From the earliest Pythagorean (~570BCE-~490BCE) view that "everything is number" [1], to the founder of modern physics, Galileo (1564-1642), who said "the book of nature is written in the language of mathematics" [2], to attempts by Hilbert (1862-1943) to mathematically "axiomatize" physics [3]，and to the symmetry principle [9], which today is considered fundamental by physics, Physics has never been separated from mathematics, but there has never been a definite answer as to the relationship between them. Thus Wigner (1902-1995) exclaimed [4]: "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. gift which we neither understand nor deserve."
CN Yang, commenting on Einstein's "On the method of theoretical physics" [5], said, "Was Einstein saying that fundamental theoretical physics is a part of mathematics? Was he saying that fundamental theoretical physics should have the tradition and style of mathematics? The answers to these questions are no "[6]. So what is the real relationship between mathematics and physics? Is mathematics merely a tool that physics cannot do without? We can interpret mathematics as a description of physical behavior, or physics as operating according to mathematical principles, or they are completely equivalent, but one thing is unchangeable, all physics must ultimately be concretely embodied in its physical parameters, regardless of who dominates whom. We need to remember the essential question, "That is, we don't invent mathematical structures - we discover them, and invent only the notation for describing them"[7]. Mathematics is abstract existence, physics is reality. We cannot completely replace physical explanations with mathematical ones. For example, ask "How do light and particles know that they are choosing the shortest path [8]. The answer is that it is determined by the principle of least action. This is the correct mathematical answer, but not the final physical answer. The final physical answer should be, "Light and particles are not searching for shortest paths, they are creating and defining shortest paths". Why this can be so is because they are energy-momentum themselves. The ultimate explanation is just math*，if we can't boil it down to specific, well-defined, measurable physical parameters. Following Pythagoras' inspiring vision that the world can be built up from concepts, algorithms, and numbers [9]. When we discuss the relationship between math and physics, do we need first ask:
1) What are numbers? Shouldn't we first attribute numbers to "fundamental quantities" in mathematics and physics? Are scalars, vectors, and spinors complete expressions of such fundamental quantities? All other quantities are composites of these fundamental quantities, e.g., tensor.
2) Do mathematics and physics have to have these fundamental quantities in common before we can further discuss the consistency between their theorems? That is, the unification of mathematics and physics must begin with the unification of fundamental quantities.
3) Where do these fundamental quantities come from in physics? In what way are they represented?
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Notes：
* And then what do imaginary numbers in physics correspond to? [10][11]
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References：
[1] McDonnell, J. (2017). The Pythagorean World: Why Mathematics Is Unreasonably Effective In Physics Springer.
[2] Kosmann-Schwarzbach, Y. (2011). The Noether Theorems. The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century. Y. Kosmann-Schwarzbach and B. E. Schwarzbach. New York, NY, Springer New York: 55-64.
Einstein, A. (1934). "On the method of theoretical physics." Philosophy of science 1(2): 163-169.
[3] Corry, L. (2004). David Hilbert and the axiomatization of physics (1898-1918): From Grundlagen der Geometrie to Grundlagen der Physik, Springer.
[4] Wigner, E. P. (1990). The unreasonable effectiveness of mathematics in the natural sciences（1960）. Mathematics and Science, World Scientific: 291-306. 【这个说法本身可能是存在问题的，不是数学在物理学中的有效性，而是不能够区分物理学准则和数学算法。】
[5] Einstein, A. (1934). "On the method of theoretical physics." Philosophy of science 1(2): 163-169.
[6] Yang, C. N. (1980). "Einstein's impact on theoretical physics." Physics Today 33(6): 42-49.
[7] Russell, B. (2010). Principles of mathematics （1903）, Routledge.
[8] https://www.researchgate.net/post/NO4_How_do_light_and_particles_know_that_they_are_choosing_the_shortest_path
[9] Wilczek, F. (2006). "The origin of mass." Modern Physics Letters A 21(9): 701-712.
[10] Chian Fan, e. a. (2023). "How to understand imaginary numbers (complex numbers) in physics." from https://www.researchgate.net/post/NO6_How_to_understand_imaginary_numbers_complex_numbers_in_physics.
[11] Baylis, W. E., J. Huschilt and J. Wei (1992). "Why i?" American Journal of Physics 60(9): 788-797. 【复数、虚数、波函数】
Dont agree
For example,
One can clearly distinguish formulas based on science (Newton, Schrodinger) from the simple identities based
on Mathematics (Euler formula say). The former is based on physical evidence, the later are internal to math.
Math is just a tool for Physics, along with other things, Computation, measurement, experiment, observation.
The relation between Maths and physics is all about finely-tuned subjectivity swallowing up finely-tuned objectivity (everything is maths).
Physically, the same way a given form of energy can be transformed into another form of enrgy, so are physical quantities (in other words, each and every physical quantity can be fondumental in its own frame and can therefore evolve).
Mathematically, the fact that ''numbers, vectors and so forth'' can be associated with physical quantities shouldn't be mistaken for a fundamental aspect of their nature (a mathematical quantity only exist in context of a physical model or interpretation).
The unification of those two disciplines leads to a single infinite concept of
INFORMATION (the only thing which can be dubbed of being fundamental in this universe, to not be confused with computation or any process).
They're determined by the symmetries; and the symmetries are discovered by the hints provided by experiment, which are completed by mathematical analysis.
Math and physics are like best friends. In physics, we deal with fundamental things like length and time. Math helps express these ideas through equations, helping us understand and predict how things work in the world. So, math and physics work together to unravel the mysteries of the universe.
“The Relation Between Mathematics and Physics (1) - What Are the Fundamental Quantities in Physics?”
- a short and correct answer to the question in the quote is in the Juan Weisz post above
“…Math is just a tool for Physics, along with other things, Computation, measurement, experiment, observation.…..”
- though, of course, the quite natural at that question – however, “Why is that so?” – really remains, and fundamentally cannot be answered in mainstream physics [and the again quite natural at that questions – what are “Computation, measurement, experiment, observation”, why they exist?, and why they are as they are?”, of course, also]
Really scientific answers to the questions above are possible, and are given, only in framework of the Shevchenko-Tokarevsky’s 2007 “The Information as Absolute” conception, recent version of the basic paper see
https://www.researchgate.net/publication/363645560_The_Information_as_Absolute_-_2022_ed
- where evidently necessary in this casefundamental phenomena/notions, first of all in this case “Matter”, “Consciousness”, “Space”, “Time”, “Energy”, “Information”,
- since to answer to the questions above it is necessary before to understand – what is that makes “Computation, measurement, experiment, observation”?; and - makes about what?; and so why the ,results are as they are?
- which [phenomena/notions] in the mainstream philosophy and sciences, including physics, all really fundamental phenomena/notions, are fundamentally completely transcendent/uncertain/irrational, and so the answers above in the mainstream are fundamentally impossible.
More concretely answer to the questions in physics is given in the Shevchenko-Tokarevsky’s Planck scale informational physical model
https://www.researchgate.net/publication/354418793_The_Informational_Conception_and_the_Base_of_Physics
https://www.researchgate.net/publication/367397025_The_Informational_Physical_Model_and_Fundamental_Problems_in_Physics
https://www.researchgate.net/publication/369357747_The_informational_model_-Nuclear_Force,
- where, basing on really outstanding breakthroughs in physics in XX century – von Weizsäcker 1954-55 “Ur-hypothesis” and Fredkin-Toffoli finding — it is rigorously scientifically shown that Matter is based on binary [von Weizsäcker] and reversible [Fredkin-Toffoli] logics.
So Matter is some automation that runs using ultimate “hardware” – the (at least) [4+4+1]4D dense lattice of primary elementary logical structures – (at least) [4+4+1]4D binary reversible fundamental logical elements [FLE]. The lattice is placed in the corresponding Matter’s fundamentally absolute, fundamentally flat, fundamentally continuous, and fundamentally “Cartesian”, (at least) [4+4+1]4D spacetime with metrics (at least) (cτ,X,Y,Z, g,w,e,s,ct), FLE “size” and “FLE binary flip time” are Planck length, lP, and Planck time, tP,
- and so all what exists and happens in Matter is reduced really to some Boolean algebra, which is well described and analyzed by mathematics, etc.
However Matter is as it is only because that was specially– completely independently on mathematics [besides as the tool, of course] – designed and created, and so all fundamental laws/links/constants that govern everything in Matter by no means follow from mathematics really.
Including, say, that
“…They're determined by the symmetries; and the symmetries are discovered by the hints provided by experiment, which are completed by mathematical analysis. …..”
- really is a strange claim. “Symmetries”, etc., are because of they are implemented in the design, and in the design a number of other things are implemented, which would be arbitrarily true at some symmetries; but, again, have concrete parameters’ values in Matter. First of all that are Planck units and the strengths of fundamental Nature forces.
More see the links above.
Cheers
Theories of physics can be represented in mathematical form to understand properties clearly. That's why mathematics is required to understand all the theories of physics.
Recent SS posts in https://www.researchgate.net/post/Most_weird_Physics_factsparticles_are_dually_such_and_waves_things_that_dont_turn_have_spin_traits_charge_cant_be_found_seperately_of_entities are relevant to this thread question.
Cheers
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