[ogogmad]

The situation is worse than I thought. The term "function", as used in foundations of mathematics, includes functionals as a special case. By contrast, the term "function", as used in mathematical analysis, explicitly excludes functionals. The two definitions of the word "function" are both common, and directly contradict one another.

[skhunted]

By contrast, the term "function", as used in mathematical analysis, explicitly excludes functionals. The two definitions of the word "function" are both common, and directly contradict each other.

This is incorrect. In mathematics there is a single definition of function. There is no conflict or contradiction. In all cases a function is a subset of the cross product of two spaces that satisfies a certain condition.

What changes from subject to subject is what the underlying spaces of interest are.

[ogogmad]

> What changes from subject to subject is what the underlying spaces of interest are.

I'm not sure I understand what you mean here. I need some clarification. How does this have any bearing on whether functionals count as functions or not? What is the "underlying spaces of interest" in this example?

In some trivial way, every mathematical object can be seen as a function. You can replace sets in axiomatic set theory with functions.

[skhunted]

Everything I wrote was assuming set theory as the foundations for mathematics and applies only to that setup. At any rate a functional is function since the definition starts with: a functional is a function from…

Some books will say: a functional is a linear map….

Note that a linear map is a function.

[ogogmad]

You genuinely don't know what you're talking about. The word "function" means different things in different areas. So does the word "map" or "mapping". In analysis, what you personally call a "function" instead falls under the term "mapping". In foundations - which is a different area with incompatible terminology - the terms "mapping" and "function" are defined to mean the same thing.

This situation is a consequence of how mathematicians haven't always been sure how to define certain concepts. See "generating function" for yet another usage of the word "function" that's in direct contradiction with the last two. So that's three incompatible usages of the term "function". All this terminology goes back to the 1700s when mathematics was done without the rigour it has today.

I find it aggravating how you're so confidently wrong. I hope it's not on purpose.

[edit] [edit 2: Removed insults]

[practal]

I am looking at the whole development of this thread with amusement, but I also find it somewhat shocking.

I see that you are desperately trying to distinguish "foundational" and "analysis" contexts from each other. If you are writing a book about analysis, it might be helpful to clarify that in this context you reserve "function" for mappings into ℂ or ℝ, for example [1] defines "function" exclusively as a mapping from a set S to ℝ (without any further requirements on S such as being a subset of ℝⁿ). Note that even under this restricted definition of function, a distribution still is a function.

In a general mathematical context, "function" and "mapping" are usually used synonymously. It is just not the case that such use is restricted to "foundations" only.

It seems to me that squabbles about issues like this are becoming more frequent here on HN, and I am wondering why that is. One hypothesis I have is that there is an influx of people here who learn mathematics through the lens of programs and type theory, and that limits their exposure to "normal" mathematics.

[1] Undergraduate Analysis, Second Edition, by Serge Lang

[ogogmad]

I learned mathematics the regular way. So you're wrong - and not just about this.

> I see that you are desperately trying to distinguish "foundational" and "analysis" contexts from each other

They literally are different. The proof is all the people here saying that distributions aren't functions, while displaying a clear understanding of what a distribution is. Maybe no one's "wrong" as such, if they're defining the same word differently.

I think you're the naive one here. Terminology is used inconsistently, and I tried to simplify the dividing line between different uses of it. I agree it's inaccurate to say it's decided primarily by Foundations vs Analysis, but I'm not sure how else to slice the pie. It's like how the same word can mean slightly different things in French and English. I agree it's quibbling, but it's harder to teach maths to people if these False Friends exist but don't get pointed out.

I never expected some obsessive user to make 6 different replies to one of my comments. Wow. This whole thing thread was a bit silly, and someone's probably going to laugh at it. I need to take another break from this site.

[skhunted]

You genuinely don't know what you're talking about. .... I find it aggravating how you're so confidently wrong.

This is a fine example of irony.

Let V be a vector space over the reals and L a functional. Let v be a particular element of V. L(v) is a real number. It is a single value. L(v) can't be 1.2 and also 3.4. Thus L is a function.

A function is simply a subset of the product of two sets with the property that if (a,b) and (a, c) are in this subset then b=c.

Can you find a functional that does not meet this criterion? If so then you have an object such that L maps v to a and also maps v to c with a and c being different elements.

Find me a linear map that does not meet the definition of function. Give an example of a functional in which the functional takes a given input to more than one element of the target set.

I think you are not a mathematician and you also don't appear to understand that a word can have different meanings based on context. "generating function" isn't the same thing as "function". Notice that generating is paired with function in the first phrase.

Example: Jellyfish is not a jelly and not a fish. Biologists have got it all wrong!

[ogogmad]

I'll try one last time.

> I think you are not a mathematician

Guess again.

> Example: Jellyfish is not a jelly and not a fish. Biologists have got it all wrong!

You have a problem with reading comprehension. I never said any mathematician was wrong.

Think about namespaces for a moment, like in programming. There are two namespaces here: The analysis namespace and the foundations namespace.

In either of those two namespaces, the word "mapping" means what you're describing: an arbitrary subset F of A×B for which every element of a ∈ A occurs as the first component in a unique element (x,y) ∈ F.

But the term "function" has a different meaning in each of the two namespaces.

The word "function" in the analysis namespace defines it to ONLY EVER be a mapping S -> R or S -> C, where S is a subset of C^n or R^n. The word "function" is not allowed to be used - within this namespace - to denote anything else.

The word "function" in the foundations namespace defines it to be any mapping whatsoever.

Hopefully, now you'll get it.

[skhunted]

[edit] I've finally blown it. You're a moron. Your definition of "function" as some subset of AxB is how it's defined in foundations. It's not how it's defined in analysis. In analysis, your definition would describe the term "mapping". What a crackpot and idiot. I'm done wasting time and sanity on this.

Interesting. So you think there are functions in real analysis that are studied that don't meet the definition I gave? Is there a functional that does not meet the definition I gave?

In all contexts a function is a subset of the product of two sets that meets a certain condition. Anything that does not meet this definition is not called a function.

Every functional meets the definition of function.

[skhunted]

The word "function" in the analysis namespace defines it to ONLY EVER be a mapping S -> R or S -> C, where S is a subset of C^n or R^n. The word "function" is not allowed to be used - within this namespace - to denote anything else.

In real analyis one is interested in functions from R^n to R. They don't define function to be only something from R^n to R. It's just that these are the functions they wish to study. They don't define function to exclusively be a map from R^n to R. It’s just that these are the types of functions they care about.

No mathematician can possibly think function is anything other than a subset of the product of two spaces that meets a certain condition.

[skhunted]

In general, instead of resorting to name calling it's best to just walk away. It makes you look bad and unreasonable.