If you do not 'read A to Z', then how can you discover the idea? [closed]

Question

The following is from an article in The New Yorker on Y. Zhang and his proof on gaps between primes:

Rutgers University Professor [Henryk] Iwaniec and his friend, John Friedlander, a professor at the University of Toronto, read with increasing attention. “In these cases, you don’t read A to Z,” Iwaniec said. $\color{green} { \text { “You look first at where is the idea." } }$ There had been nothing written on the subject since 2005. The problem was too difficult to solve. As we read more and more, the chance that the work was correct was becoming really great. Maybe two days later, we started looking for completeness, for connections. A few days passed, we’re checking line by line. The job is no longer to say the work is fine. We are looking to see if the paper is truly correct.”

I am confused: If you don’t read A to Z, then how is it possible to $\color{green} { \text { “look first at where is the idea" } }$?

Answer 1

I think this question, with editing, has a lot to do with mathematics education, in that it points to the problem of learning to read and interpret mathematical writing. I would instead ask something like: "How does one glean the ideas in a mathematical paper without reading it line by line?"

Now for a somewhat rambling answer to this question:

Note that mathematical work is steeped in context, and that without familiarity with the standard ideas and tricks surrounding the paper in question, it is nearly impossible to find the idea in the paper.

Reading line by line can allow the reader to perhaps, with a LOT of work, reconstruct a consistent mental model that reproduces the results of the paper (perhaps this is what mathematics is, making such models). What is meant by "first finding the idea" is that much of what is written in a paper is standard in the community of persons working on the problems in question. The new idea in the paper stands out against the standard "background noise". In order to find it, you have to first know what is standard.

Stated another way: If you are using only standard techniques tried by many experts to solve a reticent problem, then you are very likely to be wrong. So your proposed solution is first evaluated by whether it is novel against the "background noise" and whether it hangs together logically as a strategy. A mathematical proof has a basic logical outline that contains "the idea", and much of the detail needed to complete the idea is necessary tinkering with standard techniques to make it all work out. I hope this helps.

As for mathematics education, this is very important insofar as it encodes "big picture thinking", as opposed to "algorithmic nose-following" that many think is mathematics. See Paul Zeitz's book "The Art and Craft of Problem Solving" for a discussion of strategy, tactic and tool.

Without a sense of the living ideas encrypted in the writing of a mathematical paper, which sadly are not found in traditional mathematical writing, one is flying blind. The belief that reading papers line by line will yield an understanding of a subject is pernicious, and this question (if properly asked) has the educational benefit of dealing with that pernicious belief.

(Sorry for the behemoth comments…which now appear as the present answer.)

Answer 2

As Jon Bannon says in his answer, to the expert there should be a lot of "standard" techniques that are very familiar and can be at least recognized at a glance. Then there should be new techniques that are unfamiliar and give new leverage. You can sort of imagine this if you were reading a text in English, and then suddenly there's a part in italicized Latin -- it would stand out to you as important.

The best-written papers should make the "new ideas" plain up front, possibly in the abstract, introduction, or header statement(s). Richard Lipton writes about this frequently and has called this the "Elevator Pitch". Some articles on the subject that I found interesting to read:

• "Proofs and Elevator Rides" (Richard Lipton)
• "Proving a Proof is a Proof" (Richard Lipton)
• "Eight Signs a Claimed P≠NP Proof Is Wrong" (Scott Aaronson)