Is physics rigorous in the mathematical sense?

Question

I am a student studying Mathematics with no prior knowledge of Physics whatsoever except for very simple equations. I would like to ask, due to my experience with Mathematics:

Is there a set of axioms to which it adheres? In Mathematics, we have given sets of axioms, and we build up equations from these sets.

How does one come up with seemingly simple equations that describe physical processes in nature? I mean, it's not like you can see an apple falling and intuitively come up with an equation for motion... Is there something to build up hypotheses from, and how are they proven, if the only way of verifying the truth is to do it experimentally? Is Physics rigorous?

It is not inherently rigorous.. That is, you may try an experiment and get the same result that verifies your equation for 1 million times, but who knows, chances are still that the 1 millionth 1 time could prove you wrong. Can we prove beyond doubt, equations in physics? — A New Guy, Nov 14 '12 at 08:36
If we are going to build Physics from ground-up, then there should be a set of axioms that we should lay by, isn't it? Or else, we could be building Physics on unstable structures. There is a famous saying, whom I forgotten the author, by which he says that, the idea of a proof is to prove beyond doubt, your own argument. — A New Guy, Nov 14 '12 at 08:38
But how would you know what constitutes a good set of axioms without first analyzing empirical data gathered by careful experimentation? In fact, mathematicians are in pretty much the same predicament. The common axiom sets found in formal mathematics were chosen to maximize theorem-proving power/efficiency, not because they are somehow "self-evident" truths. — David H, Nov 14 '12 at 08:52
On the other hand, you might say physicists try to rigorously adhere to the principles and procedures of mathematical statistics when it comes to quantifying their uncertainty in the reliability of a particular result. — David H, Nov 14 '12 at 08:56
I do feel that most of the axioms in mathematics are so intuitive and self-evident like 1+1=2, etc etc, except for the axiom of mathematical induction, and euclid's 5th axiom. Then again, I am just beginning my undergraduate studies, so I haven't seen much of what there is to see. Are there similar, self-avident axioms in Physics too? — A New Guy, Nov 14 '12 at 09:22
Ah, I see what you mean, but I think you're missing my point. It's not at all obvious which sets of self-evident truths can be used to derive all others. Take Euclidean geometry for example. There are countless propositions proven in Euclid's works which are every bit as "self-evident" and simple to state precisely, and Greek geometers, in particular Pythagoras, where aware of the vast majority of them before Euclid. Euclid isolated 5 of these from which all others can be proved. But wait! Mathematicians discover new proofs all time, so what constitutes good axiom sets inevitably evolves. — David H, Nov 14 '12 at 10:04
I highly recommend you read the "Axiom" wiki page for a primer on different usages of the word in different academic settings, as well as how interpretations have evolved since Euclid: http://en.wikipedia.org/wiki/Axiom — David H, Nov 14 '12 at 10:08
The ideal of rigor for science is the hypothetico-deductive method. You make a hypothesis, deduce its consequences, and then test them against reality. The mathematical formalization of this process would be something like the AIXI algorithm in computer science, which uses data to make causal models. Also see the whole field of statistics, and its methods for establishing the likelihood of a hypothesis. The difference between mathematics and physics is that in physics you use empirical data as an input. But you can still be rigorous in your methods. — Mitchell Porter, Nov 14 '12 at 10:39
@DavidH I disagree about "The common axiom sets found in formal mathematics were chosen to maximize theorem-proving power/efficiency, not because they are somehow "self-evident" truths". Firstly, if theorem power proving were given higher priority than being "self-evident truths" then the set of statements ${\forall x [x\in x],\lnot\forall x[x\in x]}$ could have been used as axioms for math instead of ZFC. It certainly has a higher theorem proving power since it can prove any statement as it is inconsistent. — Amr, Dec 19 '13 at 21:35
@DavidH Thus, the axioms were not just chosen to maximize theorem-proving power. I believe the axioms were chosen to describe our own intuition about sets. In my opinion, there are philosophical reasons why sets were chosen the way they are — Amr, Dec 19 '13 at 21:49
@DavidH Secondly, I don't like to think of mathematics as if it asserts the theorems. I like to think of it as asserting the implication. For example, mathematics does not show us that the fundamental theorem of calculus is true, but rather that the of axioms of ZFC PROVE the fundamental theorem of calculus. — Amr, Dec 19 '13 at 21:52
@DavidH In group theory for example, I view the theorem that identities are unique as: The set of statements of group theory proves formally the statement: identities are unique. — Amr, Dec 19 '13 at 21:56
@DavidH I probably bothered you with a lot of comments, but I think what I will write now is worth reading. Even if physics becomes axiomatized, I still think it is different from mathematics in the following sense: You prove a theorem from a certain theory $T$, it does not matter if the universe satisfies theory $T$ or not, we are proving theorem s for worlds satisfying theroy $T$ only, it doesn't even matter if no such world exists. mathematics could be viewed as plain manipulation of symbols, physics is not just manipulation of symbols about. — Amr, Dec 19 '13 at 22:19
@DavidH If physics ever becomes axiomatic, and we use the axiom to prove a certain statement P. It is not only a string of characters on paper. It is an assertion about the physical universe that can be checked using EXPERIMENTS, there is no analouge of this in mathematics. There is nothing to check against, just symbols on paper. However, I like mathematics not for manipulating symbols, but for its beauty (That is something that would take a lot to talk about). Sorry for the many comments — Amr, Dec 19 '13 at 22:22

Stan Liou · Accepted Answer · 2012-11-14T08:43:13.857

No, physics is not rigorous in the sense of mathematics. There are standards of rigor for experiments, but that is a different kind of thing entirely. That is not to say that physicists just wave their hands in their arguments [only sometimes ;) ], but rather that it does not come even close to a formal axiomatized foundation like in mathematics.

Here's an excerpt from R.Feynman's lecture The Relation of Mathematics and Physics, available on youtube, which is also present in his book, Character of Physical Law (Ch. 2):

There are two kinds of ways of looking at mathematics, which for the purposes of this lecture, I will call the the Babylonian tradition and the Greek tradition. In Babylonian schools in mathematics, the student would learn something by doing a large number of examples until he caught on to the general rule. Also, a large amount of geometry was known... and some degree of argument was available to go from one thing to another. ... But Euclid discovered that there was a way in which all the theorems of geometry could be ordered from a set of axioms that were particularly simple... The Babylonian attitude... is that you have to know all the various theorems and many of the connections in between, but you never really realized that it could all come up from a bunch of axioms... [E]ven in mathematics, you can start in different places. ... The mathematical tradition of today is to start with some particular ones which are chosen by some kind of convention to be axioms and then to build up the structure from there. ... The method of starting from axioms is not efficient in obtaining the theorems. ... In physics we need the Babylonian methods, and not the Euclidean or Greek method.

The rest of the lecture is also interesting and I recommend it. He goes on (with an example of deriving conservation of angular momentum from Newton's law of gravitation and having it generalized):

We can deduce (often) from one part of physics, like the law of gravitation, a principle which turns out to be much more valid than the derivation. This doesn't happen in mathematics, that the theorems come out in places where they're not supposed to be.

I see, that is a very helpful answer Stan! I will check out the lecture. However, is there absolutely no way, in no area of Physics, can we apply the idea of Axiom-Definition-Speculation-Theorem-Proof structure of mathematics? — A New Guy, Nov 14 '12 at 08:40
There's a sliding scale; some theoretical physics treatments to come closer to the style of mathematics. For example, there are axiomatic treatments of the formalism of quantum mechanics. Generally these areas are also influenced by mathematicians (e.g., von Neumann). — Stan Liou, Nov 14 '12 at 08:47
I guess mathematical constant arising in Euclidean geometry, the number $\pi$ a human could in theory compute more digits of it entirely mentally even if they have total sensory deprivation and have never learned what any of its digits are whereas the fine structure constant, the dimensionless constant arising in nature, people need to use the Babylonian method to figure out the digits of. — Timothy, Sep 22 '18 at 23:42

score 14 · Answer 2 · edited Jun 22 '22 at 08:14

Physics is usually not rigorous. But there is a branch of physics, called mathematical physics, in which physics is treated with full mathematical rigor. There everything begins with formally stated assumptions (axioms) from which everything else is rigorously deduced.

In particular, there are fully rigorous treatments of phenomenological thermodynamics (see, e.g., my paper http://arnold-neumaier.at/ms/phenTherm.pdf), of classical mechanics, of fluid mechanics, and of quantum mechanics.

A possible set of axioms for quantum mechanics is given in my ''Postulates for the formal core of quantum mechanics'' from Chapter A4: The interpretation of quantum mechanics of my theoretical physics FAQ at
http://arnold-neumaier.at/physfaq/physics-faq.html
This chapter also contains a discussion of ''What is the meaning of axioms for physics?''. See also ''Why bother about rigor in physics?'' in Chapter C2: Some philosophy of physics.

score 6 · Answer 3 · answered Nov 14 '12 at 13:23

Some parts of physics are rigorous in the sense of mathematics - i.e. they are treated as mathematics, with physically-motivated entities. And it is not that uncommon that for some papers to make it rigorously, explicitly stating assumptions and proving things.

However, most of the time physics in not mathematically rigorous. It stems from a few things:

Physics, typically, work the other way around than mathematics. That is, knowing some effects you try to figure out assumptions so the effects can be explained.
Physics is related to the real world. And many times it is tricky to relate a pure mathematical concept to (reasonably) objectively measurable quantities.
In physics, there is not much difference if you fail at predicting because of error in mathematics, or using unphysical assumptions.
Most of things in physics started as a hand waving arguments, which were mathematically dubious, but "they worked in most cases". In that way it was possible to explain or predict many phenomena. Their mathematical grounding often went later (a few years, decades or... is still an open problem); and more than often, in a utterly unpractical form for any physical calculations.

Think of things like Lebesgue integral (still to get the actual numerical value you need to do summation or Riemann integration), delta Dirac as a distribution (for physical calculations it is treated as "narrow enough" function), formalization of path integrals (well, not suitable for calculations), ...

How does one come up with seemingly simple equations that describe physical processes in nature?

It is a big question.

There are some answer in the spirit of emergency like:

"Physics is this part of the reality that is easily described with mathematics."

Or even more cynically (it's more-or-less a quotation, but I've forgotten the author):

"Physics is this part of the reality that can be approximated as coupled harmonic oscillators."

Also, it is highly recommendable to read a classical text on that issue: Eugene Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

score 3 · Answer 4 · answered Nov 14 '12 at 07:14

Well, you generally have a few fundamental equations, and you try to build a theory upon them. For example, one can consider the gravity equation $F=\frac{Gm_1m_2}{r^2}$ to be a fundamental. How was it derived? It wasn't -- it was experimentally determined to a known accuracy. Newton's laws can be considered to be "axioms" as well, though the Lagrangian formulism of Physics is more pleasing to look at axiomatically (some very simple statements involving energy are taken as axioms, and the rest is pretty much mathematically derived).

More modern theories like the Standard Model assume entities behave according to some mathematical relation -- again, this is your axiom (I'm not too familiar with SM, so I'm not too clear about this).

In the end, the only way to check is via experiments. It's technically the same with mathematics-- you "test" your axioms against your logical facilities. The axioms are pretty simple and sometimes silly at first glance, so you never know that you're doing this. On the other hand, in Physics, the axioms are more complex and don't sound as "silly". The experiments required to verify/come up with these are correspondingly more complex, and they don't give a completely definite result -- we can only say that "equation X is valid to Y accuracy" or something like that.

If we call the fundamentals "axioms", then yes, Physics is rigorous. But these axioms are on pretty shaky ground themselves. For the past century we've been trying to tweak these axioms/postulates so that we get a system consistent with the real world.

Another way to look at it is that we just port over all axioms from Mathematics, postulate a few things (what a force is, what momentum is, etc), and everything else is a conjecture that's been verified to a certain degree. Like Goldbach's conjecture.

I think that the second interpretation is the one commonly accepted -- I've never seen anything labelled as "axioms of physics", though certain things like special relativity can be derived in an axiomatic fashion (start with space and time, then go on)

Hi. I don't mind taking the gravity equation as an axiom. What I want to ask (perhaps different from what the OP would like to ask) is: Suppose we assume certain axioms to describe the universe and use them to derive a statement P. Is the derivation of P from the axioms rigorous ? This is like the question: Is physics axiomatic ? — Amr, Dec 19 '13 at 22:06

score 1 · Answer 5 · answered Nov 14 '12 at 13:25

1

Not, always .. many times they used un mathematical models like 'dimensional regularization' or use some 'curve fitting' for some properties in several branches of physics or use 'conjectures' so the things fit

answered Nov 14 '12 at 13:25

Jose Javier Garcia

4,719

score 1 · Answer 6 · answered Mar 27 '21 at 01:56

First of all, whilst it is usually said that mathematics is a deductive science where deductions proceed from a given set of axioms, a little thought shows that this can't be completely true. For example, where do the axioms come from? We don't deduce these, they are rather induced from a set of examples. Hence, mathematics like all the other sciences is a mixture of induction and deduction.

This is just as true today as in Euclid day. For example, rigourously speaking, there is no field with one element yet people are looking for such a field. This is because they have a set of examples which has a pattern that would make sense if such a field existed. This means of course that the notion of a field has to be expanded and one of the possible candidates is a sheaf of multiplicative monoids a la Grothendieck.

Thus we see that there isn't such a clear cut division between mathematics and physics as far as methodology goes: there are experiments, induction and deduction.

Now, as far as rigorous physics goes, there is mathematical physics which riffs of a question that Hilbert suggested: to axiomatise physics. There are rigorous developments of classical field theory and of classical mechanics, this includes relativity and GR. These work in a similar way to mathematics, that is by deduction from a small set of accepted axioms. Likewise, there is an axiomatic development of quantum mechanics. Whilst for QFT, this is an ongoing research topic with AQFT (algebraic QFT), FQFT (functorial QFT), LCQFT (locally covariant QFT), the Osterwalder-Schrader and Wightman axiomatics being amongst the contenders.

score 1 · Answer 7 · answered Oct 28 '13 at 01:36

Since you are a student of mathematics with little knowledge of physics, I strongly urge you to take a few courses in modern physics before you finish your mathematics education (General Relativity and Quantum Mechanics the very least). This is a must if you are specializing in geometry/topology.

Having said that, the "axioms" of the two disciplines are not the same kind, and therefore the "rigor" of proofs is different.

Physical "axioms" are those that are invariant in every experiment. E.g., the "axiom" of conservation of energy; or that the speed of light cannot be exceeded, and so on. Newton's laws were once "axioms" but they had to be modified in extreme conditions. But that does not mean Newton's laws are false: they are still true to the world we normally experience.

Mathematical axioms can be "abstract" but not based on any experiment: parallel lines do not intersect is an ancient axiom. Whether that is true or not in our universe is not important to a mathematician.

score 0 · Answer 8 · answered Jun 05 '18 at 18:31

This answer of mine is relevant to this question also:

Ever since the time of Newton physics is about observing nature, quantifying observations with measurements and finding a mathematical model that not only describes/maps the measurements but, most important, it is predictive. To attain this, physics uses a rigorous self-consistent mathematical model, imposing extra postulates as axioms to relate the connection of measurements to the mathematics, thus picking a subset of the mathematical solutions for the model.

The mathematics is self-consistent and rigorous by the construction of a mathematical model. Its usefulness in physics is that it can predict new phenomena to be measured.

It is the demand for self-consistency that allows for falsification of a proposed mathematical model, by its predicting numbers found to be false.

The consistent euclidean model of flat earth is falsified on the globe of the earth, for example. This lead to spherical geometry as the model of the globe.

The whole research effort of validating the standard model at LHC, is in the hope that it will be falsified and open a window for new theories.

In this sense physics as a discipline is as rigorous as mathematics.

The processes of designing experiments and getting real numbers to test the models introduces errors and statistical distributions, which define the accuracy of the "rigor" expected when testing a model, so in this sense physics is not absolutely rigorous.

Daddy Kropotkin · Answer 9 · 2021-03-27T01:23:54.423

While I do not necessarily disagree with most of the points addressed in the other answers here, I do not think they actually answer the OP's question, which was clearly stated in their comment to @Manishearth's answer:

Hi. I don't mind taking the gravity equation as an axiom. What I want to ask (perhaps different from what the OP would like to ask) is: Suppose we assume certain axioms to describe the universe and use them to derive a statement P. Is the derivation of P from the axioms rigorous ? This is like the question: Is physics axiomatic ?

The answer to this question requires a nuance that I think is not being apprecaited in the other posted answers. Namely, what do we actually mean by "axioms" and "is the derivation of P from the axioms 'rigorous'$\,$"? These notions THEMSELVES were being developed in their modern form in the 20th century, with the rise of Hilbert's (among others) axiomatic framework, and exploded until today (Goedel's theorems, Zermelo-Frankel set theory, modern topology and graph theory, matroid theory, etc....). At the same time in the 20th century, physics was utilizing these new sophisticated mathematical machineries: for example, various approaches to quantum theory emerged, which Dirac showed to be equivalent algebraically, and the theory of general relativity used tensor theory, differential geometry, and Minkowski's spacetime formulation.

This historical context is important, I think, because it gives better meaning to the comparison between mathematics and physics. The two work together to advance each other - pure mathematicians probably won't agree, but the history of progress in math proves otherwise.

Therefore, I would say that YES theoretical physics - the part of "physics" where people build conceptual models using math - is axiomatic sometimes, not always, but it has been like this since Newton (there are possibly others that preceded him, but thats a question for History of S&M SE).

Newton's Principia is axiomatic and mathematically rigorous. He begins from well defined axioms (e.g., law of universal gravitation and laws of motion) and derives theorems and lemmas using calculus. For example, he showed that Kepler's heuristic rules of the solar bodies (which were built on Copernicus') are consequences of his laws. Here, Newton's laws act as the "axioms" of the theory.

But sometimes "axioms" in physical models are not called laws, and instead are principles, or even postulates. For example, Einstein postulated the "axioms" of special relativity, and uncovered a very enlightening model of reality. He then extended the special theory to the general theory with the equivalence principle, which is the conceptual core of general relativistic physics.

The laws of quantum mechanics are "axioms" in the exact sense that they are assumed to be true and then one deduces logical conclusions from them (i.e. theorems, etc...). As you say, if P is the proposition that "an electron has spin state $|\psi\rangle$" then you can use the laws of quantum mechanics to derive all kinds of other logical implications (i.e. theorems).

As @anna v stated, the axiomatic approach found use in physics mainly due to its ability to predict new phenomena, as well as generalizing already known information. Newton's laws were used by many in the centuries following him to predict the existence of many moons and planets. This predictive utility is easily seen in modern theories such as relativity and quantum theory: e.g., theory of general relativity's predictions of gravitational radiation, gravitational lensing, light bending, black holes, etc...., and Dirac's prediction of antimatter, the prediction of other fundamental particles, etc....

I think it's worth responding to the Feynman quote by @Stan Liou. This quote is being taken out of context, since Feynman is talking about the ways he views $mathematics$, which do not necessarily apply to physics. Indeed math and physics are related, but that does not mean they are classifiable identically! As others have noted here, physics is driven by empirical consensus, and people who make theories of physics try to stay ahead of the empirical/experimental progress by speculating and sometimes by generalizing. IT is in this generalizing vein that physics is axiomatic. But, lots of iterative, provisional progress is usually required in a field of physics before reliable generalizations are available to make axiomatic theories. For example, renormalization and regularization were ad-hoc fixes to issues in physics theories, but over time they have been made mathematically rigorous and there exist axiomatic approaches to these now, too. I hope my answer helps to clarify the nuances in the, seemingly simple, question "is physics axiomatic and rigorous?"

score -2 · Answer 10 · answered Jun 05 '18 at 16:17

Pythagoras theorem was true once upon a time when the great philosopher discovered it. It is still true, and will always be, as far as Euclidean geometry is concerned. The laws of motion, too, were considered perfectly valid in the times of Newton. But, now we know that Relativity is far more general, and in a sense, much more truthful. So the basic difference lies here: mathematics, an abstract science built upon axioms laid down by former mathematicians, seems to describe reality in an elegant and all the more rigorous way; whereas, physics, a natural science built upon the axioms laid down by God himself, is itself a manifestation of the reality we live in. The difference lies in that, one is abstract, and the other real; but only scientists can understand why we cannot dispense with either of them, both being fundamental to the understanding of the world we live in.

Is physics rigorous in the mathematical sense?

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