Re:Number theorist fears many proofs widely considered to be...
by Silhouette on Sunday September 29, 2019 @04:59PM
(#21109606)
Most advanced math concepts are complex and can take months to understand for someone with an unrelated math background (e.g., say, Lebesgue measure theory or Galois theory).
Certainly some advanced math concepts require a lot of foundations and therefore take time to build up to them, but usually the individual steps along the way should be quite accessible if well presented.
To follow up on one of your own examples, suppose we have someone who understands the basic ideas of abstract algebra, say someone who had done a first course in topics like groups and rings as part of their undergraduate studies. (I note in passing that even a keen school student considering studying maths at university might be able pick up these concepts with good presentation in a few days.) Then suppose that person are curious about why there is a quadratic formula but people claim that not only do we not know of any corresponding quintic formula, in fact we can prove that no such formula is possible.
Explaining the concepts and key results from Galois theory needed to get to that proof shouldn't take months. There are lots of technical terms like solvability by radicals and field extensions and quotient rings and so on, but if these concepts are properly explained, none of them would in itself be more than a typical undergrad student could understand after a few minutes. Building up the argument would of course take longer than understanding any given step, but it's not prohibitively difficult or complicated.
Unfortunately, in many university lectures and textbooks, those concepts are introduced using only a mixing pot of other technical terms and notation, without any sort of motivation or examples to put them in context. There's no intuition behind them. There's no frame of reference. So then of course the final proof of the unsolvability by radicals of a general quintic just feels like word and symbol soup that you have to learn by rote, because how could anyone possibly have derived it in any sort of deliberate or systematic way? And worse, not only does this result become "magic", the student lacks the ability to use the mathematical tools that support the result for any other purpose either.
Certainly some advanced math concepts require a lot of foundations and therefore take time to build up to them, but usually the individual steps along the way should be quite accessible if well presented.
To follow up on one of your own examples, suppose we have someone who understands the basic ideas of abstract algebra, say someone who had done a first course in topics like groups and rings as part of their undergraduate studies. (I note in passing that even a keen school student considering studying maths at university might be able pick up these concepts with good presentation in a few days.) Then suppose that person are curious about why there is a quadratic formula but people claim that not only do we not know of any corresponding quintic formula, in fact we can prove that no such formula is possible.
Explaining the concepts and key results from Galois theory needed to get to that proof shouldn't take months. There are lots of technical terms like solvability by radicals and field extensions and quotient rings and so on, but if these concepts are properly explained, none of them would in itself be more than a typical undergrad student could understand after a few minutes. Building up the argument would of course take longer than understanding any given step, but it's not prohibitively difficult or complicated.
Unfortunately, in many university lectures and textbooks, those concepts are introduced using only a mixing pot of other technical terms and notation, without any sort of motivation or examples to put them in context. There's no intuition behind them. There's no frame of reference. So then of course the final proof of the unsolvability by radicals of a general quintic just feels like word and symbol soup that you have to learn by rote, because how could anyone possibly have derived it in any sort of deliberate or systematic way? And worse, not only does this result become "magic", the student lacks the ability to use the mathematical tools that support the result for any other purpose either.