I think there's a particular problem with mathematical articles in wikipedia for the non mathematician. The lede - the section at the start of the article - is meant to be accessible to as wide an audience as possible.
Wikipedia:Manual of Style/Lead section
But in mathematical articles, the lede seldom is as accessible as this.
See for instance Riemannian manifold
As it is right now it is highly technical, at least to most readers if you haven't reached at least first or second year undergraduate level maths:
There is no way that any of that is going to be accessible to you unless you know the following concepts and more - just list all the words you don't understand:
Or be a very enthusiastic and dedicated amateur mathematician.
Look at the next section, the introduction, though, and you should find it more accessible:
You only need a minimal understanding of maths there. Only concepts you need to understand are:
Still quite a lot of concepts needed to fully understand the introduction. But they are concepts you can expect to understand with only high school maths, most of them anyway. (The concept of "embedding" and of a "path" get complicated when you define exactly what they mean, the complexity is in the details, but your intuitive idea of what they mean is likely to be close enough to get a pretty decent rough idea of what it is all about).
In this article if they simply swapped the introduction for the lede, it would comply with the guideline on ledes much better.
For some reason many of the maths articles don't comply with this guideline that the lede should be accessible to as wide an audience as possible.
It's worth looking further down the page for an introduction or history section as that may be more accessible.
Some discussion of this issue here: Wikipedia talk:WikiProject Mathematics/Archive 70
Some concepts in maths simply can't be stated in a non technical way. A good example is the Riemann hypothesis. Basically you need to have at least a second or third year undergraduate level of maths to have enough background to have a reasonable grasp of what the hypothesis is.
Indeed, not sure there is anything in that entire page that would be accessible to anyone less than undergraduate level and much is only readable by someone at graduate level who has specialized in this particular topic area.
It would be possible to explain a bit about the Reimann hypothesis in non technical language. At least to say a bit about its history (don't bother with the history section there - it's totally unreadable by a non mathematician), and why mathematicians are so interested in it etc.
But for some reason in this case nobody has done that. It could maybe do with an introduction section like the Reimann surface article.
But as for really understanding it, I think it just isn't possible to explain it in that way in this case. I wouldn't say I have a good grasp of it myself :).
That's just the nature of maths. You already know many concepts in maths such as decimal notation, probably most people know what a ratio is and how it works, how to multiply numbers and divide them, negative numbers, etc. If you've ever tried to teach those concepts in a school, people don't learn them easily. It takes a long time to learn a new mathematical concept, which then seems quite simple once you understand it. And historically, whole civilizations lasted for centuries, thousands of years even, without these concepts, without anyone even thinking of them. Historically all those concepts challenged the very best mathematicians of past civilizations, who gradually, over centuries, developed them into the ideas that most people nowadays find simple (as adults anyway, not as children learning them for the first time).
Since it took centuries for even the best mathematicians in the ancient world to develop and understand these concepts - it is no surprise really that it also takes us a long time to understand them as children, even today.
So, is the same for more advanced maths, sometimes they depend on concepts that it will take you some years of study to thoroughly understand. Even though the concepts are not really intrinsically difficult, any more than e.g. the concept of a negative number is. But they seem so at first and it just takes a long time to get familiar with them, even for the brightest mathematicians, it can take a fair while to thoroughly master new concepts in maths. You understand the definition quickly enough but to fully understand how it is used and all the implications etc takes much longer.
So - as they say in that discussion, if it was like a text book, you'd need to organize a course of study which introduces you to all the concepts you need in order to understand the mathematics.
But as Jimmy Wales said, that's not Wikipedia's remit. It might be interesting though if it could somehow partner with another site. Say, if you could jump from a maths page in wikipedia to the relevant page in a study course for that area of maths... But I don't know how practical that is.
Meanwhile, it would help if the ledes were written for a wider audience. I can understand why they are done the way they are - for a mathematician, then the lede is much more readable if it just sates what the mathematical concept is in terms of other familiar concepts they already know. Any more words make it harder to read.
But if you don't have that background, then many more words are needed.
It's a bit like if every time you used negative numbers in wikipedia, you had to start the article with a short primer on negative numbers. And every time you used a square or cube or square root or cube root, or logarithm, or sin, the article had to explain what those things are. It wouldn't be practical.
So for more advanced mathematical concepts, these things like vector fields, manifolds, inner product etc are like the cube root, and square root that everyone understands, except that they are only understood by students at at least undergraduate level of understanding of maths. And for them, explaining them in every article they are used is as tedious as it would be for everyone else if every article that used a square root or negative number had to include a primer on those concepts.
So I think that's the main source of this problem. Not sure how you can solve it.
Wikipedia's guidelines don't say that every article has to be understandable by everyone. It's recognized that specialist articles are only understood by specialists. But the guideline is, as I understand it, that you write your article to be as accessible as possible to as wide an audience as possible. And that particularly the lede section should be accessible to as wide an audience as possible.
In many mathematical articles, though, especially the more technical ones, that guideline about the lede section seems to be interpreted as saying that the lede needs to be accessible to anyone who has reached about the second or third year of an undergraduate course in maths.
In some cases at least the lede could be made accessible to anyone with high school level of maths or less. But this would make the article somewhat harder to read for mathematicians who want to understand it in mathematical detail right away. So that's the dilemma I suppose.
For another example, see the article on embedding. I'll add that as a comment as this answer is already quite long.
Wikipedia:Manual of Style/Lead section
But in mathematical articles, the lede seldom is as accessible as this.
See for instance Riemannian manifold
As it is right now it is highly technical, at least to most readers if you haven't reached at least first or second year undergraduate level maths:
"In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real smooth manifold Mequipped with an inner producton the tangent spaceat each pointthat varies smoothly from point to point in the sense that if X and Y are vector fields on M, thenis a smooth function. The familyof inner products is called a Riemannian metric (tensor). These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry."
There is no way that any of that is going to be accessible to you unless you know the following concepts and more - just list all the words you don't understand:
- differential geometer
- manifold
- smooth manifold
- inner product
- tangent space
- vector field
- metric
- tensor
- smooth function
Or be a very enthusiastic and dedicated amateur mathematician.
Look at the next section, the introduction, though, and you should find it more accessible:
In 1828, Carl Friedrich Gauss proved his Theorema Egregium (remarkable theorem in Latin), establishing an important property of surfaces. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring distances along paths on the surface. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space.See differential geometry of surfaces. Bernhard Riemann extended Gauss's theory to higher-dimensional spaces called manifolds in a way that also allows distances and angles to be measured and the notion of curvature to be defined, again in a way that was intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces. Albert Einstein used the theory of Riemannian manifolds to develop his general theory of relativity. In particular, his equations for gravitation are constraints on the curvature of space.
You only need a minimal understanding of maths there. Only concepts you need to understand are:
- Theorem
- Curvature
- 3-dimensional space
- higher-dimensional space
- measuring distances
- paths on a surface
- angles
- embedding
Still quite a lot of concepts needed to fully understand the introduction. But they are concepts you can expect to understand with only high school maths, most of them anyway. (The concept of "embedding" and of a "path" get complicated when you define exactly what they mean, the complexity is in the details, but your intuitive idea of what they mean is likely to be close enough to get a pretty decent rough idea of what it is all about).
In this article if they simply swapped the introduction for the lede, it would comply with the guideline on ledes much better.
For some reason many of the maths articles don't comply with this guideline that the lede should be accessible to as wide an audience as possible.
It's worth looking further down the page for an introduction or history section as that may be more accessible.
Some discussion of this issue here: Wikipedia talk:WikiProject Mathematics/Archive 70
Some concepts in maths simply can't be stated in a non technical way. A good example is the Riemann hypothesis. Basically you need to have at least a second or third year undergraduate level of maths to have enough background to have a reasonable grasp of what the hypothesis is.
Indeed, not sure there is anything in that entire page that would be accessible to anyone less than undergraduate level and much is only readable by someone at graduate level who has specialized in this particular topic area.
It would be possible to explain a bit about the Reimann hypothesis in non technical language. At least to say a bit about its history (don't bother with the history section there - it's totally unreadable by a non mathematician), and why mathematicians are so interested in it etc.
But for some reason in this case nobody has done that. It could maybe do with an introduction section like the Reimann surface article.
But as for really understanding it, I think it just isn't possible to explain it in that way in this case. I wouldn't say I have a good grasp of it myself :).
That's just the nature of maths. You already know many concepts in maths such as decimal notation, probably most people know what a ratio is and how it works, how to multiply numbers and divide them, negative numbers, etc. If you've ever tried to teach those concepts in a school, people don't learn them easily. It takes a long time to learn a new mathematical concept, which then seems quite simple once you understand it. And historically, whole civilizations lasted for centuries, thousands of years even, without these concepts, without anyone even thinking of them. Historically all those concepts challenged the very best mathematicians of past civilizations, who gradually, over centuries, developed them into the ideas that most people nowadays find simple (as adults anyway, not as children learning them for the first time).
Since it took centuries for even the best mathematicians in the ancient world to develop and understand these concepts - it is no surprise really that it also takes us a long time to understand them as children, even today.
So, is the same for more advanced maths, sometimes they depend on concepts that it will take you some years of study to thoroughly understand. Even though the concepts are not really intrinsically difficult, any more than e.g. the concept of a negative number is. But they seem so at first and it just takes a long time to get familiar with them, even for the brightest mathematicians, it can take a fair while to thoroughly master new concepts in maths. You understand the definition quickly enough but to fully understand how it is used and all the implications etc takes much longer.
So - as they say in that discussion, if it was like a text book, you'd need to organize a course of study which introduces you to all the concepts you need in order to understand the mathematics.
But as Jimmy Wales said, that's not Wikipedia's remit. It might be interesting though if it could somehow partner with another site. Say, if you could jump from a maths page in wikipedia to the relevant page in a study course for that area of maths... But I don't know how practical that is.
Meanwhile, it would help if the ledes were written for a wider audience. I can understand why they are done the way they are - for a mathematician, then the lede is much more readable if it just sates what the mathematical concept is in terms of other familiar concepts they already know. Any more words make it harder to read.
But if you don't have that background, then many more words are needed.
It's a bit like if every time you used negative numbers in wikipedia, you had to start the article with a short primer on negative numbers. And every time you used a square or cube or square root or cube root, or logarithm, or sin, the article had to explain what those things are. It wouldn't be practical.
So for more advanced mathematical concepts, these things like vector fields, manifolds, inner product etc are like the cube root, and square root that everyone understands, except that they are only understood by students at at least undergraduate level of understanding of maths. And for them, explaining them in every article they are used is as tedious as it would be for everyone else if every article that used a square root or negative number had to include a primer on those concepts.
So I think that's the main source of this problem. Not sure how you can solve it.
Wikipedia's guidelines don't say that every article has to be understandable by everyone. It's recognized that specialist articles are only understood by specialists. But the guideline is, as I understand it, that you write your article to be as accessible as possible to as wide an audience as possible. And that particularly the lede section should be accessible to as wide an audience as possible.
In many mathematical articles, though, especially the more technical ones, that guideline about the lede section seems to be interpreted as saying that the lede needs to be accessible to anyone who has reached about the second or third year of an undergraduate course in maths.
In some cases at least the lede could be made accessible to anyone with high school level of maths or less. But this would make the article somewhat harder to read for mathematicians who want to understand it in mathematical detail right away. So that's the dilemma I suppose.
For another example, see the article on embedding. I'll add that as a comment as this answer is already quite long.
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