How can a Fields Medallist be 'not very good at logic'?

Question

_{Source 2: Sir Michael Atiyah on math, physics and fun.}

I think the way a lot of people think about mathematics, since it's all based on logic, it can't be unpredictable.

Well, the idea that mathematics is synonymous with logic is a great ridiculous statement that some people make. Mathematics is very difficult to define, actually, what constitutes mathematics. Logical thinking is a key part of mathematics, but it's by no means the only part. You've got to have a lot of input and material from somewhere, you've got to have ideas coming from physics, concepts from geometry. You've got to have imagination, you're going to use intuition, guesswork, vision, like a creative artist has. In fact, proofs are usually only the last bit of the story, when you come to tie up the... dot the i's and cross the T's. Sometimes the proof is needed to hold the whole thing together like the steel structure of a building, but sometimes you've stopped putting it together, and the proof is just the last little bit of polish on the surface.
  So the most time mathematicians are working, they're concerned with much more than proofs, they're concerned with ideas, understanding why this is true, what leads where, possible links. You play around in your mind with a whole host of ill-defined things.
  And I think that's one thing the field can get wrong when they're being taught to students. They can see a very formal proof, and they can see, this is what mathematics is.** My story I can tell. When I was a student I went to some lectures on analysis where people gave some very formal proofs about this being less than epsilon and this is bigger than that. Then I had private supervision from a Russian mathematician called Bessikovich, a good analyst, and he'd draw a little picture and say, this -- this is small, this -- this is very small. Now that's the way an analyst thinks. None of this nonsense about precision. Small, very small. You get an idea what is going on. And then you can work it out afterwards. And people can be misled, if you read books, textbooks or go to lectures, and you see this very formal approach and you think, gosh that's the way I gotta think, and they can be turned off by that because that's not an interesting thing, mathematics, you see. You aren't thinking at that point imaginatively.
  But you mustn't get carried away by the other extreme. You mustn't go all the time with airy-faery ideas that you can't actually write and solve a problem. That's a danger. But you've got to have a balance between being able to be disciplined and solve problems and apply logical thinking when necessary. And at other times you've got to be able to freely float in the atmosphere like a poet and imagine the whole universe of possibilities, and hope that eventually you come down to Earth somewhere else. So it's very exciting to be a practicing mathematician or a physicist. Physicists in principle have to tie themselves down to Earth more than mathematicians, and one day look at experimental data. Well mathematicians have to tie themselves down in other ways too. A proof is one of the things that instantly ties them down. But it's a mistake to think that mathematics and logic are the same. They overlap in important ways, but it's a big mistake. $\color{red}{\text{I'm not very good at logic.}}$

1. Per the italicised sentences above, I understand that math needs some logic, more than logic; but not all mathematicians need specialise in Mathematical Logic.

2. Assumption: Sir Michael means the red (last sentence) literally, and not humbly or self-effacingly.

3. Then how is the red true? An excellent mathematician must have mastered the fundaments of logic, e.g. undergraduate Mathematical Logic. So does the red instead mean research-level Mathematical Logic?

Answer 1

I think the sentiment expressed by the author is pretty clear: being a good mathematician is being able to go back and forth between being able to be really precise (logic), and being able to explore (and maybe even 'intuit') interesting relationships and connections, play with new ideas, speculate, hypothesize, etc.

But, as such, I do believe Sir Michael did (probably intentionally) overstate his case a bit: as he says himself, at some point you need to take these new ideas and suspected theorems and make them hard, and that requires being good at logic. So I agree with you there that this seems like a strange statement of his. Indeed, Sir Michael is of course perfectly capable in logical thinking!

Then again, he does refer to some of the high technical formalizations of logic, and how that seems almost too 'extreme', too 'restricted' ... how indeed it is hard to see how, say, a deeply formal logical axiomatization of, say, arithmetic, gives us any novel ideas about arithmetic (if you work with the Peano Axioms, I think you'll get the idea). So that kind of logic is something he says that he resists doing.

But in the end you're right: to say that he is not good at logic at all, seems overstated, though most likely intentionally so and for dramatic effect: I think it's his way of trying to get those of us who do tend to stay in the purely 'precise' domain to 'venture out' a little more!

Answer 2

One can be an excellent mathematician and be helpless in face of even the 1st order logic (aka predicate theory). Just take a look at https://en.wikipedia.org/wiki/Principia_Mathematica (admittedly, this is a bit extreme). I remember taking logic classes and having to face this staff: it felt completely foreign to me, even though it describes (to some extent) how mathematical proofs are structured.

Answer 3

If he's a mathematician then he's very good at the kind of logic that's taught in secondary school, which is the only part of logic that's used in actual mathematical proofs. The purpose of that kind of logic is to assure correctness of proofs.

Now suppose someone asks whether the ratio of lengths of proofs to lengths of theorems is bounded. That is also a question of logic. There are many questions of that kind, and that's probably what he was talking about.

PS: It's not bounded.