A common theme in discussions online of the problems of fundamental theoretical physics is that the subject has gotten “lost in math”, losing touch with “physical intuition”. In such discussions, when people refer to “math” it’s hard to figure out what they mean by this. In the case of Sabine Hossenfelder’s “Lost in Math” you can read her book and get some idea of what specifically she is referring to, but usually the references to “math” don’t come with any way of finding out what the person using the term means by it. Here I’ll mostly leave “math” in quotation marks, since the interesting issue of what this means is not being addressed.
“Physical intuition” is also a term whose meaning is not so clear. Sometimes I see it used in an obviously naive way, referring to our understanding of the physical world that comes from our everyday interaction with it and the feeling this gives us for how classical mechanics, electromagnetism, thermodynamics work. Some people are quite devoted to the idea that this is the way to understand fundamental physics, sometimes taking this as far as skepticism about subjects like quantum mechanics.
Usually though, the term is not being used in this naive sense, but as meaning something more like “the sort of understanding of physical phenomena someone has who has spent a great deal of time working out many examples of how to apply physics theory, so can use this to see patterns and guess how some new example will work out”. This is contrasted to the person lacking such intuition, who will have to fall back on “math”, in this situation meaning writing down the general textbook equations and mathematically manipulating them to produce an answer appropriate for the given example, without any intuitive understanding of the result of the calculation. This is what we expect to see in students who are just learning a new subject, haven’t yet worked out enough examples to have the right intuition.
If the question though is not how to apply well understood fundamental theory to a new example, but how to come up with a better fundamental theory, I’d like to make the provocative claim that “physical intuition” is not going to be that helpful. New breakthroughs in fundamental theory have the characteristic of being unexpectedly different than earlier theory. The best way to come up with such breakthroughs is from new experimental results that conflict with the standard theory and point to a better one. But, what if you don’t have such results? It seems to me that in that case your best hope is “math”.
Here’s a list of the great breakthroughs of fundamental physics in the 20th century, with some comments on the role of “physical intuition” and “math”.
- Special relativity: According to physical intuition, if I’m emitting a light ray and speed up in its direction, so will the the speed of the light ray. The crucial input was from experiment (Michelson-Morley), which showed that light always travels at the same speed. Finding a sensible theory of mechanics with this property was largely “math”.
- General relativity: There’s a long argument about the role of “math” here, but I think the only way to develop “physical intuition” about curved spacetime is to start by learning Riemannian geometry (which Einstein did).
- Quantum mechanics: Here again, a crucial role was played by experimental results, those on atomic spectra. A large part of the development of the subject was applying “math” to the mysterious spectra for which there was zero “physical intuition”. Later on, a better understanding of the theory and better calculational methods involved bringing in a large amount of new “math” to physics, especially the theory of unitary representations of groups.
- Yang-Mills theory: This was pretty much pure “math”: replacing a U(1) gauge theory by an SU(2) gauge theory.
- Gell-Mann’s eight-fold way: Pure “math”.
- The Anderson-Higgs mechanism: The funny thing here is that Anderson did get this out of “physical intuition”, based on what he knew from superconductivity. Particle theorists ignored him (especially when it came time for a Nobel Prize), and their papers about this were often mainly “math”, more specifically argumentation about how the mathematics of gauge symmetry could give a loophole to a theorem (the Goldstone theorem).
- The unified electroweak theory: Looks to me more like “math” than “physical intuition”.
- QCD and asymptotic freedom: David Gross famously had the “physical intuition” that the effective coupling grows in the ultraviolet for all QFTs, based on experience with a wide range of examples. He set a mathematical problem for his student (Frank Wilczek), and when the “mathematics” was finally sorted out, they realized the usual physical intuition for QFTs had to be replaced by something completely different.
Making a list instead of the great disasters of 20th century theoretical physics, there’s
- Supersymmetry: OK, this one is “math”. I suspect though that the problem here is that the “math” is not quite right, but missing some other needed new ideas.
- String theory: As we’re told in countless books and TV programs, this starts with a new “physical intuition”: instead of taking point particles as primitive objects, take the vibrational modes of a vibrating string. Developing the implications of this certainly involves a lot of “math”, but the new fundamental idea is a physical one (and it’s wrong, but that’s a different story…).
Let’s expand some of these a little bit though:
Special relativity–>Logic shows that relativity (the Galilean kind) is in conflict with electromagnetism and Galilei’s transformations, intuition shows the third must go and be substituted by something that respects the first two, math was used to derive that.
General Relativity–>Logic shows gravity must be updated to take special relativity into account and the most naive implementation will break the equivalence principle, intuition says the equivalence principle is fundamental and one must find a way to preserve it, logic and some thought experiments (Ehrenfests’s paradox) say that because of time dilation this inevitably involves Riemannian geometry, maths follows
Quantum mechanics—>Experiment gives funny results, intuition shows that to explain them one needs some seemingly counter-intuitive ansatze (Bohr’s momentum quantization, the photoelectric effect), which are then progressively systematized using more and more sophisticated maths.
Yang Mills/Gell-mann/Electroweak theory—>Intuition shows that perhaps it is worth to extent the techniques we used to classify spin to systems where there seem to be approximate degeneracies analogous to spin (isospin/strangeness), group theory follows.
Intuition also guided the analogy between strong and weak isospin
Asymptotic freedom—->Before Gross there was the parton model, with a mix of intuition and phenomenological maths by Feynman,Bjorken etc showing that “partons” weakly coupled at high Q² exhibit the sort of scaling seen experimentally.
Higgs—>The necessity of renormalizability follows the deeply intuitive work of Wilson, through there is quite a lot of sophisticated maths to connect this to Gauge symmetry.
The result that a theory broken by the Higgs mechanism is still renormalizeable is intuitive (the Higgs is in the IR, gauge symmetry needs to operate in the UV) but needs a lot of sophisticated maths to prove.
In each case intuition has a crucial role in selecting what math to pick. The set of consistent mathematical systems is almost certainly infinitely larger than the set of consistent mathematical systems relevant to physics. So it is an interplay between the two.
The path integral, Feynman diagrams, the Parton model are more physical intuition than math.
Rohrlich, F. The unreasonable effectiveness of physical intuition: Success while ignoring objections. Found Phys 26, 1617–1626 (1996). https://doi.org/10.1007/BF02282125
Abstract
The process of theory development in physics is a very complex one. The best scientists sometimes proceed on the basis of their physical intuition, ignoring serious conceptual or mathematical objections well known to them at the time.The results soon justify their actions: but the removal of these objections is often not possible for a very long time. Four examples are presented: Newton, Schrödinger, Dirac, Dyson. Some thoughts on this “unreasonableness≓ are offered.
“Steven Weinberg, who was awarded a Nobel Prize for unifying the electromagnetic and weak interaction, likes to make an analogy with horse breeding: “[The horse breeder] looks at a horse and says “That’s a beautiful horse.” While he or she may be expressing a purely aesthetic emotion, I think there’s more to it than that. The horse breeder has seen lots of horses, and from experience with horses knows that that’s the kind of horse that wins races.”
But like experience with horses doesn’t help when building a racing car, experience with last century’s theories might not be of much help conceiving better ones. ”
That’s a quote from my book “Lost in Math” to say I agree with you.
“Math” v “physical intuition” is surely about choosing a strategy when confronted with a challenging problem in theoretical physics. The choice obviously depends on the background and experiences of the chooser, and history shows that there is no ‘best’ choice. The real issue is the potential (or lack thereof) for a proper interplay between theory and experiment. All the examples of successful theoretical developments, no matter what strategy was adopted, involve experiment, because experiment in physics is the arbiter of ‘success’. In the cases of supersymmetry and string theory, these are unsuccessful either because these approaches make no contact with experiment, or because the ‘predictions’ they have made have simply not be upheld by experiment. The more substantial issue then arises from theorists justifying their continued commitment to an unsuccessful theory by seeking to change the definition of ‘success’.
Doesn’t the canonical history of general relativity involve a rather “naive” form of physical intuition? “I am falling in an elevator”, “I am moving on a rotating disk”, etc.
Pardon me for being facetious but many years ago I decided that “intuition” is just a word one uses for a body of knowledge learned so deeply and so many years ago that one doesn’t recall not knowing it. E.g., much of at least non-rel. QM seems superficially to be intuitive to me–which is patently ridiculous.
This is a fascinating topic but I come down on the other side. Quantum mechanics, SR and GR were all driven by physical observation and intuition. “Math” was the available tool to be used. If math was dominant, Hilbert would have figured out GR.