Huge personal issue with math articles

[Robert_W_King]

This is merely my opinion here:

For me, one of the primary difficulties for understanding complex math is that I've never been subject to a model to which the particular principle of math is used, and I think it follows also in the current articles we have.

It's nice that the theorem is there, and so are the formulas, but for some people I imagine there is a fundamental total confusion that goes on.

Is there any chance that within mathematics articles there can be a section that explains the types of problems that particular concept could be applied?  Not just an example, but essentially what you would apply the theorem to find out, and potential real-world applications.  I think that this was brought up in another thread that stated we could have a "prerequesite" section, but I'm not so sure that would go far enough.

[Greg Woodhouse]

This is merely my opinion here:

For me, one of the primary difficulties for understanding complex math is that I've never been subject to a model to which the particular principle of math is used, and I think it follows also in the current articles we have.

It's nice that the theorem is there, and so are the formulas, but for some people I imagine there is a fundamental total confusion that goes on.

Is there any chance that within mathematics articles there can be a section that explains the types of problems that particular concept could be applied?  Not just an example, but essentially what you would apply the theorem to find out, and potential real-world applications.  I think that this was brought up in another thread that stated we could have a "prerequesite" section, but I'm not so sure that would go far enough.

My reaction here is essentially twofold: I basically agree with you, and what I had originally wanted to do with mathematics articles was try and explain the basic ideas and show how the concepts fit together. In fact, that's why I proposed the idea of integrative articles some time ago. What happened, though, is I constantly found myself writing and rewriting sections based on objections such as "This is too difficult for someone without a college education", or, "This is not encyclopedic." I didn't plan it this way, but I pretty much gave up on trying to write mathematics articles. It's just too hard to make everyone happy. And frankly, writing a compelling article is a lot more work than just writing down a bunch of definitions and theorems. Anyone who knows a little mathematics can do that. But just try adding a little context or elaboration and other authors will be after you like a pack of wolves.

[a.a.s.]

Robert, just two remarks

1)Here and there I've always advocated the "what for" question to be answered in maths articles. Not easy, this one, BTW.  Bear in mind, however, that maths is often 'justified' by itself: a mathematical construction finds application in another part of maths. And, for example, for long centuries it was thought that the number theory  is the Queen of the pure math and would never have interactions with the real world. Surprisingly enough, recently it became the base of the internet security. You know better that me ;-) that it would be virtually impossible to have the on-line banking without preceding long-lasting purely intellectual "exercises" with numbers.
Why do I write this? Just to indicate that there is more to maths  than answers to the contemporary real world problems.

2) As far as I can tell the "prerequisite" section was meant to indicate the articles to read before attacking the problem in a given article. This has nothing to do with "applications" and its status is "on hold" or "under consideration / modification " (well, I'm not sure).

[Robert_W_King]

Let me try to elaborate on the "what for" problem.  Example:

Let's say we're talking about differential equations.

what kinds of problems are differential equations meant to solve?
what types of answers do you get from diff. eqs?  What do they mean in context?
how are those results or problems or formulas used?  What could you tell from the way an answer is formulated?

What I'm getting at is that I think there should be in the article a way to translate the math that exists into something meaningful other than just more equations or numbers.

[a.a.s.]

Let's say we're talking about differential equations.

Biased choice, differential equations  have plenty of applications and the questions are pretty easy to answer 
More seriously - below I fully agree with you. Just note that there are many topics with no apparent/direct applications in the real world. Lie algebras, for example, are much more delicate to link to the real world as we would like. Just one important and vital topic of modern math, learned by math students.

what kinds of problems are differential equations meant to solve?
what types of answers do you get from diff. eqs?  What do they mean in context?
how are those results or problems or formulas used?  What could you tell from the way an answer is formulated?

It's helpful to have it in mind. I agree that those practical questions could and should be answered in maths articles whenever possible. However, it's not always easy to do...
Consider also that some of these questions have no straightforward "meaningful" non-philosophical answers for some other math topics, the above example of Lie algebras partially included.

What I'm getting at is that I think there should be in the article a way to translate the math that exists into something meaningful other than just more equations or numbers.

Again, I agree, whenever possible we should do it. Note that "translate math into something meaningful" looks offensive to mathematicians
More seriously, sometimes the only answer for the "what for" question would still be in the world of pure mathematics. For example, the famous continuum hypothesis or zeros of the Riemann zeta function... If there were real-world consequences we could "physically" test it to see whether it is "better" to assume the continuum hypothesis hold "true" or whether the all zeros are there where we expect. Nonetheless, the zeros Riemann zeta function is often considered to be (one of) the most important and difficult problem of maths. Just its role is far more sophisticated that can be easily explained in layman's terms. And we can do nothing about it.

In some sense, mathematics is not a science, see [[Scientific method]] article

[Robert_W_King]

I don't doubt that there are mathematics which lie in the realm of hypothetics; I don't expect "real world" models for those types.

But I think that there might be enough (including equations for physics, chemistry, engineering, etc etc) that this warrants actual demonstration.

In the case where a theorm only provides an answer for one specific problem, then do you think there should be a section that in a nutshell describes "why we care" about the result?

[Greg Woodhouse]

I don't doubt that there are mathematics which lie in the realm of hypothetics; I don't expect "real world" models for those types.

But I think that there might be enough (including equations for physics, chemistry, engineering, etc etc) that this warrants actual demonstration.

In the case where a theorm only provides an answer for one specific problem, then do you think there should be a section that in a nutshell describes "why we care" about the result?

Well, Lie algebras are far from hypothetical, they play an important role in differential geometry, non-Euclidean geometries, classical mechanics, quantum theory, symmetries of differential equations and reduction of order, the theory of finite groups,  the classification of regular solids, and more. The trouble is that the mathematics involved is less elementary, and it's harder to "see" why Lie algebras play a basic role in applications than differential equations (which, after all, make their appearence in the first term of freshman physics, if not mathematics!)

[a.a.s.]

In the case where a theorm only provides an answer for one specific problem, then do you think there should be a section that in a nutshell describes "why we care" about the result?

Sure. (a bit more difficult to actually do it, though ).