The German mathematician Leopold Kronecker explained: “Whole numbers were created by the Lord God, everything else is created by people.” The language is subtle (“Lord God” is used in German to speak to children) and suggests that he thought it was all human creation. Did he actually criticize his compatriot Ferdinand von Lindemann for proving that? it is a transcendent number: “Why study such questions when there are not even irrational numbers?”
It is a typical stance of the nominalist view that mathematical concepts do not exist, they are mere creations of mankind and will disappear when we are extinct. But most mathematicians, including myself, believe that mathematical ideas have their own existence, and our job is to discover them in the world around us. It is the Platonic view that believes that the universe is written in a special code and that in mathematics we discover parts of that code.
This view goes back to Classical Greece – Pythagoras already stated that “everything is numbers” – and his main argument is that mathematical ideas work when applied to the real world, even in situations very different from those in which they were introduced. It is what the Hungarian-American physicist Eugene Wigner called “the unreasonable effectiveness of mathematics in the natural sciences”.
If that? It was invented to study the circumference of the circle. So why does it appear in Gauss’s law of normal distribution, one of the basic principles of statistics? What does the scope of the circle have to do with population studies and opinion polls?
If complex numbers were invented to solve the cubic equation, as I discussed recently in this column, then why are they fundamental to describing the world of quantum physics? And what is “invented” in Euler’s formula FA + V = 2, which relates the number F of surfaces, A of edges and V of vertices of a convex polyhedron? Is this formula a discovery or not ?!
But Platonism also has difficulties: If mathematics makes statements about the real world, how can it be precise and its absolute and eternal truths, in contrast to what happens in the other natural sciences?
Probably the question will not be solved until we find some extraterrestrial intelligence: if their math is significantly different from ours, that will be a point for nominalism. But I make my bets that give Platonism!
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