Difference between Physics and Mathematics

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Our forums are full of physics and mathematics. One cannot talk about the former without getting into calculations and mathematics sooner or later: Hilbert spaces, differential operators and linear algebra all over the place. Students of both faculties have normally a similar motivation, foundation and interests in both - more or less. Yesterday I saw an article about the Cauchy horizon. Is that physics or is it mathematics already? The boundaries are not clear. So what is the difference?

Here is an article how mathematics often looks like:
https://link.springer.com/article/10.1007/s00012-020-00667-5
Although it began with a real world example, it quickly developed in something which isn't directly applicable to the real world. However, it describes how mathematics works, and in how far those strategies won't work in physics. Usually, let's see whether the string theorists will ever find something which at least is a bit of evidence.

So for all students who are curious about that difference, have a look at the article. There is no need to understand or even read it in detail: just have a look to get an impression. It might also help to decide whether you are more of a physicist or more a mathematician.

And here is an example for an article in physics:
https://www.sciencedirect.com/science/article/pii/S2405428320300022
although it might not be representative (too easy)
.
 
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  • #2
ZapperZ
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Biggest difference between Mathematics and Physics : there are no experiments in mathematics.

Experimental work is a huge and significant part of physics. Everything in physics is validated by experiments, despite what String Theorists might think.

Zz.
 
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  • #3
symbolipoint
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Biggest difference between Mathematics and Physics : there are no experiments in mathematics.

Experimental work is a huge and significant part of physics. Everything in physics is validated by experiments, despite what String Theorists might think.

Zz.
LIKE! LIKE!
 
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  • #4
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Biggest difference between Mathematics and Physics : there are no experiments in mathematics.
This is what everyone sees on the surface: descriptive versus deductive science. But behind those words lies a different way of thinking and a different approach to problems. The two papers give an intuition of that difference: both start with an observation and develop on completely different paths, although both papers are theoretical, no lab reports. Of course does the Higgs paper need evidence, whereas the math paper needs consistency. I wanted to demonstrate what this means in scientific life, rather than state the obvious.
 
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  • #5
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Of course does the Higgs paper need evidence, whereas the math paper needs consistency.
Didn't some areas of mathematics start out with a huge data set of evidence, leading to formalized equations describing the evidence.
Pythogorean theorem isn't self evident ( until perhaps explained in the below diagram ). But does it work for all right triangles? The theorem describes the accumulated data I would think.
1595651918288.png


Probability theory, ( AFAIK in France amonsgt the betting nobilty ) , started out as a way to describe games of chance, card games or rolling dice for example.
Again a large set of data collection to be used to test the theory of against.
 
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  • #6
robphy
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Biggest difference between Mathematics and Physics : there are no experiments in mathematics.

"no" is a little too strong.



"What is Experimental Mathematics?" (Keith Devlin , March 2009)
https://www.maa.org/external_archive/devlin/devlin_03_09.html
Experimental mathematics is the name generally given to the use of a computer to run computations - sometimes no more than trial-and-error tests - to look for patterns, to identify particular numbers and sequences, to gather evidence in support of specific mathematical assertions, that may themselves arise by computational means, including search.

https://www.tandfonline.com/action/journalInformation?show=aimsScope&journalCode=uexm20
https://www.emis.de/journals/EM/
Experimental mathematics said:
Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses.
Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was "through systematic experimentation.") Yet this tends to be concealed by the tradition of presenting only elegant, fully developed, and rigorous results.
Experimental Mathematics was founded in the belief that theory and experiment feed on each other, and that the mathematical community stands to benefit from a more complete exposure to the experimental process. The early sharing of insights increases the possibility that they will lead to theorems: An interesting conjecture is often formulated by a researcher who lacks the techniques to formalize a proof, while those who have the techniques at their fingertips have been looking elsewhere. Even when the person who had the initial insight goes on to find a proof, a discussion of the heuristic process can be of help, or at least of interest, to other researchers. There is value not only in the discovery itself, but also in the road that leads to it.
...

https://en.wikipedia.org/wiki/Experimental_mathematics
Experimental mathematics is an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns.[1] It has been defined as "that branch of mathematics that concerns itself ultimately with the codification and transmission of insights within the mathematical community through the use of experimental (in either the Galilean, Baconian, Aristotelian or Kantian sense) exploration of conjectures and more informal beliefs and a careful analysis of the data acquired in this pursuit."[2]

As expressed by Paul Halmos: "Mathematics is not a deductive science—that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork. You want to find out what the facts are, and what you do is in that respect similar to what a laboratory technician does."[3]

...

https://mathworld.wolfram.com/ExperimentalMathematics.html
Experimental mathematics is a type of mathematical investigation in which computation is used to investigate mathematical structures and identify their fundamental properties and patterns. As in experimental science, experimental mathematics can be used to make mathematical predictions which can then be verified or falsified on the bases of additional computational experiments.
 
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  • #7
ZapperZ
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Wow! They have truly bastardized the meaning of the word "experimental".

In physics, a "simulation" isn't an experiment. It is an application to see if the theoretical model may match physical reality. That isn't even done in what has been described. Using computers to solve a problem isn't an experiment. That's like saying I used a calculator to find the value of my equation, thus I did an experiment.

Zz.
 
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  • #8
robphy
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Wow! They have truly bastardized the meaning of the word "experimental".

In physics, a "simulation" isn't an experiment. It is an application to see if the theoretical model may match physical reality. That isn't even done in what has been described. Using computers to solve a problem isn't an experiment. That's like saying I used a calculator to find the value of my equation, thus I did an experiment.

Zz.
Thanks. Your opinion has been noted.
I'm not pro or con about the quoted uses of "experimental"
I just wanted to point out their use in the mathematics community.
I'm not going to argue about the meaning of the word "experimental".
 
  • #9
symbolipoint
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I just wanted to point out their use in the mathematics community.
Call it faith, but even mathematicians know how to tell the difference between an experiment and a simulation.
 
  • #10
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Wow! They have truly bastardized the meaning of the word "experimental".
I agree. Although if I think about the ERH, a calculation of a billion cases is still better than remain empty handed. And IIRC the four colour theorem wouldn't have been proven without computers.

My intention was to emphasize the different methodology. If you create a building upon a mathematical idea, then it is still mathematics even without applications. And applications often came decades after those findings.

If a cosmological model doesn't result in how our universe looks like, then it isn't physics anymore and at best suited to prove that some parameters of the model have to be false.

These two points of view are in my opinion the real difference between the two.
 
  • #11
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I'm okay with "experimental" to be a search for counterexamples.
 
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  • #12
martinbn
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Our forums are full of physics and mathematics. One cannot talk about the former without getting into calculations and mathematics sooner or later: Hilbert spaces, differential operators and linear algebra all over the place. Students of both faculties have normally a similar motivation, foundation and interests in both - more or less. Yesterday I saw an article about the Cauchy horizon. Is that physics or is it mathematics already? The boundaries are not clear. So what is the difference?
There will be no agreement on that. Different people would disagree on the differences. V.I. Arnold have said more thatn once that mathematics is that part of physics where experiments are cheap. He probabply never considered any of Bourbaki style work mathematics. Others will not agree with that.

As to the Cauchy horizon paper it probably is mathematics to begin with. This is a notion from PDE, whether the PDE has any relation to reality or not.

Here is an article how mathematics often looks like:
https://link.springer.com/article/10.1007/s00012-020-00667-5
Although it began with a real world example, it quickly developed in something which isn't directly applicable to the real world. However, it describes how mathematics works, and in how far those strategies won't work in physics. Usually, let's see whether the string theorists will ever find something which at least is a bit of evidence.
In my opinion (see, I told you there will be disagreement) this is not a real world example. Rock paper scissors has man made rules, which are completely arbitrary, they are not concequences of physical laws.
 
  • #13
rsk
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Physics is maths with toys.

Or Physics is to maths what...
 
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Biggest difference between Mathematics and Physics : there are no experiments in mathematics.

Experimental work is a huge and significant part of physics. Everything in physics is validated by experiments, despite what String Theorists might think.

Zz.
What about probability theory?
 
  • #15
rsk
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We do experiments and investigations with our maths students in school. OK, so they're only 'discovering' facts/theories which are already known, but that's the case with most school science experiments as well,
 
  • #16
ZapperZ
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What about probability theory?
Depends on how likely it is.

Zz.
 
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In my opinion (see, I told you there will be disagreement) this is not a real world example. Rock paper scissors has man made rules, which are completely arbitrary, they are not concequences of physical laws.
Which is why it serves as an example of a mathematical paper in contrast to a physics paper. Good, you slowly begin to understand what I wanted to demonstrate.
 
  • #18
Stephen Tashi
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Physics is an example of an application of mathematics. There are what might be called "mathematical applications" of mathematics that are careful to take their style and terminology from the mathematical theory they use, but physics is not (always) such an application. For example, statistical physics predates the formal theory of probability and expositions of it usually don't pay attention to contemporary mathematical terminology used in courses on statistics.
 
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  • #19
S.G. Janssens
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Physics is an example of an application of mathematics.
In more cases than I could stand, application could be replaced by abuse, and that is a pity.
 
  • #20
mathwonk
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I admittedly don't understand what experiments are in physics, but in math we think we are doing experiments when we do calculations. I.e. we make a conjecture about what we think should be true in all situations of a certain type, and then we start checking whether this actually happens, in as many cases as we are capable of calculating. So to us examples are the counterpart of experiments. I think we are at a disadvantage wrt physics since all experiments in physics are apparently instructive, if done correctly. I greatly envy and admire physicists since their experimentation seems to lead often to useful mathematics.

Some outstanding mathematicians are skilled at both, e.g. Riemann, who seems to have been inspired by the behavior of flows associated to incompressible irrotational fluid flows, to state his mapping theorem and even Riemann Roch theorem. At least this is suggested by the heuristic discussions in the books of Hermann Weyl, Carl Siegel, and George Springer, as well as the classic lectures of Felix Klein.
 
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  • #21
atyy
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I always thought Arnold's use of "experiments" was idiosyncratic, but I guess not.

https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html
"Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap."
 
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My (naive) take is that for something to be asserted as true in physics is done by experiment. (Reproducable) experiments that match some kind of numeric expectation up to accuracy of some [itex]\varepsilon[/itex]. This implies there is a lot of computation involved. This system may often be sufficient for physicists who work on describing laws of nature.

One of my students tried to prove Fundamental theorem of arithmetic to me by starting giving examples of natural numbers that break into products of primes (it was their first semester) E.g [itex]100 = 2\cdot 50 = 2\cdot 2\cdot 25 = 2^2 \cdot 5^2[/itex] etc etc. So I explained to them that doing this kind of "experiment" will only prove that there exist such naturals no matter how many times we repeat it. But we want it to be true for all [itex]n>1[/itex] and we don't have infinitely much time.

The take-away to me is that for physicists a statement could be regarded as true if some finite amount of relevant experiments match expectation. But in mathematics, that is not sufficient.

I don't mean to say that we don't experiment at all. Of course we do. How else do we come up with solutions to different problems? The point is, the experiments alone need not constitute a proof.
 
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  • #23
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Yes, there is of course the obvious difference of proof versus evidence. The apple will fall from the tree, but there is no guarantee that it will be so tomorrow. In mathematics we have to use logic and prove statements under the assumption that our logic is free of contradictions. If not we can adapt the logical system. The according principle in physics could be viewed as the assumption that physical laws are true everywhere in the universe. Hence there is a similarity in the foundations, although proof and evidence are distinct.

The papers quoted should show something different: In mathematics we can take an arbitrary object under consideration and build a system of mathematical statements around it, here the generalization of the kids' game. Btw., did anybody check whether the paper covers "Rock, Paper, Scissors, Lizard, Spock"?

However, this is a fundamental difference to physics. It is not physics anymore, if we assume the apple would not fall from the tree. But physicists have also this mathematical tendency to assume something and hope they will find evidence later on. I think this is a contradiction, in the sense that it isn't physics. String theory is a mathematical construction, models of the dark sector, too. I think physicists chasing such new models are doing mathematics, not physics, since it isn't descriptive anymore. We could have a perfect string theory without any physical relevance. And that is exactly the difference to mathematics, where such constructions can live without being challenged by experiments.
 
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  • #24
benorin
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Biggest difference between Mathematics and Physics : there are no experiments in mathematics.
Monty Hall Problem (Experiement in mathematics).png


This is a canonical counter example. (Here's the link in case you can read that, cntl+F (for find on page) "Even when given").
 
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ZapperZ
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View attachment 267283

This is a canonical counter example. (Here's the link in case you can read that, cntl+F (for find on page) "Even when given").
How is this an "experiment" in mathematics? That's like saying I did an experiment because I measured the length of the hypotenuse of a right triangle and verified that it obeys Pythagoras theorem.

As an experimentalist, I find it rather surprising that a lot of people do not seem to have a concept of what an "experiment" is!

Zz.
 
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