# How do mathematicians come up with new proofs?

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• docnet
In summary, Yitang Zhang worked alone and did not collaborate with others during his research time. After looking up the proof, it is surprising that it was done by one person. However, mathematicians can understand the proof based on a one-page summary. The key ideas of a proof can be summarized much shorter than the fully worked-out proof. Zhang had already spent time working on the problem and had an idea of how to combine his results with the help of other colleagues. The mind can work unconsciously and ideas can come seemingly out of nowhere after thinking about a problem for a long time. In the case of Zhang, the solution came to him during a dream after spending hundreds of hours on the problem.
docnet
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TL;DR Summary
How are research-level proofs written?
I watched an interview of Yitang Zhang and he said "the way to prove a finite limit of bounded gaps between primes came to him during ##30## minutes in an afternoon", and he worked alone and did not collaborate with others during his research time.

After looking up the proof, I am in disbelief he worked alone. What baffles is how one person could write ##50## pages of what feels like an enormously complicated and difficult mathematical maze to end up with the final result "so and so is the lower limit of so and so". I can't believe that so much work is done just to prove the final result because so many independent steps are taken, that don't seem to be obviously connected to the final result at all. But every step is nit-picky, deliberate, and brings the logic one step closer to the desired result. Are there mathematicians who could even read the entire proof and understand everything in it?

My main question is, do research-level proofs in mathematics such as "bounded gaps between primes", or "Harnack's Inequality for the Ricci Flow" or "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere." really come as a result of a one-person's genius, like everyone makes them out to be?

The key ideas of a proof can be summarized much shorter than the fully worked-out proof. You can give a mathematician working in that field a one-page summary (or maybe even shorter) and they'll be able to reproduce the proof based on that summary.

At the time of the "30 minutes in an afternoon", Zhang had already spent quite some time working on that problem and related problems. He had some preliminary results, he was obviously aware of the older results by his colleagues, had discussed the problem with them many times and so on. At the end of these 30 minutes he didn't have the 50 pages of proof written down, but he had an idea how to combine all these things, plus a few things he expected to be able to proof later, in a way that would lead to a proof.

If one thinks hard and continuously about something for a long time, it seems that the mind works on its own unconsciously. Ideas just pop up seemingly out of nowhere. One "sees" the relationship. However working out the details may take a huge amount of work.

DaTario, weirdoguy, docnet and 3 others
lavinia said:
Ideas just pop up seemingly out of nowhere.

I'm not a "working scientist" example, but I had numerous situations, where the solution to a problem simply came to me during a... dream.

weirdoguy said:
during a... dream.
In a jacuzzi for me...

(but as pointed out before, I'd put in hundreds of hours on that problem before the final aha moment)

## 1. How do mathematicians know where to start when coming up with a new proof?

Mathematicians often start with a problem or question that they are interested in solving. From there, they use their knowledge and understanding of mathematical concepts and techniques to develop a strategy for approaching the problem.

## 2. Are there specific techniques that mathematicians use to come up with new proofs?

Yes, there are various techniques that mathematicians use to come up with new proofs. These may include using logical reasoning, making connections between different mathematical concepts, and utilizing creative problem-solving strategies.

## 3. Is there a standard process that mathematicians follow when developing a new proof?

While there is no set formula for developing a new proof, mathematicians generally follow a systematic approach that involves carefully analyzing the problem, identifying key concepts and techniques, and constructing a logical and well-supported argument.

## 4. How do mathematicians ensure that their proofs are valid and reliable?

Mathematicians use rigorous methods to ensure the validity and reliability of their proofs. This may involve double-checking calculations, seeking feedback from colleagues, and subjecting their work to peer review.

## 5. Can anyone come up with a new proof, or is it a skill that only mathematicians possess?

While having a deep understanding of mathematics certainly helps, anyone can potentially come up with a new proof. It requires critical thinking, creativity, and perseverance, which are skills that can be developed and honed by anyone with an interest in mathematics.

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