the role of logic in math and education

Question

My question is somewhat related to this discussion:

Is mathematics one big tautology?

I have a computer science background and I have always approached math from the logic point of view (formalism?). In the past, whenever I'd try to tackle a proof, I would use the principles I had learned from my discrete math logic course. When I later took more serious math courses, this approach was reinforced because rigor was expected. I always thought of rigor as the ability to justify each one of my steps logically. I really felt that learning basic logic helped me tremendously.

When I read the first answer to the question I linked to above, I was almost astonished. Logic is only a tool? It is the least relevant skill for doing mathematics? Have I had a wrong perspective all this time? It seems math is treated as this intuitive matter, whereas I have always approached it with logical rules (and in the process have not developed much of an intuition for the objects I'm studying). The beginning rigorous math classes are treated axiomatically, so I've never really had to rely on intuition.

I have several questions:

1) How much logic do most (pure) math students actually learn? Do they take these discrete math courses, which introduce logic, or do they just develop the intuition as they're going along?

2) Is my perspective a product of a bad math education? I almost feel like discrete math is an extension of the computation level calculus courses.

Answer 1

I shared the same point of view. It wasn't until I took a class called complex analysis did I open up my view. Logic helps to show an argument is true, but not necessarily $why$ a proof is true. You might have come across a couple proofs that you've done via induction. If so, it might not have made sense all the time why the theorem you proved is true. For example in combinatorics, there are many mathematical identities used to count things. Is it intuitive why such an identity holds? Logic allows us to bypass intuition and get and check results, sometimes new discoveries follow logically, but in my opinion mathematics is also an art built out of creative and original ideas.

I should also add that algorithmic constructions and the probabilistic method are being used to prove things in non-traditional ways and with surprising results.

Answer 2

Logic and math are tools, yes.

Think about it by analogy with natural language. If a friend asks you to argue that, say, one political system is correct from a given group of premises (read axioms) such as "each person should have freedom x" etc, you will not simply see how you can manipulate those premises based on valid rules of language manipulation to find a conclusion. You will build some sort of picture in your head and try to put it into words post hoc.

Logic is an attempt to extract the rules of language manipulation which have been manifest in our natural language since day 1. For example, Socrates would always argue that if every element of $B$ has some property, and both $x \in A$ and $A \subseteq B$ are known, then $x$ has the property too. Formal logic had not yet been conceived, but the application of it was already going on.

Answer 3

Think of the following scenario - you have never programmed in your life, and you are taught the rules ("axioms") of a programming language, and get to decide which commands are "allowed" without ever writing a line of code. Then sure - you would end up thinking programming is nothing more than a bunch of axioms.

But the reality is, that programming languages were written with a purpose of solving real life problems, and when a programming language no longer helps you get to the desired results or that a given task becomes too complex in that language, you invent a new language in which that task is simpler and more intuitive.

It's the same in math - the axioms were built for a purpose, and when they no longer serve to describe the things you want to create, new axioms and new notations are invented to facilitate this.

Answer 4

From a logician's point of view, logic is far more than just a tool!! But that's probably a topic for another time...

Certainly, I think people would (or should) agree that logic is at least at very useful tool. The first answer to the question you linked to has a fair point: for most mathematicians, the logical abilities are not as important as creativity. Often times, in constructing proofs on your own, it's not sufficient to just try every possible combination of inference rules that could be used on those axioms. A lot of proofs require you to construct certain mathematical objects with special features, or you need to set up definitions which will be useful later on, and this process is (arguably) a creative process. Once you have the idea behind a proof, constructing the logical argument to follow is more like filing paperwork: it needs to be done, and students need to learn how to do it, but once you know it, there's nothing technically difficult about its implementation.

Furthermore, logic can only take you so far in terms of belief revision (as is well known in theories of belief revision). If you start with axioms which intuitively seem to capture the concepts you're after, and it turns out someone churns out a proof of some highly counter-intuitive result about that concept, logic won't tell you whether you should do a modus ponens or a modus tollens; that is, it won't tell you whether you should trust your axioms and keep the counter-intuitive result, or whether you should reject the counter-intuitive result and revise your axioms (or even your logic! but let's not go there). So when you're doing serious mathematics, you need a kind of intuition regarding whether the results you obtain seem to make sense, given what you're trying to describe.

With that said, there are probably plenty of cases where by now we've dropped the modus tollens option entirely, and just work axiomatically. There might be cases, e.g. set theory or computability theory, where we've changed our intuitions regarding the original concepts to a point where we've simply come to accept or even define our intuitions to be just what the axioms say. Even in these cases, of course, one still needs some kind of intuition about how the axioms work, but the above dilemma occurs less frequently.

To answer 1): in my experience, most math undergrads are required to take a discrete math course where they cover certain basics about logic, number theory, etc. So they learn some, but often times they're only expected to learn just enough.

To answer 2): it's true, I think, that a lot of people have the wrong impression of what mathematics is. Most people I've talked to who haven't studied serious mathematics think we just crunch numbers and do complicated integrals all day (which may be true of some, but certainly not all). Of the people who have taken serious mathematics courses, or who have taken courses which talk about serious mathematics (e.g. philosophy of mathematics courses) tend to get the impression you seem to get. Most mathematicians, I would think, get the intuitive content of the axioms/concepts, and work with those first and foremost; then they translate this intuitive content back down into the level of rigor. But that's not to say that all there is to logic is filing paperwork!