Pure mathematics is an abstraction of the real world and is a subjective art-form like music, art and literature; as was correctly defined by Aristotle, “The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful”: Aristotle, Metaphysics, M3, 1078b.
But has modern physics, since Albert Einstein, taken over mathematics and inverted the real world into the abstract world of pure mathematics and eliminated its "limitation" – the prize quality that Aristotle was so much fond of?
Well, If I may say so, I think I'm functionally right that many practicing physicists don't often question the fundamental precepts of their mathematical models, and they really should do this, but I don't think Einstein had much to do with this phenomenon. It's human nature to hold onto a paradigm and use it a little bit past it's sell-by date. Check out Thomas Kuhn's "The Structure of Scientific Revolutions" for a classic study of how paradigm shifts occur in science. What is true is that we very well may be on the cusp of a modern revolution in physics. And at the heart of that potential revolution is a reanalysis of what a mathematical model of a real process or event actually is. The role of mathematics in science is now a very hot subject of debate, particularly in physics.
The work of Lee Smolin, the cosmologist whom I've discussed in my "Time Reborn" review, brings to a head a fundamental problem with current mathematical models of reality. We have an embarrassment of riches in that we have not one, but TWO, perfectly cromulent theories that have been experimentally supported by several generations of scientists:
1) The Standard Model of quantum physics which predicts with stunning accuracy the interactions of fundamental particles; and
2) General Relativity which explains gravity and the motion of macroscopic objects.
The problem is that both of them can't be correct in the same universe. If you don't believe me, explain why the Standard Model breaks down when faced with a singularity (like in the real world). A bit of a puzzler, that. Proposed solutions, most recently the multiverse idea, have been far from satisfactory. Most annoyingly, the fundamental problem is that modern physics can't seem to construct testable, falsifiable assertions about the universe as a whole (as opposed to an experimental space in a lab). The method breaks down.
That's what Kuhn means by a paradigm failure/shift.
As I said, this all very annoying. That's why this is a very, very exciting time to be alive. Smolin has his response: an end to the mathematical 'spatialization' of time that is characteristic of most attempts to reconcile gravity with quantum, and a return to the concept that time is more than just a '4th' spatial dimension, but is rather something very different in kind. That concept is quite annoying to many math heads, who really, really, really don't like Smolin and his ilk of physicist. It's a debate to watch, much like the debate in genetic biology between Dawkins and the kin-selection gang, and Wilson and his hive of social biologists (hey, that's a damn good pun that is).
So yes, there is a hot debate going on in physics as to the efficacy of mathematical models in the real world (and even if such a thing is actually possible), but don't blame Uncle Albert. He would almost certainly be among the modern revolutionaries, storming the barricades now, the same as he did in his age.
Coming back to quantum mechanics, the 'laws' of physics may not be true everywhere and everywhen. For example, in a singularity (as in a black hole), the symmetries of quantum mechanics break down completely as I said before, and the equations produce nonsense. This implies two things:
1) The Standard Model is not a theory with a one-to-one correlation to the Universe. There are places now where the 'laws' of physics (as you've defined them) don't hold.
2) Our entire universe was once in a singularity. That's worth contemplating. At one time on our past, the laws of quantum mechanics cannot give us insight about any aspect of the universe at all.
In fact, the 'laws' of quantum mechanics had no object, at this early stage of the universe, on which to operate. So how did a universe that didn't conform to the laws arising from the Standard model become a universe which does conform to the laws arising from the Standard model? Do we need 'meta-laws' to explain the current laws? What would determine these meta-laws? And why these particular laws and physical constants instead of other laws and constants? Why this universe and not some other with different laws/constants?
So when you say the laws of physics are 'true' you need to qualify that statement. In fact, it's no longer entirely clear what a 'law of nature' really is. If a law is not true everywhere and for all time, isn't it really just a strong suggestion?
Kuhn got one thing wrong though: it is counter-revolutionary opposition to paradigm shifts that turn out to be irrational not the revolutionary paradigms shifts. Unfortunately some scientists have developed a material interest in the continued dominance of the old paradigm and irrationally reject the new. Often however the new concepts are extremely difficult to grasp especially by those who have been refining the old concepts so it's not just a material interest based on grants, reputation and the like. Kuhn was doing the sociology of the science community. His work is not about the philosophy of science (apart from the one misguided thought that paradigms are irrational rather than historical as science progresses) but unfortunately some post-modernists have abused it as `proof' of the subjectivity' of science. It is no such thing. It's just an interesting study of a group of people and the way they reach or are prevented from reaching their goals by the social relations between each other and the rest of society. It is itself a scientific work and I don't think Kuhn thought he was being irrational.
Despite being sometimes wrong, I can forgive him easily. Kuhn's book (or his earlier "The Copernican Revolution") came as a shock to philosophical thinking about science. I’ve read elsewhere that his career shift came when, as a trained physicist, he was obliged to fill in education he'd missed as an undergraduate and read Aristotle. He was introduced at a late age to the routine content of philosophy courses: that people think with conceptions, and that conceptions vary in different places and times. So do the problems that thinkers face, e.g. Marx, but not Locke, was faced with the industrial revolution and capitalism. So do the constraints on their thinking: Aquinas and Galileo had to deal with Revelation; Aristotle, Epicurus and the Stoics didn't. We also didn't even have to read Koestler's 1959 “Sleepwalkers” to know how much the history of science can depend on the odd traits of individuals--or the vagaries of research funding. Why didn't Galileo read Kepler's book? Why did he neglect Tycho's hypothesis in his World Systems dialogues? Philosophy education, unlike that in physics, entails critically but sympathetically rethinking issues as well as we can in the terms available. Modern science comes from Aristotle, a biologist contending with Platonic mathematicians on one side and Empedoclean or atomist materialists on the other. It's meaningful to say that Galileo, Newton, Darwin and Einstein are Aristotelians: and you don't get one without the others.
Next, it's typical that later theories can represent earlier ones; and 2) basically Newtonian physics is not only still taught but still used--which leads to another subtlety: 3) that scientists and engineers use various theories--or their parts--to do their work. All these things and more have been said by philosophers re Kuhn's book.
Certainly I view the field of physics through that lens. My own reading of the phrase "incommensurable" is best explained by analogy to watching a movie. Before you've seen the movie all sorts of question - "what's going to happen?" and so on - make sense. Once you've seen the movie those kind of questions make no sense and there are new questions that are important - "how did the director set up scene one to contribute to the twist in the ending?". Try as you might once you know the full plot, you can't see the movie in the same light again. Incommensurability is a lot like that - once you accept a new way of viewing the problem then the old way of looking at things just doesn't make sense. In physics, this is clearly evident in quantum mechanics and Newtonian physics (or even between relativity and Newtonian mechanics). Its true that in terms of the raw mathematics then Newtonian physics can be viewed as somehow inherent in quantum mechanics, but the way in which you view the equations and what they mean has totally changed. Being steeped in the methodology of relativity, I can't even begin to conceive of thinking in terms of absolute time true for all observers, but that is at the heart of Newtonian mechanics. That Descartes could have described Newtonian physics in terms of vortices or Maxwell describing electromagnetism in terms of fluids and mechanical systems just seems patently absurd.
I wouldn't agree that incommensurability means that "there exists no objective way of assessing their relative merits." Relativity is objectively a better theory than Newtonian mechanics, but to be persuaded you need to look at the data and a part of mind set and Kuhn's idea of paradigms is that people sometimes willfully ignore the data in front of them.
Today, we see physics as being inseparable from mathematics and that's our paradigm.
There's much more to say about Kuhn’s book, but rather than continuing, I leave it as an exercise for readers to answer a question that it, like almost Kuhn commentary, fails even to ask: "What does Kuhn mean by 'a paradigm', and how is that an important concept in his account of the history of science?" For that you'll have to read the book, and I recommend it.
Such a Boring Topic Kuhn; I know it inside out already, and I have “me” own theorem:
“John has 32 Chocolate bars, he eats 28,
what does he have now,....
..........Diabetes.”
