Congratulations to UC Berkeley mathematician Edward Frenkel whose book Love and Math: The Heart of Hidden Reality is in the top five science books for the year at Amazon! I wrote about Frenkel in a different context recently when he participated in the expression of some dangerous reservations about Darwinian theory in, of all places, the New York Times Book Review ("Someone at the New York Times Wasn’t Being Sufficiently Vigilant About Stealth ‘Creationism’ When This One Got Through").
The philosophical issues raised by Dr. Frenkel in his book are not only fascinating but very relevant to subjects we touch on often here. Math, he argues, is not only beautiful and worthy of our love. It also gives access to another, ultimate reality that transcends our own.
He says it briefly and eloquently in an interview in The Economist.
Does maths exist without human beings to observe it, like gravity? Or have we made it up in order to understand the physical world?
I argue, as others have done before me, that mathematical concepts and ideas exist objectively, outside of the physical world and outside of the world of consciousness. We mathematicians discover them and are able to connect to this hidden reality through our consciousness. If Leo Tolstoy had not lived we would never have known Anna Karenina. There is no reason to believe that another author would have written that same novel. However, if Pythagoras had not lived, someone else would have discovered exactly the same Pythagoras theorem. Moreover, that theorem means the same to us today as it meant to Pythagoras 2,500 years ago.
So it’s not subject to culture?
This is the special quality of mathematics. It means the same today as it will a thousand years from now. Our perception of the physical world can be distorted. We can disagree on many different things, but mathematics is something we all agree on.
The only reason the theory means the same is that it describes the reality of the physical world, so mathematics must need the physical world.
Not always. Euclidian geometry deals with flat spaces, such as the three-dimensional flat space. For millennia people thought we inhabited a flat, three-dimensional world. It was only after Einstein that we realised we lived in a curved space and that light doesn’t travel in a straight line but bends around a star. Pythagoras’ theorem is about geometric shapes in an idealised space, a flat Euclidian plane which, in fact, is not found in the real world. The real world is curved. When Pythagoras discovered his theorem there were, of course, inferences from physical reality, and a lot of mathematics is drawn from our experience in the physical world, but our imagination is limited and a lot of mathematics is actually discovered within the narrative of a hidden mathematical world. If you look at recent discoveries, they have no a priori bearing in physical reality at all.
The na�ve interpretation that mathematics comes from physical reality just doesn’t work. The other interpretation that mathematics is a product of the human mind also has serious issues, because it seems clear that some of these concepts transcend any specific individual.
Math isn’t something we imagine or make for ourselves, it’s something we discover. It points to a realm of objective reality beyond ours. Our reality is also objective but it is distorted, in our perception, by subjectivity. Not so with math.
I love the point he makes about Tolstoy versus Pythagoras. Tolstoy had he never lived or had he died young would never have revealed Anna Karenina. What if Pythagoras never lived? Pythagoras’ theorem, just differently named, would have been revealed in any event.
It would be interesting to apply the same test to Darwin. (Michael Flannery has considered the question here.)
The Russian-born Frenkel is not just a brilliant mathematician — he’s also an infectiously effervescent personality. Go here and look for the charming interview he did with our friend Dennis Prager.
Photo source: Edward Frenkel, UC Berkeley/Elizabeth Lippman.