Is God a Mathematician_ - Mario Livio [105]
At this point, I believe that we have gathered enough clues that we should at least be able to try answering the questions we started with: Why is mathematics so effective and productive in explaining the world around us that it even yields new knowledge? And, is mathematics ultimately invented or discovered?
CHAPTER 9
ON THE HUMAN MIND, MATHEMATICS, AND THE UNIVERSE
The two questions: (1) Does mathematics have an existence independent of the human mind? and (2) Why do mathematical concepts have applicability far beyond the context in which they have originally been developed? are related in complex ways. Still, to simplify the discussion, I will attempt to address them sequentially.
First, you may wonder where modern-day mathematicians stand on the question of mathematics as a discovery or an invention. Here is how mathematicians Philip Davis and Reuben Hersh described the situation in their wonderful book The Mathematical Experience:
Most writers on the subject seem to agree that the typical working mathematician is a Platonist [views mathematics as discovery] on weekdays and a formalist [views mathematics as invention] on Sundays. That is, when he is doing mathematics he is convinced that he is dealing with an objective reality whose properties he is attempting to determine. But then, when challenged to give a philosophical account of this reality, he finds it easiest to pretend that he does not believe in it after all.
Other than being tempted to substitute “he or she” for “he” everywhere, to reflect the changing mathematical demographics, I have the impression that this characterization continues to be true for many present-day mathematicians and theoretical physicists. Nevertheless, some twentieth century mathematicians did take a strong position on one side or the other. Here, representing the Platonic point of view, is G. H. Hardy in A Mathematician’s Apology:
For me, and I suppose for most mathematicians, there is another reality, which I will call “mathematical reality”; and there is no sort of agreement about the nature of mathematical reality among either mathematicians or philosophers. Some hold that it is “mental” and that in some sense we construct it, others that it is outside and independent of us. A man who could give a convincing account of mathematical reality would have solved very many of the most difficult problems of metaphysics. If he could include physical reality in his account, he would have solved them all.
I should not wish to argue any of these questions here even if I were competent to do so, but I will state my own position dogmatically in order to avoid minor misapprehensions. I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our “creations,” are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it.
Mathematicians Edward Kasner (1878–1955) and James Newman (1907–66) expressed precisely the opposite perspective in Mathematics and the Imagination:
That mathematics enjoys a prestige unequaled by any other flight of purposive thinking is not surprising. It has made possible so many advances in the sciences, it is at once so indispensable