[cryptica]

Math symbols are a minor issue for me. What confuses me the most are descriptions of mathematical concepts.

For example, Wikipedia describes a 'field' like this:

"In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do."

It doesn't make sense to me. What does it mean if an operation 'is defined' on a set? Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set? But if that was the case then "behave as the corresponding operations on rational and real numbers do" would mean that the fields would always need to be of infinite size (have an infinite number of elements) wouldn't it? Because if the field had a limited number of elements and you added the last two (highest) elements together, the property which requires that the result also be present in the same set could not be met because the result would be greater than the highest element in that set...

The problem is that if you start with a highly abstracted math concept and you dig through all the links and definitions of sub-concepts, they all have huge gaps like this... So when you try to combine all the definitions together to make sense of that original highly abstracted concept, you end up with tens or hundreds of possible interpretations. But in fact, Math should only have 1 interpretation for each concept so this is a very bad situation to be in.

I think math definitions should be more elaborate and repetitive if necessary. They should not try to sound terse and clever. They should not assume that the reader can fill in the gaps. The most rational readers will not be able to fill in the gaps because rational people know the dangers of making assumptions.

[impendia]

I'm a math researcher, and I'll explain why I like these sorts of definitions.

In the first place, what you quoted is not a formal, precise definition; it is not a substitute for such a definition, nor is it intended to be one. The Wikipedia page you mention has a precise definition further down the page.

So what, then, is the purpose of the description you quoted? Why include it at all?

Because it's how mathematicians conceptualize of what a field is. It is the peg we hang our hat on; it is what we remember. A mathematician who has seen fields would be able to fill in the details; and if not, they would know to look up the precise definition in a textbook.

In short, these definitions are how we keep track of the forest at the same time as the trees.

I should note that taste differs among mathematicians, and you can find different styles of exposition in math books. Some are very formal and precise; whereas others are more informal and have lots of handwavy statements along the lines of the one you quoted.

[jhanschoo]

I'd also like to point out that it is also frequent that one encounters equivalent but different formal definitions for the same mathematical structures, and this is why the informal descriptions are important as well.

[joshuaissac]

> What does it mean if an operation 'is defined' on a set? Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set?

For a binary operation f to defined on a set, f(x,y) must exist for every x and y in the set. There is no requirement that f(x,y) itself is in the set. Adding that requirement would mean that the set is "closed" under the operation f. So if we take Z+ = {1, 2, 3, ...}, ordinary division is defined on Z+, but Z+ is not closed under division, since we can get results like 2/3 that are not in Z+. Whereas division is not defined on Z0+ = {0, 1, 2, ...} because we can get undefined results like 2/0.

However, some definitions of "binary operation" include the "closed" property, so under such definitions, division would not be considered a binary operation on Z0+.

>Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set?

Specifically in the case of a field, yes; addition needs to be defined, closed and invertible for the set; multiplication needs to be defined, closed and invertible for the set excluding the additive identity (zero).

[cryptica]

This is a proper definition, thanks. The first sentence here is about the same length as the one on Wikipedia but it fully encapsulates the meaning without ambiguity.

[lapinot]

You're quoting the introduction of the article, which is notoriously a fuzzy abstract in all the wikipedia articles about mathematical concepts. Let's quote the actual (textual) formal definition (sec. 1.1) as it would stand in a textbook:

> Formally, a field is a set F together with two binary operations on F called addition and multiplication. A binary operation on F is a mapping F × F → F, that is, a correspondence that associates with each ordered pair of elements of F a uniquely determined element of F. The result of the addition of a and b is called the sum of a and b, and is denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, and is denoted ab or a ⋅ b. These operations are required to satisfy the following properties, referred to as field axioms. In these axioms, a, b, and c are arbitrary elements of the field F. [...]

Still, i don't know how to say it in another way but you probably don't have much experience in mathematics, even the first quote is arguably quite accurate.

> Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set?

Yes, unless told otherwise an operation is an internal binary operation, it's really the most common form. When it is not the output is notable enough to be specified.

> But if that was the case then "behave as the corresponding operations on rational and real numbers do" would mean that the fields would always need to be of infinite size wouldn't it? Because if the field had a limited number of elements and you added the last two (highest) elements together [...]

No, the important point here is your use of "highest". A set by default only has equality (and mappings, in and out) but no order relationship. So the most conservative interpretation of "behave as the corresponding operations on rational" would be to only include stuff that can be written using the 4 operations and equality, not ordering.

---

Maybe i'm biaised by the fact that i know what a field is, but still, this particular intro is also how i would present a field: give the most common example and say which operations it has. It sure can create false intuitions like yours about the size, but this will always be the case when we use non-normalized language.

[omaranto]

If informal descriptions confuse you, skip them and read actual definitions instead.

I actually like informal definitions a lot. I think they serve two different purposes:

1. For beginners they usually soften the blow of a fully rigorous definition, letting them get an idea of the concept before getting it exactly.

2. For experts they can often suggest what the exact definition is faster than it would be to read a precise definition!

But if you don't get anything out of them, you can skip 'em. Definitions are more important: informal descriptions are there to help you grasp the definition faster and to help you develop an intuition for the concept being defined. If you can do those things faster from just a rigorous definition, you don't need an informal description. (But as I said, I feel informal descriptions benefit both beginners and experts, so I'd also suggest practicing reading them more to get a feeling for what kind of details people tend to omit or emphasize.)

[cryptica]

It's not about 'informal definitions'. I also really like informal definitions but not the way that most mathematicians currently tend to write them.

[omaranto]

Wait, are you complaining about how mathematicians currently tend to write informal descriptions of concepts or about the informal descriptions in the intros to Wikipedia articles? I think those things are pretty different.

[gspr]

It sounds very much to me like you would like mathematicians to change our notation to accommodate someone who has not put in the effort to learn mathematics. Do you wish the same from structural engineers? Programmers? Physicists? Medical doctors? Musicians?

> "In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do."

This isn't a mathematical definition of a field. This is an encyclopedia's intuitive definition. It is a good one, in my opinion, as it'll evoke the correct definition in a person who is trained at mathematics, and hopefully convey the gist of the idea to someone who is not. Very good for one sentence! But a mathematical definition it is not!

A mathematical definition of a field would be as follows:

BEGIN DEFINITION

A field is a set F together with two functions A:F×F→F and M:F×F→F and two elements z∈F and o∈F that together satisfy the following properties:

ASSOCIATIVITY OF A: A(A(a, b), c) = A(a, A(b, c)) for all a,b,c∈F.

ASSOCIATIVITY OF M: Same as above with M in place of A.

COMMUTATIVITY OF A: A(a, b) = A(b, a) for all a,b∈F.

COMMUTATIVITY OF A: Same as above with M in place of A.

NEUTRALITY OF z WRT A: A(a, z) = a for all a∈F.

NEUTRALITY OF e WRT M: M(a, o) = a for all a∈F.

INVERSE FOR A: For all a∈F, there exists an element na∈F such that A(a, na) = z.

INVERSE FOR M: For all a∈F such that a≠z, there exists an element ra∈F such that M(a, ra) = o.

M DISTRIBUTES OVER A: M(a, A(b, c)) = A(M(a, b), M(a, c)) for all a,b,c∈F.

END DEFINITION.

(This definition presupposes that one knows what a set is under a standard framework.)

Since this notation is cumbersome, it is common to write A(a,b) as a+b and M(a,b) as a·b or ab. Likewise, z if often written 0 and and o is often written 1 (or e). Similarly, na is often written -a and ra is often written 1/a, but do take care to recall that "-a" and "1/a" are just symbols. One typically compresses notation even further, and writes "a + -b" as "a - b" (not to be confused for the juxtaposition of a and -b).

Do you feel better about this definition than Wikipedia's informal one? I invite you to scribble out a verification that for example the reals form a field under this definition (with ordinary addition as A, ordinary multiplication as M, ordinary 0 as z, ordinary 1 as o).

> Because if the field had a limited number of elements and you added the last two (highest) elements together, the property which requires that the result also be present in the same set could not be met because the result would be greater than the highest element in that set...

You're reading too much into "behave like the corresponding operations on real numbers do". One does not demand that addition preserves order. Notice how there is no reference to ordering or elements being "larger" or "smaller" in the definition above. It is for example the case that in the field of two elements, 1+1=0. That's fine.

> The problem is that if you start with a highly abstracted math concept and you dig through all the links and definitions of sub-concepts, they all have huge gaps like this

Not at all. You are trying to do rigorous mathematics using informal statements (not to detract from the informal statement; someone who has seen the formal definition of a lot of mathematical structure can almost surely reconstruct the correct formal definition of a field in a second from the informal one).

> I think math definitions should be more elaborate and repetitive if necessary.

So mathematicians should make communication between ourselves more cumbersome in order to please outsiders? I'm sorry, I don't mean to sound like a gatekeeper, but this is ridiculous.

> They should not try to sound terse and clever. They should not assume that the reader can fill in the gaps.

Do you demand this of musicians and engineers and chefs and mechanics and pilots and doctors and nurses too?

> The most rational readers will not be able to fill in the gaps because rational people know the dangers of making assumptions.

The most rational readers who have studied mathematics will. To a point, of course. There is often a tradeoff to be made, but your blanket statement is just plain wrong.

[cryptica]

The user joshuaissac gave a very good description of a 'field' as a response to my comment and it was about the same length as the definition on Wikipedia. It shows that it's possible.

I don't see why certain knowledge should be out of reach of those who are not involved directly in that field. I could explain complex software engineering concepts to a layman. They wouldn't be able to use that knowledge to implement the software themselves, but they would be able to use the knowledge to make good high level decisions about it; for example to decide which of two solutions is better given a specific problem.

[gspr]

> The user joshuaissac gave a very good description of a 'field' as a response to my comment and it was about the same length as the definition on Wikipedia. It shows that it's possible.

Sure, his definition is also a good one. It leaves out a lot, though. Which is fine, if one can assume the reader knows the context. My definition, too, leaves out a lot (it assumes set theory), and rests on the informal language known as English.

> I don't see why certain knowledge should be out of reach of those who are not involved directly in that field.

It isn't. The content of mathematics research papers may well be out of reach, but that's quite natural, don't you think? I, as a mathematician, do not expect to be able to read research papers on chemistry without putting in a lot of work.

> I could explain complex software engineering concepts to a layman.

OK. It does not follow from that that everything can be explained to a layman. Some things are easier to explain with layman analogies and mental images than others. However, to keep this fair, I think you should see how many laymen can follow Wikipedia articles on complex software engineering topics with ease! That is, afterall, where we started this discussion.