motivation of abstract math

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saus
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motivation of abstract math

Postby saus » Fri Dec 09, 2011 4:46 am UTC

In my first year of grad school, I feel like I'm learning about shittons of random abstract generalizations and special cases.. to the point where things seem less interesting in their own right. Things in algebra seem to be more natural. You have a handful of central objects - groups, rings, etc, and you can look at certain kinds of them, and they all have several very natural examples. In differential topology, manifolds... i just feel like, who cares? Lp spaces - okay, that exists, but who cares? The whole development of the fundamental theorem of calculus for lebesgue integrals - seems like a lot of extra work just for a slightly more general theorem. Point set topology - okay we can prove things, but does this ever actually come up? And usually the exercises are insightful but contrived.

I know this stuff was invented for a reason, as a development of natural questions. I'd love to know why I'm supposed to be learning all of this random stuff. It seems like there's an infinite number of objects you could invent and ask questions about, so why do we consider these ones? There are plenty of theorems in the books that use the ideas presented - but only to prove things about what we just made up. Are there any accessible proofs that use this stuff?

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antonfire
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Re: motivation of abstract math

Postby antonfire » Fri Dec 09, 2011 5:55 am UTC

saus wrote:In differential topology, manifolds... i just feel like, who cares?
Really? Every "nice" shape and its dog is a manifold. Manifolds are things that can be locally described by a coordinate system. Not every such thing arises as a subset of a euclidean space.

saus wrote:Lp spaces - okay, that exists, but who cares? The whole development of the fundamental theorem of calculus for lebesgue integrals - seems like a lot of extra work just for a slightly more general theorem.
The point is not a the fundamental theorem of calculus. The point is to have a nicer system of functions to work with. This allows you to actually formalize and have theorems about, say, Fourier transforms, which are important for all sorts of things.

saus wrote:Point set topology - okay we can prove things, but does this ever actually come up?
Absolutely. Point set topology is what allows us to talk about things like limits and shapes in a very general setting. Oftentimes you'll have something which has a topological structure that doesn't come from a metric structure. For example, the product of infinitely many metric spaces. (Which comes up in all sorts of contexts.) Without point set topology, we wouldn't be able to talk about the "shape" of this thing at all. This also comes up quite naturally in algebra, for example the Zariski topology on an algebraic variety, or the natural topology on the p-adics.


Here are some concrete examples:

The Fourier series proof of the fact that 1 + 1/22 + 1/32 + 1/42 + ... = pi2/6.

The topological proof that there is no algebra on R3. If there were, multiplication by non-scalar element near 1 would induce a non-vanishing vector field on the unit sphere.

The solution of the heat equation using Fourier series, including a generalization to non-smooth initial conditions.

For fun, the topological proof of the infinite of primes. Exercise: work out how this is the same as Euclid's proof.
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Proginoskes
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Re: motivation of abstract math

Postby Proginoskes » Fri Dec 09, 2011 9:13 am UTC

Topology: The Banach-Tarsky paradox. (You can cut a sphere into a finite [IIRC] number of pieces, reassemble them, and get two spheres of the same size.)

Maybe the OP'er shouldn't go into analysis then. There's lots of neat stuff in Algebra and Combinatorics. (Godel's Incompleteness Theorem, and other "true but unprovable in system S" theorems, like Goodstein's Theorem, seem to lurk only in the algebaic part of mathematics.)

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Re: motivation of abstract math

Postby doogly » Fri Dec 09, 2011 3:48 pm UTC

Motivation is a big problem in math often times. Mathematicians like to cover their tracks. Tie fox tails to their boots.

One major reason for this is that if you define and object in a certain way, then prove that parts of the definition aren't necessary, you redefine it. So curvature of a surface is defined based on the particular embedding, a nice easy thing to visualize, but then you proof the Theorem Egregium, then you come up with a more elegant definition of curvature. By now, curvature is defined in terms of a section on a bundle. What a terrible place to start though! And this is just geometry. Analysis may get worse.

I love problem solving where the solution requires more mathematical structure than the definition. The most interesting results in group theory are the ones that answer questions which can be phrased without the word "group." same goes for every other field. It is worth noting at this point that I am not a real mathematician, just a physicist who has been hanging out in this department for a while.

For topology, point set topology was something I never really had much interest in. Differential geometry was a natural thing to study, and algebraic topology answers lots of questions you can phrase in other ways. "Classify surfaces," what a fantastic problem. "Can magnetic monopoles exist," makes a physicist excited. So I kept a point set book handy when I did algebraic topology and differential topology. And then when you find out the Zariski topology, which is a nice thing to consider, is actually not Hausdorff, some of these "ridiculous, pointless" structures are a little less so. But if you pick up a book like counterexamples in topology, you are talking all the time about spaces that are constructed for the purpose of topology. Good catalog, but Zariski, that is a meaty topology right there.
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Re: motivation of abstract math

Postby Yakk » Fri Dec 09, 2011 4:26 pm UTC

antonfire wrote:Manifolds are things that can be locally described by a coordinate system.
And man, where in the universe could we find an example of something like that?
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saus
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Re: motivation of abstract math

Postby saus » Sat Dec 10, 2011 8:44 am UTC

As far as this affects me, I've realized that it doesn't really matter and it can all be fun in a puzzley way, even if I don't really know what I'm talking about.

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Re: motivation of abstract math

Postby antonfire » Sat Dec 10, 2011 11:28 am UTC

That sounds like the wrong lesson to take away from this.
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Re: motivation of abstract math

Postby Yakk » Sat Dec 10, 2011 1:21 pm UTC

Step 1: Ask a question about why analysis and topology seems so abstract to me.
Step 2: Get answers pointing out concrete connections to a bunch of analysis and topology, including most of the things you decry as being hopelessly abstract.
Step 3: Give up, and say that they are just abstract puzzles, and can be enough.
Step 4: ...
Step 5: Profit!
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Re: motivation of abstract math

Postby Talith » Sat Dec 10, 2011 2:59 pm UTC

As a mathematician and not a physicist, I've always seen high end pure maths as a subject worth pursuing in its own right. The fact that there are strong connections between what we study and what physicists can use is just a happy bonus that i'm glad they noticed. That said, any reasons that can be found to convince people to fund pure maths research is a a big deal, so I think it should certainly be a point that's emphasised.

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Re: motivation of abstract math

Postby saus » Sat Dec 10, 2011 10:38 pm UTC

Yakk wrote:Step 1: Ask a question about why analysis and topology seems so abstract to me.
Step 2: Get answers pointing out concrete connections to a bunch of analysis and topology, including most of the things you decry as being hopelessly abstract.
Step 3: Give up, and say that they are just abstract puzzles, and can be enough.
Step 4: ...
Step 5: Profit!


Hyperbole. The kind that makes me look like an idiot and asshole.

The examples given were reassuring, but I think I won't be able to truly appreciate the importance of things until I learn more.

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Re: motivation of abstract math

Postby Yakk » Sun Dec 11, 2011 12:17 am UTC

You can understand the usefulness of manifolds without learning (much) more.

The surface of the earth is an example of a geometry where there is a local Euclidean coordinate system, but not much in the way of a global one. Manifolds and charts and atlases need little more than this to inspire them -- when you note that you can do the same with much more exotic surfaces (like Einstein's warped space-time), many more rather concrete applications fall out.

Much of the point set topology lets you go from knowing very little about a structure to knowing a whole bunch. The axioms and tiers of said topology let you figure out what properties you need to prove about the space you are talking about to get ridiculously powerful proof hammers out of your back pocket and make entire kinds of problems trivial -- or, similarly, if you can show that there is a property you are missing in your space, you get information about what other weird quirks fall out.

Lp spaces end up topologically collapsing, but that is the path down which you form a foundation for Fourier Analysis -- if you have ever seen a noise filter (visual or auditory), or seen an image zoomed, or blurred, or sharpened, or used a camera that electronically de-blurs the image you take, that is using Fourier Analysis. Or at least I took a trip through Lp spaces along my route to the foundations of Fourier Analysis. That branch of mathematics is one of the more useful ones in modern communication and signal theory.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Re: motivation of abstract math

Postby imatrendytotebag » Sun Dec 11, 2011 7:25 am UTC

Another thing is that these abstract structures (for instance, the notions of point-set topology) allow you to draw connections between different topics, and find fundamental similarities. So, we know what it means for a real-valued function to be continuous (the epsilon-delta definition). But the more abstract notion exposes a new, interesting way to view continuous functions as sort of structure-preserving maps, the same way homomorphisms are structure-preserving maps of groups.

Math is less "hey, here's a cool set of axioms, let's play around with them and look at some examples and see what's up", and more "Hey, all of these things seem to exhibit similar properties, are there some underlying rules that govern the behavior of both of these equations?"

In the end, I agree that as you learn more, you begin to see the value in the various abstractions. If you do some basic number theory before you study group theory, it's very interesting to see quotient groups as a generalization of modular arithmetic (or perhaps more fittingly, quotient rings).
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Re: motivation of abstract math

Postby saus » Mon Dec 12, 2011 12:36 am UTC

Yakk wrote:Lp spaces end up topologically collapsing, but that is the path down which you form a foundation for Fourier Analysis -- if you have ever seen a noise filter (visual or auditory), or seen an image zoomed, or blurred, or sharpened, or used a camera that electronically de-blurs the image you take, that is using Fourier Analysis.


What do you mean topologically collapsing? Noise filter stuff sounds really cool. I'll look into that.


imatrendytotebah wrote:Math is less "hey, here's a cool set of axioms, let's play around with them and look at some examples and see what's up", and more "Hey, all of these things seem to exhibit similar properties, are there some underlying rules that govern the behavior of both of these equations?"


I actually always thought of it as the former >_> I'm going to take a second look at some old material with this new perspective. This actually might make everything seem a lot less random. Instead of a ton of random things that happen to have familiar examples, I have a few familiar examples that can be looked at from a ton of different angles. Motivation.

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Re: motivation of abstract math

Postby doogly » Mon Dec 12, 2011 12:51 am UTC

You thought it was the former because almost every math book is written with the former perspective. Grrr, Bourbaki, hate you so.
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Re: motivation of abstract math

Postby Yakk » Mon Dec 12, 2011 1:26 am UTC

I'm unable to remember why I think L_p spaces collapse topologically -- my analysis is getting really rusty. Now that I think about it, this might only be on the less interesting finite dimensional cases. /shrug. I'm probably wrong!

---

So the approach I was introduced to fourier analysis and L_p spaces was a little story about two mathematicians who both published a result at the same time, and the two results contradicted each other. And both arguments seemed reasonable. So they had to go back and figure out foundational reasons why what they thought of as functions was wrong.

The professor sketched out one of the proofs (on 3-5 blackboards), and then the class pointed out the parts where the logic used in the proof had holes, and then the professor pointed out where we missed holes.

We then spent most of a semester correcting that proof -- which required introducing extra assumptions in the hypothesis (because the theorem, as stated, wasn't true) and the like.

It was a relatively good way to motivate the definitions and proofs involved -- we had a goal, I think it was to show that you could approximate any function with an infinite series of polynomials? And we proceeded to remove the "any function" clause and define what "approximate" means, and play around that. Going from there to Fourier analysis wasn't that big of a step.

On the other hand, this was in an undergraduate course. So maybe a grad school might want to power through it faster. :)
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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