What's uninspiring is your ignorance of game theory.
Anyone who puts their name on that list might potentially be a target. On the flip side, there is no signaling value in putting your name on the list anonymously. Therefore anonymous names on the list believe in it (tho some people might make the calculation that they can't handle being a target but they might still resist and obstruct in other ways.)
So: It's inspiring that a lot of people are ready to obstruct or delay even if they're not ready to deal with personal consequences.
> Anyone who puts their name on that list might potentially be a target.
My first inclination is to read letters like this as a threat from employees to the employer. It says hey boss-men, this shite is not on. Signing anonymously undermines that message. I tend to read those signatures as as, I don't like this but it's not worth my job. I have no faith in the efficacy or even existence of "obstruct or delay" tactics from folks like that.
> It says hey boss-men, this shite is not on. Signing anonymously undermines that message.
No it doesn't. It says "Hey boos I'm telling you this shit is not cool, and there's nothing you can do to me personally because you don't know who I am."
Let me put it differently. Suppose YOU are the boss. You company has 1000 employees and you receive a letter with 500 anonymous signatures saying "we fucking hate what you're doing" (so, 50% of your employees, 100% anonymous). Do you get a little bit worried? Or do you get not worried at all because everybody signed anonymous? Actual question, let me know how you think.
> "Hey boos I'm telling you this shit is not cool, and there's nothing you can do to me personally because you don't know who I am."
Why does this change the calculus for management? They don't pay folks to be happy, they pay them to do their jobs. Threaten to take away the labour however and you create a bargaining position. That's how strikes and threats of strikes work. This letter is fundamentally different. For a start, have you considered the veracity of a list of anonymous petitioners? How do you differentiate the real thing from a made up list?
> Terrence Tao was a good example of what happens when an exceptionally smart person stops getting funded by an American University: not moving to another country, but got VC money and created a new company.
We got here because wealthy ideologues spent half a century creating a biased media ecosystem and attacking the idea that objective truth exists. Calling someone a Nazi because they made a Nazi salute isn’t causative, it’s just recognizing a problem which should have been tackled more aggressively 20 years ago.
What value do you think comes from trying to finely parse whether the “I was racist before it was cool” guy calling for a “eugenic immigration policy”, who “would not mind at all if Gaza and Israel were both wiped off the face of the Earth” has a slightly different self-identity than his boss who threw the Nazi salutes? “Anti-Semitic bigot who at the very least doesn’t mind being around people who throw Nazi salutes” is longer but doesn’t seem very different in terms of how decent people would view them.
The same value as finely parsing whether you are the same person as Stalin or whatever. You are viewing your enemy not as a bunch of flawed or even mentally ill people, but as some movie villain or even not-reall-humans. This is a problem. That's how you start down the path of justifying genocide. When the right becomes crazier, the solution is not for the left to become even crazier and starting some arms race of who can be most insane.
I think the Icelandic data is a bit skewed upwards. It’s showing senior and principal level salaries in large companies, most “normal” devs earn maybe 70% of that.
That's pretty cool. Unfortunately school holidays mean I can't take time off whenever, but I can definitely use the idea to plan time off around those.
The term "function" sadly means different things in different contexts. I feel like this whole thread is evidence of a need for reform in maths education from calculus up. I wouldn't be surprised if you understood all of this, but I'm worried about students encountering this for the first time.
Don’t know if you are a mathematician or not but mathematically speaking “function” has a definition that is valid in all mathematical contexts. Functional clearly meets the criteria to be a function since being a function is part of the definition of being a functional.
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.
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.
> 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.
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….
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.
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.
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.
Yeah, the whole argument felt somewhat unhinged and silly. It is fine to point out that sometimes "function" is used in a more specific manner than "mapping", particularly in analysis, but I doubt any mathematician would think that a functional is not a function, in a general context such as a HN comment.
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!
> 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.
If one has a “thing” that “maps” elements of one set to another that satisfies the condition I previously gave then that thing is a function. Every functional satisfies that definition. Therefore every functional is a function.
[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.
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.
For any smooth function (like the conjugate) it makes sense to ask whether there exist holomorphic functions that approximate it arbitrarily well.
However: Suppose that for every n > 0 there exists a holomorphic function f_n such that |f_n(z) - z| < 1/n for all z. Then |f_n(z)| <= |f_n(z) - z| + |z*| = |z| + 1/n by the triangle inequality. A consequence of Liouville's theorem is that any entire holomorphic function with polynomial growth is a polynomial; here in particular we would need to have f_n(z) = a_n z + b_n for some complex numbers a_n and b_n. For real x we would have |(a_n - 1)x + b_n| < 1/n for all x, so a_n = 1. For imaginary iy we would have |(a_n + 1)iy + b_n| < 1/n for all y, so a_n = -1, which is a contradiction.
In fact, if a sequence of holomorphic functions converges uniformely on compact sets, the limit is itself holomorphic because of Cauchy's theorem.
It's astonishing to me that there are people defending nazis as "just another political ideology", or "just a normal reaction to leftwing people". The very ideology that the free world spent 5 years at total war to defeat. The people who defeated fascism in WW2 are literally called The Greatest Generation but nowadays we have people thinking "oh I bet we can make fascism work for us. It's never backfired on its adherents!"
I wonder if that's due to WW2 veterans dying, or just plain time passing. And will we see another Axis?
> The very ideology that the free world spent 5 years at total war to defeat. The people who defeated fascism in WW2 are literally called The Greatest Generation but nowadays we have people thinking "oh I bet we can make fascism work for us. It's never backfired on its adherents!"
This is a historical anachronism. "Nazi ideology" (by which we typically mean things in particular like eugenics, racism and antisemitism, but perhaps also things like "strong nationalism" or "leader-centralized authoritatianism") was not the reason the US or other Allies went into the war, nor was it the reason why most people volunteered to fight, assuming they were not drafted. (Of course, there were exceptions.) The mythologization of the war in modern times has led to a much, much greater modern hatred of Nazi ideology than what existed in the 1940s, even in the postwar period. If you read newspaper articles and private letters from the period, it's really quite shocking the sentiments the average American, Frenchman, or Englishman had.
It is also of course always worth mentioning the enormous Soviet contribution to the war effort against Germany, without which Allied victory would have almost certainly been impossible - and the Soviet Union under Stalin was not exactly "the free world", to put it mildly.
German totalitarism did not cause Britain to go to war but it was why Britain did not negotiate peace in 1940.
German totalitarism was why they never held any serious legitimacy among conquered Poles and later Russians. Terror and more terror. It says something that, after long hesitation, Russians solidified around Stalin. Is it an anachronism as well? Did they see any hope for a prosperous post-war future with/under Hitler?
