Mathematical truth

by Neil Rickert

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.


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.

Within classical mathematics, we find that some mathematicians hold the philosophical view known as “Mathematical Platonism” while others call themselves “Fictionalists”. Both Platonism and fictionalism are described HERE.  I’ll note that I am a fictionalist.

It is often said that most mathematicians are Platonists.  I don’t really know if that’s true.  I suspect that most mathematicians haven’t really thought much about either Platonism or fictionalism.

In any case, Platonists and fictionalists often disagree about mathematical true.

What is true?

Most mathematicians would agree on the truth of ordinary statements of arithmetic.  I expect that they all agree that 3+5=8 is true.  And most classical mathematicians would mostly agree on the truth of theorems that have been proved.

Often, a mathematical proof consists of showing that a result (or statement or theorem) can be derived from the axioms.  For ordinary arithmetic, it is usual to start with the Peano axioms, though most proofs probably do not go all the way back to that axiom set.

If you ask whether the Peano axioms themselves are true, you will begin to see some disagreement.  Some mathematicians will say that the axioms are true.  Others will say that the axioms are just assumptions, albeit very useful and important assumptions.

In my experience, Platonists are more likely to say that the axioms are true, while fictionalists are more likely to doubt whether it is meaningful to ask whether the axioms are true.

And then there are questions about statements which can be neither proved nor disproved.  The Continuum Hypothesis is an example of this.  It has been shown that the Continuum Hypothesis can be neither proved nor disproved using the standard axioms of set theory.  Some mathematicians, usually Platonists, assert that there is a fact of the matter as to whether CH is true, though we do not currently know that fact.  Other mathematicians, often fictionalists, tend to doubt that there is a fact of the matter for CH.


My main reason for this post was to indicate the different ways that people look at truth for different kinds of statements.  For statements that can be derived from axioms, or for statements that can be shown false on the basis of axioms, there is widespread agreement on whether those statements are true and on what are the criteria by which we determine that they are true.

For the axioms themselves, there is some disagreement as to whether those should be considered to be true.  And there is disagreement on whether it is meaningful to say that the axioms themselves are true.  And, likewise, there is disagreement about whether it is meaningful to talk of the truth of statements that have been shown to be undecidable (neither provable nor disprovable from the axioms).

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