# In what ways is physical intuition about mathematical objects non-rigorous?

I'm asking this question as a mathematician who is very far removed from the Physics world, and has little to no knowledge of what math goes into it, and what math comes out of it. What I do hear is that people have "physical intuition" about mathematical objects (especially in algebraic geometry), and that they then try to prove it mathematically.

So out of curiousity, my question, then, is this. Which combination of the following is true for why "physical intuition" isn't already rigorous:

1. They assume the existence of objects that they don't construct.

2. Their logic is flawed.

3. They experiment (with particles and such) and assume that if it works enough times then it is true.

4. They assume that "reasonable" mathematical conjectures are true without bothering to be sure.

5. They don't have axiomatized definitions, and rely on vague notions.

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I think the reason one says physics is not rigorous is, essentially, that while one's tools are valid, no one really knows the axioms, so to speak. So physics is not rigorous in the sense of 'logically valid reasoning from axioms', as mathematics is. –  Ketil Tveiten Jun 12 '10 at 16:56
I've emended the title to better reflect the motivation in the first paragraph. –  Victor Protsak Jun 12 '10 at 22:22
Here is an example of physical intuition which can in principle be upgraded to a rigorous proof: mathoverflow.net/questions/19649/… –  Qiaochu Yuan Jun 12 '10 at 23:19

Mathematics is virtually the only profession that has the luxury of insisting on near-100% certainty. In physics, one is always making approximations and idealised assumptions, and even the known laws of physics may themselves only be approximations of a more accurate set of laws. In such a context, it is far more efficient to rely on heuristics that are accurate 99% of the time, than to insist on rigorous proof that is correct 100% of the time. (In principle, even a mathematically rigorous argument could be subject to scrutiny by the logicians and philosophers, but this is still a far stricter standard than what is needed for a physically convincing argument.)

A typical example of a heuristic is this: if a dynamical system has no obvious conservation laws or other structure to it that would restrict the dynamics, then it is likely to be mixing. With this heuristic one can do a huge amount of statistical mechanics; but it is completely non-rigorous. And indeed, the rigorous theory of statistical mechanics lags far behind the heuristic theory, and is thus far less useful for physics. In physics, it's OK if Maxwell's demon (or some other troublemaker) steps in to ruin everything $10^{-100}$ of the time; but this is unacceptable by the standards of rigorous mathematics.

A mathematical instance of the above heuristic in action would be the assertion that the digits of pi are uniformly distributed because there is no obvious reason why they should not be so. This is extremely convincing at the heuristic level, but falls well short of a rigorous proof, which is still completely out of reach of known methods (we can't rule out the bizarre possibility that pi has attached to it its own "Maxwell's demon").

Note though that these types of heuristics are routinely used in certain areas of mathematics, such as cryptography; and conversely there are plenty of physicists who work on rigorous mathematics; and there are people who would classify themselves as both mathematicians and physicists, or who do a mixture of rigorous and heuristic work. So the distinction is not really so sharp on closer inspection.

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Mathematics and physics are joined at the hip;they are both expressed using the same language (i.e. structures built from logical expressions and/or set theoretic constructs). But the driving motivation of each has become very different in the last 2 centuries. Mathematics is driven by logical consistency while physics needs empirical validation that's far less certain. While it is clear they draw from each other vigorously-particularly in algebra and geometry-they cannot be intechanged. –  The Mathemagician Jun 13 '10 at 2:06
I think the example of cryptography is a little bit more complicated. Mathematicians working in cryptography are usually very careful to state as a hypothesis that certain problems are difficult (or just that there are difficult problems) much in the same way that certain results are conditional on the Riemann hypothesis. On the other hand physicists working in quantum cryptography seem to use "prove" in the sense of physics. Hence one sees statements to the effect that quantum cryptography is proved to be secure whereas RSA is not. –  Torsten Ekedahl Jun 13 '10 at 5:36
Ah, I was referring to the "offensive" side of cryptography (code-breaking) rather than the "defensive" side of cryptography (code design). The two sides of cryptography have opposite standards of rigour (and for good reason). I discuss this at google.com/buzz/114134834346472219368/FXab3PJgWhu/… –  Terry Tao Jun 13 '10 at 10:02
Mathematics [...] has the luxury of insisting on near-100% certainty. The question is about rigor, not certainty. Physicists submit themselves to the rigor of needing to make correct predictions about the outcomes of experiments, and this is simply a different kind of rigor than the kind involved in verifying a preexisting mathematical proof (which is only a small portion of what's done in the profession of mathematics). Rigor and certainty don't go hand in hand. We can prove rigorously from the axioms of quantum mechanics that there is uncertainty about when a nucleus will decay. –  Ben Crowell Jun 18 '14 at 15:07
In physics, it's OK if Maxwell's demon [...] steps in to ruin everything $10^{−100}$ of the time; but this is unacceptable by the standards of rigorous mathematics. The probability is more like $P=10^{-10^{100}}$. This is a far better certainty than anything in mathematics. $P$ is much smaller than the probability that Nelson will fix up his proof that Peano arithmetic is inconsistent. $P$ is also much smaller than the probability that a proof assistant like Coq will make an error due to a software bug. More importantly, there is no way, even in principle, to make these probabilities $<P$. –  Ben Crowell Jun 18 '14 at 15:15

As I see it, the situation is a combination of all of the reasons listed, but I would frame the issues differently:

1. There are many derivations and topics in physics that are entirely rigorous in principle, but where physicists don't consider it worthwhile to dwell in detail on the issues of rigor. For instance, what is a "Dirac delta function"? It's not really a function, but rather a dual vector in the space of continuous functions, but this distinction is often not important in a physics discussion. This situation is simply a matter of division of labor between mathematicians and physicists.

2. Many derivations in physics involve ad hoc approximations whose strength is not well understood. The final equations that are solved are often entirely rigorous, but their relevance to physical reality is negotiable. For instance, in astronomy, when is it fair to approximate planets as points or spheres? Or to pick a more serious example, the shallow water equations are an ad hoc simplification of the Navier-Stokes equations. This is also a division of labor, but a more troublesome one than in case 1. A better justification of these ad hoc approximations could certainly be valuable in physics, and some of them are also interesting mathematics conjectures.

3. Quantum field theory is a special case. There is very strong evidence that quantum field theory, as understood by methods such as Feynman diagrams but also new methods, is an island of mathematical consistency checks that would ideally be connected to ordinary rigorous mathematics. Quantum field theory methods can be used for many "theories" that look important for pure mathematics. Many of these theories have only an abstract resemblance to the true quantum field theories of physics and can't be checked with experiments. Instead satisfy a vast array of consistency checks and lead to many interesting conjectures. On the other hand, a few of these theories are realistic and do agree with experiments. There is no good division of labor to explain why quantum field theory isn't rigorous. It is unfinished business for mathematicians and physicists.

On the other hand, the mathematical status of quantum field theory has at least improved over time. Quantum field theory in 1 dimension is rigorous, and significant pieces of conformal field theory in two dimensions (or one complex dimension) are rigorous.

4. There are also some non-examples that are entirely rigorous but hard to believe. Relativity is hard to believe and quantum mechanics (in the sense of quantum probability) is hard to believe, but they are entirely rigorous. They aren't even known to be ad hoc approximations to something else. (Well, non-quantum general relativity is surely an approximation to some theory of quantum gravity, but let's set that aside for now.) Sometimes they are explained in a non-rigorous way as in case 1, but that shouldn't fool mathematicians. It is important not to conflate these rigorous victories with non-rigorous extensions such as quantum field theory and string theory. Indeed, the disease of case 3 does not strictly require quantumness; some of the difficulties of rigor already occur for classical stochastic field theories.

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In what sense is the shallow water approximation -- or any approximation, for that matter -- a flaw in logic? And in what sense would it constitute a lack of rigour? –  José Figueroa-O'Farrill Jun 12 '10 at 20:35
José: It is of course a leap of faith to make approximations without error bounds. For instance, no one can prove that the Navier-Stokes equations even have full solutions, but some of the simplifications of these equations do have solutions. Arguably a more serious case is condensed matter theory (and even molecular physics), where people have a great deal of faith in the Schrodinger equation for N particles, but a great deal less faith in the zillions of approximate models that are inspired by it. –  Greg Kuperberg Jun 12 '10 at 21:10
I am not sure I agree that "quantum mechanics is entirely rigorous". There is an axiomatic conceptual scheme for quantum mechanics, but most interesting physical computations aren't rigorous. For example, look at the derivation of tunneling probabilities using quasiclassical approximation in any QM textbook. It required a major effort on mathematician's part to "rigorize" even some of that theory (Maslov index, Lere's lagrangian analysis, Guillemin-Sternberg's geometric asymptotics). –  Victor Protsak Jun 12 '10 at 22:09
Well yes, Victor, but such examples fall under case 2. The Copenhagen/von Neumann model of quantum probability is the part that is hard to believe, and it is rigorous. –  Greg Kuperberg Jun 12 '10 at 22:15
Greg: A leap of faith in Physics is not the same thing as a flaw in logic, simply because Physics is not a deductive science. Physics is an empirical science and in modelling physical phenomena assumptions are often made which allow one to arrive at predictions (perhaps by simplifying the analysis of the model). In the end, such assumptions either live or die based on their ability to conform to experiment. –  José Figueroa-O'Farrill Jun 13 '10 at 1:42

Regardless of whether or not it is possible to make specific arguments from physics rigorous, they are often not taught rigorously. I'll give some specifics from my own education.

An obvious example is to consider the content of a basic quantum mechanics course. A course might start by considering wavefunctions. These are asserted to be functions $\mathbb{R}^4\rightarrow\mathbb{C}$ without bothering to state precisely which functions are allowed. Do they need to be continuous, differentiable, smooth, behave in a certain way as we go to infinity? In an elementary course, nobody will bother to say.

We'll be taught that wavefunctions are solutions to PDEs. So we implicitly have to start assuming differentiability to some degree or other. Except we'll be asked to consider potential wells that result in wavefunctions that occasionally have discontinuous derivatives. And then we'll be asked to consider wavefunctions that are Dirac delta functions which are clearly not functions in the usual sense. If you object the lecturer will mutter something about Hilbert spaces under their breath, despite the fact that it's trivial to prove we're being asked to consider spaces of wavefunctions that don't form a Hilbert space.

These kinds of elementary quantum mechanics can be made rigorous. But it's not part of the education of many physicists. And this means that actual arguments made by physicists are often not rigorous even when there is no fundamental problem with making them rigorous.

(And much of this sounds like a complaint. But the truth is, back when I was doing physics, I wouldn't have wanted to "waste" time on making these arguments rigorous.)

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I recently attended a talk by Graeme Segal in which he pointed out that you should never ask physicists to decide what type of functions they want. –  Theo Johnson-Freyd Jun 13 '10 at 3:54
The specific classes of functions we want rarely matter, aside from "functions which somehow satisfy this equation" where somehow may include distributionaly, or in some cases "not at all." Though from a theory-building point of view, they're more critical; general relativity is built off of diffeomorphisms after all... –  jeremy Jun 13 '10 at 3:59
Jeremy, whilst physicists typically do not care to declare a class of functions, I contest your assertion that the class rarely matters. In wave mechanics, the probabilistic interpretation of overlap integrals dictates that wavefunctions should live in a subspace of $L^2$ (this is not guaranteed by the Schroedinger equation alone), with its standard inner product. If I write down a different inner product, you will have to figure out an isometry to the standard $L^2$ product before you can do physically meaningful calculations. –  Tim Perutz Jun 13 '10 at 5:03
Absolutely; what I should have said was: "it does not matter to physicists." As in, personally, it's not something they explicitly think about. Clearly, in terms of the actual structure, everything matters, whether they know it or not! But the average physicist is, really, only barely aware of why the delta function is not a function, let alone what would tell them what kind of functions their framework tells them they can use. For example, in quantum mechanics, knowing only "conservation of probability" and "things are normalized" is typically enough to settle these kinds of issues. –  jeremy Jun 13 '10 at 9:15
Similarly, when I took mechanics, it disturbed me a little that we'd use things like delta-function "impulses." The argument given for why that's okay was "in real-life the impulse isn't really infinite" but to me that seemed to make the problem even worse! Of course, the real "physical" reason (and the one a good professor would've given!) is in terms of thinking about what it means in terms of Newton's laws being satisfied, position being continuous, it being okay if e.g. velocity isn't, etc. That's not good enough for an ODE class, but if your careful it leads to the real proofs. –  jeremy Jun 13 '10 at 9:21

I think some of what's been said is a little misleading. A lot of people have implied that because physicists have the ability to rely on experiments, they don't have to worry as much about formal proofs. But this isn't really right, and although ultimately if a particular theory is "correct" or not depends on this, physical intuition can still lead to good ideas and good math, even if it leads to empirically incorrect ideas (in fact, many good theoretical physicists care as little for experiments as they do for proofs!). The history of physics is actually full of "recycling" good ideas from places where they didn't work, to new places where they do! These ideas are so re-usable not because they were based in experiment, but because the kind of reasoning behind them was qualitatively good, even if it did not end up being quantitatively good (compare to similar instances in math!).

I'm a theoretical physicist, but have an undergrad degree in math, and that's caused me to put a lot of thought into exactly how my thinking in my physics training differs from my thinking in my math training. And I really think that the kind of qualitative thinking that I see a lot of mathematicians use to reason with before building a formal argument is almost exactly the kind of thinking physicists use.

I know one of the things made me first realize this, was a few years ago, reading a blog post by Terry Tao. The post was about some analysis topic that I wasn't familiar with (which I no longer recall specifically) and I had stopped to think carefully about what he was saying for a few minutes after seeing a confusing statement, and tried to reason through it using my physics intuition. After getting some idea of what was going on, I finished reading the article, and, after finishing it, I realized that the logic behind the article as a whole (as opposed to behind each individual statement) was basically identical to what my physics explanation was.

In terms of the points you mention above, "physics intuition" would correspond roughly to 1, 5, and a lesser extent 4. But this (from my point of view) seems to correspond pretty well to exactly how mathematicians think both before they formalize an argument carefully, and in the "big picture" point of view (which is really partly inherited from the former).

So in a sense, physical intuition is everything you do in math, up to, but not including the final step when you make your arguments careful. Although we usually go "most" of the way to making an argument careful, ultimately we do have to bring things to the level of being able to make a calculation which one could compare with experiment, and this requires being fairly careful about the reasoning we use being mathematically sensible (although, from the point of view of most theorists, this is not the interesting part).

We also like to break our arguments up into "fundamental" pieces, but not in the same way as mathematicians do, in terms of axioms/definitions/theorems/lemmas, but in terms of "physically reasonable" pieces, since they are easier to get a handle on in terms of theory-building. But the problem is that, while these physically reasonable pieces usually correspond to simple physical statements, they usually correspond to very complicated statements when spelled out axiomatically, which makes that form of them too cumbersome to work with.

It's difficult to explain specifically what the similarities I'm thinking about are, so if you want to see some specific examples, that would be more amicable from a mathematician's point of view, it could be valuable to grab a text on the calculus of variations and go through some of the proofs of things that you already know through other means (e.g., geodesics) since this specifically is one kind of reasoning that's used all over physics. There're also a number of such books written from a solving physics problems point of view.

There is also "quantum fields and strings: a course for mathematicians" which is a bit tougher, but written by actual physicists, and I think could give a good deal of insight into how we actually think. I would avoid anything called "quantum mechanics for mathematicians" for this because they tend to not be written by people who are primarily physicists.

You could also go back to Euler or Gauss or Riemann, since a lot of their arguments are very "physical" and are highly recognizable for physicists. I believe Spivak's volumes on differential geometry contain some of Riemann's papers, along with discussion translating them into modern language which could be useful to see. The MAA also has a "how Euler did it" column that could be interesting in this regard, too.

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I was once in conversation with John Polkinghorne, a representative mathematical physicist working in quantum field theory. He told me he didn't see why proofs mattered. So I would say "in any way that rigorous is defined via formal proof, that's something physicists are not into". Certainly physics expositions include types of bootstrapping up from simple assumptions by analogy, "proof" by apparent self-consistency, and abuse of notation.

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1. They assume the existence of objects that they don't construct.
2. Their logic is flawed.
3. They experiment (with particles and such) and assume that if it works enough times then it is true.
4. They assume that "reasonable" mathematical conjectures are true without bothering to be sure.
5. They don't have axiomatized definitions, and rely on vague notions.

To varying degrees, all five of these are the case. However, it's important to note that by and large physicists get the right answers. This is what makes physics (and all the other sciences, really) such a vital source of inspiration for mathematics -- results which are false on a straightforward reading of the scientists' language may actually hold for every "reasonable" or "nice" situation that a scientist is interested in. This means that there is often a lot of really cool mathematics lurking in the the formal characterization of what "nice" means.

A few examples:

• Eg, nonstandard analysis and synthetic differential geometry can be seen as ways of taking the language of infinitesmals that physicists pervasively use and putting it on a formal footing.
• People often say that a string like 0101001011000110111 is "more random" than one like 0000000000000000000. This statement doesn't make sense interpreted in the usual terms of probability theory, but does make sense when interpreted in terms of Kolmogorov complexity (and leads to ideas like the minimum description length principle of statistics).
• In computer science, people often say things like "merge sort and bubble sort are different functions". This doesn't make sense when you think of a function as a graph, but does make sense when you think of them as operations -- and taking that viewpoint seriously yields intuitionistic logic.
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I don't think it makes sense to say that even in terms of Kolmogorov complexity, because with the freedom of what abstract computation model you use, you can't say anything about the Kolmogorov complexity of any few fixed strings. –  Zsbán Ambrus Jun 12 '10 at 18:37

A physicist viewpoint:

A proof is a (very careful) explanation why something works. A physicist cannot always wait until one is constructed! Has someone proved that it is possible to bring color to the "light theater"?

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Could you explain what you mean by "light theater"? Do you mean the electromagnetic spectrum? –  S. Carnahan Jun 12 '10 at 19:49
I meant cinema. My point is that you can use science to build gadgets that work, without fully understanding why. Feynman once wrote that, although $E=mc^2$ explains why nuclear bombs work, without this formula the Manhatten project would still have succeeded in building a nuclear bomb: People would still have figured out how to do it, but noone would have understood where all the energy comes from. –  Tim van Beek Jun 14 '10 at 7:07

Once you make a mobile phone using quantum mechanics for the transistors, electromagnetics for the antenna, filters etc. and it actually does work, then you have proved something about nature. Falsification of QM and EM is now impossible (falsification would make the phone stop working), but the theories can still be generalized.

Carl (math student :-)

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