Beauty, Passion and Computation
I listened to a couple of interviews with Gregory Chaitin on the Closer to Truth website. They may have been part of TV episodes that I haven’t seen but I was actually invigorated by some of the things he said, and it made me want to share them.
One of the interviews (in two parts) is found under the heading Is Mathematics Eternal, and is contained within the topic What are the Deep Laws of Nature? Here, Chaitin seemed most intent on dispelling the standard view of mathematics, where mathematics is seen, perhaps, as the epitome of rationality, or as thought without feeling, or logical structure without any inherent meaning. The discussion begins with a discussion of beauty.
We mathematicians are not machines,” he says, “we’re not calculating machines at all. We’re human beings and we’re emotional. Why are we devoting our lives to mathematics?
His reply is essentially that mathematics seduces them, that mathematicians are passionate about mathematics. He tells us that what many mathematicians are really looking for is beauty. And here he uses a phrase that is likely filled with a number of thoughts worth exploring. “It’s something sensual,” he says, “it’s the sensuality of ideas.” The whole body is somehow participating. And I believe that too, but how does one understand the sensuality of ideas? What does it say about how mathematics is happening. This sensuality may only be reached when one participates in mathematics at a certain depth.
Chaitin compared the effect that pure mathematics had on him as an adolescent boy, to the effect that women had on him. “It was confused in my case,” he tells us. The reason for the confusion, as he sees it, may be that beauty is in some way connected to the life force. “One wants to create something beautiful, one wants to be illuminated by it.” Chaitin uses the word illuminated more than once, and it is a particularly good choice of word.
When asked about what makes a proof beautiful, one of the things he said is, “at first it seems surprising and then it seems inevitable. And you ask yourself, how come you didn’t see it sooner?” Even when speaking about a beautiful painting, Chaitin uses the phrase, “it illuminates you,” like a light coming out of the canvas. He says the same of a beautiful proof, “it illuminates you.” I think the image (in both cases), that something becomes lit, is a provocative image, referring to both light and understanding. “A proof that isn’t illuminating is useless.” The purpose of a proof, he explains, is not to prove something, it’s to give you understanding. It should give you some intuition about why the result is right.
As the first segment nears its end, Chaitin brings up the mathematician Ramanujan.
I think Ramanujan is great because he contradicts everything that mathematics is supposed to be.
Chaitin tells us that Ramanujan believed that an equation is of value only if it expresses one of God’s thoughts. Ramanujan said of his own ideas that they were given to him, in his sleep, by the goddess Namagiri. Then Chaitin moves on to Cantor. “He is the most weird contradictory person.” Cantor was trying to understand God. He saw the contradictions inherent in his ideas but was not discouraged by them because, perhaps, his ideas were more important. Cantor was learning something about the infinite. Chaitin also reminds us that Poincare referred to Cantor’s worked as a disease from which he hoped future generations recovered. But Cantor’s work is foundational now. Having been cleansed of some of its difficulties, and with Cantor’s more irrational inspiration hidden, it strongly influenced the course mathematics took in the 20th century.
Chaitin is making the argument that these peculiar aspects of Ramanujan’s and Cantor’s experiences are ignored because they contradict the way people expect to think about mathematics.
I prefer to take the other extreme. I believe mathematics is based only on emotion and inspiration and it’s totally irrational and we don’t know where it comes from especially in cases where it’s an obsession.
Chaitin also recalled the work of Leonard Euler:
He created a lot of the mathematics that physicists and engineers use, but it was like a river, like a torrent of creative thought. Every week he would do another wonderful paper. He would give the whole train of thought and you would think when you read it, oh, I could do that. But no. Today a lot of his proofs are not considered rigorous. But even though his proofs are not what today is considered a valid proof, he discovered all this mathematics. Where did all those ideas come from?
In another segment, under the heading Is Information fundamental, Chaitin is addressing the question of whether information is more fundamental to the universe than matter. The inspiration for these ideas may be the computer. But what if the computer isn’t just a metaphor? Chaitin asks. A theory can be thought of as a computation. You input your theory and the output is the physical universe, or mathematical theorems. And when is a theory good? This goes back to Leibniz, he says. A theory is good when it’s a compression, when what you put into the computer is simpler or smaller than what you get out. Then you understand. And that understanding can be mathematical or it can be physical. And maybe the right way to think about the universe is that the universe is a computation – computing its future state from its current state. One can think of everything as a computation – understanding, the physical universe, DNA, as well as current technology.
Chaitin suggests that this provides a whole new way of looking at epistemology, which reach back to ideas presented by Leibniz after 300 years of development. We have reinvented an old question: Is the universe built of matter or of mind?
I wrote a bit on Leibniz a couple of years ago. I will revisit Leibniz and Chaitin for more on this.