Lost in Math: A Book Review

I recently finished reading a thought-provoking book by theoretical physicist Sabine Hossenfelder, Lost in Math: How Beauty Leads Physics Astray. Hossenfelder writes in an engaging and accessible style, and I hope you will enjoy reading this book as much as I did. Do we have a crisis in physics and cosmology? You be the judge. She presents convincing arguments.

The basic premise of Hossenfelder’s book is that when theoretical physicists and cosmologists lack empirical data to validate their theories, they have to rely on other approaches—”beauty”, “symmetry”, “simplicity”, “naturalness“, “elegance”—mathematics. Just because these approaches have been remarkably successful in the past is no guarantee they will lead to further progress.

One structural element that contributes to the book’s appeal is Hossenfelder’s interviews with prominent theoretical physicists and cosmologists: Gian Francesco Giudice, Michael Krämer, Gordon Kane, Keith Olive, Nima Arkani-Hamed, Steven Weinberg, Chad Orzel, Frank Wilczek, Garrett Lisi, Joseph Polchinski, Xiao-Gang Wen, Katie Mack, George Ellis, and Doyne Farmer. And, throughout the book, she quotes many other physicists, past and present, as well. This is a well-researched book by an expert in the field.

I also like her “In Brief” summaries of key points at the end of each chapter. And her occasional self-deprecating, brief, soliloquies, which I find reassuring. This book is never about the care and feeding of the author’s ego, but rather giving voice to largely unspoken fears that theoretical physics is stagnating. And an academic environment hell-bent on preserving the status quo isn’t helping matters, either.

Anthropic Principle

Do we live in a universe fine-tuned for life? If so, is it the only possible universe that would support life? Recent work indicates that there may be more than one set of parameters that could lead to a life-supporting universe.

Beauty is in the Eye of the Beholder

Is our sense of what is “beautiful” a reliable guide to gaining a deeper understanding of nature? Or does it sometimes lead us astray? We know from history that it does.

In the past, symmetries have been very useful. Past and present, they are considered beautiful

When we don’t have data to guide our theory development, aesthetic criteria are used. Caveat emptor.

Experiment and Theory

Traditionally, experiment and observation have driven theory. Now, increasingly, theory drives experiment, and the experiments are getting more difficult, more expensive, and more time consuming to do—if they can be done at all.

Inflation

The rapid expansion of the universe at the time of the Big Bang is known as cosmic inflation, or, simply, inflation. Though there is some evidence to support inflation, that evidence is not yet compelling.

Mathematics

Mathematics creates a logically consistent universe all its own. Some of it can actually be used to describe our physical universe. What math is the right math?

Math is very useful for describing nature, but is math itself “real”, or is it just a useful tool? This is an ancient question.

Memorable Quotations

“I went into physics because I don’t understand human behavior.” (p. 2)

“If a thousand people read a book, they read a thousand different books. But if a thousand people read an equation, they read the same equation.” (p. 9)

“In our search for new ideas, beauty plays many roles. It’s a guide, a reward, a motivation. It is also a systematic bias.” (p. 10)

On artificial intelligence: “Being unintuitive shouldn’t be held against a theory. Like lack of aesthetic appeal, it is a hurdle to progress. Maybe this one isn’t a hurdle we can overcome. Maybe we’re stuck in the foundations of physics because we’ve reached the limits of what humans can comprehend. Maybe it’s time to pass the torch.” (p. 132)

“The current organization of academia encourages scientists to join already dominant research programs and discourages any critique of one’s own research area.” (p. 170)

Multiverse

The idea that our universe of just one of a great many universes is presently the most controversial idea in physics.

Particles and Interactions

What is truly interesting is not the particles themselves, but the interactions between particles.

Philosophy

Physicists and astrophysicists are sloppy philosophers and could stand to benefit from a better understanding of the philosophical assumptions and implications of their work.

Physics isn’t Math

Sure, physics contains a lot of math, but that math has traditionally been well-grounded in observational science. Is math driving physics more than experiment and observation today?

Quantum Mechanics

Nobody really understands quantum mechanics. Everybody’s amazed but no one is happy. It works splendidly well. The quantum world is weird. Waves and particles don’t really exist, but everything (perhaps even the universe itself) is describable by a probabilistic “wave function” that has properties of both and yet is neither. Then there’s the many-worlds interpretation of quantum mechanics, and quantum entanglement

Science and the Scientific Method

In areas of physics where experiments are too difficult, expensive, or impossible to do, some physicists seem to be abandoning the scientific method as the central pillar of scientific inquiry. Faith in beauty, faith in mathematics, faith in naturalness, faith in symmetry. How is this any different than religion?

If scientists can evaluate a theory using other criteria than that theory’s ability to describe observation, how is that science?

Stagnation

Some areas of physics haven’t seen any new data for decades. In such an environment, theories can and do run amok.

Standard Model (of particle physics)

Ugly, contrived, ad hoc, baroque, overly flexible, unfinished, too many unexplained parameters. These are some of the words used to describe the standard model of particle physics. And, yet, the standard model describes the elementary particles we see in nature and their interactions with extraordinary exactitude.

String Theory

String theory dates back at least to the 1970s, and its origins go back to the 1940s. To date, there is still no experimental evidence to support it. String theory is not able to predict basic features of the standard model. That’s a problem.

Triple Threat: Crises in Physics, Astrophysics, and Cosmology?

Physics: Sure, the Large Hadron Collider (LHC) at CERN gave us the Higgs boson, but little else. No new physics. No supersymmetry particles. Embarrassments like the diphoton anomaly. Do we need a bigger collider? Perhaps. Do we need new ideas? Likely.

Astrophysics: We’ve spent decades trying to understand what dark matter is, to no avail. No dark matter particles have been found.

Cosmology: We have no testable idea as to what dark energy is. Plenty of theories, though.


See Hossenfelder’s recent comments on the LHC and dark matter in her op-ed, “The Uncertain Future of Particle Physics” in the January 23, 2019 issue of The New York Times.


The book concludes with three appendices:

  • Appendix A: The Standard Model Particles
  • Appendix B: The Trouble with Naturalness
  • Appendix C: What You Can Do To Help

Hossenfelder gives some excellent practical advice in Appendix C. This appendix is divided into three sections of action items:

  • As a scientist
  • As a higher ed administrator, science policy maker, journal editor, or representative of a funding body
  • As a science writer or member of the public

I’m really glad she wrote this book. As an insider, it takes courage to criticize the status quo.

References
Hossenfelder, S., Lost in Math: How Beauty Leads Physics Astray, Basic Books, New York (2018).
Hossenfelder, Sabine. “The Uncertain Future of Particle Physics.” The New York Times 23 Jan 2019. https://www.nytimes.com/2019/01/23/opinion/particle-physics-large-hadron-collider.html.

Where Cosmology Meets Philosophy

We continue our series of excerpts (and discussion) from the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.

The physical explanatory power of inflation in terms of structure formation, supported by the observational data on the fluctuation spectra, is spectacular.  For most physicists, this trumps the lack of identification and experimental verification of the underlying physics.  Inflation provides a causal model that brings a wider range of phenomena into what can be explained by cosmology, rather than just assuming the initial data had a specific restricted form.  Explaining flatness (Ω0 ≅ 1 as predicted by inflation) and homogeneity reinforces the case, even though these are philosophical rather than physical problems (they do not contradict any physical law; things could just have been that way).  However claims on the basis of this model as to what happens very far outside the visual horizon (as in the chaotic inflationary theory) results from prioritizing theory over the possibility of observational and experimental testing.  It will never be possible to prove these claims are correct.

Inflation is one compelling approach to explaining the structure we see in the universe today.  It is not necessarily the only one, but it currently has the most support.  Basically, a tiny fraction of a second after the Big Bang, the universe expanded dramatically.  Around 10-36 seconds after the Big Bang the universe had a diameter on the order of 1.2 × 10-27 meters.  To put that size in perspective, the diameter of a proton is between 0.84-0.87 × 10−15 meters.  So, when inflation began, the entire universe had a diameter almost a trillion times smaller than a single proton!  10-34 seconds later when the inflationary period was coming to an end, the size of the universe was a little over half the distance to Alpha Centauri!

The basic underlying cosmological questions are:
(1)  Why do the laws of physics have the form they do?  Issues arise such as what makes particular laws work?  For example, what guarantees the behaviour of a proton, the pull of gravity?  What makes one set of physical laws ‘fly’ rather than another?  If for example one bases a theory of cosmology on string theory, then who or what decided that quantum gravity would have a nature well described by string theory?  If one considers all possibilities, considering string theory alone amounts to a considerable restriction.
(2)  Why do boundary conditions have the form they do?  The key point here is, how are specific contingent choices made between the various possibilities, for example whether there was an origin to the universe or not.
(3)  Why do any laws of physics at all exist?  This relates to unsolved issues concerning the nature of the laws of physics: are they descriptive or prescriptive?  Is the nature of matter really mathematically based in some sense, or does it just happen that its behaviour can be described in a mathematical way?
(4)  Why does anything exist?  This profound existential question is a mystery whatever approach we take.

The answer to such questions may be beyond the limits of experimental science, or even beyond the limits of our intellect.  Maybe, even, these questions are as meaningless as “What lies north of the north pole?1because of our limited intellect.  Many would claim that because there appears to be limits to what science or human intellect can presently explain, that this constitutes evidence for the existence of God.  It does not.  Let’s just leave it as we don’t know.

Finally, the adventurous also include in these questions the more profound forms of the contentious Anthropic question:
(5)  Why does the universe allow the existence of intelligent life?
This is of somewhat different character than the others and largely rests on them but is important enough to generate considerable debate in its own right.

Well, a seemingly flippant answer to this question is we wouldn’t be here if it didn’t, but that begs the question.  Perhaps intelligent life is the mechanism by which the universe becomes self-aware, or is this just wishful thinking?  In the end, I am willing to admit that there may be some higher power in the universe—in the scientific pantheist and humanist sense—but I will stop short of calling that “God” in any usual sense of the term.

The status of all these questions is philosophical rather than scientific, for they cannot be resolved purely scientifically.  How many of them—if any—should we consider in our construction of and assessments of cosmological theories?

Perhaps the limitations of science (and, therefore, cosmology) is more a manifestation of the limitations of our human intellect than any constraint on the universe itself.

One option is to decide to treat cosmology in a strictly scientific way, excluding all the above questions, because they cannot be solved scientifically.  One ends up with a solid technical subject that by definition excludes such philosophical issues.  This is a consistent and logically viable option.  This logically unassailable position however has little explanatory power; thus most tend to reject it.

Let’s call this physical cosmology.

The second option is to decide that these questions are of such interest and importance that one will tackle some or all of them, even if that leads one outside the strictly scientific arena.  If we try to explain the origin of the universe itself, these philosophical choices become dominant precisely because the experimental and observational limits on the theory are weak; this can be seen by viewing the variety of such proposals that are at present on the market.

And let’s call this metaphysical cosmology.

1Attributed to Stephen Hawking

References
Ellis, G. F. R. 2006, Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.
[http://arxiv.org/abs/astro-ph/0602280]

Ryden, Barbara. 2003.  Introduction to Cosmology. San Francisco: Addison Wesley.

Theory and Observation

We continue our series of excerpts (and discussion) from the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.

Thesis F1: Philosophical choices necessarily underly cosmological theory.
Some cosmologists tend to ignore the philosophical choices underlying their theories; but simplistic or unexamined philosophical standpoints are still philosophical standpoints!

Cosmology, and indeed all human inquiry, is based on (at least) two unproven (though certainly reasonable) assumptions:

  1. The Universe exists.
  2. The human mind is at least to some degree capable of perceiving and understanding the Universe.

Any cosmological theory will have additional foundational unproven assumptions.  These are called axioms.  Ellis admonishes us to at least be aware of them, and to admit to them.

8.1 Criteria for theories
As regards criteria for a good scientific theory, typical would be the following four areas of assessment: (1) Satisfactory structure: (a) internal consistency, (b) simplicity (Ockham’s razor), and (c) aesthetic appeal (‘beauty’ or ‘elegance’); (2) Intrinsic explanatory power: (a) logical tightness, (b) scope of the theory—the ability to unify otherwise separate phenomena, and (c) probability of the theory or model with respect to some well-defined measure; (3) Extrinsic explanatory power, or relatedness: (a) connectedness to the rest of science, (b) extendability—providing a basis for further development; (4) Observational and experimental support, in terms of (a) testability: the ability to make quantitative as well as qualitative predications that can be tested; and (b) confirmation: the extent to which the theory is supported by such tests as have been made.

As you can see, a theory is not an opinion.  It must be well-supported by facts.  It must be internally consistent.  It must have explanatory power.  The Russian physicist A. I. Kitaĭgorodskiĭ (1914-1985) put it succinctly: “A first-rate theory predicts; a second-rate theory forbids, and a
third-rate theory explains after the event.”  Einstein’s special and general relativity are spectacular examples of first-rate theories.  In over 100 years of increasingly rigorous and sophisticated experiments and observations, relativity has never been proven to be incorrect.

Ellis emphasizes the importance of observational and experimental support in any scientific theory.

It is particularly the latter that characterizes a scientific theory, in contrast to other types of theories claiming to explain features of the universe and why things happen as they do.  It should be noted that these criteria are philosophical in nature in that they themselves cannot be proven to be correct by any experiment.  Rather their choice is based on past experience combined with philosophical reflection.  One could attempt to formulate criteria for good criteria for scientific theories, but of course these too would need to be philosophically justified.  The enterprise will end in infinite regress unless it is ended at some stage by a simple acceptance of a specific set of criteria.

So, even our criteria about what makes a good scientific theory rest upon axioms that cannot be proven.  But unlike religion, scientific theories never posit the existence of any supernatural entity.

Thesis F3: Conflicts will inevitably arise in applying criteria for satisfactory cosmological theories.
The thrust of much recent development has been away from observational tests toward strongly theoretically based proposals, indeed sometimes almost discounting observational tests.  At present this is being corrected by a healthy move to detailed observational analysis of the consequences of the proposed theories, marking a maturity of the subject.  However because of all the limitations in terms of observations and testing, in the cosmological context we still have to rely heavily on other criteria, and some criteria that are important in most of science may not really make sense.

String theory? Cosmic inflation?  Multiverse? If a theory is currently neither testable nor directly supported by observations, is it science, or something else?

References
Ellis, G. F. R. 2006, Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.
[http://arxiv.org/abs/astro-ph/0602280]