Mathematics and logic

by Neil Rickert

Yesterday, Massimo Pigliucci posted on the relation between mathematics and logic:

so I though I would offer my opinion on that topic.  I see things differently from Massimo, but that’s probably just the different perspective as see by a mathematician (me) and a philosopher.

Massimo cites Peter Cameron (a mathematician) and Sharon Berry (a philosopher – actually a student of philosophy of mathematics).  Check Massimo’s post for the links.

What is logic?

If we ask the questions “Is mathematics a branch of logic?” or “Is logic a branch of mathematics?” then we will need to start with an agreement on what we mean by “logic.”  And I don’t think there is a clear answer to that.  Sharon Berry gives 5 possibilities, which are discussed by Massimo.  Here is my own list of possibilities:

  1. Ordinary clear thinking (such as used by Sherlock Holmes)
  2. Structured reasoning about ordinary things, typically based on formulating premises and arguing to conclusions from these premises.  I’ll call that “philosophic logic”, because it seems to be used a lot within philosophy.
  3. Formal logic, which is similar to philosophic logic, except that the rule following (of rules of inference) is done more strictly, and it is often used more abstractly.  This is sometimes called “propositional logic”.
  4. Mathematical logic, which goes beyond propositional logic by potentially allowing reference to infinities with the universal and existential quantifiers.

In ordinary speech, people often use the term logic for (1) in that list.  I don’t really consider that to be logic at all, for it appeals to ordinary intuition about the world.  It seems to me that there should be some sort of structuring before we can talk of logic.

Is mathematics a branch of logic?

To me, the answer is a clear NO.

Mathematics makes a lot of use of ordinary clear thinking.  But “ordinary clear thinking” is not the name of any field of study, so it can’t have branches.  So it does not even make sense to talk about mathematics as branch of logic in sense (1) of that term.  The other three listed senses of logic are sufficiently well defined that one could talk of them having branches.  But I don’t see that mathematics is a branch of any of them.  Mathematical reasoning goes beyond what is permitted by those understandings of logic.

Is logic a branch of mathematics?

Again, my clear answer is NO.  I do consider sense (4) of logic, namely mathematical logic, to be a branch of mathematics.  And sense (3) is amenable to mathematical study, so could be considered a branch of logic, though some logicians might prefer not to be included as a part of mathematics.  But senses (1) and (2) are clearly outside of mathematics.

Some proponents of AI (artificial intelligence) believe that all is computation.  But most will admit that this is as yet unproven.  I tend to be a skeptic of some of the stronger claims of AI proponents, and it is my sense that many mathematicians are similarly skeptical.

Mathematicians and logic

Having disposed of the question of how mathematics relates to logic, we can look at how mathematicians relate to logic.

There’s no doubt that mathematics requires clear thinking, which is sense (1) of logic, as described above.  Most mathematicians also find the quantifiers from logic in sense (4) to be notationally useful.  But that’s about it.

As best I can tell, many mathematicians have never actually studied logic.  My own best recollection is one brief class lesson at 4th grade of elementary school, where I found that I could solve all of the logic questions by just applying clear thinking and without learning any of the inference rules of logic.  That’s it for my classroom study of logic.  Peter Cameron, in the post cited by Massimo, writes “I didn’t learn logic as a student.”  I have not tried to survey other mathematicians, but this seems quite typical.

Next semester, I will be teaching a class on computing theory.  One of the topics to be covered will be finite automata.  This is usually considered part of mathematical logic.  As I discuss automata in class, I will go through the pumping lemma.  The usual proof of the pumping lemma depends on the pigeonhole principle.

To a first approximation, the pigeonhole principle says that if you have a bunch of pigeons in pigeonholes, and if there are more pigeons than pigeonholes, then at least one of the pigeonholes has to contain more than one pigeon.  This seems obvious on the basis of ordinary clear thinking.  And it seems obvious that this is a very general true statement about finite sets, and not limited to sets of pigeons.  Yet I have never attempted to give an entirely formal statement of the principle, nor to give an entirely formal logical proof.  The idea of doing so seems too tedious, and not worth the effort for something that is so obvious on the basis of ordinary clear thinking.

I mention the pigeonhole principle to illustrate that mathematics is mostly dependent on clear thinking, or sense (1) of logic, rather than on any formal meaning of logic.

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