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Re:Division By Three (2006) by anatoly on Sunday April 27, 2014 @01:59PM (#7656039)
(cont'd)

OK, so who needs this? Well, this paper is beautiful on its own, but it's not particularly important to many other mathematicians. The reason is that in set theory there's something famously called an "Axiom of Choice" that basically allows you to handle infinite sets and their pairings much more easily than without it. The Axiom of Choice used to be somewhat controversial in the beginning of the 20th century. What happened during that time was that people realized they could look at set theory (and in fact at all of mathematics) much more rigorously than before by coming up with some basic axioms and showing how everything else follows from them. This is something that had been practiced in geometry ever since Euclid, but not really in other parts of mathematics, and around the beginning of the 20th century people started doing the same thing not just with points and lines, but also with numbers and sets and functions. When they tried to _formalize_ set theory - that is, take the usual stuff people have been doing with sets and figure out how to bring it all down to a handful of very obvious axioms - they discovered that all the axioms were completely obvious (as wanted!) except one, the Axiom of Choice, that sort of seems necessary to do all kinds of advanced math, but is not as obviously true as others. See, the Axiom of Choice is not _explicit_: it says something like - under certain conditions - "a matching in pairs between these two sets exists, but I'm not showing you what it is explicitly, I'm just telling you it exists". When it tells you that, it kinds looks reasonable that these two sets should be of the same "size", intuitively, so you're glad that the Axiom of Choice settles the matter by just claiming that a matching exists; but at the same time you may feel a little uneasy about the fact that the matching is not described with some formula or algorithm or anything, you're just told it exists and that's it.

Anyway, with time mathematicians pretty much accepted the Axiom of Choice as a vital part of their mathematical toolkit. Even though some very non-intuitive things can be proved using it (look up the Banach-Tarski paradox if you're interested), it's much better to have it than not. And with the Axiom of Choice, the whole problem of proving that bags of marbles B1 and R1 are of the same size becomes trivial, even when they're infinite. You don't need to carefully engineer a B1-R1 matching out of the larger one between all the six bags; the Axiom of Choice sort of gives you the firepower to just claim that this B1-R1 matching exists, and call it a day. Nearly all mathematicians are satisfied with this, because for them set theory is no more than a convenient tool to carry out their investigations into _other_ things, like numbers, geometric spaces and what not. It's only people who study _set theory itself_ (and call themselves, appropriately enough, set theorists) that are still interested in understanding just how much the Axiom of Choice is needed. These people appreciate much more than other mathematicians a proof of some fact in an _explicit_ way, not requiring the Axiom of Choice, even if it's trivial when using the Axiom. So the paper about "dividing by 3" is an example of this sort of investigation: do we really need the Axiom of Choice to prove that B1 and R1 are of the same size? Turns out we don't, and we can do this explicitly with a rather complicated, yet beautiful, construction. To mathematicians as a whole, this result is largely meaningless; to set theorists, it's a minor theorem of some interest, though even they, I think, probably would look at it more as an interesting puzzle with a beautiful solution. And that, by itself, is nothing to sneer at.

Re:Division By Three (2006) by sgentle on Monday April 28, 2014 @01:44AM (#7658390)
Thank you for a very clear and thorough explanation of a complex concept. Comments like this make me wish HN had some form of super-upvote equivalent to Reddit's gilding.

Assuming I can summon dang just by using his name, I feel like part of the reason behind the rise of "dumb and/or mean" is that as HN has gotten larger the community has also attenuated. Commenters become a faceless crowd and the quality of our interactions degrades accordingly.

I wonder if having a way of saying thanks other than a (humanity-less) counter increment or a (discouraged and poorly scaling) public comment would encourage more humane behaviour?

My first instinct is some kind of private comment system, but obviously that would enable less-accountable forms of harassment. Maybe that combined with a karma threshold and an aggressively enforced flag button?

Another less radical idea would be some way to mark an exceptional comment, which would also highlight that person's future comments to you in some way.

Re:Division By Three (2006) by dang on Monday April 28, 2014 @02:27AM (#7658500)
I do like the idea of being able to endorse a comment as particularly exceptional. One special and wonderful case of this which happens pretty often is when a person with first-hand knowledge about a story shows up in a thread. (Edit: ran across an example five minutes after I wrote that: https://news.ycombinator.com/item?id=7657553). It's not clear exactly how such a feature would work, though.

About this:

> I feel like part of the reason behind the rise of "dumb and/or mean" is that as HN has gotten larger the community has also attenuated. Commenters become a faceless crowd and the quality of our interactions degrades accordingly

... my main question is how we could test that. It might be true, it might not; hard to know.

Re:Division By Three (2006) by sgentle on Monday April 28, 2014 @07:43AM (#7659296)
Hm, good point.

I suppose there are two parts to my assertion:

1. When you think of yourself as talking to an actual person, you are less likely to be dumb and/or mean.

2. As HN has grown, the degree to which the average commenter considers themselves to be talking to an actual person has decreased.

I was pretty sure I'd be able to find a psychology or sociology study somewhere giving some evidence on (1) or a more general case of (2) for large groups. Wildly, it looks like all of the interest is on how groups reduce your personal identity (deindividuation), not how they reduce the identity of others to you.

So I guess without an appeal to prior research, the best thing would be to develop a measure of "does this comment seem like it's addressed to a real person". You could do that by manually going through and rating them, as (I believe?) you already did with quality metrics.

You could also come up with some heuristics and cross-check them with your own assessment. Here are some I can think of:

* Use of parent's username

* Use of first names

* Use of other usernames

* Use of "you", "your".

* Email addresses, phrases like "contact me", "email me", etc.

* Questions.

* Comment reply chains (ie people replying to a descendent of their own comment)

* "Please", "Thanks" and "Thank you"

I'm not saying that I think comments with this quality are necessarily good, but my prediction is that per (2) the proportion of such comments correlates inversely with the number of active users and per (1) that proportion correlates with a drop in comment quality.

Re:Division By Three (2006) by maxerickson on Monday April 28, 2014 @09:08AM (#7659609)
A lot of times when people use first names or user names they are personalizing a conversation in a context where that doesn't make sense (unfortunately, they are often doing the personalization as part of an attack).

I guess the positive signal still outweighs the negative, but it isn't always a positive signal.

Re:Division By Three (2006) by lisper on Sunday April 27, 2014 @05:09PM (#7656892)
Wow, that was a terrific explanation. Thanks for taking the time to write it up!
Re:Division By Three (2006) by judk on Sunday April 27, 2014 @10:50PM (#7657993)
http://en.m.wikipedia.org/wiki/Zorn's_lemma

Minor major nitpick: Axiom of (Uncountable) Choice (AUC) is controversial. It isn't intuitive, but it is independent from the intuitive axioms, which means that you don't have to accept it, and if all you care about is things that might ever be computable during the existence of the Universe, you never need AUC, and AUC never tells you anything useful since it is nonconstructive in the worst way. (It doesn't even say, like NP does in computing, that if you brute-force, you can solve any problem to any desired level of accuracy). AUC just asserts an answer exists. The interestingness is in knowing what other statements are equivalent to it , but it is and forever will be confined to purely abstract math (really! Unlike RSA, there will provably NEVER be a computational or physical application in the accessible world of countable infinities.)