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.

Mathematical truth

While this post is about mathematical truth, it is really intended as part of a series of posts about truth.  The mathematics here will be light.  I am choosing to discuss mathematical truth because some of the distinctions are clearer in mathematics.  But I do intend it to illustrate ideas about truth that are not confined to mathematics.

Mathematicians actually disagree about mathematical truth.  But the disagreements are mostly peripheral to what they do as mathematicians.  So they usually don’t get into intense arguments about these disagreements.

Philosophy

First a little philosophical background.

There is a school of mathematics known as Intuitionism.  This differs from the more common classical mathematics, in that it has a more restrictive view of what is allowed in a mathematical proof.  And, consequently, it has a more restrictive view of truth.  In particular, Intuitionists do not accept Cantor’s set theory.

The mainstream alternative to Intuitionism is usually called “Classical Mathematics“.

This post mainly has to do with truth in classical mathematics.  I mention Inuitionism just to acknowledge its existence and indicate that it is not what I will be discussing.

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.

Searle on direct realism

As I hinted in my previous post, I want to discuss some aspects of Searle’s theory of perception.

Searle makes a good start with:

I believe the worst mistake of all is the cluster of views known as Dualism, Materialism, Monism, Functionalism, Behaviorism, Idealism, the Identity Theory, etc. The idea these theories all have in common is that there is some special problem about the relation of the mind to the body, consciousness to the brain, and in their fixation on the illusion that there is a problem, philosophers have fastened onto different solutions to the problem. (page 10).

I agree that those are mostly mistakes.  Searle continues with:

A mistake of nearly as great a magnitude overwhelmed our tradition in the seventeenth century and after, and it is the mistake of supposing that we never directly perceive objects and states of affairs in the world, but directly perceive only our subjective experiences.

That is Searle’s statement about his direct realism.  I do support the view that perception is direct, but I avoid the term “direct realism” because the word “realism” seems to carry some unnecessary metaphysical baggage.

Constrained invention

This will mostly be a copy of what I recently posted in a Yahoo groups discussion.  And, incidentally, Yahoo badly mangled that post (stripped out most of the formatting).

As background, I’ll note that in an earlier Yahoo groups post, I had indicated that I was opposed to the view that perception is passive.  This seemed to puzzle some participants in the discussion.  So my post — the one I am quoting — was intended to explain what I mean when I say that perception is active.

The quoted post

You guys need to get out more. You are trapped in a world of logic, and unable to think outside that box.

You both seem committed to God’s eye view thinking, though you may be in denial over that. So you see perception as a system to report to you what is seen by the hypothetical God. But how could that ever work?

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).

Convention (2) – word usage vs. behavior

This continues the series that I started at

• Convention (1) — introduction

Today’s post distinguishes between conventions of word usage, and conventions of other kinds of behavior.  Word use is, of course, a kind of behavior.

I’ll give an example of each.

A word use convention

In his “Truth by convention,” Quine writes:

A contextual definition sets up indefinitely many mutually analogous pairs of definienda and definientia according to some general scheme;  an example is the definition whereby expressions of the form ‘sin —/cos —‘ are abbreviated as ‘tan —‘.

Truth and axioms in mathematics

There’s been some discussion of truth in mathematics in the comments to my previous post.  Here, I want to expand a little on my view and express puzzlement at the idea that axioms are themselves true or false.

In response to a question, said “Actually, I take axioms to be neither true nor false, and I take the truth of mathematical theorems to be relative to the assumed axioms.”  Let me restate that in terms of the Peano axioms for ordinary arithmetic.

1. The Peano axioms are neither true nor false.  Rather, they are definitional statements.  They define that part of mathematics known as Peano Arithmetic (or PA, or simply arithmetic).
2. Theorems proved in PA are true in a relative sense.  Their truth is relative to the PA axioms.  They are true as used within PA, but perhaps not even meaningful outside of PA.