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To what extent are mathematical formalisms an extension of intuitive reasoning to grasp the world such as in the fields of Quantum Physics and Relativity?

My first thought is that when intuitive reasoning fails or when it is difficult for a human mind to apprehend the world, e.g in those two cases, mathematical reasoning becomes a powerful tool to pursue the investigation of the world.

And would you know some references that treat this topic?

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    Phenomenology traditionally treats natural sciences and mathematization as extensions of perception and intuition by instruments, physical and intellectual, that can succeed where they fail unaided. See IEP, Phenomenology and Specific Sciences on interpreting relativity and quantum mechanics in this vein.
    – Conifold
    Nov 23, 2021 at 21:00
  • Have added some interesting references to my answer. Check them out.
    – Nikos M.
    Dec 9, 2021 at 19:01
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    See Sabine Hossenfelder's video on why the maths of the standard model can appear useless. Essentially Mathematicians have got so good at adapting the standard model to accommodate new discoveries that it can essentially accommodate anything and is therefore next to useless.
    – Richard
    Dec 9, 2021 at 19:39
  • @Richard Got a link?
    – J Kusin
    Dec 10, 2021 at 2:46
  • @JKusin Sorry.. It's a video I watched a few weeks ago, Sabine is a guest on stage (outdoors somewhere) with 3 people, and one guy from Sweden is online. They discuss problems with the standard model amongst other things. I looked in my youtube history and couldn't find it :(
    – Richard
    Dec 10, 2021 at 10:30

4 Answers 4

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Short Answer

Your question is excellent, because it inquires after the intersection of math and physics philosophically. Be mindful this is a broad topic because one has to know math, physics, and philosophy, ideally in equal proportions with history somewhat mashed in there for perspective.

Mathematical physical theories greatly enhance and help to define what is meant by the physical universe and supplement our basic intuitions which are sometimes referred to as naive physics, but note also the psychology of naive physics is used by some to ground the origins and effectiveness of mathematical theory, a perspective known as mathematical empiricism.

Long Answer

Scope

The topic you reference is the relationship between mathematics and the physical world, which is called the mathematization of science (Google ngram) which is both a topic of interest in the philosophy of science and the philosophy of mathematics particularly relevant to the question of math foundations. Obviously, the technicals of mathematical physics are best asked about in PhysicsSE. But I'll try to give you some pointers to ideas for the philosophical grounding of their intersection.

Historical Perspective

The systematic application of mathematics to the physical world starts at least as early as the Babylonians, which goes back to around 1,800 BCE who used it for astronomy and measurement. Babylonian math was very practical and very limited and might be considered the bare minimum needed to be considered the start of the mathematization of physics. About 1,500 years later, a big step forward was taken by Euclid of Alexandria with his axiomatization of mathematics known as Euclidean Geometry which took for granted certain presumptions about the physical world, such as the physical reality of parallel lines that largely went unchallenged until the 19th century. Gauss considered the mathematics of Archimedes, who was active around the same time, to also be one of the most important contributors to mathematics making him an important contributor to math and physics.

In this abridged history, the next outstanding contribution to the mathematization of physics undoubtedly was the work Galileo Galilei who used rigorous measurement in his experiments, a relatively new practice as science began to grow out of natural philosophy. The ancient Greeks had their sciences, as discussed by GER Lloyd, but around the 16th century, science entered a new phase of mathematical rigor, the next leap forward not occurring for about 300 years, with the advent of mathematical logicism and formalism of whom both Gottlob Frege and David Hilbert are considered major contributors.

But, before we get to the end of the 1800's and the start of the 1900's where the birth of the theories of QM and relativistic physics radically transformed mathematical physics, we need to take a step back and offer two other famous contributors of mathematical physics, Isaac Newton and his mechanics of motion and gravitation and Johannes Kepler and his astronomy. The early science of Galileo was done in a time when Aristotelian physics had been defended for a very long time within the Catholic church (and was part of the reason that the Church burned Giordano Bruno and imprisoned Galileo); by the time of Kepler, mathematics irrefutably killed the geocentric model of the universe.

Next is Big Al. This notion that mathematical theories of physics face resistance by intellectuals of the day didn't end with Kepler by any means. Albert Einstein won his Nobel Prize for physics for the photoelectric effect because the scientist who wrote his recommendation thought relativity wasn't much of a theory. In Albert Einstein, we also see a physicist who did foundational work at the quantum level as well as the relativist level, and ironically didn't consider his math skills very good. Here are some quotations to show his sometimes wry humor, of which, I like:

"So far as the theories of mathematics are about reality, they are not certain; so far as they are certain, they are not about reality."

Since then mathematical physics has churned out some theories about strings, dark matter, and multiverses, which some physicists see as being very non-scientific raising questions relevant to Einstein's insight above. For an amusing challenge to the notions that some of contemporary physics today is actually science, watch this short video by Sabine where she (rightly, IMNSHO) calls the multiverse a mathematical religion.

Philosophy, Math, and Science

What's going on here with this math and science stuff? Well, that's a little involved but we can offer quickly some philosophical ideas that can serve as a starting point to explore mathematical physics.

First, mathematical physics is an example of mathematical application and mathematical models. Second, in the models of mathematical physics, one is exploring quantitative relationships between the primitives of the physical ontology, such as energy, spacetime, and matter. But there's more to mathematical physics than ontological primitives, what might be thought of as physical entities, there are epistemological aspects. How do we know what we know in physics? This is one of the primary concerns in the philosophy of physics as well as the philosophy of science more broadly.

Let's just give an example of a transformation in philosophical thinking. After Newton simultaneously invented the calculus and mathematized force, mass, acceleration, and gravitation, scientist-philosophers saw the universe as being characterized by the mechanistic interaction of particles within a Euclidean space governed by the passage of absolute space and absolute time. What does that mean? It means that physics now had a rigorous footing on mathematics extending our naive physics into a very rational and empircal physics. In fact, by the end of the 19th century, some scientists thought there wasn't much new to learn about the laws of nature.

But there had been some developments in geometry, known as non-Euclidian, that had set the mathematical communities of the world abuzz, since Euclid's geometry was thought to justify the mathematics of physics. When the non-Euclidian geometries were accepted, mathematical philosophy was transformed because it turned out that grounding of mathematical theories in physics wasn't clear cut. It is on the basis of these theories that Albert Einstein reformulated a vision of the universe where time and space were related, relative, and the shape of the universe might be considered fundamentally curved. Later, this space was shown to be something akin to Minkowski space. In fact, there are now a plurality of models of space and time, keeping alive the question of the Ancient Greeks, what exactly is space and time?

This answer is already too long, but I'll leave off with the results of the mathematization of physics which continues to renew philosophical speculation and theorization. If math goes on in the mind of people, and mathematical theories are linguistic constructions of societies (constructivism), is the universe inherently real (scientific realism) or is it only an instrument of our minds (instrumentalism)? These are the sorts of philosophical considerations that arise when people combine mathematical thinking with physics.

Summary

You ask a tough question, because of the broadness of it. The logical positivists of the 20th century spent a great deal of time theorizing about math, physics, language, and philosophy, and we can say with certainty that we are still learning from their failures. This question is a great first step in the direction of following their thinking. About the only thing that can be said for certain is that anyone who claims to have all the answers probably doesn't, and the best thing to do is keep reading and thinking. Good luck!

References

  • Relativity and Common Sense, Hermann Bondi - I love this book because it examines relatively conceptually, and is a classic.
  • The Philosophy of Quantum Theory, Tim Maudlin - This is an excellent introduction starting with a short list of physics results in experimentation and a subsequent series of philosophical explanation including an introduction to Bell's Theorem as well as the mathematics involved. Written by a philosopher instead of a physicist, it's very clear.
  • Causality: Models, Reasoning, and Inference, 2nd Ed., Judea Pearl - This book is for you if you are interested in modern methods of mathematical causality. There are formal systems of non-quantitative and non-experimental approaches to causality, but this book has some mathematically rigorous methods, easiest to understand if pursued after a math education.
  • A Companion to the Philosophy of Science, Ed. W.H. Newton-Smith - The 1st edition has over 80 brief articles on important topics including "Mathematics, Role in Science". Tough to get through if you haven't studied some philosophy, such as epistemology, ontology, metaphysics, etc.
  • Logical Positivism, A.J. Ayer - Written by one of their own after the movement had its heydey, an anthology of logical positivist thought.
  • The Conceptual Roots of Mathematics, J.R. Lucas - To me, this represents the clearest exposition on the topic because it's written at the level a non-mathematician undergraduate can understand.
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  • Thanks for your long answer. Reading your answer, it seems that mathematization of science is also connected to the empirical checking of the theory. So that the intuition and understanding of the human physicist is not anymore important, and that the main point is to check wether the mathematical theory fits the reality. What do you think about this ?
    – J.A
    Dec 12, 2021 at 16:00
  • My pleasure! I would say that there aren't any pat answers, and that there's a building of comprehensive worldview that is involved that rests a lot on the psychology of the thinker. See your follow-up question is rooted in philosophy of mind, because it asks how do humans do philosophy and build and accept explanation, and for me, cognitive science is the only basis to answer your question. Having given my framework, I'd claim, that the physicist's intuition is important for physics, and the mathematicians for mathematics, but for a philosophy that integrates math and science...
    – J D
    Dec 12, 2021 at 16:11
  • one has to have a thinker whose intuitions are capable in history, philosophy, language, mathematics, and science, and so broadly construed, represent a charitable interpretation of scientism, which is to say that the intuitions and theories have to be consistent across a range of theorization... some philosophers of math have very unscientific views, and some scientists don't quite get what math is... so...
    – J D
    Dec 12, 2021 at 16:13
  • the genius of Einstein was that he learned the theory of math and the theory of physics, and sat down and asked himself, what do my intutions tell me? And then used his Gedankspielen, his questions about flashlights on trains, for instance to rethink what might be called the first principles of physics, leading him to a very novel path. I would argue that the thoughts on this by Paul Feyerabend offer some insight the messy business which is mathematical physics.
    – J D
    Dec 12, 2021 at 16:16
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    @J.A: I don't see any obvious issues with this answer, but in my opinion there is something very very big missing from it. Specifically, all known applications of mathematics to the real world follow from theorems of a very very weak system ACA0, or at most ACA. This covers all empirically verifiable scientific theories, including relativity and quantum mechanics. And ACA is a very intuitive formal system about natural numbers and sets of them, so its effectiveness in applications is not at all unreasonable, at least at human scales.
    – user21820
    Dec 14, 2021 at 19:01
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this is a complex topic, but I want to present a counterexample to one of Nikos M.'s points.

When Maxwell formulated his four equations of unified electromagnetism, he noticed afterwards that they hinted at the possibility of wavelike behavior. He then rewrote those equations in such a form as to break out the wavelike character for study, and discovered that the propagation speed of those waves could be directly calculated. When he did the calculation, he was stunned to discover that the propagation speed of those "electromagnetic waves" was almost exactly equal to the known speed of light.

In this sense, Maxwell got a lot more out of the math than he had put into it.

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  • Since you reply to my answer, I can say the following: 1) That light is wave-like and has electromagnetic properties was already an experimental fact (before Maxwell's Synthesis), 2) The wave equation is already included in the 4 equations of Maxwell (it is almost a trivial combination) 3) To calculate the speed of electromagnetic waves one has to introduce experimental values for some constants. Surely one (approximate) coincidence of values would be of minor or no importance if the previous points were not already satisfied.
    – Nikos M.
    Dec 9, 2021 at 18:09
  • Light's electromagnetic properties, like polarization, were studied by Faraday, and in fact inspired Maxwell (en.wikipedia.org/wiki/…). As for the wave equation from Maxwell's 4 equations, is really simple, one substitutes one equation into the other, without sources.
    – Nikos M.
    Dec 9, 2021 at 20:36
  • ok, will delete my comment. Dec 9, 2021 at 20:40
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In physics, scale matters. This was already understood in ancient Greek philosophy when Aristotle demarcated the celestial sphere from the terrestrial and with Zeno, the microscopic from the terrestrial.

And so it is today, with cosmology marked out from the physics of the everyday and that of quantum mechanics.

Our intuition has been extended by instruments into these very different scales and as intuited by the ancients we have found very different worlds there.

Although mathematics is seen to be very different from ordinary language and thought, the truth is that it's merely a highly refined language with its own argot and specialised techniques. This is no different, in essence, to say carpentry, which will also have its own language and techniques. And in fact, matter, a physical term par excellance, is derived from the Greek word for timber.

To say that mathematics is used to grasp the world beyond human intuition is to misunderstand both mathematics and intuition. It was human intuition that discovered and developed mathematics for a variety of ends. Mathematics is not outside of human intuition but inside of it. Poincare said, that although deduction is important in mathematics, it is the intuition that discovers amd progresses the mathematical art. Hilbert agreed, that is why he wrote a book called Geometry and the Imagination. For imagination, read intuition.

It is the world that is not immediately apparent to our senses that our mathematical technology gives insight into. This is the world that early thinkers called the intelligible world. In this, they are no different to telescopes or microscopes. They are instruments that expand our understanding of this world. And mathematicians, in some sense, are like engineers crafting new machines to peer into this world.

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  • +1 I love the appeal to the phrase "inside of human intuition". It's very meaningful.
    – J D
    Dec 16, 2021 at 8:56
  • Nicely said, up vote from me
    – Nikos M.
    Dec 17, 2021 at 14:05
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I am of the opinion (expanded below) that we get out of mathematics, what we already put there in the first place, one way or another (in the same sense one cannot win nor lose money by mere bookkeeping).

Specifically:

(Special) Relativity theory is about the finite speed of light (and that is a constant upper limit on any velocity).

That things happen in finite speed, not instantaneously, was already an intuition, if not a fact, centuries before Einstein, postulated it as such. In fact Galileo, tried to measure the velocity of light using a very simple method and others followed him later.

When Newton formulated his theory of Mechanics, out of experiments, a notion of space and time arose as well. Where do things happen and how long it takes. Centuries later, when new technology was available, electromagnetic phenomena were studied in more detail. This led to Maxwell's synthesis of electricity and magnetism. Both Newton's synthesis and Maxwell's synthesis were born out of experiments that were possible through the advancment of the technology.

But now a new problem arose. Both theories referred to same space and time, but each theory had a different relation to that space and time. For example, Newtonian mechanics has no concept of absolute velocity (Newton's 1st law), only relative velocity and has Galilean symmetries. Whereas Maxwell's theory has that electromagnetic waves (eg light) travel with the same absolute velocity and has Lorentzian symmetries.

So now there was a conflict. Either the phenomena take place in different space and time, for each theory or some modification needs to be made.

So far all the above are driven by formalisms derived by experiments.

Physical reality cannot take place in different space and time for each phenomena. So this was from the start ruled out (in contrast to what the mathematics was telling us). So one option remains: one or both theories needs to be modified.

In fact modifications were tried both ways, usually on Maxwell's theory, as Newton's was considered more stable. So far Mathematics alone does not tell us which theory nor how to modify it. So all attempts were made. If Einstein had not derived Relativity theory, someone else would surely derive it. It was the nature of the problem that needed that.

So all attempts were made on the mathematical theories. What prevailed was what was corroborated by experiment. Mathematics alone could not derive that.

Quantum Mechanics became possible when the technology was advanced to study atomic phenomena.

Max Planck studied the black-body radiation, which classical theory fails to predict correctly (again mathematics is false). So he made some assumption about quanta of energy, which was really a heuristic, which helped derive the correct distribution (which was known experimentally already). So far Planck still thinks that this is a heuristic that will be explained in the classical framework.

Later on, Einstein again, used the same heuristic, to explain the photoelectric phenomenon. The heuristic is related to concrete experiments.

Further experiments and failure of the (very beautiful) mathematical theory of classical physics to account for the new phenomena, led to a theory which was initially a mess (all heuristics here and there and no beauty), BUT predicted very well the experimental data.

Later on, a new formalism was created (matrix mechanics/wave mechanics) which incorporated all these heuristics into a new framework. But it derives what is already put there by the drive of experiments.


Nowadays, there are all kinds of beauties and symmetries (eg super-symmetry) in modern day theories, which by all accounts are very beautiful, yet fail to correctly predict the experiments (or break the symmetries when needed).

Hermann Weyl (a famous mathematician and physicist) had often claimed that he followed mathematical beauty when nothing else could be used to derive a physical theory. Unfortunately, Weyl again points out that Nature is very distinct in saying yes and no regardless of beauty of the theory.

Hope the previous historical notes and analysis, sheds some light, on whether mathematical beauty or symmetry is more useful than what one already knows.

References:

  1. History of Special Relativity
  2. History of Quantum Mechanics
  3. Historical theories about light
  4. The Unreasonable Effectiveness of Mathematics in the Natural Sciences
  5. On "The Unreasonable Effectiveness of Mathematics in the Natural Sciences"
  6. The Applicability of Mathematics as a Philosophical Problem
  7. Steiner on the Applicability of Mathematics and Naturalism
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