Truth and axioms in mathematics

by Neil Rickert

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.

Mathematical logicians sometimes say that PA is true.  And that kind of talk puzzles me.

Of course, PA is true about everything within PA.  However, that seems tautological, and not worth saying.  If that is all that is meant by saying that PA is true, then why say it?

If saying PA is true means more than that trivial tautology, then I wonder about modular arithmetic.  For if I do arithmetic mod 5, then 3+2 = 0.  But that’s false in PA.  So if the axioms of PA are true, does that mean that modular arithmetic is false?  If the axioms of Euclidean geometry are true, does that mean that non-Euclidean geometry is false?

This is why I take axioms to be neither true nor false.  I see them as defining principles for a particular area of mathematics.  But it should mean more than that to say that they are true.  Again, to put it in terms of PA, nothing seems to depend on whether the Peano axioms are true.  Arithmetic depends on our taking PA as providing basic definitions and principles.  And, of course, that makes the Peano axioms tautological truths within PA.  However as best I can tell, it does not make the Peano axioms true in anything other than that tautological sense.

Perhaps this is why I am not a mathematical platonist.

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2 Comments to “Truth and axioms in mathematics”

  1. Your ideas on the nature of mathematics and mathematical principles seem to be far off base. You claim that Peano Axioms are ‘definitional’ and thus not true, what exactly is that supposed to mean? “Bachelor = unmarried male” certainly seems like a true statement to me.

    In addition, your idea of the independence of axiomatic systems fails as a challenge against Platonism for two reasons. 1) Many mathematicians still believe that there could be a single mathematical universe that would solve the ‘independence’ problem associated with several mathematical problems, including the continuum hypothesis (Hugh Woodin has done interesting work on this with Ultimate L). 2) In addition, one could think that there are many different systems of mathematics and still be a Platonist with regard to ontology (Joel David Hamkins is one such person). These individuals might tend to think that different axiomatic systems are as real as the different non-euclidean geometries that currently exist.

    Another thing I would like to point out is the inability for fictionalists (which I think is a view that borders on the absurd) and nominalists to ascertain why mathematics is so useful to science without simply stating it as miraculous. Yea, I often hear the retort of “Math is simply representation and we create it.” That type of argument fails for two reasons. 1) If we think of math as a representation, then the thing it is describing should have an existence that is equivalent to some sort of mathematical structure (the one represented). Did we invent the patterns in nature? Do the predictions that pure math has made in the sciences (especially theoretical/particle physics) that were later empirically verified really make one feel comfortable with the assertion “Don’t worry, its still all a constructed narrative.” Why doesn’t anything else labelled as fictitious (novels, movies, etc.) describe the world so accurately, what gives math a special distinction?

    You even seem to point to some form of structural realism (easily compatible with mathematical realism) when you make this statement: “And if they can find a sufficiently structure mathematical representation space, then they can use that mathematical structure to analyze representations, and to make plausible inferences about the world that is represented.” Mathematical Platonism doesn’t really refer to a heaven with perfect, timeless, abstract entities. There are several different branches of mathematical realism that are determined why the ontological nature of mathematical entities themselves. Many structural realists actually forgo mentioning particular objects and instead believe that the mathematical structures themselves are all that exist.

    Regardless, your arguments against Platonism fail to deliver in a pretty severe way. In fact, I think fictionalism is extremely hard to justify/take seriously relative to the history, practice, and results that mathematics has garnered humanity over the centuries.

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    • You claim that Peano Axioms are ‘definitional’ and thus not true

      That seems like a rather exotic reading of what I wrote.

      In addition, your idea of the independence of axiomatic systems fails as a challenge against Platonism for two reasons.

      That’s an odd comment. I have not attempted to challenge platonism. I have merely explained why I am not a platonist. I do not criticize others for being platonists.

      Another thing I would like to point out is the inability for fictionalists (which I think is a view that borders on the absurd) and nominalists to ascertain why mathematics is so useful to science without simply stating it as miraculous.

      May I suggest that you tone down the rhetoric.

      I have explained, in other posts, why mathematics is useful in science. Try “Why mathematics is useful to science” as one such post. You might not agree, but I have at least provided an explanation which did not suggest anything miraculous.

      Regardless, your arguments against Platonism fail to deliver in a pretty severe way.

      I’m sure that is true. As we mathematicians might say, it is vacuously true. That is to say, I have not made any arguments against platonism.

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