## The (Un?)reasonable Effectiveness of Mathematics

1 10 2005

The phrase, “the unreasonable effectiveness of mathematics” goes back to the title of an essay by the physicist Eugene Wigner in 1960. He points out that mathematics is developed largely on aesthetic grounds, and yet large parts of it eventually get co-opted by physics and the other natural sciences to formalize parts of their theories. There seems to be no reason to believe that mathematics (especially the limited fragment of mathematics that humans actually get around to developing) should have anything to do with the physical world. He then goes on to point out how surprising it should be that it’s even possible to formulate laws of physics in the first place, let alone that they should be mathematical. And he spends the last little bit of the essay discussing the conceivability both of finding a unified theory to which all our scientific theories are approximations, or of the impossibility of such a theory, which would leave us with multiple contradictory theories, each good for its own domain. The fact that we’ve managed to come so far seems to cry out for explanation.

Wigner seems to miss some aspects of the development of mathematics though. He suggests that mathematicians find something beautiful and develop it, but doesn’t point out that these theories are actually very often developed just to explain things in already-established areas of mathematics. For instance, complex numbers were developed to fill in the steps in the solution of certain cubic equations over the real numbers. At least some of the theory of groups was first developed specifically by Galois and Abel to show why there was no corresponding method for solving quintic equations. If all of mathematics was developed for motivations resembling these (as I think plausible), then once we realize that the very basic parts of mathematics are applicable, it may be no surprise that the rest of it is as well. If the natural numbers apply to some phenomenon, and some other theory was developed to explain the natural numbers, then it seems plausible that this theory would be applicable to the explanation of the phenomenon the natural numbers apply to.

Of course, this still leaves open the question of why so much mathematics seems to apply in contexts other than these. If group theory was developed to explain properties of real numbers and other fields, then why should it apply to the fundamental particles of physics in a context independent of any such field?

Greg Frost-Arnold has a fascinating post suggesting that in fact in pre-Galilean astronomy, the effectiveness of mathematics might not have seemed so unreasonable. After all, they believed then that the “heavenly bodies” had similar properties of permanence and perfection to the objects of mathematics. And if they were all created by the same God, then it would make sense that mathematics and astronomy had a lot of overlap. The effectiveness only started seeming really unreasonable when Newton showed that there were mathematical theories unifying earthly and astronomical motion.

At any rate, this contemporary effectiveness of mathematics, which seems so unreasonable, for some reason hasn’t been a very central question in the philosophy of mathematics. Instead, people have focused on more foundational questions about mathematics, like what the nature of mathematical truth is, and how it is that we have access to it. But I think Hartry Field’s program in Science Without Numbers gives the closest thing to an explanation for the effectiveness of mathematics. His main goal is to prove a certain claim about the ontology of mathematics (namely, that there is none), but I think it’s more successful as an extension of the methods of Krantz, Luce, Suppes, and Tversky in their Foundations of Measurement to explain how mathematics can be applied in a rigorous manner. He formulates the axioms of Newtonian mechanics in a way that the mathematics that is applied to it can be straightforwardly seen to be a conservative extension. Thus, he justifies this application.

Michael Dummett, in “What is Mathematics About?” criticizes this program, saying that “Field envisages the justification of his conservative extension thesis as being accomplished only piecemeal.” Dummett suggests that this would be unsatisfying, because it would never make mathematics completely justified, but only justify particular applications of particular theories. Whether or not he’s right that this is all that Field would accomplish (Field seems to claim to have shown that all of mathematics is conservative over any non-mathematical theory), I think that this is actually almost exactly the goal we should want to achieve. It wouldn’t do to suggest that any mathematical theory can be applied to any aspect of the world – there’s only certain applications that make sense, and only those should be justified. We would still face some puzzles as to why it is that so much mathematics ends up applying to so much of the physical world, but at least each particular application would no longer seem so unreasonable.

### 3 responses

7 10 2005

I have never accepted the argument that mathematics is unreasonably effective in physics. This argument has the following flaws, IMO:

1. We are looking backwards, with the advantage of hindsight, at the history of mathematics and saying some of it models the world very well. But we are looking at the mathematics which models the world well, not all the mathematics there could be, nor even, just all the mathematics that has been articulated. Newton’s Principia, for example, proceeds through progressively refined models of planetary motion, with the later models being more accurate than the former. Is it any wonder that the later models are better fitting than the former? That’s what Newton was doing, making better-fitting models.

2. Our only way, now, to access knowledge about the world studied by physicists is through advanced mathematics, since there is no non-mathematical theoretical physics. Of course, the maths fits the physics well, because we are looking at just one object, not two.

3. The history of theoretical physics has often been as follows:

Step 1: Mathematicians create some concept, without any application in physics in mind.
Step 2: Theoretical physicists think this concept might be useful.
Step 3: The physicists (maybe with help from mathematicians) modify the concept to fit the physics domain.
Step 4: The modified concept may be taken up again by mathematicians.

Examples: probability theory (invented by mathematicians 1660s, adopted by physicists late 1800s); non-Euclidean geometry (1850s, early 1900s); category theory (1940s; 1990s).

Of course, the applicability of these examples may seem unreasonable. To which I would reply: its applicable because the physicists selected the mathematical concepts which are applicable.

The best response to Wigner’s ideas, I think, is this: Isn’t it amazing that all our trousers have two legs, so that we can wear them! Imagine if all our trousers had 1 only leg or had 3 legs!

9 10 2005

There are times when I feel like this – but at the same time it would be nice to find some way to verify that this is in fact the explanation. Some of it seems not to require much modification at all.

15 10 2005

I am very happy to find people who are not at all impressed by Eugene Wigner’s wonder at “the unreasonable effectiveness of mathematics”.

Just to emphasize a few points, first, as to Step 1, mathematicians mostly did not just create some concepts. Approximately, while creating something, they often kept in mind some application (or some use of a concept already) and then developed formal apparatus for using a concept in question. Arithmetic, ancient geometry, etc. was created that way, as well as probability theory in the 17th century (using concepts from gambling). Newton was also looking for computational apparatus for dealing with continuous quantities in the analysis of motion. As far as I understand, pure mathematics is more or less the invention of 19th and 20th century (maybe except number theory).

Second, there is another way of motivation in mathematics. As Kenny pointed out, some parts of math were developed “to explain things in already-established areas of mathematics”: complex numbers, etc., even non-Euclidean geometry was being built in the context of Euclidean geometry, etc.

All this should be kept in mind to counter the speculative statement that “mathematics is developed largely on aesthetic grounds”.

In general, I think the best answer to Eugene Wigner is given by Peter: “it’s applicable because the physicists selected the mathematical concepts which are applicable”. But perhaps should be added that our world is such that we can count things, measure them, there are interesting quantitative regularities in the world and the like.
Although I would admit, that the case of group theory might need some further reflection (I am not sure about it.)

PS. I have noticed that mathematicians and mathematically-minded physicists are prone to various speculations, e.g. about “aesthetic grounds”, ideal spaces in the mind or ideal mind-independent world, etc. One physicist has told me that at school there are often a problem with pupils who are very good at mathematics but who have difficulties with physics, biology. It looks like that good formal thinking leads to a kind of inability to evaluate down-to-earth stuff.