Why does mathematics work in the physical sciences?

I looked at reddit, and they said that it's not surprising it does, just because that's what it's there for.

But there's definitely a question there, and I don't think it's how does mathematics work. We're not going to get a scientific explanation that doesn't use mathematics, and that seems like an infinite regress, however informative it is. And asking for a purely philosophical explanation of any sort of process, including the history and successes of science, or even pure mathematics, seems foolish.

But saying it works because it's meant to seems completely question begging. People of questionable sanity can spend their entire lives trying to get something to work without doing so. I'm asking why it is we got it to work, not why we spent so long getting it to work -- which is probably because it works.

By asking 'why' I'm just trying to make sense of the success of applied mathematics, asking what it means. So I think in effect I'm asking whether its success suggests anything. Beyond, I mean, its usefulness.

One way to understand its success is the Quine-Putnam indispensability argument for mathematical Platonism. Aside from that and arguments for scientific realism, what else works from the assumption that applied mathematics is phenomenally useful?

This question might be closed as too broad, in which case what is the term for this question, who is debating it?


7 Answers 7


I looked at reddit, and they said that it's not surprising it does, just because that's what it's there for.

Well, that is the correct answer... just stated very simply.

How did math start? It started with some basic problem that needed solving, probably of the type "I need to describe the size of some collection, because I need to convey this size to someone without actually showing them the physical collection itself". Add many dumb ideas and a couple of bright ones, and before long we've invented numbers and counting.

If you abstract this, you can say that mathematics was invented because we need better tools to describe certain things (quantities and dimensions) in our world, and later, the relationships between different quantities and dimensions.

So math "works" in the physical sciences because math and the physical sciences work in the same domain - they try to describe certain (coinciding) aspects the world.

Furthermore, early math was round-aboutly "invented" by examining the physical world & figuring out ways to conceptualize in language the realities that were observed. As an example, the number "1" (or any number, I guess) would most likely never exist if our physical reality didn't contain distinctness as a describable feature (or if humans weren't able to comprehend it). Some animals can count, while some can't - some animals can distinguish between Rock 1 and Rock 2, while some animals can only see a rock - and not have the cognitive ability to understand that it's possible to have two different rocks, just... rock. Applying this knowledge to humans and how we interact with the world, it is pretty clear that our understanding of math depends on our ability (or at least capacity) to understand the physical world first.

So math works in the physical sciences because math stems from the physical world - it's a language-description of what the world looks like in certain contexts. It works because that's what it was invented to do.

TL;DR - a wheel is round because the purpose of a wheel is to be round.


Let me expand on this, since the comments seem to contain a lot of confusion.

We'll look at this line of numbers:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

To use the same example I did in a comment, let's say I have 12 oranges that I want to pack into groups of 3. How many boxes do I need? Brute-forcing this quickly yields the answer of 4. With these numbers in hand (3, 4 and 12), and looking at the sequence of numbers above, eventually even someone unlearned in mathematics will notice the following:

  • The distance from 1 to 3 has length 3
  • The distance from 3 to 6 has length 3
  • The distance from 6 to 9 has length 3
  • The distance from 9 to 12 has length 3

The number 12, or the numerical "length" of 12, can be described as 4 equally long pieces of length 3. Which is a different way of saying 12 can be distributed into 4 equally large groups of 3. Looking back to the number line, you realize that you can swap numbers 4 and 3. You note that this means you can do 3 lengths of 4 and end up at 12 - or 4 lengths of 3, and still end up at 12. Maybe you've at this point realized it, or maybe you haven't, but you've now discovered multiplication, division and the commutitative law.

Curious if this is just lucky happenstance, you repeat the experiment. You gather 10 rocks, and you decide to split them into groups of 2. Again you notice the same as last time - you get 5 groups of 2, and each group is as large as the others. Taking 5 groups of length 2 - or vice versa - yields 10. But in this instance, you also notice that 5 is exactly on the middle between 0 and 10 - you've found a way to reach the half of something. This "something" could either be a stick (which in this example would be 10 units in some physical dimension) or a quantity.

This is essentially how early math was discovered/invented - whichever side of that question you land on - but the basic truth about it remains the same; it works because we stole it from the physical world. Since these principles essentially describe aspects of the physical world, it only makes sense that they would also "work" within the studies of the physical world - the reverse would actually be positively absurd.

  • i find this teleological argument unconvincing
    – user38026
    Sep 3, 2019 at 21:02
  • for the reasons in my question. a better analogy would be a wheel helps move things because that's its purpose. but that's either trivial or in need of something more: why is it that when we try to move things (with a wheel) we can? like i said, i don't think just because is an answer to that. you have to work to dissolve a question, not just say you don't have an answer
    – user38026
    Sep 3, 2019 at 21:32
  • A wheel is round because that minimizes the quantity of lifting and friction it imposes on forward motion. Likewise, I might suggest, mathematics does what it does for physics because lines in space and succession in time are the parts of our experience that get other details out of our way (the uneavenness of the terrain, for instance) when we discuss movement of things with other people.
    – user9166
    Sep 3, 2019 at 23:32
  • This comes down hard on the side of 'math is invented' using some magical thing called 'abstraction'. Half the world take the opposite view that math is discovered, that it is sewn into reality by the nature of being and so it is there to be found. But math is neither invented nor discovered: like music, it is expressed.
    – user9166
    Sep 3, 2019 at 23:40
  • @another_name "why is it that when we try to move things (with a wheel) we can?" - why is it that when we try to move things with jagged slab of rock we can't? - the round, centrally suspended object works, so we call it wheel. In another reality, a jagged rock might work, so they'd call that wheel. In another reality, nothing works, so there's no one to name anything. 'Math' is huge, we use the stuff that works in sciences for work in sciences. The math that doesn't , we leave to the mathematicians. In a reality without concepts, there'd be no conceptualizers (us) thus no one would notice.
    – bukwyrm
    Sep 4, 2019 at 7:33

Mathematics works only to the extent that it is logical.

There is in this respect nothing specific to mathematics as compared to our other modes of representation. They will all work as long as our modelling remains logical. Language works. Diagrams work. Pre-linguistic thought works. Any model works, as long as it is kept logical.

Thus, the value of mathematics is entirely in the fact that it is a more formal, and therefore more rigorous, mode of representation than our other modes of logical thinking.

All is said in Wikipedia's article on Mathematics:

Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. https://en.wikipedia.org/wiki/Mathematics

Mathematics works because, and only to the extent that, it is logical.

We got our logic through natural selection so, presumably, it was thoroughly tested over something like the 525 million years of the evolution of neuronal systems over the entire biosphere.

This doesn't mean that it should therefore work in all situations, only that it could be difficult for us to find one where it doesn't work.

Mathematicians can also invent theories that don't "work" because it just happens that there is nothing in the universe that works like that.

When a mathematical theory works, it can be thought of as a model of something real. For any such mathematical model, there is no good reason to claim that we know that it will work for ever, as if it was somehow a perfect model. In effect, we may believe that it will work for ever when in fact it won't because at some point in the future the model will be falsified by new facts. And we don't know the future.

In this case, we just don't know when it will stop to work. So, we can only believe that mathematical models will work. And then, that a model works doesn't mean that it is correct. Newton's laws of gravitation worked beautifully but then were effectively falsified by the more precise observation of Mercury's orbit.

Thus, we don't really know whether mathematics works since we don't know if it works for things we haven't been able to observe yet.

It may well be that we won't find anything ever for which mathematics doesn't work. However, this should be no surprise. I don't know of anything in nature that would somehow be illogical. So, again, as long as mathematics is logical, we should be safe.

So, again, this isn't specific to mathematics. Any model, as long as it is logical, will work. The specificity of Mathematics is that it is a more formal and therefore a more rigorous mode of thinking.

Pre-linguistic thought also works as long as it is logical. For example, you can try to think of Russian dolls. No mathematics. No language. Just your mind's power of imagination. Think of three dolls: doll A, doll B, doll C. Try to imagine a situation where doll A would be inside doll B and doll B inside doll C, while doll A wouldn't be inside doll C. Me, I can't. Our mind seems a pretty good model of reality and this before any mathematics at all.

So, the question of why mathematics works is trivial. It works because human logic works, and mathematics works only to the extent that it is logical.

The reason that logic works is less trivial. It works because it has been thoroughly tested by nature itself and finding a flaw in it is probably not easy at all. It seems safe to believe that finding a flaw in logic is beyond our current technological powers and will remain so for a very long time.

However, here too, there is no eternal guaranty. Only the future will tell.


I don't think I need to dwell on the question of the role played by abstraction in mathematics.


From the point of view of Intuitionism in mathematics, it is because if it did not work, the mathematics would be different. It would not just be the same mathematics differently encoded, because that is not different. It would be essentially different.

There are two different philosophies of mathematics that cover the vast majority of people's views of mathematics (often in combination). Either 1) mathematical objects have a life of their own, they are built into our world and they are out there to be discovered or 2) we have invented mathematics the same way we invented all other science, by observing the environment and abstracting from it.

Both of these, to my mind, do not try to give a real answer here.

Position 1) begs the question, it has already decided that math describes everything, and that is why science works. They solve the problem with circular logic and mysticism.

Position 2) does too, but with a language trick. They inject this notion of 'abstraction', which is really a veiled synonym for 'thinking like mathematics'. Then they decide that mathematics is one of the ways of thinking like mathematics and we are done.

But what is abstraction? What makes one statement more abstract than another? How do we as humans identify a good abstraction? Let's work backward through that set of questions.

Humans identify a good abstraction by feeling. It clicks and we experience a feeling that E.O.Wilson named 'consilience'. We feel a certain kind of calm clarity and we tend to assume that given enough time, everyone else who has to make the same decision can be brought to feel the same way about it.

So what makes one statement more abstract than another is this feeling. An abstract principle is psychologically "nicer" than its more realistic forms. It feels less like thinking, and more like knowing. There is an in-built sense of this "niceness", which often goes by the name of intuition. We can see this in action in babies.

So abstraction is a psychological effect that guides us as to what is and what is not simple. We observe that some things are exceedingly simple. For Brouwer, these start with "There is always a next instant of time." "Between to instants of time, there is always another." These are the basic intuitions of arithmetic and analysis. Others would continue with geometry "The shortest path to something is straight at it."... and then other basic notions. (He pointedly wouldn't, but he is notoriously fussy.)

These most undeniable simple propositions are what makes up mathematics. They may do so as axioms, names or simply as implied background, but that is what math is. It is extreme abstraction, built up by taking all combinations of those abstractions until they absolutely conflict.

The reason all other abstract approaches fall back on mathematics is that they are abstract, and mathematics is the set of abstractions so simple that without them, we as a species simply could not survive, combined only in the simplest possible ways, but most thoroughly.

So we can adopt a variant of position 2 that admits a real definition of abstraction. If you are going to define mathematics as abstraction, do it by observation and not linguistic trickery, and include a defensible notion of what abstraction is. The result is the intuitionist theory of mathematics.

So whereas the basic form of position 2 would predict that aliens would come to the same math, just in a different form, this one says that mathematics expresses human intuition, and that alien intuition can be expected to differ, and human intuition can be expected to evolve. Ergo the part of Intuitionism nobody likes: We should be conservative about what rules out other kinds of mathematics, and demand 'constructive' proofs.

Aliens would also need to survive. So they would have equally indispensable intuitions, which would give rise to an essentially different mathematics, which would have to be equally effective.

  • i agree with your first sentence, but i've never encountered the claim before, and it seems far from trivial. for that reason, i found the rest of the answer too difficult to read
    – user38026
    Sep 3, 2019 at 21:41
  • Intuitionism as a philosophy of mathematics is unpopular, in ways that are far from trivial...I will try to make this less dense.
    – user9166
    Sep 3, 2019 at 23:42
  • Thanks @Jobermark for that neat overview of intuitionism. I however remember a more forceful version of that — perhaps here? philosophy.stackexchange.com/q/64039/37256 Not sure since its deleted. Can we resurrect that version? My request for it on meta
    – Rushi
    Sep 5, 2019 at 5:31
  • @Rusi I have looked there, and that is not really about intuitionism, more a defense of primitive math as more meaningful than axiomatics. I find the viewpoint of 'math as a branch of rational psychology' casts light on topics well wide of math. So there are capsule summaries of it all over. I am something of a Steven Kleene/Everitt Bishop 'Stan'.
    – user9166
    Sep 5, 2019 at 9:52

I'm just trying to make sense of the success of applied mathematics... what it means... I'm asking whether its success suggests anything

According to James, success suggests something is true, in agreement with reality:

What does it mean to call a proposition or belief “true” from the perspective of pragmatism? This is the subject of James’s famous sixth lecture. He begins with a standard dictionary analysis of truth as agreement with reality. Accepting this, he warns that pragmatists and intellectualists will disagree over how to interpret the concepts of “agreement” and “reality,” the latter thinking that ideas copy what is fixed and independent of us. By contrast, he advocates a more dynamic and practical interpretation, a true idea or belief being one we can incorporate into our ways of thinking in such a way that it can be experientially validated. For James, the “reality” with which truths must agree has three dimensions: (1) matters of fact, (2) relations of ideas (such as the eternal truths of mathematics), and (3) the entire set of other truths to which we are committed. To say that our truths must “agree” with such realities pragmatically means that they must lead us to useful consequences.

The SEP quotes him:

you can say of it then either that “it is useful because it is true” or that “it is true because it is useful”. Both these phrases mean exactly the same thing.

James seems to argue that success shows its truth and that adds nothing to what we've said. That's a parsimonious way to "make sense" of mathematical successes, which could even be seen as an argument for pragmatism, if that making sense -- inferring something -- is necessary for usefulness. James described himself as a "realist", and many philosophers think that realism is compatible with pragmatism. In which case we may not need to give up on mathematical realism to "make sense" in that way.


This is a perennial question in philosophy, which should be taken as an indication of its importance. Note that to find a satisfying answer to some extent on your metaphysical presumptions. Needless to say, whatever your blend of rational and empirical thinking is will determine the approach you take. In analytical philosophy, statements from math and science often play a role in the expression of an answer. Science, after all, is recognized as being as tremendously successful as mathematics, the two being intertwined.

In metaphysics, the question arises about creating ontologies and epistemologies, and so one could rephrase your question as:

Why do the ontologies and epistemologies of math and science overlap and rely on each other so effectively? Why do these two ways of knowing provide such great justification about beliefs about "external reality", "objectivity", "the physical universe", etc.?

Well, let's set aside the evolutionary aspect of why thought, whatever that is, and focus on a specific philosophical idea: causality.

Math and science are both processes by which the brain understands cause and effect. The human brain is generally engaged in intuition and inference aligned to the goals of the person, and that is to determine modality and probability which refer to certainty. Some types of modality are the alethic, epistemic, and deontic, and all are important in individual and collective decision making, a necessary component of action. And action of course determines to some extent reproductive fitness.

But why math and science together? Because the primitives of thought are such that math and science are not truly distinct ontologically, but are two different types of problem solving. Mathematics is primarily a deductive practice when practiced analytically, and science is an inductive and abductive activity. But, outside of intuitive inference, all inference is generally inductive, abductive, or deductive! Thus, math and science together encompass a complete skill set of inference to allow certainty in thought.


"Why does mathematics work in the physical sciences?"

I don't know if this is what the OP was looking for, but I think the existing answers are missing an important interpretation of this question.

Mathematics provides only an approximation; it makes no claims of having a perfect correspondence with the real word. In Fourier analysis for instance, any arbitrary curve can be approximated by adding together sine waves of various sizes. The more sine waves that are used, the closer the approximation is to the original curve.

One would expect a similar situation when looking at the fundamental behaviour of the universe. A simple formula gives a not-so-good approximation; a more complicated formula gives a good approximation; an incredibly complicated formula gives an excellent approximation; etc. The more accurate the result we want, the greater the complexity of the equation will need to be.

Consider gravity, where F is the force of gravity between two bodies having masses m₁ and m₂, separated by a distance R:

  • Simple approximation: F = G₀×m₁×m₂ / R²
  • Good approximation: F = (G₀×m₁×m₂ / R²) + (G₁×m₁²×m₂² / R⁴)
  • Better approximation: F = (G₀×m₁×m₂ / R²) + (G₁×m₁²×m₂² / R⁴) + (G₂×m₁³×m₂³ / R⁶)

The values of G₀, G₁, G₂, etc. can be calculated by observation. As our technology improves, we can add more and more terms to the equation, and get more and more accurate approximations of how the universe behaves.

G₀ is known to be approximately 6.67430/10¹¹ m³/(kg⋅s²).

The "problem" though is that no matter how well we measure the real world, the values of G₁, G₂, G₃, etc. all appear to be zero. All of the terms except for the first have no effect.

That is, the simplest, crudest approximation to the behaviour of gravity just happens to be (as far as we can measure) a perfect match to the actual behaviour. There is no reason why this should be. There is no rational explanation for it.

But wait, there's more. It's not only gravity that has this amazing coincidence between reality and simple mathematics. Every aspect of the universe, every relationship that physics can measure, also has the same property.

The entire universe appears to behave according to very simple mathematical rules. Rather than mathematical equations approximating the behaviour of the universe, it appears that the universe itself is actually following those rules.

The question is, why?

Was the universe designed? Did it have a great architect? Is it all only a simulation running in a giant computer? Is it simply an almost impossible coincidence? Is there some reason, yet unknown, that requires this relationship?

  • This is one of the reasons that all the answers suggesting that mathematics by observation is not a good theory. We never observe a match, we choose something simpler than what might actually match better because we like the math more.
    – user9166
    Sep 4, 2019 at 17:29
  • 1
    @jobermark, my point was that we don't "choose something simpler than what might actually match better". The simplest formula already matches perfectly. Sep 4, 2019 at 17:46

In his article, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", Eugene Wigler offers an explanation: (page 230)

A possible explanation of a physicist's use of mathematics to formulate his laws of nature is that he is somewhat an irresponsible person. As a result, when he finds a connection well known from mathematics, he will jump at the conclusion that the connection is that discovered in mathematics simply because he does not know of any other similar connection.

Not all mathematical connections have a physical interpretation. Connections between transfinite cardinals may be an obvious example. Also just because a mathematical connection has been found useful in physics does not mean that science will not later reject it such as with Ptolemy's cosmology.

The underlying problem, given that we empirically observe patterns and connections, is why do those patterns exist at all? Mathematics offers languages to help describe those patterns well enough to predict what will (hopefully) happen later. One can even describe the pattern of the sun rising with natural language and make a very reliable prediction knowing no mathematics whatsoever: The sun will rise tomorrow.

It seems unreasonable to assume that natural language, any more than the currently accepted, mathematically based gravitational theory, used to assert that the sun will rise tomorrow is the reason why the sun will rise tomorrow. Both natural language and the mathematically formulated theory (another language) merely describe the pattern. From those descriptions one can make good predictions. However, being able to predict the pattern doesn't explain why the pattern is there at all.

Here is the question:

One way to understand its success is the Quine-Putnam indispensability argument for mathematical Platonism. Aside from that and arguments for scientific realism, what else works from the assumption that applied mathematics is phenomenally useful?

The underlying problem is the existence of patterns. Why are there patterns at all? Natural language and mathematics can describe patterns by reducing physical phenomena to these languages. Mathematical Platonism tries to give a reason why for these patterns by assigning ontological status to these languages (Platonic Forms) rather than to some deity setting up and/or sustaining the patterns.

It seems like a circular argument to claim that mathematical Platonism is indispensable simply because some mathematical theory is useful at the moment to make predictions. It also seems analogous to a Christian argument claiming the existence of God is necessary given the empirical evidence of the world around us. However, if one is looking for something besides Platonism or scientific realism one could always turn to a deity of some sort to explain both the existence of these patterns and our ability to use language to make good predictions.

Wigler, E. P. Symmetries and Reflections. Retrieved on August 27, 2019 from Internet Archive at https://archive.org/details/SymmetriesAndReflections/page/n1

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