Are we finding the mathematical structure of reality?
I’m intrigued by Max Tegmark’s conviction that the universe is, itself, a mathematical structure. He presented his ideas, again, on February 15 at the recent annual meeting of AAAS, in a symposium called Is Beauty Truth? He said that he has just completed a book on the same topic. I listened to the entire session and I suspect that I won’t be able to get a really good sense for the meaning and the implications of what he proposes until I read the forthcoming book. He did a brief interview for a short article in Science (published by AAAS) but the interview didn’t do much to clarify things.
I agree that, as he said, if you look closely at our current working assumption (which he called the external reality hypothesis) it is equally suspect. This external reality hypothesis assumes that the existence of the physical world is fully independent of us, that it doesn’t require an observer, and that in itself devoid of anything human. But there is little doubt that our reality is a perceived reality, built from our interaction with it. I want to hear more about what he means when he says that mathematical existence and physical existence are the same thing. He describes mathematical structure easily, as “abstract entities with relations between them.” They “don’t exist in space and time,” he says, rather “space and time exist in them.” My hunch is that this is true, but how? He also said that he doesn’t believe that mathematics is a human creation. He believes that we discover mathematical structure. What may be human about it are just the names we give things.
I will admit that the fact that he is a cosmologist at MIT influences my expectations, so I want to know more clearly how he comes to this view, and what he expects such a view will change about how we imagine ourselves and the world around us. My own view is that the same structure exists everywhere, in us and in everything of which we are a part. And we have exploited this by formalizing it (in mathematics for example) then using it to see the things that extend far beyond our perceptual range.
The introduction to the symposium included the following:
In 1939, Paul Dirac observed that “the physicist, in his study of natural phenomena, has two methods of making progress”: experiment and observation, and mathematical reasoning. Although he said, “there is no logical reason why the second method should be possible,” nevertheless it works, and to great effect. The key, Dirac felt, was beauty, leading him to his principle that successive theories of nature are characterized by increasing mathematical beauty. The results of this were rich and included some predictions not confirmed until after Dirac’s death. Nevertheless, the powerful guidance Dirac found in mathematics did sometimes lead him astray, as he rejected the principle of “renormalization,” developed by Feynman, Schwinger, and Tomonaga, to remedy the nonphysical infinities that kept cropping up in Dirac’s equations for quantum electrodynamics. Even as other physicists accepted it, Dirac never did, saying it was “just not sensible mathematics.” Nevertheless, it was powerful physics.
Sylvan Schweber of Brandeis University was the first to speak at the symposium. He provided participants a number of enlightening facts in his brief survey of the history of mathematics’ relationship to physics. His survey was fairly dense with information, and so hard to paraphrase. He quoted Einstein responding in 1933 to the question: How does the physicist know that he can find the right way? Einstein’s reply was this: “Nature is the realization of the simplest conceivable mathematical ideas.” For Einstein, creativity resided in mathematics. Schweber cited the emergence of mathematical physics as its own discipline, and commented that there was little talk about philosophies of science and mathematics (or about beauty and truth) by mathematical physicists and experimentalists, after World War II. The prevailing concerns became “getting the numbers out,” tackling the complexity of accelerator experiments, and the expanding use of computers. But he also made the observation that prolific advances in computing inspired views like that of physicist-turned-biologist John Hopfield who sees physical and biological processes as hierarchies of computations and computational devices.
It could seem like Schweber would be tempted to agree with Tegmark, but I’m pretty sure he doesn’t.