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Originally Posted by Robh
I can see where you're coming from. Pure mathematics can appear to be so removed from physical reality that one can only wonder at the amount of time being wasted on some of these studies.
All the sciences and indeed a great majority of careers rely on varying levels of mathematical understanding and usage, whether it be all-encompassing as in physics and engineering or as an occasional acquaintance in social work in the form of statistics.
However, there is always a fuzzy area between pure and applied mathematics. Once considered pure and "useless" mathematics can all of a sudden find a revolutionary application in the real world e.g. complex analysis and electronics. It begs the question whether things are only to be only studied for their practical application or is there reason to research it for its esoteric value. For example, Goldbach's conjecture (unproven) states that every even number greater than two can be expressed as the sum of two prime numbers (4=2+2, 6=3+3, 8=3+5, 10=3+7 or 5+5 etc). Pretty much useless but a lovely problem nonetheless. There are many other things which have no real practical use but are valued and appreciated by the human mind e.g. art, music, a good novel.
I can't comment on the monstrous moonshine conjectures other than to wonder how many mathematicians would be familiar with them. Perhaps, there is a present or future application in the real world, perhaps not. Perhaps, there are some revolutionary lines of thinking that might inspire other areas of mathematics.
So, I guess in the final analysis it comes down to how many areas of pure mathematics have diverged and progressed along specialised lines of study that are pretty much isolated from mainstream mathematicians.
It is to be hoped that mathematics does not evolve and separate into a myriad of isolated studies that to all intents and purposes form a random pile of unintelligible and useless conjectures. After all, one can invent and formulate theorems around any structured set of axioms.
Interesting topic for debate.
Regards, Rob.
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I did 3 years of Pure Maths as an undergraduate before moving over to Applied Maths. My senior lecturers who also researched Pure Maths considered themselves artists rather than mathematicians. Their "art" was based on creativity, simplicity and logic.
One can argue that theoretical physics is based on the same principles.
Physics however is first and foremost an experimental science hence apart from creativity, simplicity and logic, there is also reality to consider. The reality component is expressed through experiment and observation.
When pure mathematicians start taking on the role of theoretical physicists the reality component can go missing.
This is highlighted by String Theory. String Theory is not verifiable through experiment or observation.
Steven