Question

Is it possible to suck at rigor of math but great at intuition?

Answer 1

Maybe someone can throw you an example (I'm sure Ramanujan will show up in the comments), but if the real question is "can I successfully do math by bypassing having a rigorous understanding?" the answer is almost certainly no.

It's not entirely correct that one completely eschews rigor when they get older, either. When you write papers suddenly you're faced with the realization that some of your intuitive pictures were dishonest, and you need to make them more clear. In doing so, you are forced to have a better understanding of the problem (and perhaps the subject) than you did before you wrote down the argument. Sometimes I spend a week trying to prove some irritating obviously true technical lemma before it becomes clear that it's false, and false for important structural reasons I need to account for. Suddenly I have learned more about my problem because of the need to prove this stupid technical lemma.

The need for rigor obligates us to make our ideas more clear, and I don't believe one can operate purely at a "post-rigorous" level and not eventually develop faulty intuition. I don't think Terry believes this either.

Answer 2

I think if you do stage 3 without fully passing stage 2, then you *might* actually just be doing stage 1 real fancy-like.

Answer 3

Google Italian school of algebraic geometry for a good time.

Answer 4

If you think you are doing 3 when you haven't done 2, you are doing 1.

Answer 5

You can win a Fields Medal without doing rigorous mathematics - ask Ed Witten (though I would never say he "sucks" at rigor). In the words of the man himself:

"What’s a little funny about my relation to the math world is that although some of my papers are of mathematical interest, they rarely have the detail of math papers. And I can’t provide that detail. I simply don’t have the right background. What I bring to the subject is an ability to understand what quantum field theory or string theory have to say about a math question. But quantum field theory and string theory are not in the precise mathematical form where such statements can usually be rigorous."

Answer 6

In my experience, over 90% of students who think they are great at understanding math but bad at doing exercises don't really understand the math, they have severe conceptual errors as well as bad approaches to solving problems. Similarly, the overwhelming majority of mathematics students who say they have great intuition but are bad at proving things don't actually have great intuition, they just find talking about connections and speculating about analogies fun, but don't like the part where people read what they write and say that their argument has gigantic gaps or is complete nonsense or is an attempt to prove something false.

Of course, there are some mathematicians who are better or worse at recalling or finding and proving key lemmas. Some mathematicians are better or worse at writing things up in a clear and interesting fashion. There are mathematicians who have done great research after mediocre results in coursework, and vice versa.

Answer 7

A possible candidate for this could be Taniyama Yutaka, known for the Taniyama-Shimura-Weil conjecture which was proved by Andrew Wiles. In the BBC Horizon documentary on the proof of Fermat’s last theorem from 1996, the filmmakers interview Shimura Goro talked about hus friend, and explained that he often suffered from bouts of depression with self-doubt. I seem to recall that Shimura mentions that Taniyama often made careless mistakes, but that these mistakes were «good» mistakes which led him towards promising results. Shimura speaks of trying to emulate Taniyama’s style, but found that it was quite difficult to make these «good mistakes». Taniyama sadly took his own life in 1958.

Answer 8

A lot of people are mentioning Ramanujan, which I think is incorrect.

Ramanujan used rigor, he just wasn't classically taught.

Answer 9

That's exactly how I feel in analysis.

The Intuition makes perfect sense. But actually proving anything is a bitch.

Answer 10

I would argue that to be impossible, excepting the most exceptional of cases like Ramanujan.

The concept of three, where intuition and the big picture and connections come through happens most often through the arduos work of learning and understanding rigour. It is moreso a case of having worked for a long enough time with these topics that they become ingrained, so much so that to be able to skip over the details and see connections that are more likely true than not becomes easier since the intermediate logic is something you can (sometimes) work out earlier.

Building that intuition takes a large amount of work, and for most people only comes from repeated practice, reading, and research. Hence the purpose of the PhD and postdoctoral studies. Moreover, having a good understanding of rigour and structure in your area of expertise makes it less likely to draw incorrect relationships between objects.