Over at The Economist, we have this brief interview with Edward Frenkel, a mathematician at Berkeley. We met Frenkel in this post from last week. Frenkel has a new book out called Love and Math, one copy of which is currently residing on my Kindle.
One part of the interview caught my attention:
Symmetry exists without human beings to observe it. 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.
This is a common view among mathematicians, but it doesn’t make much sense to me. I don’t know what it means to say that mathematical concepts exist outside of the physical world and outside of the world of consciousness.
To make it specific, consider the notion of a continuous function. The idea of a continuous function is an abstraction developed by mathematicians because it is useful for modelling a great many processes in the physical world. In what sense would this abstraction continue to exist if humanity suddenly winked out of existence? We could imagine that the physical processes themselves would continue to exist. But what could it mean to say the abstract model itself would still exist?
Of course, when you undertake mathematical research it certainly feels as though you are making discoveries about objects that exist independently of anyone’s ideas about them. That’s because you are. But you’re not so much making discoveries about the objects themselves, but about the logical relationships among the concepts you have chosen to define. It is like chess. The rules of chess are invented. But then you discover that a consequence of those rules is that a king, knight and bishop can force checkmate against a lone king while a king and two knights cannot. The rules of chess are so rich that even after centuries of analysis people continue to play the game and to make new discoveries about it. And so it is with the objects mathematicians choose to define and study.
So it seems to me at any rate. But Frenkel takes a different view:
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 naive 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.
Take Evariste Galois, who was killed in a duel at the age of 20. He came up with a beautiful theory on symmetry called Galois theory. His contemporaries didn’t get it but this theory now forms the core of modern mathematics. But what if the work had been burned? Would we never have known Galois theory? No. Someone else would have discovered it because it is inevitable.
I don’t see how that paragraph about Euclidean geometry justifies Frenkel’s claim that mathematics doesn’t come from physical reality. It is certainly true that Euclidean geometry is not adequate as a description of the large-scale structure of the universe. But the fact remains that it is a wonderfully accurate model for the world of our day-to-day experience. We hardly needed modern physics to tell us that Euclidean geometry is about an idealized space that does not exist in the physical world. That fact does not weaken in the slightest the suspicion that mathematical objects come from physical reality.
Indeed, the whole point of developing abstract models is to strip away the extraneous details of the real world so that we can focus solely on what’s important. That our abstract constructions do not actually exist in the real world is precisely the point.
As for Galois theory, of course someone else would have discovered it had Galois never been born. But that’s because the logical relationships among the objects Galois studied were out there waiting to be discovered, as per my earlier chess analogy. But all of the abstract objects Galois studied, like fields, groups and polynomials, ultimately owe their existence to abstractions drawn from physical reality.
I cannot think of a single example of a mathematical object which doesn’t owe its existence ultimately to some real-world consideration. Of course, the mathematical world is vast and I have only studied a small part of it, so I’m certainly willing to consider possible counterexamples I have overlooked. But I’d like to know what specific examples Frenkel has in mind. I guess I should read his book!
Time to wrap this up, so here’s one last excerpt:
Because it is simply true?
Yes. It’s a difficult philosophical question, to which we still don’t have the answer, but it’s an important question to be aware of. It’s not the same as the mathematics we use to calculate a tip—it goes to the heart of reality and of consciousness. It is all around us, with smart phones and computers and GPS devices and the algorithms that control our lives. The Amazon recommendations we are offered are based on very sophisticated algorithms, which analyse our past purchases, correlating us with other users. Mathematics is invading our world more and more and it communicates timeless, persistent and necessary truths which transcend time and space. The Langlands Programme should be as familiar to us as the theory of relativity.
I’ve gotten a little disenchanted with talk of “the heart of reality and consciousness.” It’s too grandiose for my taste. I think it’s enough just to say that certain abstract models help us order our experiences and certain other models are not helpful. To go beyond that seems, well, unhelpful.
And that’s how I view talk about the existence of mathematical objects. It’s not helpful. What matters about mathematical objects is what you can do with them, not the sense in which they can be said to exist.