How do you evaluate mathematical discoveries?

[bostonnew]

Hi all,

Can someone help me understand how mathematical discoveries are evaluated? If a new finding is made, how do you know if it's significant or not? What differentiates the content of the prestigious journals from others?

I can understand that solving problems that have been unsolved for longer periods of time perhaps merit more respect. But what other criteria exist? Also, do mathematicians generally agree on what is impressive and what isn't?

Thanks!

 

[Jamma]

A lot of the time, mathematicians don't really care about what is important and what isn't- if it is important it will be known to be so from how many citations it gets.

But I suppose in practicality the following can contribute:

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originality/bringing into use or suggesting new and potentially useful techniques or ideas

broadness of application (perhaps comes into the above if new techniques are born)

difficulty (? perhaps not actually relevant, although may often be a sign that something has been achieved)

like you say, setting to rest an old postulate (e.g. if someone came up with the question "does x^n+y^n=z^n have integer solutions for n>2" only a few years before its solution, I doubt it'd have got so much attention)

perhaps (actually, definitely) aesthetic beauty comes into it. A well written paper that uses some ingenious and beautiful techniques will definitely be bonus points

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I'm sure there are other things, but mathematicians don't really get too hung up on the question: "is this a good paper or not?" they tend to just try and release as many as they can and hope that some turn out to be useful for people.

 

[Anonymous217]

Usually something genuinely interesting, an intuitive property that is just very hard to prove, or something that can lead to significant contributions to mathematics (i.e. the method of proof or the tools used in the proof lead to a new development within the field itself). It's hard to describe, but that is what first comes to mind.

 

[bostonnew]

These answers are quite helpful. I'm very curious as I don't know much about this field.

Do mathematicians think differently than scientists about what constitutes a good discovery?

 

[Jamma]

In that their discoveries don't need to be instantly applicable to the real world, yes!