Mathematics and science

In his blog post today:

• Renaissance Science — XXV

thonyc gives some interesting history on the use of mathematics in science. I found this quite interesting, and it corrects some of my own misunderstandings. Popular books which touch on the history of mathematics tend to gloss over much of the detail.

We tend to see the use of mathematics in science as relatively sudden. But thonyc’s account shows that it was actually more gradual. In a way, that makes a lot of sense and is perhaps what we should have expected.

The experimental method

Discussions of the scientific method usually emphasize the idea of experimental testing. That’s how I was introduced to science in elementary school. Many internet discussions of science emphasize the experimental method. This can be a way of distinguishing between science and religious creationism, because the so-called scientific creationists do not use the kind of experimental testing that we see in science.

In reality, though, experimental testing is not limited to science. A good cook tests her concoctions. A tennis player tests his strokes. Experimental testing is ubiquitous in life, and is better thought of as part of pragmatism. Even a religious creationist tests his ideas by seeing how his intended audience responds to his stories.

Mathematics and reality

I have recently posted a couple of comments relating to whether mathematics is real. These were blog comments or forum comments. And, to be clear, these posts were elsewhere (not on this blog). This post will expand on those.

Discovery or invention

A blogger asked whether mathematics is a matter of discovery or invention, and I commented. But I’m not sure where that was. So I am reconstructing and expanding my comment.

My first remark was that mathematics is an invention. I’m reminded that Kronecker famously said “God gave us the natural numbers; all else is the work of man”. But it has long seemed to me that Kronecker gave too much credit to God. It is all the work of man.

I did not want to dispirit the blogger, so I tried to connect invention with discovery.

Nobody doubts that Lego blocks were invented. But give children some Lego blocks, and they will quickly discover all kinds of interesting things that they can do with those blocks. You really cannot separate invention and discovery. If you invented something that was useless and boring, you would quickly forget it. Any important invention involves the discovery that what you invented is useful or interesting or both. We might say that “invention or discovery” is a false dichotomy. They go together hand in hand.

Getting back to mathematics, presumably at some time it was discovered that counting was a useful practice. But maybe the practice had to be invented before you could discover that it was useful. Again, discovery and invention are intertwined.

Numbers are not just counting. For counting, you need a sequence of names to use. But numbers themselves are abstract entities, presumably derived from the practice of counting. So the use of numbers in this way was a separate invention. But again, it was an invention that depended on discovering the usefulness of numbers.

Truth in mathematics

I’m tentatively planning several posts on “truth”, and mathematics seems a good place to start — partly because I’m a mathematician, and partly because some of the distinctions seem clearer in mathematics than in other areas.

I can think of at least two different ways that we use “true” in mathematics. The most basic of these is with ordinary statements such as or the Pythagorean theorem. Both are generally recognized as true. The other way that some people use “true” is when they talk about whether the axioms are true. That could refer to the Peano axioms for arithmetic, the Zermelo-Fraenkel or about whether the axiom of choice is true or about whether the continuum hypothesis is true. As we shall discuss below, the way we think about the truth of axioms is different from the way that we think about the truth or ordinary mathematical statements.

Ordinary statements

When I talk of “ordinary statements” in mathematics, I am talking about statements such as in arithmetic or the Pythagorean theorem in geometry. We normally have a system of axioms that we use in our mathematics. For ordinary arithmetic, these are the Peano axioms. For geometric questions, we normally use Euclid’s axioms, supplemented by some version of the parallel postulate. For set theory, we most commonly use the Zermelo-Fraenkel axioms, possibly supplemented by the axiom of choice.

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.

Generalization in mathematics

Generalization is an important part of mathematics, and I shall discuss that here.  My discussion will mostly consist of examples with commentary on those examples.

I’m planning a future post on generalization in science.

Numbers

The simplest example has to do with numbers.  And our use of numbers presumably started with counting.  By assigning names, from a fixed sequence (1,2,3, …), we could count objects.  And then we could compare the results from counts of different collections of objects. This turned out to be useful for keeping track of quantities.

Rules were developed to deal with counting of groups of objects.  If we knew the counts of each of two groups, we could combine those with addition rules, to get the combined count.  And if we has several groups of the same count, we could combine those with multiplication rules.

Platonism, fictionalism and all that

There was recently an interesting discussion of platonism and fictionalism as philosophies of mathematics.  This was at “The Electric Agora” blog.  I added a couple of comments myself.

Yesterday, I went back to take another look.  That was mostly to see if there were any additional comments.  And there were two, both by Robin Herbert.  But comments are now closed for that post.  So I’ll say something here.

First some links:

• First comment by Robin Herbert;
• Second comment by Robin Herbert.

Both comments add to the discussion and are worth reading.

Is fictionalism true?

In his first comment, Robin says:

So the argument that fictionalism must be true because the axioms are only conventions appears to make the same mistake as saying the truth or falsity of “if A then B” depends on the truth or falsity of A.

To me, this seems weird.  I have said that I am a fictionalist.  I have never said that fictionalism is true.  I’m not at all sure that I know what it would even mean to say that fictionalism is true.

Penelope Maddy interviewed

Penelope Maddy was recently interviewed by 3am magazine (h/t Brian Leiter).  I found the interview interesting.

Maddy is a philosopher of mathematics.  In the past I have read some of her work related to set theory.  I very much disagreed with her philosophy, but her work was still worth reading.  She was very much into questions such as whether axioms are true.  That’s sometimes called “mathematical realism” because it is based on the idea that mathematics is saying something about the real world.

For myself, I could never make sense of mathematical realism.  As I saw it, axioms were neither true nor false.  I saw axioms as just useful assumptions whose consequences interested the mathematician.

It seems that Maddy has now moved away from that realism, so is adopting a view a bit closer to mine.  She now also doubts the Quine-Putnam indispensability thesis (that mathematical platonism is indispensable to physics).  Again, that is closer to my view.

I’m not at all sure that one’s philosophy of mathematics affects how one does mathematics.  But I did find it interesting to read of this evolution in her thinking.  And I’ll put this down as recommended reading for those interested in the philosophy of mathematics.

Against ontology – part 2

My second example of why I don’t like ontology, is a TEDx talk by Kit Fine (h/t Brian Leiter).  In that talk, Fine discusses what is the fundamental nature of the being of numbers.

It’s a puzzle to me that anyone would suppose that numbers have any fundamental being.  It seems obvious that they do not.

Fine gives three possible versions of the nature of numbers.  The first is due to Frege and Russell, the second to von Neumann, the third to Cantor.  The only one of those that I find useful is von Neumann’s.  But I do not take it as being about the nature of numbers.  Rather, I take it as a useful way to model arithmetic within set theory.  I have always assumed (perhaps wrongly) that was why von Neumann proposed that definition.

Kit Fine seems to think that there are puzzles about numbers and mathematics, that can be resolved by understanding the nature of numbers.  He suggests that there is a puzzle as to why mathematics is so useful in science.  Others apparently also see that as a puzzle.  Fine asks (about numbers):

How can they be so far removed from the familiar world, yet so intimately connected to it?

Presumably, he thinks that understanding the fundamental nature of numbers will answer that question.

Numbers have no fundamental nature.  Perhaps knowing that will help Fine.

The usefulness of numbers and of mathematics is explained by how we use them, not by what they are.  The usefulness of numbers in science is explained by how scientists use them.

Coffee cups and donuts

There’s a saying among mathematicians, that a topologist is someone who cannot tell the difference between a coffee cup and a donut.  I’ll discuss that in this post, and I’ll suggest implications beyond mathematics.

Usually, when we say this, we are thinking of the donut and the coffee cup as two-dimensional surfaces.  Once we go to the three-dimensional objects, nobody denies that the donut has a soft and spongy texture which makes it clearly different from a coffee cup.

Topology

Let’s start with a brief rundown on what is topology.  It is a branch of mathematics where we discuss ideas such as continuity, convergence, etc.  A classic example of convergence is with the sequence 0.9, 0.99, 0.999, …  We can see that the sequence gets closer and closer to 1, and we say that it converges to 1.  So topology has something to do with the geometric ideas of getting closer.  But it does so without needing a notion of metric (or distance).

Is mathematics natural?

While perusing the Uncommon Descent blog, I noticed a post

• Mathematics challenges naturalism, says math prof

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.