In response to “Physics on Edge” (Vol. 3, No. 2).
To the editors:
I thoroughly enjoyed “Physics on Edge.” One is reminded of a famous quote from Richard Feynman, “It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are. If it doesn’t agree with experiment, it’s wrong.” It is largely on this basis that George Ellis opposes theories of the multiverse. For a long time, I wholeheartedly supported this notion of the scientific method. Being slightly more motivated by the actual mathematical nature of the equations of physics in my own work, I have recently debated the applicability of the rigorousness of the scientific method to some of the theories that Ellis discusses, especially with respect to the multiverse. On the one hand, a stringent requirement for theories to be accepted as scientifically valid is that they must agree with experiment according to the classical and rigorous definition of science. But, on the other hand, so much of theoretical physics has arisen and developed from mathematics. General relativity, Maxwell’s equations, and quantum field theory all have at their basis a principle of least action, which in itself is entirely a geometric notion independent of any physical reality. Much of the work of JosephLouis Lagrange, William Hamilton, Leonhard Euler, and Carl Jacobi involved reformulating physical notions of Newtonian mechanics into purely geometric forms that found more abstract descriptions in theories of differential equations and dynamical systems, as expressed so beautifully in the works of Vladimir Arnold for example.^{1} The implication is that physical theories can, in principle, be discovered without actually appealing to physical motivations. What does this say about the rigorous definition of science and the scientific method? From these arguments one can imagine a class of theories that while mathematically rigorous—although it seems that what mathematicians and physicists consider to be mathematically rigorous physical theories tends to differ—have no hope of ever being tested because they fall outside the physical domain of experimental possibility. We return, once again, to the classical question: is a theory that is grounded solely in mathematics and that cannot be tested a scientific theory? If one were to adhere to the strictest criteria, the answer is no. As Ellis, quoting Carlo Rovelli, points out in his essay: “The very existence of reliable theories is what makes science valuable to society.”^{2} That measure of reliability can only come from experimental verification.
How then can one address scientific theories that are purely mathematical and cannot be tested, such as the many multiverse theories Ellis discusses in his essay? I am of the firm belief that such theories are still valuable, but we must accept that these theories belong in the realm of philosophy rather than the science. Multiverse theories, it should be noted, are being presented as rigorous scientific theories because they seem to follow from known scientific principles, such as general relativity and quantum field theory, as opposed to being presented as potential scientific theories that have a largely philosophical basis. The problem is compounded if proponents of the popular science movement such as Neil deGrasse Tyson, Lawrence Krauss, and Bill Nye then present these philosophical notions as actual science to a general public that, generally speaking, are unable to decipher the various issues with many of these theories. Consider an interview Tyson granted to The Huffington Post several years ago in which he presented the multiverse as the rigorous outcome of scientific theories.^{3} If these theories were presented in the form of philosophical debate precisely because they have no experimental basis, the landscape of science and scientific discussion would undoubtedly be much richer than it currently is. In that regard, one must accept that such theories are philosophical or part of a branch of mathematics and not pure science, which in itself is not a negative thing. It is just, as they say, “calling it for what it is.”
A question not discussed in Ellis’s essay, but which I believe is of the utmost importance is what it means for a theory to be even purely mathematical at its basis? In the field of pure mathematics, a claim or theorem is only considered to be valid if it can be proven, and a mathematician’s notion of proof is perhaps the strongest notion of proof in human society. No physical theory, even those physical theories supported by accepted experimental results, can be held to this standard of proof. We must automatically exclude physical theories from this level of mathematical consideration. If not pure mathematics, do physical theories then belong in the realm of applied mathematics? Even here there is an issue, as mentioned above. All physical theories are partial differential equations—they are, strictly speaking, hyperbolic partial differential equations because of the requirement to obey causality—but applied mathematicians take a very different approach to most physicists when it comes to solving such equations. At their heart are the fundamental concepts of existence and uniqueness of solutions, stability of solutions, boundary conditions (Dirichlet, Neumann, Robin, etc.), and initial conditions. This is the main point. Multiverse theories are largely generated by eternal inflation and string theory. I have been hardpressed to find any paper on eternal inflation that specifically addresses these questions with respect to solutions of the EinsteinKleinGordon field equations in the context of inflation except for the specific case of stochastic eternal inflation.^{4} In this particular case it was shown that eternal inflation in this context is, in fact, not eternal because of the nonexistence of unique solutions to these equations. If all multiverse theories are grounded in known physical theories, which themselves are grounded in partial differential equations, how can one take seriously the mathematical viability of such theories? That is, if the concepts of existence and uniqueness of solutions, boundary conditions, and the stability of solutions are not being addressed on any serious level?
If such multiverse theories cannot be considered as purely scientific in the absence of scientific testability, cannot be considered as purely mathematical in the absence of rigorous mathematical proof, and cannot be considered as a pure branch of applied mathematics because of these reasons, then where exactly do they belong? I believe that such theories belong in the realm of philosophy. Such theories can then be debated and discussed, perhaps attaining a level of mathematical viability according to the criteria above. Even if this were the case, then such theories, if one adheres to the strict definition of science, will always fall outside the scientific realm because they cannot be experimentally tested.
Ikjyot Singh Kohli
George Ellis replies:
Ikjyot Singh Kohli gives an interesting analysis of the relations between mathematical theories, physical theories, and philosophy. I agree with the tenor of his letter.
Ikjyot Singh Kohli is a Postdoctoral Fellow in the Department of Mathematics and Statistics at York University in Toronto.
George Ellis is Emeritus Distinguished Professor of Complex Systems in the Department of Mathematics and Applied Mathematics at the University of Cape Town in South Africa.
 Vladimir Arnold, Mathematical Methods of Classical Mechanics, trans. Karen Vogtmann and Alan Weinstein, 2nd edn. (New York: Springer, 1989). ↩
 George Ellis, “Physics on Edge,” Inference: International Review of Science 3, no. 2 (2017). ↩
 David Freeman, “Why Revive ‘Cosmos?’ Neil DeGrasse Tyson Says Just About Everything We Know Has Changed,” The Huffington Post, March 3, 2014. ↩
 Ikjyot Singh Kohli and Michael Haslam, “Mathematical Issues in Eternal Inflation,” Classical and Quantum Gravity 32 (2015): 075001. ↩
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