Physics is Grounded in Mathematics

by Eric Steinhart

Mathematics is effective in science. Wigner (1960: 14) regards this effectiveness as magical: “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” The prudent reply that it is surely not very scientific to base scientific reasoning on miracles. A more rational alternative says that mathematics is effective in science because physical reality is grounded in mathematical reality.

The Effectiveness Argument goes like this: (1) Mathematics is effective in science. (2) The best explanation for this effectiveness is that physical reality is grounded in mathematical reality. (3) So, by inference to the best explanation, all physical reality, including our universe, is grounded in mathematical reality – in pure mathematics.

The second premise in the Effectiveness Argument is supported by a variety of writers. Dipert (1997: 332) argues that “the very possibility of a clear understanding of the world requires the possibility that it is a simple mathematical structure”. Steiner (1998: 4 – 5) puts it even more powerfully like this:

The strategy physicists pursued . . . to guess at the laws of nature, was a Pythagorean strategy: they used the relations between the structures and even the notations of mathematics to frame analogies and guess according to those analogies. The strategy succeeded. . . . The success of the Pythagorean strategy might lead the reader to conceptual Pythagoreanism, the view that the ultimate properties or ‘real essences’ of things are none other than the mathematical structures and their relations. More radically, one might adopt metaphysical Pythagoreanism, which simply identifies the Universe or the things in it with mathematical objects or structures. (Some physicists write as though an elementary particle just ‘is’ an irreducible group representation, or even that the entire universe is.)

Steiner (1998: ch. 4) brilliantly discusses many examples in which the pythagorean strategy of identifying physical things with mathematical things is successful. His cases include: Maxwell’s study of electromagnetism; Schroedinger’s study of wave mechanics; Dirac’s study of the positron; Schwarzschild’s solution for the equations of general relativity (i.e. black holes); Heisenberg’s study of the symmetries of nucleons; Kemmer’s study of pions; Gell-Mann’s and Ne’eman’s study of particle systems with unitary spin and the consequent discovery of quarks; Einstein’s inference of the field equations for general relativity; the Heisenberg-Born-Jordan derivation of matrix mechanics; Schroedinger’s derivation of the Klein-Gordon equation; the derivation of the Yang-Mills equation; the study of analytic continuations in crossing symmetries.

As a continuation of Steiner’s reasoning, Tegmark (1998: 44) says: “the usefulness of mathematics for describing the physical world is a natural consequence of the fact that the latter is a mathematical structure.” Accordingly, Tegmark (1998: 46-47) simply collapses the distinction between mathematical and physical existence:

One might say that wherever there is light, there are associated ripples in the electromagnetic field. But the modern view is that light is the ripples. One might say that wherever there is matter, there are associated ripples in the metric known as curvature. But Eddington’s view is that matter is the ripples. One might say that wherever there is physical existence, there is an associated mathematical structure. But according to our TOE [theory of everything], physical existence is mathematical existence. (The italics are Tegmark’s.)

Dipert, R. (1997) The mathematical structure of the world: The world as graph. Journal of Philosophy 94 (7), 329-358.

Steiner, M. (1998) The Applicability of Mathematics as a Philosophical Problem. Cambridge, MA: Harvard University Press.

Tegmark, M. (1998) Is ‘the Theory of Everything’ merely the ultimate ensemble theory? Annals of Physics 270, 1-51.

Wigner, E. (1960) The unreasonable effectiveness of mathematics in the natural sciences. Communications on Pure and Applied Mathematics 13, 1-14.

Guest Contributor Eric Steinhart is an associate professor of philosophy at William Paterson University. Many of his papers can be found here .

About Daniel Fincke

Dr. Daniel Fincke  has his PhD in philosophy from Fordham University and spent 11 years teaching in college classrooms. He wrote his dissertation on Ethics and the philosophy of Friedrich Nietzsche. On Camels With Hammers, the careful philosophy blog he writes for a popular audience, Dan argues for atheism and develops a humanistic ethical theory he calls “Empowerment Ethics”. Dan also teaches affordable, non-matriculated, video-conferencing philosophy classes on ethics, Nietzsche, historical philosophy, and philosophy for atheists that anyone around the world can sign up for. (You can learn more about Dan’s online classes here.) Dan is an APPA  (American Philosophical Practitioners Association) certified philosophical counselor who offers philosophical advice services to help people work through the philosophical aspects of their practical problems or to work out their views on philosophical issues. (You can read examples of Dan’s advice here.) Through his blogging, his online teaching, and his philosophical advice services each, Dan specializes in helping people who have recently left a religious tradition work out their constructive answers to questions of ethics, metaphysics, the meaning of life, etc. as part of their process of radical worldview change.

  • James Sweet

    I am not sure exactly what it means to say that “the physical world is a mathematical structure”. Certainly I agree that mathematics describes something fundamental, i.e. it is not just some arbitrary description. But I don’t think that means that because it describes something fundamental, it is that fundamental thing.

    When I say, “My coffee mug is empty,” while it is true that the notation I use is arbitrary (it could well have been said in Spanish rather than English) and it is true that there are all sorts of implications (what I really meant is “My coffee mug contains a gaseous mixture composed of approximately 3/4 nitrogen, as well as a number of other molecules — but no damn coffee!”), it is still true that I am describing something real and non-arbitrary. I am making a linguistic assertion about the present nature of my coffee mug, one which accurately and non-arbitrarily describes a fundamental truth about my coffee mug… but I would never deign to argue that “the emptiness of my coffee mug is a linguistic structure”! I’m not even sure what that statement would mean.

    Mathematics describes reality in a fundamental and non-arbitrary way, but I don’t think that from this we can assert that reality is a mathematical structure. Again, I’m not even sure what that would mean…

    I’d be interested if you would care to address my point about Incompleteness in the comments to the previous post. Any mathematical system must be either inconsistent or incomplete. This seems incompatible with the assertion that the universe is a mathematical structure…?

    • Eric Steinhart

      Godel’s Theorems concern the limits of proof within a formal system (e.g. given sufficiently powerful axioms, there are theorems that have no proofs). Godel’s Theorems are not relevant in this context at all. (Just like Turing’s proof that there is no way to compute the halting function isn’t relevant to the design of your computer.)

  • mikespeir

    I’m inclined to see things James Sweet’s way, although I’m a babe in the woods when it comes to this kind of thing.

    Still, I’m wondering, what would be the alternative to math for describing reality with precision? Is there any other conceivable construct that would accomplish the same thing? Somehow, I have this intuitive sense that if math is unique in this quality, there may not be any real difference between it and reality. I’m not sure that follows at all, even after leaping logical gorges. It just kind of struck me.

  • James Sweet

    I’m also a “babe in the woods,” but luckily (unluckily?) my ego is big enough to keep me blathering on about it anyway :)

    Perhaps paradoxically, I feel like I agree in some sense with the statement that “there may not be any real difference between [math] and reality”, while disagreeing with “reality is a mathematical object.”

    Say we have some fundamental physical law — let’s pretend Newtonian mechanics is perfect and fundamental, just so I can type “F = ma”, which has fewer characters. I feel that Eric Steinhart wants to say that “force really is the product of mass and acceleration,” as opposed to someone else who might say, “force is modeled by the product of mass and acceleration, but it really is something distinct and separate.” I am not sure either of those statements are necessarily meaningful. It seems to me that either there is nothing to be said, and that it only seems like there is something to be said because of a linguistic and/or cognitive artifact; or that there is something to be said but that we can’t possibly say it because of either a linguistic and/or cognitive limitation, or perhaps because of fundamental limitations on what can ever be said about reality by a being existing in reality. (Which would be all beings, since anything that exists must do so in reality, right?)

    Which brings us right round back to Eric’s original post. I am not sure whether “the ground” is inherently inconceivable, practically inconceivable, or if it is just nonsense. I suspect the latter, but I am quite convinced it is one of the three — which is why I see no need to affirm or deny its existence. Either it’s bullshit; we could discuss it in principle, but we can’t because of fundamental limitations of being human; or it can’t even be discussed in principle.

  • Brian

    “…physical existence is mathematical existence,”

    is not:

    “all physical reality, including our universe, is grounded in mathematical reality – in pure mathematics,”

    just as it is not:

    all mathematical reality, including our universe, is grounded in physical reality – in pure physicality.

    • Eric Steinhart

      Mathematical structures can be grounded in other mathematical structures – e.g. the natural numbers in pure sets. But you’re right that further clarification is needed.