In 1969, Eugene Wigner wrote what has become a famous paper, titled “The unreasonable effectiveness of mathematics in the natural sciences.” There’s a pretty good summary of the related issues in the Wikipedia article of the same name.
As you might guess from the title of this blog post, I disagree with Wigner. In my view, the effectiveness of mathematics is entirely reasonable. And it has long seemed reasonable to me. I thought about it either in high school or as a graduate student in mathematics (I’m not sure which), and came up with what I found to be a satisfactory explanation.
Perspective on mathematics
I’ll start with my broad perspective, which I have probably mentioned before on this blog. I often say that mathematics is not about reality. The mathematician Kronecker famously said “God gave us the natural numbers. All else is the work of man.” I almost agree, except that I think Kronecker gave God too much credit. As I see it, the natural numbers are also the work of man. That’s part of why I am a mathematical fictionalist.
Presumably, Wigner had a similar view, that mathematics is not about reality. That’s actually a common view among mathematicians. And that’s part of why he saw it as unreasonable, that mathematics could be so effective.
The philosophers Quine and Putnam had a different view. Because of the role of mathematics in the sciences, they came up with a version of mathematical realism, known as the “Quine – Putnam indispensability thesis.” Needless to say, I disagree with Quine and Putnam on this.
What were they thinking?
Assuming that I am right about this, where did other people go astray? What were they thinking?
Philosophers tend to think of concepts as immutable. Yes, they will agree that we do form concepts, and we do change them from time to time. But they still tend to think of them as immutable. Perhaps that’s because of the way that they use logic. And logic tends to assume fixed concepts.
When I look at advances in science, I see scientists carefully formulating their concepts. And if the scientists are able to design their concepts, why not design them to fit the need of mathematics. Of course, concepts also have to fit reality, so the scientist does not have complete freedom. However, physics deals with the most basic of concepts, and has more freedom than other sciences in how concepts are designed.
From early on, I saw Newton’s laws as reformulating concepts. Most importantly, Newton distinguished between weight and mass, so that mass was really a new concept. And Newton reformulated the concept of force. He was able to do so, in a way that allowed his laws to be expressed in a very mathematical form.
Carving up the world
I described part of this in my previous post. There, I explained that physics uses a very mathematical way of carving up the world into measurable units.
As described in earlier posts, we carve up the world and give names to the parts. We are then able to describe the world (or aspects of the world) simply by construction statements out of these names (and with a little syntactic glue). Science carves the world in a very mathematical way. And it uses numbers as names for the parts. It does so in a way that the mathematical relation between the numbers matches the geometric relation between the parts so named. This allows mathematics to be used on our descriptions or measurements. Properties such as force, mass, length, time are really just mathematical idealizations of what we measure. So theoretical physics can also use mathematics in its theorizing.
Fictionalism
I have suggested that the most important use of mathematics is with measurements. But measurements are just the names of the parts into which we carve the world. So what is important there, is using mathematics on names. And therefore nominalism, the thesis that mathematical entities are just names, ought to be adequate for using mathematics this way. I prefer to call it “fictionalism” rather than “nominalism”, but I suppose that is mostly a matter of taste.
I should be clear here. I am not suggesting that properties such as force, mass, length, time are themselves merely names. We do carve up the world into parts, so we might as well say that the parts are real. But the mathematics is only applied to the names that we give those parts. So fictionalism (or nominalism) is adequate for mathematics. But assuming nominalism for the properties from physics would be a mistake.
In particular, I see no need for the Quine – Putnam thesis.
Summary
I have discussed why I see it as entirely reasonable that mathematics should be as useful as it is in the natural sciences.
The Heretical Philosopher 