Why is mathematics so fantastically successful at describing the universe?

Question

Anyone who has studied physics will quickly see how fantastically successful mathematics is at describing the universe. The famous physicist Richard Feynman said in his book "the character of physical law" (pg.39):

"But what turns out to be true is that the more we investigate, the more laws we find, and the deeper we penetrate nature, the more this disease persists. Every one of our laws is a purely mathematical statement in rather complex and abstruse mathematics... It gets more and more abstruse and more and more difficult as we go on. Why? I have not the slightest idea..."

What does this imply about the nature of reality? And what can we say about this increasing complexity as we penetrate deeper?

Answer 1

1. Large portions of mathematics are of no use whatever in describing the universe. Thousands of theorems and proofs in pure mathematics are completely irrelevant to science. There is a famous Cambridge toast (whether apocryphal or not) : 'Here's to pure mathematics, and may it never be of use to anyone !'

2. It is not mathematics that describes the world. It is physical theories which do so, and these have indispensable mathematical components but include observation, experimentation, theorising and other conceptual components which are unconnected with mathematics. In even the most successful scientific theories, mathematics is only one element.

3. None the less it is an element, and a necessary one. Why is mathematics so important in successful science ? Its capacity for abstraction plays a part. Take the example of someone placing a ladder at an angle against a wall. Describing the situation in English or some other natural language will involve a narrative embracing the colour of the wood, the question why someone left it there, its number of rungs and whatever else comes to the narrator's mind. A mathematical description is more austere; it 'radically erases detail' (Sarukkai). There is a triangle formed by the ladder, the wall, and the intervening ground. From the length of the wall to the point where the ladder touches it, and the length of ground between wall and ladder we can calculate the length of the ladder, the third side of the triangle. Of course in this real-world situation the measurements will not exactly match those of the Law of Cosines applied to an ideal or model triangle. But they will be greatly more accurate than any alternative and will generalise to all triangles. In a way mathematics is so successful because it 'mathematicises' the real world; and the real world lends itself, within limits, to 'mathematicisation', to austere description. But within this situation of measurement, any number of non-mathematical, empirical assumptions are at work in the background. We assume, for instance, that the ruler by which we measure wall and ground is and remains straight, that it has been correctly calibrated, &c.

4. This brings me to a final point, implicit in what has just been said. If mathematics plays an indispensable part in the formulation of natural laws, in few if any situations do its equations, embodied in natural laws, enable precise description or prediction outside idealised conditions. This is because the 'initial conditions' to which natural laws are applied are seldom free from 'disturbing' elements. If we apply a law describing the rate of expansion of gas a vessel containing nothing but that gas, it is unlikely that any such vessel exists. The gas will have to be absolutely pure and uncontaminated by other ingredients, and such is not the real world of scientific practice.

Conclusion

There is no uncontestable answer to your very good question. To the references others have given you might add :

Sundar Sarukkai, 'Applying Mathematics: The Paradoxical Relation between Mathematics, Language and Reality', Economic and Political Weekly, Vol. 38, No. 35 (Aug. 30 - Sep. 5, 2003), pp. 3662-3670.

Symposium: 'Why Are the Calculuses of Logic and Arithmetic Applicable to Reality?' G. Ryle, C. Lewy and K. R. Popper Proceedings of the Aristotelian Society, Supplementary Volumes Vol. 20, Logic and Reality (1946), pp. 20-60.

Answer 2

The question itself has been already discussed by the previous answers, but I think that understanding the history and the purpose of mathematics and how it differs from other sciences would be of great help in general.

As someone studying mathematical logic/meta-mathematics. I can assure you that the purpose of mathematics is not to describe the universe, and that mathematics does not describe it. Physics does, chemistry does, biology does, natural sciences in general do — but mathematics certainly doesn't.

We consider that both writing and mathematics (through counting) were created around 8000 BC as a way to keep trace of agricultural goods. Nowadays mathematics are a way to express, understand and share complex ideas in general.

During antiquity there was no such thing as logically constructed mathematics (that does not mean that mathematics were not constructed that way but rather that they were not consciously constructed that way). For example, if someone argued that 1 + 1 does not equal 2, he would have been ignored for being stupid and not understanding 1+1=2. But no proof of 1+1=2 would have been given. Any proof would just have been a speech in natural language that would have convinced the majority. As a consequence, most of the work done until Renaissance was not more advanced than what can be done with an abacus and ruler & compass maths.

Around the 18th/19th century mathematicians realized that both mathematics and natural languages were too complex, that proceeding that way lead to errors and misunderstandings. They worked on a way to guarantee that a proof was correct and they could understand each other properly.

At that point mathematical notations and logic were created (Leibniz and Peano played a major role in this and despite being hard to read by today's standards I would still recommend to everyone to read their books/papers). Notations allows people to universally understand exactly what we are talking about, and logic is a rather simple tool that allows us to make proper proofs.

The base of formal mathematics is axioms : a set of rules that we take for granted. Axioms cannot be proven, they have to be extremely carefully chosen. As long as we assemble, mix the axioms following a set of rules, the result is guaranteed to be logical, error-less and the whole proof is a coherent thought.

But if the axioms are not chosen properly we can prove nonsense. For example if the axioms used consider two opposite propositions to be true, then everything can be proven according to those axioms.

In maths we arbitrarily define rules basic enough to build upon in a safe (logical) way. But nothing prevents you from creating a custom set of axioms, and build upon them to create a description of the opposite of our universe.

In physics (and other natural/experimental sciences) we chose a set of axioms basic enough not to allow us to make errors and try to define an equation describing some natural phenomenon. Then we do experiments trying to show that the equation we built was wrong, if we fail such a high amount of times that we can consider the equation to be sufficiently accurate and reliable we use it.

Maths is a powerful tool to describe complex things in general. But it is just a way to describe cognitive abstractions in a standardized format.

Also mathematics have a limit. Gödel's incompleteness theorem states that no matter how full and complex the set of axioms we chose, there will always be truths that we will not be able to prove.

Answer 3

This is known as the indispensability argument, because the mid-twentieth century philosopher Willard Van Orman Quine used it to argue (roughly) that mathematical entities were "indispensable" posits of empirical science, and hence we have empirical evidence that mathematical entities exist.

There are a number of objections to this argument in §4 of that Stanford Encyclopedia entry.

One of van Fraassen's arguments against scientific realism is also applicable to the indispensability argument. He's the realist argument:

the positive argument for realism is that it is the only philosophy that doesn't make the success of science a miracle. That terms in mature scientific theories typically refer (this formulation is due to Richard Boyd), that the theories accepted in a mature science are typically approximately true, that the same term can refer to the same thing even when it occurs in different theories—these statements are viewed by the scientific realist not as necessary truths but as part of the only scientific explanation of the success of science, and hence as part of any adequate scientific description of science and its relations to its objects.

And van Fraassen's objection:

The explanation provided is a very traditional one—adequatio ad rem, the ‘adequacy’ of the theory to its objects, a kind of mirroring of the structure of things by the structure of ideas—Aquinas would have felt quite at home with it.

Well, let us accept for now this demand for a scientific explanation of the success of science. Let us also resist construing it as merely a restatement of Smart's ‘cosmic coincidence’ argument, and view it instead as the question why we have successful scientific theories at all. Will this realist explanation with the Scholastic look be a scientifically acceptable answer? I would like to point out that science is a biological phenomenon, an activity by one kind of organism which facilitates its interaction with the environment. And this makes me think that a very different kind of scientific explanation is required.

I can best make the point by contrasting two accounts of the mouse who runs from its enemy, the cat. St. Augustine already remarked on this phenomenon, and provided an intentional explanation: the mouse perceives that the cat is its enemy, hence the mouse runs. What is postulated here is the ‘adequacy’ of the mouse's thought to the order of nature: the relation of enmity is correctly reflected in his mind. But the Darwinist says: Do not ask why the mouse runs from its enemy. Species which did not cope with their natural enemies no longer exist. That is why there are only ones who do.

In just the same way, I claim that the success of current scientific theories is no miracle. It is not even surprising to the scientific (Darwinist) mind. For any scientific theory is born into a life of fierce competition, a jungle red in tooth and claw. Only the successful theories survive—the ones which in fact latched on to actual regularities in nature.

This is from The Scientific Image, pp 39-40.

As I read this, van Fraassen argues that we don't need to explain the predictive success of scientific theories. We've constructed them to make good predictions, and basically if and when they no longer make good predictions we abandon them and formulate new ones. There's nothing remarkable about the phenomenon that scientific theories designed to be good for making empirical predictions are indeed good for making empirical predictions. Nothing follows about "the nature of reality."

The same argument can be made in response to Feynman's argument. We've developed and deployed mathematical formalisms that have been useful for the scientific task at hand. If and when they're no longer useful, we develop new formalisms. There's nothing remarkable about the phenomenon that mathematical formalisms designed to be good for certain tasks are indeed good for doing those tasks. Nothing follows about "the nature of reality."

Answer 4

I think it was physicist Eugene Wigner who first posed that question back in 1960, in a paper titled The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Philosophers are still trying to figure it out. Physicists couldn't care less.

Answer 5

I was drawn here from a recent similar math.se question, for which I drafted this answer before it was closed:

This is a tricky question because it's unclear whether the answer is epistemic, metaphysical or a mixture. Let's try those two options separately, each simplified but hopefully not beyond the point of usefulness:

Epistemology, i.e. how mathematical knowledge is possible

Section 7 here discusses how regularity in nature could be understood, in light of objections such as David Hume's to induction and similar forms of reasoning. Unless we have an a priori reason to expect such reasoning to work, it can't prove itself empirically, because such an argument would boil down to, "inferences about the unobserved from the observed will work in these unobserved cases because they worked in these observed cases". But maybe we do have such a reason.

For example, as long as the "reasonable" hypothesis class we start with has a finite VC dimension, it takes only a finite (in fact, surprisingly small) dataset to narrow down that class in a probably approximately correct way. The exact theorem to quote here depends on the hypothesis class we look at, but such hypothesis classes are a mathematical variant on the falsifiable hypotheses Popper considered. In short, any starting point of the form "the world works according to these equations, these finitely many parameters to be determined empirically" automatically meets the criteria for being empirically fitted to the world, and possibly falsified, in which case we can always start over.

Physicists have also explored why finitely many parameters are preferable to infinitely many ones. At least one major open problem in physics results from an infinite number of parameters we don't know how to make finite.

Metaphysics, i.e. why the nature of the universe would be conducive to mathematical exposition

You'd be amazed how much of the physics we know can be deduced from the idea that, because reality is objective, certain changes of perspective don't make a difference, because they're just coordinate transforms. We can describe this in terms of symmetries, or conservation laws if you prefer.

Also, while I can't find a reference right now, some researchers have proven theorems of the form, "if these facts about the world that don't sound mathematical at all are true, there's a way of mapping numbers to the events in space and time that makes these familiar physical equations true". This is roughly analogous to how Tarski restated Euclidean geometry in terms of lines intersecting at points, and how points are ordered on lines. When I first attended a talk on this subject a decade ago, Newtonian mechanics had already been reduced to "non-mathematics" in a more complex version of this idea, and work on doing the same thing for special and general relativity were in progress.

I've talked a lot about physics so far. OK, but what about chemistry or anatomy or economics? Well, here's where metaphysics is crucial. Let's say we accept, just for the moment, that things in the world are made of smaller things up to a point, after which we've hit the indivisible "atoms" of the ancient Greeks. If that's true, the mathematics of the smallest things will impose mathematics on the next-smallest things etc. On the other hand, completely different arguments can give us cause for optimism on different metaphysical assumptions, such as infinitely divisible fluids, particles emerging from more fundamental fields etc.

Answer 6

All we need is that the laws of physics are constant over space and time, and mostly not too complicated. Therefore, any specific event is a function of various laws of physics and how they interact. In other words, it's a logical product of the laws of the physics.

Since it's logical, we can create mathematical models of the laws. Since mathematics is about going from premises to conclusions, math will tell the scientist what happens in a given situation where certain laws interact.

We need the "mostly not too complicated" to make this fact actually useful. We don't have to use mathematical models too complex for us to solve for most things we're concerned about. If we couldn't solve the math for some part of the universe, it wouldn't be useful. Since there's enough laws of physics that are continuous (in the sense that small changes cause small differences), and positively linear in some respects, we can get useful results with fairly easy math.

The increasing abstruseness is a result of how we discover laws. We discover laws that are easy to express mathematically early, and so as research continues we always find that we've found most of the laws of a particular complexity or less, so the next we find is probably more complicated and mathematically abstruse.

Answer 7

Explaining in terms of numbers and abstract calculations and formulas are one one thing. Having those explanations validated through testing is the true proof of that leavening.

Thus, that math has been "fantastically successful" ignores the fact that we are no closer to having a handle on the true nature of the universe than were a century ago when Einstein published his Theory of General Relativity.

The fact remains that most explanations remain speculative.