The reasonable effectiveness of mathematics in the natural sciences

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.

Carving up the world with science

In previous posts, I have discussed how we carve up the world, and how that carving up is what allows us to express true statements about the world.  Science also expresses true statements about the world.  In this post I will discuss how that relates to carving up.

Yes, science also carves up the world in its own way.  And it does that in order to be able to make true statements about the world.  So the basic idea is the same.  But the method is very different.

Which science

Unsurprisingly, different sciences carve up the world in different ways.  Biology is concerned with living organisms.  So it wants to carve up the world into organisms, and then to further carve up those organisms into organs, cells, proteins, genes, etc.  At a larger scale, it wants to look at populations of organism.

In this post, I shall mainly be looking at how physics carves up the world.  That’s partly because the physics way of carving is most different from our ordinary way of carving.  And, additionally, all sciences borrow from physics, at least to some extent.

Measuring

Take a look at a ruler, such as we use for measuring length.  I have a ruler in front of me know, as I write this.