Physicists like me often think that mathematicians are unduly obsessed with rigor. We’re used to hearing that things that we use all the time (like the path integral in QFT) are not well-founded mathematically. Then we think that the mathematicians must be really dense for not accepting the path integral, when it’s obviously right (we have the experiments to prove it!). What’s the point in searching for rigor if you already know something is right?
But from what I’ve been reading recently, it seems like I’ve been too hard on the mathematicians. They aren’t interested in rigor for the sake of rigor. A couple of days ago I read Michael Harris, who says “that the basic unit of mathematics is the concept rather than the theorem, that the purpose of a proof is to illuminate a concept rather than merely confirm a theorem, and that the replacement of deductive proofs by probabilistic or mechanical proofs should be compared, not to the introduction of a new technology for producing shoes, say, but rather to the attempt to replace shoes by the sales receipts.” Less interestingly, here’s Terrence Tao talking about rigor and intuition in math. And Peter Woit is always arguing that the way to make progress in physics is by a better understanding of the mathematical structures behind the Standard Model and other physical theories. I don’t totally understand this argument (how will a deeper understanding give us new predictions?), but it’s very interesting.
The reductionist hypothesis does not by any means imply a “constructionist” one: The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe.
Earlier, I mentioned Anderson’s idea of more is different. This is a really important idea, although in some sense rather obvious. Obviously we can’t derive biology from quantum electrodynamics (obvious to anyone who knows biology or quantum electrodynamics). Obviously there is a huge gap between these subjects, despite the assumption of the reductionist hypothesis. Understanding where this gap comes from is about understanding how a quantitative difference (the difference in number between the handful of particles in a particle physics experiment compared to the many, many particles in a cat) creates qualitative differences (entirely different concepts, laws, and equations are needed to describe many-particle systems (the cat) than to describe the particles the cat is made of). In solid-state physics (many-electron physics), these qualitative differences arise through the process of spontaneous symmetry breaking. This transition is well-understood, but other transitions (between neuroscience and cognitive science, for example) are not understood in such neat terms. More is different is also true at the transition from the quantum world to the classical world. As I understand it, decoherence provides the mechanism for spontaneous symmetry breaking in many-particle systems. This is important stuff. Anyone who thinks they want to study particle physics or cosmology because they are more fundamental should consider how small the domain of “fundamental” science really is.
One of the most fundamental false economies of scale is the overgrowth of cities. At one time it looked economically attractive to cram millions of people together into huge agglomerations. The biggest cities everywhere are running into social problems which indicate that this was a false economy.
This is from Freeman Dyson again. I am big fan of cities and am not really convinced by this point. But maybe cities can grow too big. To the point where marginal returns are small or negative. Maybe two smaller cities could be better than one big one. But to talk about social problems in the cities, you have to consider social problems outside of the cities. My understanding is that people move to the slums in cities like Mumbai and Rio because life in other parts of their country is even worse than life in the cities. In the US, it seems like some social problems are more concentrated in cities, and some more concentrated outside.
Dyson spends much of the third chapter describing a supposed conflict between “unifiers” and “diversifiers” in science and other fields: “Today we still find scientists divided into two camps: the unifiers who, like Einstein, believe that nature can be reduced to a finite set of equations; the diversifiers who, like Wiechert, believe that nature is inexhaustible.” He assures us that there is in fact no contradiction between these two approaches, but he seems almost not to believe it. But there really is no conflict! There is no reason to choose sides between the unifiers and diversifiers. “More is different” and we can prove it in a way that even the unifiers will accept.
“If it should turn out that the whole of physical reality can be described by a finite set of equations, I would be disappointed.” This is a remarkable statement from a physicist. I wonder what Dyson thinks the alternative is.
Dyson also bemoans the lack of communication between cosmology and biology and writes: “Among contemporary physicists, John Wheeler is unique in taking seriously the possibility that the laws of physics may be contingent upon the presence of life in the Universe.” Things have changed.
It’s not a novel point, but reading a book written in 1985, it’s interesting how science has progressed in unexpected directions and not progressed in the expected directions.
“We should at least wait until the experts have decided whether the superstring theory has anything to do with the universe we are living in. If the theory is wrong, it should be possible to prove it wrong within a few years. If it is right, to prove it right will take a little longer.”
“If we do not stifle their interest in science, one of them has a chance to be the new Heisenberg, building Hawking’s equation into a fully coherent mathematical theory of black holes. The twelve-year-olds will be twenty-five in the year 2000, and that would be a good moment for a new theory of the universe to be born.”
“The crater associated with the 65-million-year mass extinction has not been found… Perhaps the crater associated with the dinosaur extinction is waiting to be discovered somewhere on the bottom of the ocean. Or perhaps it has disappeared into one of the ocean trenches.”
Well we found the impact crater and it wasn’t even on the bottom of the ocean. In fact the crater had already been discovered by the time of Dyson’s lectures. Meanwhile, most people seem to have given up on the idea of ever proving superstring theory right or wrong. Black holes are still a mess.
We just hope that science keeps giving us surprises.
I started reading Freeman Dyson’s Infinite in All Directions today.
The book is adapted from the Gifford Lectures Dyson gave at University of Aberdeen. Some of the other lectures look interesting. Dyson mentions (twice so far) William James’s The Varieties of Religious Experience, originally a Gifford Lecture, which sounds like something I should read.
From the preface: “As a working hypothesis to explain the riddle of our existence, I propose that our universe is the most interesting of all possible universes, and our fate as human beings is to make it so.” This must be only half-serious. I imagine a new Candide, bearing witness to the greatest boredoms, while all the while Dyson reassures his friend that we live in the most interesting of all possible worlds.
From the introduction, summing up the theme of the book: “Diversity is for me the chief source of beauty and value… Diversity is the spice of life, and the prevalence of evil in our world is the price we pay for diversity.” This I very much agree with.
In the first chapter, Dyson writes about the relationship between science and religion. He believes that science and religion are in no way incompatible. As far as I can tell, he only makes one argument for this compatibility and it’s a little difficult to follow: “We have learned that matter is weird stuff. It is weird enough, so that it does not limit God’s freedom to make it do what he pleases.” I think what he is saying is that since quantum physics is non-deterministic (putting aside foundational questions), we ascribe outcomes we cannot predict to “random chance”, but perhaps it is not random and is actually “God’s will”. In this way, we can reconcile a God with will with the laws of physics. This idea is familiar to me; I had exactly the same idea when I first learned about quantum mechanics. I think the idea is ultimately unsatisfying (perhaps later I will write why), but at least it seems consistent, as Dyson argues. But perhaps when less mystical explanations of quantum measurement (and human consciousness) are more widely accepted, even this consistency will disappear.