidence to show that the Platonic view is just plain wrong. He titled his article "Let Platonism Die."
A pretty simple way to resolve this would be to say it's both a discovery & an invention, but that would be too easy & rob people of hour and hours and... years of arguments.