Will fundamental physics eventually become an aesthetic construction and proceed on aesthetic grounds?

Question

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.

Answer 1

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.

EDIT:

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.

Answer 2

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?

Answer 3

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.

Answer 4

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.