Comment by practal
24 days ago
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
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.
I never expected some obsessive user to make 6 different replies to one of my comments. Wow.
You have 6 posts in the thread started by my top comment. I had multiple replies to one of your posts because HN requires one to wait a while to reply and I was in a hurry. The order of posts doesn’t matter. At least not to me.
Insinuating I’m obsessive has a negative connotation. Along with outright insults such comments make you look bad and unreasonable.
Terry Tao in one of his analysis books writes:
Functions are also referred to as maps or transformations, de- pending on the context.
This after defining a function in essentially the same I did.
Just to make clear, so you are saying Serge Lang is wrong, too? And as proof you cite various anonymous HN users, most of them heavily downvoted?
> 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.
Seems you agree with me after all.
> 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.
A distribution is a function, but considered on a different space.
It is even harder to teach math to people by insisting that above fact is wrong. Schwartz got a Fields medal for this insight.
It’s strange to hear a fellow mathematician say that if I’m in set theory class then a functional is a function but isn’t one in functional analysis. In Rudin’s Functional Analysis book he proves that linear mappings between topological spaces are continuous if they are continuous at 0. I’ve never heard of someone believing that a continuous mapping is not a function.
Terry Tao writes in his analysis book:
Functions are also referred to as maps or transformations, depending on the context.
Tao certainly knows more about this than I ever will.
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