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.

What is truth?

Pilate famously asked the title question (John 18:38).  I expect people have been asking that question for as long as they have been asking questions.  For a good discussion of theories of truth, check the entry in the Stanford Encyclopedia of Philosophy.

Truth is a central concept in philosophy.  But I am not at all satisfied with the way that it is used.  Hence this post.

Correspondence

If you ask about truth, you may be answered with the correspondence theory.  But the idea of “correspondence” is usually left unexplained.  I sometimes see statements similar to:

• A sentence is true if it corresponds to the facts.
• A sentence is true if it expresses what is the case.
• A sentence is true if it expresses the state of affairs.

The trouble with all of these, is that they seem to be roundabout ways of saying “A sentence is true if it is true.”  And that does not say anything at all.

Guns don’t kill people

Republican politicians kill people, with their failure to enact sensible laws

It is time to put the lives of our children ahead of the profits of the gun merchants.

It is well past time for politicians to stop accepting blood money from the NRA.

On scientism

I’m never quite sure what “scientism” is supposed to mean.  It often seems to be little more than a strawman, or a punching bag used by some theists and some philosophers.

In any case, a recent blog post by Larry Moran:

• One philosopher’s view of random genetic drift

drew my attention to an argument against scientism by Massimo Pigliucci.

I suggest you start by reading Larry Moran’s post.

Pigliucci’s questions

In his argument, Pigliucci gives a list of questions that he sees as philosophical questions rather than scientific questions.  I shall quote the questions here, and then give my comments on them.

• In metaphysics: what is a cause?

Why should any scientist care about this?  Yes, causation is important to science.  But for science, we test causation be seeing what we can cause in our experimentation.  We attempt to narrow down causes.  And we tend to extrapolate that knowledge by way of predictions.  It is not at all clear that this has anything much to do with the metaphysics of cause.

• In logic: is modus ponens a type of valid inference?

I’m not sure why any scientist would even care about this.

Where philosophy goes wrong

Here are a couple of assumptions that philosophers frequently make:

• There is a certain way that the world is.
• There is information all around us, telling us the way that the world is.

By “philosopher”, I really mean human.  We almost all philosophize to some extent.

I see those two listed assumptions as mistaken.

What is the mistake?

If your primary concern is getting around in this world, then the assumptions are not unreasonable.  The problem comes when we try to understand human cognition (or, roughly, what is it that the brain is doing?)

If we start with those assumptions, the ones that I consider mistaken, then there isn’t much for a cognitive system to do.  It just has to pick up the information that is all around us, and find out what that information tells us about the world.  That’s what leads to the idea of computationalism (“cognition is computation”).  And our experience with digital computers suggests that computation can be done in a way that is mindless and mechanical.

But a cognitive system has to be far more creative than that.