Geometry, Mathematics and the Limits of Knowledge (very dramatic)

[ForMyThunder]

I've been thinking...

I don't know that much mathematics. I'm just saying. It seems to me that nowadays, every branch of mathematics has been accounted for. Forming a basis are things like geometry, topology, algebra, analysis, etc. Then you mix all these together and you come up with algebraic topology, algebraic geometry, analytical geometry, etc. But I guess you can dump these into a pot and just swirl 'em around and you get things like differential algebraic geometry, or differential geometric topology, or differential algebraic geometric topological number theory... which probably mean nothing at all. These subjects all seem to be building on top of one another.

Will there ever be another point in the history of mathematics where another fundamental branch is created? Sort of like when Newton invented Calculus or Galois did his thing.

Do you think we will ever reach a point in human history where we have "figured mathematics out"? I don't mean to ask if there is a TOE of mathematics but just a point in mathematics where any further speculation on a topic gives nothing new or insightful?

 

[JaredJames]

Quite simply, we don't know if another branch will ever be created - because we haven't created it yet.

So far as "a point when we have it all figured out" goes, again, we don't know. Overall I'd say it's doubtful, but in so far as specific things go then yes - right angle triangles for example, we know all there is to know with them, the theory is 'perfect'. However, mathematics is only a way of describing what we observe. We can only refine this as much as possible. For some things, there's only so far we can observe and as such only so far we can refine - black holes for example, we'll never know what happens inside the event horizon for sure so a mathematical, 100% guaranteed description will never exist.

 

[llynne]

The amount we know about math is less than the tip of an iceburg when compared with mathematical relationships in nature. Here we are living in a base 10 world with four dimensions... reality may be better understood in base 144 with 12 dimensions. Or base 2 with only 2 dimensions (with a twist)

 

[JaredJames]

The amount we know about math is less than the tip of an iceburg when compared with mathematical relationships in nature. Here we are living in a base 10 world with four dimensions... reality may be better understood in base 144 with 12 dimensions. Or base 2 with only 2 dimensions (with a twist)

We may "live in a base 10 world", but not everything in science and maths uses those base units. There are many, many different systems used depending on the circumstances.

I'd also note that you can't reduce the number of dimensions lower than our current four as you then lose the ability to describe, well anything, in adequate detail.

 

[Gokul43201]

However, mathematics is only a way of describing what we observe.

I don't mean to nitpick, but I think it's important to point out that Mathematics deals with abstract objects, not observables. Mathematics does not set out to describe things that we observe, it only describes things that it defines. Some of these things may just so happen to bear a resemblance to aspects of things that we observe. When such a resemblance is found, it is the job of the physical sciences to use the appropriate mathematics as a tool to investigate these observables.

 

[JaredJames]

I don't mean to nitpick, but I think it's important to point out that Mathematics deals with abstract objects, not observables. Mathematics does not set out to describe things that we observe, it only describes things that it defines. Some of these things may just so happen to bear a resemblance to aspects of things that we observe - when such a resemblance exists, it is the job of the physical sciences to use the appropriate mathematics as a tool to investigate these observables.

I suppose I was taking it from more of a physics angle. But yeah, you're right.

It was more to demonstrate my point regarding "figuring it all out".

 

[humanino]

Given a formal system which contains at least the integers, there will be true statements which one cannot prove from within the system (yes, Godel). Those can be considered as new axioms for a larger formal system, or at least hint towards the existence of larger systems. In fact, Godel has precisely the question above when he comes up with his undecidable propositions.

 

[apeiron]

Will there ever be another point in the history of mathematics where another fundamental branch is created? Sort of like when Newton invented Calculus or Galois did his thing.

Mathematics has advanced by generalisation - seeing what happens when you relax constraints. So you start of for example with flat space, then think about worlds without that constraint, so allowing space to be curved. You then reintroduce a mechanism for definining contraints in this new situation and thus can now do mathematics in any kind of space, flat or curved, you wish to specify.

So it is not so much new branches that are discovered, as increasingly abstract levels of description that are achieved by the successive relaxation of global or axiomatic constraints.

And logically, this process could end if you have found a way to remove every possible kind of constraint that could exist, so are working with some kind of now naked potential (which you know how to systematically constrain so as to recover the worlds you might want to model).

This is what category theory claims to be doing - defining the hierarchy of increasing generalisation.

There are some natural divisions of maths, such as into geometry and algebra. Atiyah helps explain why there is this fundamental dichotomy into a local and a global description of "the same things".

http://lj.setia.ru/texts/atiyah.pdf

 

[AlephZero]

At a more prosaic level that Godel, there were plenty of new fields of math discovered/invented/developed in the 20th century. Two examples are most of the current methods for numerical computation (motivated by the invention of the computer, of course), and the chaotic and/or fractal behaviour of dynamical systems, which has applications in many areas from weather forecasting to computer-generated movies.

I don't see any reason why the math community will run out of new things to discover any time soon.

 

[humanino]

The concept of "TOE" in physics also remains an elusive dream. It is quite possible that Nature is an infinite onion. There are other recent advances which may or may not relate to physics, but I think answer the idea "they all seem to be building on top of one another" negatively : for instance, the Moonshine and conformal field theory, as they relate to superstrings. This is a possible ongoing revolution, for which we just do not have a full picture and confirmation yet. Another possible example is non-commutative geometry. Both those suggest that the interface between pure mathematics and physics is more than ever alive and well, and there is no reason to doubt that it will continue.

 

[Pythagorean]

I'm a member of the church of the infinite onion

 

[Dembadon]

Mathematics is a language, and languages tend to evolve. We develop it as needed or desired. It feels awkward to think about mathematics being "all figured out," and I'm not comfortable with the idea. There are certainly puzzles within it, but developing the language is just as important!

 

[llynne]

...I think answer the idea "they all seem to be building on top of one another" negatively : for instance, the Moonshine and conformal field theory, as they relate to superstrings. This is a possible ongoing revolution, for which we just do not have a full picture and confirmation yet...

New math, when it arrives will incorporate and explain the old. The idea that we just build more on the old foundations misses the point that math has a relationship with reality and that if there are more dimensions or whatever the reality is that the old math works within it's own parameters. Just as we use Newtonian formulas in the human scale of things and einsteinian (oops) formulas for the vast scale and we found some problems on the teeny weeny scale which got people thinking there could be multi dimensions which seems mathematically sound even if we can't yet get our heads around it and there are several schools of thought ...(some of them run by total nutters!) One day some smart-arse will decide to publish some simple little conversion which everyone can laugh at for twenty years until we all realize the intrinsic obviousness of it all and declare that we could have come up with it ourselves! (am I raving? yep, sorry :D )