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.

The Anthropic Question

George F. R. Ellis writes in Issues in the Philosophy of Cosmology:

9.1 Issue G: The anthropic question: Fine tuning for life
One of the most profound fundamental issues in cosmology is the Anthropic question: why does the Universe have the very special nature required in order that life can exist?  The point is that a great deal of “fine tuning” is required in order that life be possible.  There are many relationships embedded in physical laws that are not explained by physics, but are required for life to be possible; in particular various fundamental constants are highly constrained in their values if life as we know it is to exist:

Ellis goes on to quote Martin Rees.

A universe hospitable to life—what we might call a biophilic universe—has to be special in many ways … Many recipes would lead to stillborn universes with no atoms, no chemistry, and no planets; or to universes too short lived or too empty to evolve beyond sterile uniformity.

Physics does not tell us anything (yet) about why the fundamental constants and other parameters have the values they do.  These parameters include, for example, the speed of light, the Planck constant, the four fundamental forces and their relative strengths, the mass ratio of the proton and the electron, the fine-structure constant, the cosmological density parameter, Ωtot, relative to the critical density, and so on.  And, why are there four fundamental forces?  Why not five?  Or three?

Also, why do we live in a universe with three spatial dimensions and one time dimension?  Others are possible—even universes with two or more time dimensions.

But it appears that only three spatial dimensions and one time dimension is conducive to life (at least life as we know it), as shown in the diagram above (Whittle 2008).

In fact, altering almost any of the parameters would lead to a sterile universe and we could not exist.  Is the universe fine-tuned for our existence?

Let’s assume for the moment it is.  Where does that lead us?

  1. As our understanding of physics advances, we will eventually understand why these parameters must have the values that they do. -or-
  2. We will eventually learn that some of these parameters could have been different, and still support the existence of life. -or-
  3. God created the universe in such a way that life could exist -or-
  4. We’re overthinking the problem.  We live in a life-supporting universe, so of course we find the parameters are specially tuned to allow life. -or-
  5. There exist many universes with different parameters and we just happen to find ourselves in one that is conducive to life. (The multiverse idea.)

#4 is the anthropic explanation, but a deeper scientific understanding will occur if we find either #1, #2, or #5 to be true.  #3 is problematic for a couple of reasons.  First of all, how was God created?  Also, deism has a long history of explaining phenomena we don’t understand (“God of the gaps”), but in time we are able to understand each phenomenon in turn as science progresses.

The anthropic explanation itself is not controversial.  What is controversial is deciding to what degree fine tuning has occurred and how to explain it.

In recent years, the multiverse idea has become more popular because, for example, if there were a billion big bangs and therefore a billion different universes created, then it should not be at all surprising that we find ourselves in  one with just the right set of parameters to allow our existence.  However, there is one big problem with the multiverse idea.  Not only do we have no physical evidence that a multiverse exists, but we may never be able to obtain evidence that a multiverse exists, due to the cosmological horizon problem1.  If physical evidence of a multiverse is not forthcoming, then in that sense it is not any better than the deistic explanation.

To decide whether or not there is only one combination of parameters that can lead to life we need to rule out all the other combinations, and that is a tall order.  Recent work in this field suggests that there is more than one combination of parameters that could create a universe that is hospitable to life (Hossenfelder 2018).

Thinking now about why our universe is here at all, it seems there are just two possibilities:

(1)  Our universe has a supernatural origin.

(2)  Our universe has a natural origin.

If our universe has a supernatural origin, then what is the origin of the supernatural entity (e.g. God)?  If, on the other hand, our universe had a natural origin (e.g. something was created out of nothing), didn’t something have to exist (laws of physics or whatever) before the universe came into existence?  If so, what created those pre-conditions?

In either case, we are facing an infinite regression.  However, we could avoid the infinite regression by stating that something has to exist outside of time, that is to say, it has no beginning and no ending.  But isn’t this just replacing one infinity with another?

Perhaps there’s another possibility.  Just as a chimpanzee cannot possibly understand quantum mechanics, could it be that human intellect is also fundamentally limited?  Are the questions in the previous two paragraphs meaningless or nonsensical in the context of some higher intelligence?

1We appear to live in a universe that is finite but very much larger than the region that is visible to us now, or ever.

References
G.F.R. Ellis, 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]

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

M. J. Rees, Our Cosmic Habitat (Princeton and Oxford, 2003).

Mark Whittle, “Fine Tuning and Anthropic Arguments”, Lecture 34, Course No. 1830.  Cosmology: The History and Nature of Our Universe.  The Great Courses, 2008.  DVD.
[https://www.thegreatcourses.com/courses/cosmology-the-history-and-nature-of-our-universe.html]

Observation, Theory, and Reality

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

8.3 Limits of Representation and Knowledge of Reality
It follows…that there are limits to what the scientific method can achieve in explanatory terms.  We need to respect these limits and acknowledge clearly when arguments and conclusions are based on some philosophical stance rather than purely on testable scientific argument.  If we acknowledge this and make that stance explicit, then the bases for different viewpoints are clear and alternatives can be argued about rationally.

We human beings want so badly to be able to explain our existence and existence itself that we tend to “fill in the blanks” and treat speculation (no matter how well reasoned) as if it were something akin to fact.  This is true for both science and religion.  A more reasonable approach, it seems to me, is to reject absolute certainty—especially where physical evidence is sparse or nonexistent—while always striving to deepen our understanding.  That is the scientist’s stock-in-trade—or should be.  Each of us needs to become more aware of the limitations of our understanding!

Thesis F6: Reality is not fully reflected in either observations or theoretical models.
Problems arise from confusion of epistemology (the theory of knowledge) with ontology (the nature of existence): existence is not always manifest clearly in the available evidence.  The theories and models of reality we use as our basis for understanding are necessarily partial and incomplete reflections of the true nature of reality, helpful in many ways but also inevitably misleading in others.  They should not be confused with reality itself!

We humans create our own “realities”, but under the very best of circumstances (science, for example), our “reality” is only an imperfect model of what actually exists.

The confusion of epistemology with ontology occurs all the time, underlying for example the errors of both logical positivism and extreme relativism.  In particular, it is erroneous to assume that lack of evidence for the existence of some entity is proof of its non-existence.  In cosmology it is clear for example that regions may exist from which we can obtain no evidence (because of the existence of horizons); so we can sometimes reasonably deduce the existence of unseen matter or regions from a sound extrapolation of available evidence (no one believes matter ends at or just beyond the visual horizon).  However one must be cautious about the other extreme, assuming existence can always be assumed because some theory says so, regardless of whether there is any evidence of existence or not.  This happens in present day cosmology, for example in presentations of the case for multiverses, even though the underlying physics has not been experimentally confirmed.  It may be suggested that arguments ignoring the need for experimental/observational verification of theories ultimately arise because these theories are being confused with reality, or at least are being taken as completely reliable total representations of reality.

Absence of evidence is not evidence of absence.  But, without evidence, all we have is conjecture, no matter how well informed.  As Carl Sagan once said, “Extraordinary claims require extraordinary evidence.”

No model (literary, intuitive, or scientific) can give a perfect reflection of reality.  Such models are always selective in what they represent and partial in the completeness with which they do so.  The only model that would reflect reality fully is a perfect fully detailed replica of reality itself! This understanding of the limits of models and theories does not diminish the utility of these models; rather it helps us use them in the proper way.  This is particularly relevant when we consider how laws of nature may relate to the origins of the universe itself, and to the existence and nature of life in the expanding universe.  The tendency to rely completely on our theories, even when untested, seems sometimes to arise because we believe they are the same as reality—when at most they are descriptions of reality.

Ellis makes a pretty good case here against dogma.  Though he does not specifically mention religion (and why should he, as the subject at hand is cosmology), I do think these ideas apply to religion as well.

Always a journey, never a destination.

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]

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.