How Abstract Is Your Math Thinking?

[andytoh]

I'm just wondering about the spectrum of abstract thinking among the math lovers here. How pure do you like math? Do you insist on visualizing your math problems to be more than a collection of sets and mappings?

I couldn't phrase everything I wanted to say in my 5 choices (limited to only 100 characters). Physical application is just one of the major measurements I for abstractness of a math topic. But of course, there are others.

Let's say abstractness is measured by how applicable it is directly to real-life situations, how symbolic it is, how difficult it is to visulize, how untangible it is (whatever that means).

 

[Crosson]

I voted Quite Abstract because I feel that proofs are aesthetically enhanced by physical content.

 

[DeadWolfe]

Pure Abstract:

Ig there's no category theory, I ain't doing it.

 

[andytoh]

All the math I study has a remote hint of general relativity, no matter how abstract it may seem. So I voted quite abstract. I spend most of my time studying point-set topology and differential topolgy, but I probably wouldn't if I knew that it had no implications in general relativity.

 

[Gib Z]

I voted Pure Abstract, however I do not only do proofs. I do other things, but couldn't care less about physical interpretations.

Basically you ask, are we Pure Mathematicians or Applied Mathematicians. I would like to be the former. Pure Mathematicians don't only do proofs...

 

[Hurkyl]

Hrm. Why is "physical application" your yardstick for concreteness?

 

[Chris Hillman]

Say whaaaat?

All the math I study has a remote hint of general relativity, no matter how abstract it may seem. So I voted quite abstract. I spend most of my time studying point-set topology and differential topolgy, but I probably wouldn't if I knew that it had no implications in general relativity.

Huh?!

I beg to differ.

 

[Gokul43201]

I just don't get this thing about math and "proofs". Isn't all math essentially about "doing proofs" (though I've never used that phraseology before)? I think there's some kind of culturally derived connotation that I'm missing here.

 

[J77]

Second option.

Do people get paid to do proofs all day or are you "abstractees" students?

 

[andytoh]

Huh?!

I beg to differ.

I mean I study mostly topology, differential topology, and differential geometry the most, dealing mainly with abstract proofs. But I wouldn't be dealing with these topics so much if I couldn't apply them eventually in general relativity. It is my interest in applying the math I learn in general relativity that prevents me from voting purely abstract.

 

[christinaz]

second one. maybe this is the reason that I'm not mathy at all.

 

[1016]

I choose balance of them, where it is so far that i was learning after all...
which i could apply it to physical when only i truly acquired it knowledge..

 

[radou]

Isn't voting option #4 ('Quite abstract. I enjoy doing proofs, but enjoy it more when it has physical applications') contradictory? If one enjoys something more when it has to be supported by physical applications, then one cannot be 'quite abstract'.

 

[slider142]

I voted "Quite Abstract", since I started concentrating more on pure math after learning some general relativity. Now, I tend to find applications of the more general areas of math, like category theory, to physics both elegant and fascinating.

 

[andytoh]

Hrm. Why is "physical application" your yardstick for concreteness?

It is not my only measurement of abstractness. I couldn't phrase everything I wanted to say in my 5 choices (limited to only 100 characters). Physical application is just one of the major measurements I for abstractness of a math topic. I thought of not explaining the 5 choices at all, but that would have lead to more confusion. So I chose one major criterion. But of course, there are others.

Let's say abstractness is measured by how applicable it is directly to real-life situations, how symbolic it is, how difficult it is to visulize, how untangible it is (whatever that means)... any other suggestions to define abstractness to make the poll voting more accurate?

It's hard to measure abstractness so you should just intuit which of the five choices you are. For example, I find group theory more abstract than differential equations because it is "more symbolic" and "has less direct application to physical situations."

 

[acm]

Pure Abstract, I study math to study math.

 

[Crosson]

Isn't voting option #4 ('Quite abstract. I enjoy doing proofs, but enjoy it more when it has physical applications') contradictory? If one enjoys something more when it has to be supported by physical applications, then one cannot be 'quite abstract'.

I voted this option because, although I work on many areas of math, it is the problems that connect to physics that get me the most excited.

I feel like I've seen the best of physics, but there isn't enough of it, so I look to math to get my fix. Still, when a new physics problem comes a long, this is the most exciting of all.

 

[andytoh]

Here's an example of what I mean by abstract thinking.

Most of us regard a point (x1,...,xn) in the cartesian product (X1) x ... x (Xn) as simply a set of n-tuples. However, x = (x1,...,xn) can be regarded equivalently as a mapping from an indexing set I to the union U(Xi) such that x(i) belongs to Xi for all i in I. The obvious bijective correspondence between these two sets is x(i)=xi for all i in I. However, this more abstract notion of the cartesian product leads to some (fun?) techinicalities that are otherwise disregarded.

For example, the simple theorem:

Xi is subset of Yi for all i in I implies that (X1) x ... x (Xn) is subset of (Y1) x ... x (Yn)

requires more work to prove because strictly speaking the function x = (x1,...,xn) mapping I to the union UXi must have its range changed before it can be regarded as a function mapping I to the union UYi.

So does anyone here actually enjoy this more abstract way of looking at the simple point in the cartesion product? If you do, then you an abstract dude. If not, then you are a concrete dude.

 

[mathwonk]

well the point of the abstract way is to allow generalizations which are interesting and important.

in particular, to products over arbitrarily large index sets, and unordered ones.

furthermore, the real abstract apporoach is to view the construction as a functor, from indexed collections to indexed products. That way functions with the monic cancellation property, whether they are inclusions or any other injections, are not significantly distinguished.

I.e. thinking of an inclusion as really different from an injection is not the abstract way of thinking. thus an indexed family of injections is carried over to an injection of products, it doesn't really matter what the domain or target set is.

you have to stop thinking of what set things are defined as, and start thinking of what their properties are.

so once you go abstract you need to go all the way, to thinking 'functorially", and then there is no great difficutly, and things ARE ACTUALLY EASIER, oops.

hey hey hey i have 4000 posts! take that pete rose.

 

[arildno]

Congratulations with your post number, mathwonk!

 

[nightdove]

Here's an example of what I mean by abstract thinking.

Most of us regard a point (x1,...,xn) in the cartesian product (X1) x ... x (Xn) as simply a set of n-tuples. However, x = (x1,...,xn) can be regarded equivalently as a mapping from an indexing set I to the union U(Xi) such that x(i) belongs to Xi for all i in I. The obvious bijective correspondence between these two sets is x(i)=xi for all i in I. However, this more abstract notion of the cartesian product leads to some (fun?) techinicalities that are otherwise disregarded.

What's the point of such obvious redundancy?

Unnecessary abstraction is a waste of time.

Most of the pleasure comes in converting the abstract into concrete terms in my case, ie. terms I can visualise at least partially, by applying the obverse of induction.

 

[Crosson]

What's the point of such obvious redundancy?

The above abstraction allows what to generalize to the infinite-dimensional case, and so it is in fact your "concrete" definition that is redundant.

Unnecessary abstraction is a waste of time.

No abstraction is necessary (look at Chimpanzees). I would change your statement to "Unpleasant abstraction is a waste of my time".

 

[nightdove]

The above abstraction allows what to generalize to the infinite-dimensional case, and so it is in fact your "concrete" definition that is redundant.



No abstraction is necessary (look at Chimpanzees). I would change your statement to "Unpleasant abstraction is a waste of my time".

The former definition already describes infinite-dimensional case sufficiently.

You have no right to change my statement about "unnecessary abstraction" - it stands. There is no need to bring chimpanzees into the picture either. That would be an unneessary point of reference unless you intended it to be self-referential. )

Abstraction that does not lead to insight, unpleasant or not, is a waste of time, yours, and more importantly, mine. )

 

[andytoh]

The former definition already describes infinite-dimensional case sufficiently.

Both equivalent definitions of the cartesian product allow for infinite-dimensional products and even uncountably infinite dimensional products. However, the more abstract definition can do more than the concrete definition.

For example, if you want to find an injective map from the collection of all countable subsets of AxAxAx... to AxAxAx..., then the abstract definition of the cartesian product is required. The countable subsets of AxAxAx... will have to be taken in the form {f1, f2, f3,...}, where each fi is a mapping from N to A. An injection F:( countable subsets of AxAxAx...) -> AxAxAx... could be defined as

F({f1, f2, f3,...})(n) = fi(k),

where n <-> (i,k) is any bijection from N to NxN. The proof of injectivity of F is now immediate.

The concrete definition of the cartesian product will be useless in this case.

 

[kirstin.17]

I voted quite abstract. Some people can't appreciate beautiful proofs but some are just so clever and amazing... how would anyone have thought of that!?... It's beautiful.

 

[nightdove]

The question is, do people really have set preferences for different levels of abstraction?

You are as abstract as you need to be to solve a given interesting problem or to understand an insight with aesthetic features. I find it impossible to force myself to do ugly proofs (is there subjectivity here?), for instance, those in Optimisation Theory. Also Analysis only gets interesting once you move onto topology. Proofs regarding sequences of real numbers are absolutely banal.

But one of my favs in A-level was Group Theory while all the other students balked, so I vote for Option 5 (very abstract).

At that time, the Further Mathematics teacher was not able to give a single instance of a physical application of Group Theory (although many exist in the computer sciences), but this did not diminish my love for it. It seemed intrinsically logical and had a purity of spirit that set it apart from say, the concrete study of brutal objects like hyperbolic functions or complementary functions. I know I am not comparing like for like, but at A-level Further Math, Group Theory is the farthest you get into the abstract. The rest of it is methods based.

 

[andytoh]

So nightdove, it seems you are both concrete and abstract depending on your mood? Your opinions seem to go from one extreme to the other.

 

[nightdove]

Perhaps. I have already mentioned that I tend to be rather parsimonious with abstraction, too much conceals more than it reveals. So it isn't abstraction for its own sake, but abstraction for the sake of yielding an "einsicht", which you could describe as subjective (what is insight to one may not be insight to another), but probably not a function of mood.

Would you regard probability distribution theory (outside of measure theory) as abstract or concrete?