Proofs, the mind, and mathematics

A recent article in Quanta Magazine anticipates the publication of the 6th edition of Proofs from The Book, collected by Martin Aigner and Günter Ziegler.  The original volume was inspired by the well-known and prolific mathematician Paul Erdős, who traveled the world, participating in countless collaborative efforts, and who would say of proofs that he judged to be of sublime beauty, “This one is from The Book.” This Book was imagined as the heavenly collection of mathematics’ perfect proofs. Aigner suggested the possibility of actually making The Book in 1994. Aigner, along with fellow mathematician Günter Ziegler, and with contributions from Erdős himself, published the first volume in 1998. Unfortunately, Erdős died in 1996, at he age of 83, and never saw the volume in print. The book received the 2018 Steele Prize for Mathematical Exposition.

One of the nice things that the article points out is that there are theorems that have a number of different proofs, each one telling you something different about the theorem or the structures involved in the proof of the theorem.

An example comes to mind — which is not in our book but is very fundamental — Steinitz’s theorem for polyhedra. This says that if you have a planar graph (a network of vertices and edges in the plane) that stays connected if you remove one or two vertices, then there is a convex polyhedron that has exactly the same connectivity pattern. This is a theorem that has three entirely different types of proof — the “Steinitz-type” proof, the “rubber band” proof and the “circle packing” proof. And each of these three has variations.

Any of the Steinitz-type proofs will tell you not only that there is a polyhedron but also that there’s a polyhedron with integers for the coordinates of the vertices. And the circle packing proof tells you that there’s a polyhedron that has all its edges tangent to a sphere. You don’t get that from the Steinitz-type proof, or the other way around — the circle packing proof will not prove that you can do it with integer coordinates. So, having several proofs leads you to several ways to understand the situation beyond the original basic theorem.

This kind of discussion highlights how mathematical ideas can be multi-aspected, the very thing that makes a mathematical idea powerful and difficult to categorize in our experience. But in the lower right margin of the article were links to related articles, and it was here that I found Michael Atiyah’s Imaginative State of Mind.  This piece was written about a year ago, when Michael Atiyah hosted a conference at the Royal Society of Edinburgh on The Science of Beauty. There is a video of his introductory remarks on youtube worth a listen.  The article was built around Atiyah’s response to some questions that the authors were able to ask him on the occasion of the conference.

Roughly speaking, he has spent the first half of his career connecting mathematics to mathematics, and the second half connecting mathematics to physics….
….Now, at age 86, Atiyah is hardly lowering the bar. He’s still tackling the big questions, still trying to orchestrate a union between the quantum and the gravitational forces. On this front, the ideas are arriving fast and furious, but as Atiyah himself describes, they are as yet intuitive, imaginative, vague and clumsy commodities.

I felt encouraged by the refreshingly sensory ways Atiyah characterized his experience as a mathematician. Like here:

The crazy part of mathematics is when an idea appears in your head. Usually when you’re asleep, because that’s when you have the fewest inhibitions. The idea floats in from heaven knows where. It floats around in the sky; you look at it, and admire its colors. It’s just there. And then at some stage, when you try to freeze it, put it into a solid frame, or make it face reality, then it vanishes, it’s gone. But it’s been replaced by a structure, capturing certain aspects, but it’s a clumsy interpretation.

In response to being asked if he had always had mathematical dreams he said this:

The crazy part of mathematics is when an idea appears in your head. Usually when you’re asleep, because that’s when you have the fewest inhibitions. The idea floats in from heaven knows where. It floats around in the sky; you look at it, and admire its colors. It’s just there. And then at some stage, when you try to freeze it, put it into a solid frame, or make it face reality, then it vanishes, it’s gone. But it’s been replaced by a structure, capturing certain aspects, but it’s a clumsy interpretation.

And when asked about the two works for which he is well known (the index theorem and K-threory) he suggested this very visual way of describing k-theory:

The index theorem and K-theory are actually two sides of the same coin. They started out different, but after a while they became so fused together that you can’t disentangle them. They are both related to physics, but in different ways.

K-theory is the study of flat space, and of flat space moving around. For example, let’s take a sphere, the Earth, and let’s take a big book and put it on the Earth and move it around. That’s a flat piece of geometry moving around on a curved piece of geometry. K-theory studies all aspects of that situation — the topology and the geometry. It has its roots in our navigation of the Earth.

The maps we used to explore the Earth can also be used to explore both the large-scale universe, going out into space with rockets, and the small-scale universe, studying atoms and molecules. What I’m doing now is trying to unify all that, and K-theory is the natural way to do it. We’ve been doing this kind of mapping for hundreds of years, and we’ll probably be doing it for thousands more.

I found a nice description of how the index theorem can connect the curvature of a space to its topology (or the number of holes it has).

One of the things Atiya is committed to at the moment is reversing the mistake of ignoring the small effect of gravity on an electron or proton. He says he’s going back to Einstein and Dirac and looking at them again and he thinks he sees things that people have missed. “If I’m wrong,” he says, “I made a mistake. But I don’t think so.”

At the end of introductory remarks he made at The Science of Beauty conference he said that he found himself closer to the mystical views of Pythagoras than to those who completely rejected mysticism. ”A little bit of mysticism is important in all forms of life.”

When asked if he thought a computer could be made to recognize beauty, his response led to his characterizing the mind as a parallel universe. More than just logic, the mind has aspects that recognize states. These are not verbal or pictorial states, but conceptual states. And beauty lives somewhere in the mind. This is the kind of insight that doing mathematics can produce. And it will, I believe, lead us to completely new ideas about who we are and what it is that our minds may be producing.

A last thought on mathematics:

People think mathematics begins when you write down a theorem followed by a proof. That’s not the beginning, that’s the end. For me the creative place in mathematics comes before you start to put things down on paper, before you try to write a formula. You picture various things, you turn them over in your mind. You’re trying to create, just as a musician is trying to create music, or a poet. There are no rules laid down. You have to do it your own way. But at the end, just as a composer has to put it down on paper, you have to write things down.

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