Wednesday, November 28, 2018

Lost in Math

I recently read Sabine Hossenfelder's Lost in Math and recommend it. The backbone of the text is a series of interviews with fellow theoretical physicists, punctuated by the author's own typically gimlet observations and reflections. I don't intend to review the book here, but merely to provide a few highlights.

Some of the people interviewed:

  • Nima Arkani-Hamed
  • Steven Weinberg
  • Chad Orzel
  • Frank Wilczek
  • Garrett Lisi
  • Joseph Polchinski
  • Xiao-Gang Wen
  • Katherine "Astrokatie" Mack
  • George Ellis

Hossenfelder's thesis is that the dearth of data in high-energy physics over the past few decades has left theorists with little guidance, so they end up selecting among the many possible theories using criteria spun largely out of thin air. "Beauty" is how many refer to that inarticulate quality that supposedly characterizes a good theory. Hossenfelder particularly takes issue with "naturalness": the idea that dimensionless constants in physics should have a value of order of magnitude 1. Simplicity, elegance, and symmetry are other criteria.

Chapter 2's catalog of where "beauty" has failed physics should be required reading for aspiring theorists. A primal example is Kepler's inscribing the planetary orbits in the Platonic solids; she also cites Galileo's preference for circular orbits over elliptical. Here's a choice line:

The historian Helge Kragh concluded his biography of Dirac with the observation that "after 1935 [Dirac] largely failed to produce physics of lasting value. It is not irrelevant to point out that the principle of mathematical beauty governed his thinking only during the later period." (p. 21)

In chapter 5, she names the types of proposed multiverses: Eternal Inflation, String Theory Landscape, Many Worlds, and The Mathematical Universe. All of these seek to avoid fixing the values of mathematical parameters in theories by positing that every value of every parameter is actualized somewhere or in some way. The weakness of the justification for such theories becomes evident when she undermines the uniqueness of the present need for them: "Since every theory requires observational input to fix parameters or pick axioms, every theory leads to a multiverse when it lacks input" (106). Throughout all of physics we could have speculated about a multiverse, but we didn't.

She talks with Steven Weinberg at length. The most interesting part was this remark on quantum foundations:

If you had a theory that said that, well, particles move around and there's a certain probability that it will go here or there or the other place, I could live with that. What I don't like about quantum mechanics is that it's a formalism for calculating probabilities that human beings get when they make certain interventions in nature that we call experiments. And a theory should not refer to human beings in its postulates. You would like to understand macroscopic things like experimental apparatuses and human beings in terms of the underlying theory. You don't want to see them brought in on the level of axioms of the theory. (124)

Aristotle's dictum that form precedes matter makes perfect sense of this irreducibility. The truth of two facts is no mere coincidence: that Weinberg doesn't understand this aspect of quantum theory and that he would be one of the last people to admit that Aristotle had anything meaningful to say about the world.

I typically find George Ellis insightful, and Hossenfelder's interview with him does not disappoint. He's especially good in observing how atheists' baseless claim that science disproves the existence of God actually contributes mightily to undermining the authority of science (214). The conversation also fruitfully turns to the value of philosophy, including this exchange:

"Yes—when we have an infinity appearing in a function, we assume it's not physical," I explain. "But there's no good mathematical reason why a theory should not have infinities. It's a philosophical requirement turned into a mathematical assumption. People talk about it but never write it down. That's why I say it gets lost in math. We use a lot of assumptions that are based on philosophy, but we don't pay attention to them."

"Correct," George says. "The problem is that physicists have been put off philosophy by a certain branch of philosophers who spout nonsense—the famous Sokal affair and all that. And there are philosophers who—from a scientific viewpoint—do talk nonsense. But nevertheless, when you are doing physics you always use philosophy as a background, and there are a lot of good philosophers—like Jeremy Butterfield and Tim Maudlin and David Albert—who are very sensible in terms of the relationship between science and philosophy. And one should form a good working relationship with them. Because they can help one see what are the foundations and what is the best way to frame the questions. (218)

Recognizing a value to philosophy is certainly a significant step, but does it go far enough? I have at best a passing familiarity with the work of the three philosophers named, but I daresay that they are the typical "safe" philosophers that physicists typically turn to, the kind that never question the central idea of modernity that philosophy has no meaningful access to the natural world except through "science." This kind is a far cry from recognizing that what we call physics today is actually a subset of a much larger field called natural philosophy, whose axioms are as certain and as immediately graspable as your presence in the place where you read this; which can draw on physics without being wholly dependent on it for data about the real world; and on which physics is wholly dependent if only implicitly. I would delight to be proven wrong about these three philosophers: please comment below with citations.

Sabine Hossenfelder, Lost in Math: How Beauty Leads Physics Astray (New York: Basic Books, 2018).