Striving for a Faithful Image of Nature

I wrote a FQXi essay doubting that nature can be faithfully represented by mathematics.

H. C. Ottinger is writing a QFT textbook dedicated to proving that it is possible:

Quantum Field Theory as a Faithful Image of Nature ...

This book is particularly influenced by the epistemological ideas of Ludwig Boltzmann: “... it cannot be our task to find an absolutely correct theory but rather a picture that is as simple as possible and that represents phenomena as accurately as possible” (see p. 91 of [8]). This book is an attempt to construct an intuitive and elegant image of the real world of fundamental particles and their interactions. To clarify the words picture and image, the goal could be rephrased as the construction of a genuine mathematical representation of the real world. ...

Let us summarize what we have achieved, which mathematical problems remain to be solved, and where we should go in the future to achieve a faithful image of nature for fundamental particles and all their interactions that fully reflects the state-of-the-art knowledge of quantum field theory. Sections 3.2 and 3.3 may be considered as a program for future work.

QFT is our best candidate as a mathematical theory of everything, but this text confuses me. The title indicates that QFT is a faithful image of nature. But then he endorses Boltzmann's philosophy that rejects such an absolutely correct theory, and strives for theories that are as simple and accurate as possible. Then later he indicates that maybe we will have a faithful theory in the future.

So is QFT a mathematically faithful image of nature and an absolutely correct theory or not?

I guess not.

These theoretioal physicists all seem to assume mathematically perfect theory of nature is going to be found. The book "Dreams of a Final Theory" by S. Weinberg promotes that idea. The LHC was supposed to get the experimental evidence.

Boltzmann’s philosophical lectures attracted huge audiences (some 600 students) and so much public attention that the Emperor Franz Joseph I (reigning Austria from 1848 to 1916) invited him for a reception at the Palace to express his delight about Boltzmann’s return to Vienna. So, Boltzmann was not only a theoretical physicist of the first generation, but also an officially recognized part-time philosopher. ...

Besides we must admit that the purpose of all science and thus of physics too, would be attained most perfectly if one had found formulae by means of which the phenomena to be expected could be unambiguously, reliably and completely calculated beforehand in every special instance; however this is just as much an unrealisable ideal as the knowledge of the law of action and the initial states of all atoms.

Phenomenology believed that it could represent nature without in any way going beyond experience, but I think this is an illusion. No equation represents any processes with absolute accuracy, but always idealizes them, emphasizing common features and neglecting what is different and thus going beyond experience.

He says that the Duhem-Quine thesis, which was supposedly one of the great ideas of XX century philosophy of science, was just a rehash of what Boltzmann said.

Here are the book's postulates:

First Metaphysical Postulate: A mathematical image of nature must be rigorously consistent; mathematical elegance is an integral part of an appealing image of nature.

Second Metaphysical Postulate: Physical phenomena can be represented by theories in space and time; they do not require theories of space and time, so that space and time possess the status of prerequisites for physical theories.

Third Metaphysical Postulate: All infinities are to be treated as potential infinities; the corresponding limitlessness is to be represented by mathematical limiting procedures; all numerous infinities are to be restricted to countable.

Fourth Metaphysical Postulate: In quantum field theory, irreversible contributions to the fundamental evolution equations arise naturally and unavoidably.

There are sharp differences among physicists over whether their mission is to find better and better mathematical approximations, or to find the mathematically perfect theory valid to all scales. Lubos Motl rips Larry Krauss on this issue:

But let me get to the main question, namely whether a true theory that works at all scale does exist or can exist.

Krauss writes: We know of no theory that both makes contact with the empirical world, and is absolutely and always true.

[Motl] Well, you don't because you're just a bunch of loud and obnoxious physics bashers. But what's more important is that the state-of-the-art theoretical physicists do know a theory that both makes a contact with the empirical world and is – almost certainly – absolutely and always true. It's called string theory.

[Krauss] This theory, often called superstring theory, produced a great deal of excitement among theorists in the 1980s and 1990s, but to date there is not any evidence that it actually describes the universe we live in.

[Motl] This is absolutely ludicrous. The amount of evidence that string theory is right is strictly greater than the amount of evidence that quantum field theory is right. Because string theory may be shown to reduce to quantum field theory with all the desired components and features at the accessible scales, it follows that they're indistinguishable from the empirical viewpoint. That's why Krauss' statement is exactly as ludicrous as the statement that there exists no evidence backing quantum field theory.

Needless to say, his statement is even more ludicrous than that because the evidence backing string theory is actually more extensive than the evidence supporting quantum field theory. Unlike renormalizable quantum field theory that bans gravity, string theory predicts it. And one may also mention all the Richard-Dawid-style "non-empirical" evidence that string theory is correct.

OK, near the end, where he already admits that there exists a theory that makes his initial statements about non-existence wrong if the researchers in the subject are right, we read:

[Krauss] While we don’t know the answers to that question [whether there is a theory that is valid without limitations of scales], we should, at the very least, be skeptical. There is no example so far where an extrapolation as grand as that associated with string theory, not grounded by direct experimental or observational results, has provided a successful model of nature. In addition, the more we learn about string theory, the more complicated it appears to be, and many early expectations about its universalism may have been optimistic.

The claim that "there has been no theory not grounded by direct experimental or observational results that has provided a successful model of Nature" is clearly wrong. Some ancient philosophers – and relatively modern chemists – have correctly guessed that the matter is composed of atoms even though there seemed to be no chance to see an individual atom or determine its size, at least approximately.

But it was correctly "guessed", anyway.

Motl defines string theory to include all the successful developments of QFT, while Krauss is sticking to the string theory vision that all that will someday be derivable from some 10 or 11 dimensional geometry.

Yes, Democritus and others correctly guessed that matter was made of atoms. But they did have some evidence. They observed that water could be frozen, boiled, or polluted, and then returned to being water, without any lasting damage. Likewise with other elements. Water is not technically an element, but they would have been even more impressed if they decomposed it into gases, and then combined them to make water again. Being made of atoms was the most plausible explanation of this phenomenon.

Motl continues:

Also, both special and general theory of relativity were constructed without any direct experimental or observational results. The Morley-Michelson experiment could have "slightly" invalidated the previous statement except that Einstein has always claimed that he wasn't aware of that experiment at all and it played no role in his derivations. Similarly, general relativity was constructed by purely theoretical methods. The perihelion precession of Mercury was nice but it was just a "by the way" observation that Einstein noticed. This anomaly was in no way the "soil" in which the research of Einstein that produced GR was "grounded".

No, this is not correct. Maybe Einstein ignored Michelson-Morley, but it was crucial for the FitzGerald length contraction, Larmor time dilation, relativistic mass, Lorentz transformation, and four dimensional spacetime geometry, as the papers announcing the discovery of these concepts all cited the experiment as being crucial. Even Einstein wrote in 1909 that the experiment was crucial to special relativity, if not to his own work, which was mainly to assume what Lorentz and Poincare had proved.

And we now know from unpublished papers that explaining the Mercury perihelion precession was one of the biggest motivators for general relativity, if not the biggest.

Max Tegmark comes squarely on the side of mathematics perfectly describing the universe, and not just being approximations, and he explains in this recently-reprinted 2014 essay:

For the bird—and the physicist—there is no objective definition of past or future. As Einstein put it, “The distinction between past, present, and future is only a stubbornly persistent illusion.” ...

The idea of spacetime does more than teach us to rethink the meaning of past and future. It also introduces us to the idea of a mathematical universe. Spacetime is a purely mathematical structure in the sense that it has no properties at all except mathematical properties, for example the number four, its number of dimensions. In my book Our Mathematical Universe, I argue that not only spacetime, but indeed our entire external physical reality, is a mathematical structure, which is by definition an abstract, immutable entity existing outside of space and time.

What does this actually mean? It means, for one thing, a universe that can be beautifully described by mathematics. ...

That our universe is approximately described by mathematics means that some but not all of its properties are mathematical. That it is mathematical means that all of its properties are mathematical; that it has no properties at all except mathematical ones.

You might expect Motl to agree with this, but he attacks it fort he incoherent nonsense that it is.

Note that Tegmark says that time is reversible, while the above QFT textbook says that QFT features the irrersibility of time.

Update: Lawrence M. Krauss writes in the New Yorker, where he seems to be the resident physicist:

The physicist Richard Feynman once suggested that nature is like an infinite onion. With each new experiment, we peel another layer of reality; because the onion is infinite, new layers will continue to be discovered forever. Another possibility is that we’ll get to the core. Perhaps physics will end someday, with the discovery of a “theory of everything” that describes nature on all scales, no matter how large or small. We don’t know which future we will live in.

As usual, Feynman was a voice of sanity. Today's physicists are conceited enuf to think that they can find the core, in spite of all the contrary evidence.