While perusing the Uncommon Descent blog, I noticed a post

This seemed a strange thing to say. It perhaps even involves what Gilbert Ryle would have called “a category mistake”. Browsing through that post, I saw that it referred to an article in Aeon magazine

- Some philosophers think mathematics exists in a mysterious other realm. They’re wrong. Look around: you can see it (James Franklin)

Clearly, professor Franklin and I have very different ways of looking at mathematics. And that’s what I will be discussing in this post.

**Mathematics and naturalism**

Let’s start with that reference to “naturalism”. Franklin adopts a philosophy that he describes as Aristotelian realism. And it is in relation to that philosophy, that he makes his comment about naturalism.

Aristotelian realism stands in a difficult relationship with naturalism, the project of showing that all of the world and human knowledge can be explained in terms of physics, biology and neuroscience.

I’ve never fully understood what people mean by “naturalism”, nor have I understood the arguments about it that seem to be part of the culture wars. The idea that the world and human knowledge can be explained in terms of physics, biology and neuroscience seems to me a non-starter — again, I see it as involving something like a category mistake.

Franklin continues:

If mathematical properties are realised in the physical world and capable of being perceived, then mathematics can seem no more inexplicable than colour perception, which surely can be explained in naturalist terms. On the other hand, Aristotelians agree with Platonists that the mathematical grasp of necessities is mysterious. What is necessary is true in all possible worlds, but how can perception see into other possible worlds?

Personally, as a fictionalist, I don’t see the difficulty. We simply take it for granted that other worlds can contemplate the same fictions. There is no need for our perception to be able to see into other worlds.

Presumably, the idea that there is something mysterious about mathematics, is why Uncommon Descent posted on this.

**What is mathematics about?**

We jumped to the middle, to see what Franklin had to say about naturalism. But let’s go back to the beginning to see the picture that he is painting. Franklin ends his first paragraph with

When all the sciences and their subject matters are laid out, is there any aspect of reality left over for mathematics to be about? That is the basic question in the philosophy of mathematics.

To me, that seems a strange way of putting it. I don’t see mathematics as about reality. Most of the mathematicians that I have known would agree that it is not about reality. I suppose most mathematicians are platonists, so perhaps would take mathematics to be about platonic entities. I’m a fictionalist, which I see as not much different from platonism, in the sense that I take platonism itself to be a fiction.

There’s an old saying “If the only tool you have is a hammer, then every problem looks like a nail.” The version for philosophy becomes “If the only tool you have is logic, then every problem looks like a logic proposition.” One of the problems that I have with philosophy, and with the philosophy of mathematics, is this emphasis on propositions. For example, a theorem proved in mathematics would be consider to be a true proposition. And it is fine to see theorems that way. But when philosophers ask what mathematics is about, they are mainly interested in what the propositions are about. And if they are about fictions, then mathematics should not be important.

While the propositions of mathematics may be about fictions, or about platonic entities (depending on you philosophy), I see mathematics as implicitly about method. It’s not what we prove (the propositions) that matter; it is how we prove them (the method) that is important. So while I see mathematics as explicitly about fictions, I see it implicitly about method. And it is the latter that I take to be important. The methods used by mathematics are often idealizations of the methods used by scientists. So it is the role of method that I see as making mathematics useful to the sciences.

**Mathematical necessity**

Franklin quickly gets to the apparent necessity of mathematical truths, which he sees as a problem. Citing philosopher Peter Singer, Franklin writes:

‘The self-evidence of the basic truths of mathematics,’ he says, ‘could be explained… by seeing mathematics as a system of tautologies… true by virtue of the meanings of the terms used.’ Singer is wrong to claim that this philosophy of mathematics, called logicism, is ‘widely, if not universally accepted’. It has not been accepted by any serious philosopher of mathematics for 100 years.

The “he” in that quote is Singer.

Franklin’s reaction seems strange to me. I agree with Singer, that mathematical propositions can be seen as analytic (true by virtue of the meanings of the terms), which explains their necessity. Yet I agree with Franklin’s criticism of logicism. It seems to me that Franklin has missed something there. I reject logicism as a philosophy of mathematics. But I agree that the necessity of mathematics truth comes from those truths being analytic. Logicism, as a philosophy of mathematics, claims that mathematics arises out of logic. And I disagree with that. As I see it, accounting for mathematical truth falls far short of accounting for mathematics. To account for mathematics, you have to look at the methods that we use to address problems. It is not sufficient to just look at the propositions we derive by applying those methods.

**Nominalism, platonism or realism**

Franklin uses “nominalism” for those who say that mathematics isn’t about anything. He would probably consider my fictionalism to be a version of nominalism. In answer to “what is mathematics about”, Franklin says:

The ‘No’ answer, whose champions are known as nominalists, says that mathematics is just a language. On this view, it is just a way of talking about other things, or a collection of logical trivialities (as Singer claims), or a formal manipulation of symbols according to rules. However you cut it, it is not really about anything.

It seems to me that Franklin is really describing formalism. I see fictionalism as very different from that. A formalist looks only at the form of mathematical expression. This fictionalist (i.e. myself), looks at mathematical expression as semantically rich. While it is sometimes a useful exercise to show that a proof goes through entirely on the basis of form, my normal mathematical thinking is about the semantics (the meanings of the terms). So that description Franklin gives does not fit with how I do mathematics.

Franklin, himself, argues for some kind of realism. He claims that we see mathematics all around us.

A dozen eggs can be arranged in cartons of 6 × 2 or 3 × 4, but eggs are not sold in lots of 11 or 13 because there is no neat way of organising 11 or 13 of them into an eggbox: 11 and 13, unlike 12, are prime, and primes cannot be formed by multiplying two smaller numbers.

His Aeon article is illustrated with a picture of a carton of eggs (or a half-carton with one egg missing). I do not find that persuasive. I grew up in a house where we had a chicken coop in the backyard to give us a supply of fresh eggs. I never once saw the hens arrange eggs in neat rows. Yet that is what I would want to see before saying that the mathematics is all around us.

Instead, I see the mathematics (such as with the egg carton) arising as a result of human organization. We systematically organize our world. And mathematics, which I see as about method, gives us the systematic methods of organization that we can use. So I see mathematics as implicitly about systematic behavior. I see mathematics as important to science, because science is systematic.

Many people seem to think that Newton’s laws uncovered something profoundly mathematical about the world. That’s not my view. Rather, I see Newton as having developed a systematic way of accounting for motion. And the mathematics that we see in Newton’s laws is just the mathematics of the systematicity that Newton used.

**Final remarks**

I think I have said enough to indicate where I look at mathematics differently from Franklin. But the implications go beyond mathematics. Many philosophers and cognitive scientists take it that our minds function by finding mathematical patterns in the world. By contrast, I see the mind functioning by finding ways of systematically organizing how we deal with the world. And I see the mathematical patterns as emerging from our systematic organization, rather than as being already present in the world.