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How do a mathematician "come up with" a good definition? How long does it take? Is there any strategy to shorten the time it take?

I'm not very good at math, but from my understanding the most important part in math is definition. It took me some time to appreciate its importance after trying to prove something really hard( or at least trying to understand others' proofs of proving it), for me.

And finally, would you consider it annoying, when you found resources, maybe about other subjects not just math, using vague wording after being a mathematician?

(Although I'm not very good at it, and the issue is quite annoying for me, mathematics is still my favorite subject.)


My initial idea is that I'm thinking about how those mathematicians who are at the frontier of mathematics create new idea. I cannot create one, but I guess that must be hard but exciting, and once the "define" process is done, and with some luck, it can attack those unsolved problems, which is the biggest challenge in mathematics...

To give an example why definition is important: I once read about the C++ draft and some definitions are very hard to understand, but it does "push" the frontier of it to include more powerful features. I think math works the same way if new idea are condensed into definition.

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Typically, mathematical definitions are developed through a process of trial and error. Mathematician A publishes a naïve definition (or even starts using a mathematical concept and assumes certain properties, without a formal definition); mathematician B points out some inconsistencies or paradoxes that arise from the naïve definition; mathematician C publishes a tighter and more formal definition etc.

This process can take a long time. The concept of a set was first formalized by Cantor in 1874, but it took nearly fifty years before an axiomatic definition of set theory was developed that avoided issues such as Russell's paradox. Newton and/or Leibniz invented calculus in the 17th century but rigorous foundations for calculus were not developed until the early 19th century.

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