For centuries people have debated whether – like scientific truths – mathematics is discoverable, or if it is simply invented by the minds of our great mathematicians. But two questions are raised, one for each side of the coin. For those who believe these mathematical truths are purely discoverable, where, exactly, are you looking? And for those on the other side of the court, why cannot a mathematician simply announce to the world that he has invented 2 + 2 to equal 5.
This question pops to the surface of the math world every so often, like a whale surfacing for air. Most mathematicians will simply set aside this quandary for those from the philosophical realm, and get on with proving theorems.
However, the mathematical whale has surfaced this year, thanks to the European Mathematical Society Newsletter’s June edition, where the question will once again be raised.
If you’re looking for a side to join, then maybe the Platonic theory is your cup of tea. The Classical Greek philosopher Plato was of the view that math was discoverable, and that it is what underlies the very structure of our universe. He believed that by following the intransient inbuilt logic of math, a person would discover the 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.
And while Barry Mazur, a mathematician at Harvard University, doesn’t count himself as a Platonist, he does note that the Platonic view of mathematical discovery fits well with the experience of doing mathematics. The sensation of working on a theorem, he says, can be like being “a hunter and gatherer of mathematical concepts.”
Mazur provides the opposing view as well, asking just where these mathematical hunting grounds are. For if math is out there waiting to be discovered, what once was a purely abstract notion then has to develop an existence unconceived of by humans. Subsequently, Mazur describes the Platonic view as “a full-fledged theistic position.”
Brian Davies, a mathematician at King's College London, writes in his article entitled “Let Platonism Die” 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.
So the question remains; if a mathematical theory goes undiscovered, does it truly exist? Maybe this will be the next “does a tree falling in the forest make any sound if no one is there to hear it?”