Is there a degree of abstraction beyond which mathematicians can not cross? Some point where we're limited to working within some set of rules and can't go any further for any branch of mathematics?
Methinks you are making a hidden assumption here: the assumption that something "more abstract" is something more powerful, and more useful. Even admitting that abstraction can sometimes be extremely powerful and useful, it is, as I understand the concept, not at all the case that the "highest level of abstraction" is particularly powerful or useful. In fact, it leads to a perfectly uninteresting theory: a theory about objects that are all the same (i.e. undistinguishable), because you have thrown out the making of any distinctions between them.Is there a degree of abstraction beyond which mathematicians can not cross? Some point where we're limited to working within some set of rules and can't go any further for any branch of mathematics?
It sounds like extraneous roots from quadratic equations. Thanks for your replies.The degree of abstraction that is useful is only known by experience, unfortunately. I agree with Failure: often math gets so abstract it's not useful.
Maybe it would be a useful exercise for you to inquire a little more deeply into how some of the currently more abstract theories, like the theory of groups, rings, and fields, vector spaces, set theoretical topology, and category theory and the like have come about. This would give you an idea of how the process of finding higher, but still useful (not sterile) abstractions over time in practice works.I'm beginning to think that this thread should have been posted in the philosophy section. Anyways, Failure and others, are there any guiding rules that help out as to what degree of abstraction we should go to?
In connection with this, Simon Singh's Fermat's Enigma goes into how it was proven that modular forms are equivalent to elliptical equations which was an astonishing result (proving Fermat's Last Theorem incidentally) and appears to be the closest part of Langland's program to unify mathematics to date.Maybe it would be a useful exercise for you to inquire a little more deeply into how some of the currently more abstract theories, like the theory of groups, rings, and fields, vector spaces, set theoretical topology, and category theory and the like have come about. This would give you an idea of how the process of finding higher, but still useful (not sterile) abstractions over time in practice works.
For example, over time, people notice that a great many theorems of several existing theories could be proven for all those theories at once, if one happened to introduce a suitably chosen common abstraction of all of them. So that new abstraction would have to catch enough details to still allow substantial proofs to be formulated.
There are books about these developments, I just haven't got any references handy. So this would be under the rubric: history of mathematics (but not beginning with Babylonian mathematics, perhaps).
I am in no way able to judge "Langland's program to unify mathematics". Maybe you could explain to ignoramuses like myself what "unifying mathematics" would amount to in this case?In connection with this, Simon Singh's Fermat's Enigma goes into how it was proven that modular forms are equivalent to elliptical equations which was an astonishing result (proving Fermat's Last Theorem incidentally) and appears to be the closest part of Langland's program to unify mathematics to date.
The wording I like is when you walk with your head in the clouds, make sure you keep your feet on the ground.I am in no way able to judge "Langland's program to unify mathematics". Maybe you could explain to ignoramuses like myself what "unifying mathematics" would amount to in this case?
The reason I ask is this: the introduction of a higher level abstraction (like for example topological spaces) does not at all necessarily make the lower level abstractions it subsumes (like metric spaces) completely superfluous. Also, we have groups, rings, and fields, but we still want to be able to talk about natural or real numbers, and their particular properties, do we not?
Nor did I till I googled it just now.I am in no way able to judge "Langland's program to unify mathematics". Maybe you could explain to ignoramuses like myself what "unifying mathematics" would amount to in this case?
The reason I ask is this: the introduction of a higher level abstraction (like for example topological spaces) does not at all necessarily make the lower level abstractions it subsumes (like metric spaces) completely superfluous. Also, we have groups, rings, and fields, but we still want to be able to talk about natural or real numbers, and their particular properties, do we not?