How do mathematical realists explain the applicability and effectiveness of mathematics in physics?

Question

If mathematics is real, in what sense is it real, and how do mathematical realists account for the remarkable usefulness of mathematics in describing physical phenomena? Do mathematical realists propose a causal relationship between mathematics and physics to explain this applicability?

Notice this question bears some similarity to:

• If the Platonic world exists how would we know?
• If mathematical platonism is true, how do biological brains governed by physical laws access eternal platonic mathematical truths?

However, those questions are primarily epistemological in nature, focusing on how human minds are capable of grasping Platonic mathematical concepts. My question is distinct because it transcends the existence of human minds. Consider, for instance, the early moments of the Big Bang, when no conscious observers arguably existed. Even in such a context, my question remains relevant: if mathematical realism is true, why does the physical universe adhere to mathematical principles? This challenge persists regardless of whether or not there are minds present to contemplate it.

So, how do mathematical realists explain the applicability of mathematics to physics?

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Related: Since mathematicians are physical beings, does this mean that mathematics ultimately reduces to physics?

Answer 1

Physics doesn't follow any rules. Physics just is. If physics stops following our rules, we don't say physics is wrong. We have to adapt our rules. Many of the rules and patterns we discover in physics appear mathematical in nature. What gives?

Do mathematical realists propose a causal relationship between mathematics and physics to explain this applicability? I don't know, I am not a mathematical realist, but this is the same interaction problem as in dualism: if physical effects can only have physical causes, and I don't see how it could be otherwise, then no.

Wigner’s article (1960) has become far more popular than any of his important work in Physics. As so often, this is due to his suggestion of mystery where he thinks the effectiveness is unreasonable. He stops just short of calling it irrational. A deep mystery at the root of Physics and Mathematics. We do not know how to explain this…

Or do we?

Take his first example in the article where he wonders what pi is doing in the Gaussian distribution.

So where does pi come from? The presence of pi is linked to the circular symmetry inherent in the normal distribution, where the probability density is evenly distributed around the mean, similar to how points on a circle are distributed around its center. Not so mysterious after all.

Is it unreasonable that our physical universe has three spatial dimensions that are orthogonal? What would it be like if these dimensions were not orthogonal? Once you have that insight, some additional thinking will convince you that the fact that gravity falls with the inverse square, and exactly the square, of the distance is entirely reasonable. Even though before you could have wondered how physics knows how to exactly square the distance.. This is easy to see if you imagine a hollow sphere inside a larger hollow sphere. The same gravity that is projected on the inner sphere must be spread out over the larger surface of the outer sphere. Since the surface of the larger sphere increases with the square of the radius of the sphere, the gravity must fall with its inverse square.

I would say mathematics is reasonably effective. One could even argue that it is not quite so effective as we might have hoped when it comes to solving some simple problems like the three body problem. There is no general closed-form solution for something as simpel as the three body problem. The same goes for the Schroedinger equation which cannot be anytically solved for anything more complex than H₂⁺. For any bigger molecule (say, the entirety of chemistry) we need perturbation theory to approximate the solution.

Mathematics is just like English a language to describe the world. Nobody questions the unreasonable effectiveness of the English language when it comes to explaining the world.

I want to leave you with a reference to an article by Max Tegmark in which he describes his idea for the Mathematical Universe. On this view, the universe is entirely mathematical in nature. Of course this is bonkers, but there are some very interesting ideas in it. It is not too technical.

Answer 2

why does the physical universe adhere to mathematical principles?

It might not. The adherence is generally understood to be the other way around. We have proposed mathematical models that adhere to our observations of the world and that make useful predictions about the future.

If you are just asking why does the world behave the way it does: we do not know.

Answer 3

I think that there is an underlying assumption that we "found" mathematics and that it just "happens to" align with physical reality, but there is no reason to believe that this is the case.

Mathematics is a language that describes patterns, and only indirectly describes physics. What I mean is that the pattern of F = ma is merely a description of a pattern in which something is the product of two other things. Mathematics does not assign physical meaning to any of those letters, it describes what multiplication is (amongst other things).

The adherence to a strict ruleset (even when non-standard, the ruleset being used is part of the language) gives mathematics the additional flexibility of being able to explore and create patterns that have not been observed. This makes it somewhat more than useful as we can create patterns without having anything to describe, and then perhaps find a way in which that pattern describes something physical later.

The fact that physics "adheres" to mathematical principals is really an observation that physical reality has relatively patterned behaviour. But it is worth noting that our mathematical models are only correct to within some margin of error that we cannot reduce to 0. We can also look at history to see that our models have failed us previously, so we have changed models, made different models at different scales, or set points at which we have no working model at all.

The remainder of the problem seems to be deciding whether descriptions of any kind, including patterns that may not represent real things, "exist" or are "physical"... but this problem can be debated without the need to call upon Physics or Mathematics, or trying to collapse either into each other.

Answer 4

From a classical point of view no explanation is known, see Eugene Wigner's famous paper.

An exotic modern hypothesis is the simulation hypothesis - and its objections. See Brian Greene The Hidden Reality, section "Are You Living in a Simulation?" in chapter 10 "Universes, Computers, and Mathematical Reality."

Answer 5

Both mathematics and physics is about neccessity.

Mathematics is about that which is logically necessary. Whilst physics is about that which is physically necessary.

It's not surprising then that there is a strong relationship between the two.

In fact, Hilbert took his sixth problem as the problem of axiomatising physics and thus reducing physics to mathematics.

Moreover, the theory of numbers, which is usually thought of as a branch of mathematics, can be seen as a branch of physics. It is the theory of individuals. It can be seen as the first theory of physics that has been axiomatised.

Note that the question of mathematical realism is irrelevant to this argument.

Answer 6

The issue is thorny; see the well-known Wigner 1960 article on The Unreasonable Effectiveness of Mathematics in the Natural Sciences: in the end, the applicability of mathematics borders on the mysterious and there is no rational explanation for it.

For a fresh perspective, I suggest: Johannes Lenhard, The applicability of Mathematics as a Philosophical Problem (Found.Science, 2018).

Naive realism cannot avoid "platonism" while naive formalism cannot even recognize the applicability of a mathematical theory: "[the formalist] cannot do so because for him the formulas of the theory express no thoughts. [We] can make applications of arithmetical equations because they express thoughts (Frege)."

Thus, let start againg from Frege point of view that arithmetical statements express objective thoughts and are therefore universally applicable: "a statement of number is an assertion about a concept."

Concepts (and also the reletaed thoughts and sense) are not ideas that live in the mental world; they are objective. For Frege they are objects.

We may avoid the last step: from objective to objects and consider Wittgenstein's point of view in PI: §241: ""So you are saying that human agreement decides what is true and what is false?" —It is what human beings say that is true and false; and they agree in the language they use. That is not agreement in opinions but in form of life."

Thus, objective means inter-subjective: agreement in a "form of life".

Thoughts are "sherable" because we express it in signs and mathematical symbols are signs.

The mathematization of the world means that we express out thought about the world through mathematical symbols. If so, the applicability of the mathematical language to the world is "simply" the applicability of the language: when we apply our (human) mathematics we apply our description of the world.

In the end, we find Peirce's Pragmatic maxim: "It appears, then, that the rule for attaining the third grade of clearness of apprehension is as follows: Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object."

Maybe we can rewrite it as: "the whole of our conception of the object is the way we mathematically 'intercat' with them".

This is similar to Ian Hacking's approach to the debate between realism and antirealism in Representing and Intervening (1983): we have to shift the discussion from representation to manipulation. Electrons, for instance, cease to be hypothetical entities when they are manipulated to investigate experimentally something else.

Linking this to mathematics, we may say that electron exists because we interact with them and the interaction is through mathematics.

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Re Wigner, see Ivor Grattan-Guinness, Solving Wigner’s mystery The reasonable though perhaps limited effectiveness of mathematics in the natural sciences (2008)

The alternative view emphasizes two factors that Wigner largely ignores: the effectiveness of the natural sciences in mathematics, in that much mathematics has been motivated by interpretations in the sciences, and still is; and the central place of theories in both mathematics and the sciences, especially theory-building, in which analogies drawn from other theories play an important role. A major related feature is the desimplification of theories, which attempts to reduce limitations on their effectiveness. Significant also is the ubiquity and/or generality of many topics and notions in mathematics. It emerges that the connections between mathematics and the natural sciences are, and always have been, rationally although fallibly forged links, not a collection of mysterious parallelisms.

Answer 7

One way to explain it part way is the time reversal symmetry of physics, which according to David Albert is THE law that has survived physics the longest, is closely related to the timelessness of mathematics under the realist perspective. I'm not saying there's necessarily a causal linkage between these two properties, but there could be a confounding reason amenable to realists.

In a sense Wigner is wrong to label the connection as unreasonable as it's not a complete mystery in this light. There is a likeness in the time-agnostic nature to all of math and most of physics that explains the applicability and effectiveness.

The one law of physics at the fundamental level that is not time symmetric is the so called Past Hypothesis, which states the universe began in a very low entropy state. Every other fundamental physical law is time symmetric, meaning they treat the past an future the same.

The past hypothesis is an initial condition. The rest of fundamental physics and all of math share a time-agnostic nature where abstract mathematical objects always exist, and physical laws treat past and future the same.

The parallel is simply demonstrated by any equation with an equality (=) symbol. In both math and physics neither side comes before the other at the fundamental level. The initial condition is added to these time-symmetric laws, but doesn't change their time-symmetric aspect.

Not all philosophers of physics treat time this way. Maudlin is one who thinks time has an inherent direction, so they will be less inclined to buy this argument. But funnily enough, I believe he bakes a directness into his mathematics (he's developing a geometry). So it's a parallel argument. So, I think there is a persisting connection in that both either share a time-agnosticness or both share a directedness.

Answer 8

The question is sort of backwards, in the following sense. A certain school of philosophy argues that the applicability of mathematics in the natural sciences is an argument in favor of realism. The thesis is generally referred to as the "indispensability argument", and is variously attributed to Quine and Putnam.

Answer 9

Mathematics is the systematic study of identifiable relationships. (Like IN, WITH, NOT, SAME, ADJACENT, and MORE.)

All relationships are expressible as relationships, and the terminology that is useful for expressing them includes the terminology that is used for expressing relationships. Thus, in other words, all relationships are mathematically expressible.

Math works for physics because there exist physical relationships.

Mathematical realism is the claim that relationships (like the number 3) are real all by themselves, without relating any things (like thing WITH thing WITH thing, the way Aristotle would have defined it), but this is irrelevant to why math works for physics, since physics has things to relate, regardless of whether relationships are real by themselves or what exactly that means.

Answer 10

The physical universe does not adhere to Mathematical principles, but instead, Math was inspired by the physical universe.

If I have an apple and you give me an apple, I can think how they both look the same and I can say, 1+1=2. Math evolved like this and mathematicians tried extending it beyond what we could test in real life, and then later tests in real life confirmed what they had already theorized.

For example, the imaginary number i was made up to solve the square root of negative one problem. And many years later, in Quantum Mechanics, I'm pretty sure that the number i was confirmed to exist in nature as a "real" number, despite the number being called imaginary, because mathematicians at the time thought they were making it up.

Now consider Physics, which began as philosophy of nature, people asking questions about nature.

When a large object is moving really fast, it appears to have more power to cause damage. If a small object is moving fast, it still has power, but less power. And if a big object is moving slow, it has power but less, and if a small object is moving slow, it has very little power or strength. This is where f = ma comes from. The mass TIMES the acceleration of the object directly relate to its force. Our initial observations appear to be backed by evidence, that there is a very specific relationship between mass and acceleration and force that can be expressed in math for clearness.

We don't have to use math to explain the universe, I can explain any physical phenomenon in English, but the good thing about math is that it is clear and objective. Which makes math a better tool for explaining things.

Math is a process of gathering facts and identities and giving them names so you can reference them later when it's appropriate. It is a model that was created to be clear, explicit, and objective, and this makes it very useful for describing relationships in the universe.

E = mc^2, this is very clear and easy to understand, there is no room for vagueness. It is describing a relationship regarding the proportion of mass and speed of light and how they equate to energy. An equation like this is more precise than using words.

The fact that the universe appears to be consistent, and that it appears to be exact, is what makes Math so useful in modeling the universe physically. Because many things in the universe exist as relationships to one another, direct relationships that are consistent and precise, this kind of thing is what math is good at modeling and describing.

In summary I think math applies to nature because Math is a useful framework for explicitly defining logical relationships between different things, and creating a consistent dogma that can be extended or used when applicable. If there are no contradictions in physics, then establishing mathematical rules should apply and therefore be able to be extended to discover new answers.

Math is useful for defining objective facts which always hold true, and therefore you can consider previous facts when defining new facts. And you can consistently rely upon those relationships when performing precise calculations, because the universe naturally appears to also exist in this way. It is consistent and objective and does not contradict itself.