Infinity

George F. R. Ellis weighs in on the concept of infinity in his excellent paper, Issues in the Philosophy of Cosmology, available on astro-ph at https://arxiv.org/abs/astro-ph/0602280. He writes:

9.3.2 Existence of Infinities

The nature of existence is significantly different if there is a finite amount of matter or objects in the universe, as opposed to there being an infinite quantity in existence. Some proposals claim there may be an infinite number of universes in a multiverse and many cosmological models have spatial sections that are infinite, implying an infinite number of particles, stars, and galaxies. However, infinity is quite different from a very large number! Following David Hilbert, one can suggest these unverifiable proposals cannot be true: the word “infinity” denotes a quantity or number that can never be attained, and so will never occur in physical reality.38 He states:

Our principal result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought . . . The role that remains for the infinite to play is solely that of an idea . . . which transcends all experience and which completes the concrete as a totality . . .

This suggests “infinity” cannot be arrived at, or realized, in a concrete physical setting; on the contrary, the concept itself implies its inability to be realized!

Thesis I2: The often claimed physical existence of infinities is questionable. The claimed existence of physically realized infinities in cosmology or multiverses raises problematic issues. One can suggest they are unphysical; in any case such claims are certainly unverifiable.

This applies in principle to both small and large scales in any single universe:

The existence of a physically existing spacetime continuum represented by a real (number) manifold at the micro-level contrasts with quantum gravity claims of a discrete spacetime structure at the Planck scale, which one might suppose was a generic aspect of fully non-linear quantum gravity theories. In terms of physical reality, this promises to get rid of the uncountable infinities the real line continuum engenders in all physical variables and fields40. There is no experiment that can prove there is a physical continuum in time or space; all we can do is test space-time structure on smaller and smaller scales, but we cannot approach the Planck scale.

Infinitely large space-sections at the macro-level raise problems as indicated by Hilbert, and leads to the infinite duplication of life and all events. We may assume space extends forever in Euclidean geometry and in many cosmological models, but we can never prove that any realised 3-space in the real universe continues in this way—it is an untestable concept, and the real spatial geometry of the universe is almost certainly not Euclidean. Thus Euclidean space is an abstraction that is probably not physically real. The infinities supposed in chaotic inflationary models derive from the presumption of pre-existing infinite Euclidean space sections, and there is no reason why those should necessarily exist. In the physical universe spatial infinities can be avoided by compact spatial sections, resulting either from positive spatial curvature, or from a choice of compact topologies in universes that have zero or negative spatial curvature. Machian considerations to do with the boundary conditions for physics suggest this is highly preferable; and if one invokes string theory as a fundamental basis for physics, the “dimensional democracy” suggests the three large spatial dimensions should also be compact, since the small (“compactified”) dimensions are all taken to be so. The best current data from CBR and other observations indeed suggest k = +1, implying closed space sections for the best-fit FL model.

The existence of an eternal universe implies that an infinite time actually exists, which has its own problems: if an event happens at any time t0, one needs an explanation as to why it did not occur before that time (as there was an infinite previous time available for it to occur); and Poincaré eternal return will be possible if the universe is truly cyclic. In any case it is not possible to prove that the universe as a whole, or even the part of the universe in which we live, is past infinite; observations cannot do so, and the physics required to guarantee this would happen (if initial conditions were right) is untestable. Even attempting to prove it is future infinite is problematic (we cannot for example guarantee the properties of the vacuum into the infinite future—it might decay into a state corresponding to a negative effective cosmological constant).

It applies to the possible nature of a multiverse. Specifying the geometry of a generic universe requires an infinite amount of information because the quantities necessary to do so are fields on spacetime, in general requiring specification at each point (or equivalently, an infinite number of Fourier coefficients): they will almost always not be algorithmically compressible. All possible values of all these components in all possible combinations will have to occur in a multiverse in which “all that can happen, does happen”. There are also an infinite number of topological possibilities. This greatly aggravates all the problems regarding infinity and the ensemble. Only in highly symmetric cases, like the FL solutions, does this data reduce to a finite number of parameters, each of which would have to occur in all possible values (which themselves are usually taken to span an infinite set, namely the entire real line). Many universes in the ensemble may themselves have infinite spatial extent and contain an infinite amount of matter, with all the problems that entails. To conceive of physical creation of an infinite set of universes (most requiring an infinite amount of information for their prescription, and many of which will themselves be spatially infinite) is at least an order of magnitude more difficult than specifying an existent infinitude of finitely specifiable objects.

One should note here particularly that problems arise in the multiverse context from the continuum of values assigned by classical theories to physical quantities. Suppose for example that we identify corresponding times in the models in an ensemble and then assume that all values of the density parameter and the cosmological constant occur at each spatial point at that time. Because these values lie in the real number continuum, this is a doubly uncountably infinite set of models. Assuming genuine physical existence of such an uncountable infinitude of universes is the antithesis of Occam’s razor. But on the other hand, if the set of realised models is either finite or countably infinite, then almost all possible models are not realised. And in any case this assumption is absurdly unprovable. We can’t observationally demonstrate a single other universe exists, let alone an infinitude. The concept of infinity is used with gay abandon in some multiverse discussions, without any concern either for the philosophical problems associated with this statement, or for its completely unverifiable character. It is an extravagant claim that should be treated with extreme caution.

38An intriguing further issue is the dual question: Does the quantity zero occur in physical reality? This is related to the idea of physical existence of nothingness, as contrasted with a vacuum. A vacuum is not nothing!

40To avoid infinities entirely would require that nothing whatever is a continuum in physical reality (since any continuum interval contains an infinite number of points). Doing without that, conceptually, would mean a complete rewrite of many things. Considering how to do so in a way compatible with observation is in my view a worthwhile project.


So, given this discussion of infinities, the answer to the doubly hypothetical question, “Can God make a rock so big he can’t pick it up?” is likely a “Yes”! – D.O.

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.

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]