The Perfectly Reasonable Effectiveness of Mathematics in the Natural Sciences

A Perfectly Reasonable Explanation for the Effectiveness of Mathematics in the Natural Sciences?

Eugene Wigner, in an article whose title I shamelessly ripped off for this paper, says that “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and…there is no rational explanation for it.”[1] Indeed, on the surface, there is something extremely uncanny in the fact that a prediction arrived at by purely mathematical means could turn out to be accurate to within a few thousandths of a percent when we later have the ability to test it against empirical observations. In the words of physicist Richard Feynman, it is “quite amazing that it is possible to predict what will happen by mathematics, which is simply following rules which really have nothing to do with the original thing.”[2] But is the power of prediction really all that mysterious? The connection between math and physics may not be as flawless as some would think. Many types of predictions are only possible when idealized physical situations are considered. The mystery of the effectiveness of mathematics can thus be divided into two parts. Why can physical principles be formulated mathematically in such a way that deductions from these formulas give accurate physical predictions when such deduction is possible? And why are predictions accurate even when highly idealized physical situations are considered?

A structuralist account of the ontology of both mathematics and physics could perhaps explain why physical principles can be formulated mathematically. I believe there are several metaphysical stories about why the relevant structure would be simple enough for us to deal with in as much depth as we do. The accuracy of conclusions drawn from considerations of idealized physical situations may be as much of an “accident” of this particular universe as the simplicity of the underlying structure. Overall the applicability of mathematics to the natural sciences may not be perfectly reasonable, but it seems that most of the mysteries about application are reducible to mysteries about physics itself. These mysteries may be compounded by the fact that we as humans are unable to know beforehand what deductively follows from certain axioms, which makes many straightforward results appear surprising.

Wigner cites Newton’s law of gravity as “a monumental example of a law, formulated in terms which appear simple to the mathematician, which has proved accurate beyond all reasonable expectations.”[3] The law itself says nothing more than that the force of attraction between two massive bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them, yet it leads to extremely accurate predictions about where planets and moons and satellites will be at some time in the future, based on where they are and how fast they’re moving now. But these predictions are only possible because the masses and distances in question are such that only two objects need to be considered for a fairly accurate computation. When three masses are considered, the system of differential equations resulting from Newton’s law have no explicit solution, and all predictions based on them are thus computed by numerical methods, which are inherently inaccurate.

Of course, while Newton’s law of gravity gives accurate predictions in most cases, it turns out to be at odds with General Relativity, which has been confirmed repeatedly since its formulation. Newtonian physics fails to account at all for certain observed phenomena that follow readily from Einstein’s theories. It is not just incorrect physical theories, however, that must first be idealized in order to provide useful predictions within a reasonable amount of time. Some theories still regarded as correct are, if anything, more computationally complex than Newtonian gravitational theory. Mark Wilson brings up the example of placing a rock on top of a rigid post. The complexity of equations for the waves traveling through the post, especially when combined with factors such as friction with the air and ground that dissipate the energy of these waves, makes their use highly impractical. So instead of attempting “to predict the final state of the post by mathematically tracking its ongoing temporal evolution, watching its kinetic energy gradually ebb away[,]…we demurely turn our gaze away from the post for a decent interval and calculate what its shape might possibly be after the messy process of energy loss has been completed.”[4] This “decent interval” is the amount of time required for enough of a reduction in kinetic energy for us to be able to think of the post as being at a stationary equilibrium. The question of why predictions deduced in this way continue to be so accurate will be addressed later.

Certainly Richard Feynman, if anyone, was well aware of the difficulties of applying mathematics to physical theories. The amazement expressed in the previous quotation must therefore be at something other than the practical predictive nature of mathematical formulations of the most basic laws of physics. More likely, Feynman was amazed both by the fact that physical principles can be formulated mathematically, and by the power of idealized mathematical formulations, when applied to idealized physical situations, to predict phenomena with a very high degree of accuracy. There is the additional question of why there should be physical laws at all, which may be less of an issue for physicists than it is for philosophers.

The ability of physical principles to be formulated mathematically may be explainable if one looks at both math and physics as studies of structures. Hartry Field, in Science Without Numbers, seeks to reformulate Newtonian gravitational theory without mathematical objects. He maintains that such objects do not really exist, and that they only seem so crucial to physics because their use makes deductions much easier. While Field himself advocates a nominalistic view of mathematics, Stewart Shapiro points out that he proves theorems about the structure he articulates for space-time. “The activity of proving things about space-time is the same kind of activity as proving theorems about real numbers. Both are the deductive study of a structure.”[5] We might expect the space-time of a universe governed purely by Newtonian principles to have the same underlying structure as the set of quadruples of real numbers. It turns out that our universe isn’t governed purely by Newtonian principles, but modern physicists continue to talk about such structures as Hilbert space. Even if modern physics is nearly as far from the ultimate truth about the structure of the universe as Newtonian physics was, the consistent use of mathematics in the formulation of physical theories suggests that there is something mathematical about whatever structure ultimately turns out to be that of the universe in which we live.

This might mean that the entirety of physics is merely a subset of mathematics, in which case the fact that physical theories can be formulated mathematically would not be the least bit surprising or mysterious. But some explanation is required for how physics can legitimately be understood as a subset of mathematics, when physics is about existing physical entities, while mathematics is about…what? Leaving detailed defenses of structuralism to more skillful philosophers of math, I propose that mathematics is about structures. For the question at hand, it is not important whether math is about realized structures or possible structures, nor is it important whether structures depend on exemplifications or can exist apart from any exemplification. What is important to realize is that the collection of all physical objects in the universe, along with all relationships between them, is an exemplification of a structure.

Wilson criticizes as lazily optimistic the view that “mathematics will embody a ‘structure’ isomorphic to any conceivable physical circumstance, even if mathematics won’t be able to say anything interesting about that structure if it proves horribly unregulated.”[6] The fact is, however, that the structure exemplified by the collection of all physical objects in the universe is not horribly unregulated. Rather, it is so regulated that mere human beings, who took tens of millennia to figure out that placing heavy loads on things that roll makes them easier to move, can discover large amounts of useful information about it. So here we have this highly regulated, mathematically embodied structure. It would be quite surprising if mathematics couldn’t tell us anything useful about the physical universe. The way mathematics provides useful information, of course, is through sets of formulas, so it really shouldn’t be surprising that physical principles can be formulated mathematically.

What is surprising is that there are physical principles at all. Why is the structure of the universe so regular that simple principles govern the behavior of every particle in it? An anti-realist could respond to this question by saying simply that such principles do not exist. The structure of the universe is not regulated and we are completely unjustified in claiming that a phenomenon exhibited by matter under certain circumstances here and now will be exhibited by matter under those same circumstances everywhere else, at any time. This view does have some initial plausibility, given the incomprehensible minuteness of the portion of the universe we have observed, or even the portion we could observe with present technology and unlimited time, when compared to the (apparent) vastness of time and space. (That time and space are actually as vast as they seem could also be questioned by an anti-realist, of course.) But if natural principles didn’t exist, clearly mathematics would not be at all applicable to the natural sciences, and this paper would have been done before it started. So at all times we are supposing that science as we know it does, or at least could in the future, correspond roughly to truths about the physical universe.

There are several possible accounts of why the universe has a structure that is simple enough to be governed by a few basic rules. Most attractive to the average theist would be the notion that there is a benevolent creator who wanted any intelligent life in the creation to be able to increase its knowledge about the creation and it’s place therein. The best way to allow beings with a wide range of intelligence levels to comprehend the structure of the universe would clearly be to make that structure regular enough to be describable with a small set of basic principles. Another theistic account, which many would find less attractive, involves a creator that is either lazy and has a relatively short attention span. Such a creator wouldn’t want to spend too much time making sure everything always ran smoothly in the creation, so would create something to run entirely on its own. A short attention span would discourage this creator from waiting around to see how everything eventually turned out, so the creation would operate on principles that determined how everything would progress. (If we’re allowing for the existence of a being who can create entire universes at will, surely we can allow for this being to be capable of overcoming all the difficulties humans have with making truly accurate predictions from the most basic principles.) Being lazy, the creator also wouldn’t want to spend much time computing how everything eventually turned out, so the rules governing the creation would be as simple as possible. One possibly nontheistic account involves the existence of a large (possibly infinite) number of separate universes with different structures. In any universe with a particularly irregular structure, if intelligent life was somehow able to evolve, it certainly wouldn’t develop a study of physics that used a surprising amount of mathematics. The fact that our own physics is formulated mathematically might result from nothing more than the coincidence of our living in one of the more regularly structured physical systems. Whether any one of these accounts is the correct one is not directly relevant to the applicability of mathematics in the natural sciences. Rather, it is a metaphysical question to be possibly addressed by philosophers of physics.

What it means for physical principles to be suitably describable in the language of mathematics is precisely that applying mathematical deduction to formulations of these principles results in predictions that can be “translated” back into the language of physics, where they turn out to be incredibly accurate. But we have seen that only rarely in mathematical physics are formulations of the basic physical principles themselves applied directly to the physical circumstances under consideration. Instead, idealized reformulations, or idealizations of the physical circumstances, or both are used. Granting that from the existence of physical principles in the first place, it follows that mathematical formulations of those principles are possible and give accurate predictions, there is still a significant question to be answered about the effectiveness of mathematics. In addition to the idealizations scientists knowingly use, there is the most basic set of principles we have formulated thus far. If the history of science has any lessons to teach us, we have every reason to believe that the current state of physics is itself just an idealized version of the actual situation, just as Newton’s law of gravity is now seen as little more than a convenient simplification of Einstein’s theories for situations when forces and velocities are not too “extreme”. Why should these idealizations still give suitably accurate predictions?

Predictions based on idealized versions of reality are accurate because the part of the universe under consideration, as it actually is, behaves much as it would if the universe as a whole behaved in the idealized way. For instance, calculations treating the earth and the moon as a pair of point-masses in an otherwise empty universe are accurate because the earth and the moon behave, in actuality, much as they would if they were a pair of point-masses effectively isolated from the rest of existence. There is a physical part of the explanation for this, and a mathematical part. On the physical side, the effectiveness of idealization results from the nature of the principles at work. One can imagine a universe in which the force of mutual attraction between massive objects depended differently on masses and distance than in our own universe. If gravitational force didn’t depend on mass, then none of the small satellites we’ve put into orbit could be ignored in computations without getting significantly different results. If the force didn’t decrease so rapidly with increasing distance, the effects of more distant planets would be too great to leave out of equations. The fact that the principles at work are as easily idealized as they are is a question within physics, and can possibly be “answered” through the same accounts as were proposed to explain the existence of these principles in the first place.

On the mathematical side, the question can be answered more directly, rather than being pushed aside into another field: Idealized mathematical formulations give accurate predictions precisely because we choose them to do so. We decide on some desired degree of accuracy, and then proceed, within mathematics itself, to work out simpler equations that provide that much accuracy. It can often be proven, again entirely within mathematics itself, whether two different formulas will give results within a specified distance of each other. The fact that accurate simplifications are ever possible may seem surprising, but I think the notion of “implicational opacity” is important here. Jody Azzouni describes it at the sentential level as “our inability to see, before a proof is designed, the implications of mathematical propositions.” This inability often leads to “our surprise at how certain concepts are intimately connected to other concepts they otherwise seem irrelevant to.”[7] Implicational opacity stems from the limitations of the human mind, and is probably required of any interesting deductive system. If it were immediately obvious what ultimately followed from what, we would have no reason to carry out investigations of the system.

Undoubtedly, there is reason to question the relationship between implicational opacity and how mathematics really works. But this mystery behind apparent mysteries within pure mathematics is perhaps better addressed by the philosophy of mind. Similarly, other surprising aspects of the relationship between mathematics and physics stem from mysteries within physics itself. If there is anything unreasonable about the effectiveness of mathematics in the natural sciences, it therefore stems from unreasonableness elsewhere in philosophy, physics, and metaphysics. On the surface, this may seem like a completely unsatisfactory “solution” to the problem, because it really doesn’t solve anything. But it does suggest that we would do well direct our efforts away from the philosophy of math and towards the philosophy of physics and perhaps the philosophy of mind. Solutions to the apparent problem of unreasonableness, if they are to be found at all, seem likely to lie in these other fields.

[1] Wigner, Eugene, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” in Symmetries and Reflections, 1967, p. 223.

[2] Quoted in footnote 2 of Jody Azzouni’s “Applying mathematics: an attempt to design a philosophical problem,” 2000.

[4] Wilson, Mark, “The unreasonable uncooperativeness of mathematics in the natural sciences,” 2000.

[5] Shapiro, Stewart, Thinking About Mathematics, 2000, p. 261.

[6] Wilson, “Unreasonable uncooperativeness.”