Consider the following argument:

Proposition 1: The language of physics (as an empirical science) is mathematics.

I think this should be uncontroversial to the majority of working physicists.

Proposition 2: That the mathematics used in physics will eventually be formalised in the way that mathematicians use mathematics.

There is no generally accepted formalism that makes sense of the Feynman Path Integral, though there are special cases that have been formalised. But I think most physicists would accept that this will be a matter of time & human ingenuity. When Newton invented calculus to investigate problems in dynamics, he was famously criticised by Berkeley for his fluxions, they were only put on a more certain (cumbersome) basis a century later.

Whereas its not a priori certain that a question wholly mathematical necessarily has a theory behind it, I think it is generally accepted in the community that the mathematics behind physics should - I would argue that its this certainty that allows physicists to take the shortcuts they do. I should clarify that formal here means that all foundational questions have been cleared up (I'm not going into Godel now).

Proposition 3: That these formalisms will form a self consistent whole.

Again, I don't think this should be controversial. Currently we have GR & QFT. I take it as generally accepted that there is a further theory that will combine both.

Proposition 4: That this theory will not be subject to Popperian falsifiability (though we cannot verify this).

Popper suggested that theories progress by falsification. I'm proposing once we reach a 'true' theory, by definition it will not be falsifiable by definition. Of course only some 'Oracle' can verify this by comparing the 'true' underlying mathematical reality to the one we've reached by our unaided efforts. (Note that I am putting the word true in quotes as I'm not sure what true means in these circumstances.)

Proposition 5: That a self-consistent whole is capable of further internal development.

I can't see how this can be controversial.

Proposition 6: That this internal development must proceed on aesthetic grounds—what mathematicians call mathematical intuition, elegance; and what physicists call physical intuition.

Given that the theory can no longer be tested, meaning that there can be no experimental evidence to force a change, the only development must be internal. I'm assuming physical/mathematical intuition can be characterised as a certain form of aesthetic. I don't see this as controversial, given some of the famous pronouncements by physicists & mathematicians of all stripes.

Where does this argument fall down?
I suggest at step 4, because there can be no final underlying mathematical reality. This seems a bit presumptuous, considering that the whole scientific project relies on this, but if there is, we go to step 5, which again states there isn't a final underlying mathematical reality.

As Dorfman points out below, its not a given that underlying reality can be expressed in mathematical form. Verlinde, inventor of Entropic Gravity has stated same in an interview. (I'd provide a link but I forget where I saw it).

The point of my argument is to demonstrate even accepting that there is will lead to contradictions.

  • The question in the headline is very interesting, but I worry it doesn't seem to capture this question as it currently is written very precisely. In other words I think there would be some interest in answering the primary concern robustly, but the way the issue is approached "propositionally" perhaps makes it somewhat difficult to address the core issues at stake. Is there any chance I might be able to persuade you to try to formulate your concern a bit more directly/concisely? – Joseph Weissman Sep 7 '12 at 23:36
  • I fear that you are right. I'll give it another go. – Mozibur Ullah Sep 23 '12 at 18:29

The argument fails on Proposition #1.

"The language of physics is mathematics" is true only in the sense that mathematics is the language used by physicists to model the empirically observed effects measured in the physical world. This does not mean that all physical systems can necessarily be modeled mathematically, or that any set of mathematical formulas will completely describe the functioning of the physical world.


I fear I may not have made my point clearly enough, so let me try again.

The relationship between mathematics and physics is approximately the same as the relationship between baseball statistics and baseball. (Substitute "cricket", or some other local sport if necessary.)

We can say that baseball statistics is the language of baseball, and can expect that the formalisms of the statistics will form a coherent system (in that the numbers balance, and there are no conflicts or contradictions.)

However, this system of statistics is not the equivalent of baseball. To think so is confusing the map for the territory.

The "self-consistent whole" and "theory" in steps 3 and 4 is a map. No amount of further manipulation will teach us much about the territory we have not already explicitly put into the map.

  • I've modified proposition 1 considering your first statement, so it should pass. Would you agree that your remaining statements means it fails at 4? – Mozibur Ullah Apr 17 '12 at 10:44
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    You made your first point clearly enough, I both understand and agree with it. (But I do think proposition 1 is the working hypothesis of the majority of physicists. Do you not?) I'm not so sure about your second argument. I think you're conflating physics with the phenonomenal world. Surely to model the phenonomenal world we must use a language. This can only be a representation. (I don't want to get into the argument that the phenomenal world as presented to us is also a representation). Physics is a language to model (that is to both describe & predict) the phenomenal world. – Mozibur Ullah Apr 19 '12 at 23:58
  • Plus, it has explanatory power: its predictions are derived from general laws. That is its tending to the condition of an axiomatic theory. I can't agree that no further manipulation will teach us much..., consider the reformulation of Newtonian Mechanics into its 'equivalent' Lagragian Mechanics. – Mozibur Ullah Apr 20 '12 at 0:02

Proposition 1 is generally regarded as true, and proposition 2 is probably (depending on what you mean) equivalent to saying that physicists are actually using math, as opposed to something else, even if they take some shortcuts and don't define everything.

But I don't think it's a given that the formalisms will form a self-consistent whole. It would be nice if they could, and it's certainly worth trying, but I don't see a priori why it should be possible to have formalisms developed in different areas be fully consistent with each other, except in the trivial way where they basically become lookup tables telling you what happened in each case. And anyway, self-consistency doesn't imply consistency with the natural world.

Proposition 4 is almost certainly wrong: a true theory could still be falsifiable (e.g. in thought experiments) without being fasified if it is a true theory. (Also, that it is a true theory is a separate proposition; it doesn't follow automatically from earlier propositions.).

Proposition 5 seems straightforward enough; even if it's all tautologies, we don't necessarily know them.

Proposition 6 also seems to be wrong--what about developing things on practical grounds because we want to make better iPads and download videos from YouTube faster?

  • I've probably not made myself clear enough: The question is about fundamental physics, the project to understand the material world ie theory-building. So your critique of proposition 6 isn't relevant. I agree that formalisms do not form a self-consistent whole now, the question is whether they ever will be, and I think that very much depends on your commitment to what form underlies reality. I'm supposing that the majority of physicists do think that the underlying reality is consistent. – Mozibur Ullah Apr 17 '12 at 6:05
  • :I don't follow your critique of proposition 4: Popper characterises a theory as being falsifiable by experiments carried out in the real world, not thought experiments. These are generally used to identify where a theory may go wrong, ie the EPR thought experiment. My assumption that we have a true theory is a metaphysical one. From a Popperian perspective we can never be fundamentally certain, though we're entitled to pretend to do so. – Mozibur Ullah Apr 17 '12 at 6:12
  • @Mozibur Ullah - Falsification experiments carried out in the real world generally test between two (or more) hypothetical worlds, in order to rule out at least one of them. In that sense--that one can carry out tests that a theory true-in-the-real-world would fail in a hypothetical world--the theory would be testable. Anyway, fundamental physics has had plenty of important implications in the past (e.g. atomic weapons), so it seems an error to reject practical concerns when one is understanding the world at its most fundamental level. – Rex Kerr Apr 17 '12 at 17:56
  • @kerr: Its an easy jump to make the claim I'm rejecting practical concerns, but this doesn't follow at all from my argument, although I can see why it may do so. – Mozibur Ullah Apr 18 '12 at 3:42
  • @MoziburUllah - "We must proceed on aesthetic grounds" vs. "we had better study this so we can build a reactor" sounds very different to me. Therefore, proposition 6 rejects practical utility. Mathematics is no different: much of applied math (e.g. approximation theorems for PDEs) is also driven by utility, not elegance or enjoyability. – Rex Kerr Apr 18 '12 at 12:03

I would say that fundamental physics will eventually become a human construction and then proceed on aesthetic grounds.

Our process of "knowing the world" has co-evolved with the universe, and our math coincides with physics because of this, not from something outside of it. One should remember that every science instrument encodes within it already an ontology about the world.


Once we know the complete theory of our universe, and I happen to be pretty confident that this is modern string theory, and this theory is defined in all achievable circumstances, then there is no further developments possible, other than pedagogical. You can rewrite Maxwell's equations in pretty form, but you aren't getting more insight unless you have a new idea. So point 5 is not really true.

Once you know the laws, you can imagine new situations where the laws lead to interesting consequences, but this is not a modification of the laws, since the prediction algorithm is already known. At best it is a slight simplification.

Popper's notion of "falsifiability" doesn't mean that a true theory isn't constantly tested--- it just passes all the tests, and that makes it true. What physicists call "aesthetic development" is an activity of creating new ideas. The activity people think of as "aesthetic development" of rewriting old ideas in prettier form, is usually done by people who don't have any new ideas (or else they wouldn't waste time) and so is a progress of regress.

What physicists call "physical intuition" is not what is called "elegance" by mathematicians--- it is the collection of insights regarding the behavior of physical systems that is found from known behavior of familiar systems, formalized into principles. The physical intuition is a collection of prejudices that is fundamentally justified by the belief that there is a collection of comprehensible laws that give a description of nature. We have found a candidate for these laws in string theory, and I am certain the correct candidate, but to complete the mathematical development requires more physical intuition and mathematical development.

Part of the disconnect between physicists and mathematicians is that mathematicians by and large abandoned logical positivism, as manifested in Hilbert formalism, after world war II. This means that they accept Banach Tarsky measure paradoxes. These measure paradoxes are relatively harmless in the real number case, but they make problems with defining the path-integral formally. Physicists have kept logical positivism, and reinforced it, by using it in the last 5 decades to achieve insight that other fields cannot achieve. This is the main schism between physics and mathematics and between physics and philosophy, the positivism, and it won't go away until everyone accepts positivism.

  • Euclidean geometry was taken as a pre-eminent example of a perfected axiomatic system. It's a historical fact that people were not satisfied with the parallel postulate and that this led to the invention of non-euclidean geometry, with the consequences that you know of. Surely this is an example of internal development that you reject? – Mozibur Ullah Apr 19 '12 at 22:11
  • I agree that physical intuition is different from mathematical elegance. But they can be thought of forms of aesthetic. After all some of the best physicists have found physical laws beautiful. – Mozibur Ullah Apr 19 '12 at 22:16
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    Intellectual history is not as straight-forward as you're making out. Some mathematicians rejected Hilberts formalism from the beginning, in fact one would suppose Godel did, otherwise he wouldn't have embarked on his attack. Brouwer espoused intuitionism, which attracted people of the calibre of Weyl, with the general acceptance by the mainstream of ZFC it went underground only to feed into Category Theory itself an outgrowth from Algebraic Topology and which is a major driving force in contemporary mathematics. – Mozibur Ullah Apr 19 '12 at 22:28
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    Mathematics & Physics periodically drift apart and then come together again. Their concerns, methods, language are different. A recent example was the recognition that fibre-bundles provided a global geometric picture of non-abelian gauge theory in the 70s. – Mozibur Ullah Apr 19 '12 at 23:04

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