cokernel

(Edited to try to answer the objection more clearly.)

The intervals overlap heavily. Each of the intervals R_i contains infinitely many rational numbers, and all but finitely many of those will have corresponding intervals which are completely contained within the interval R_i. As a result, the estimate obtained by the construction described is always strictly smaller than the chosen e.

Slogan: "rational numbers exist but do not take up space."

This is in a measure-theoretic sense, so literally 100% (not just 99.5%). On the number line, the measure of an interval is the (absolute) difference between its endpoints (the "length" of the interval). For any measurable set, you can approximate its size from above by covering its points with intervals and shrinking them as much as possible.

But because there are only countably many rational numbers, you can cover all of the rationals with a set of intervals with a finite total sum of lengths.

To be a bit more specific, if your rationals are r_1, r_2, r_3, ..., then for any small number e > 0, you can form the set of intervals

(r_1 - e/2^2, r_1 + e/2^2), (r_2 - e/2^3, r_2 + e/2^3), (r_3 - e/2^4, r_3 + e/2^4), ...

which have a total sum of lengths e/2 + e/4 + e/8 + ... = e and contain all of the rational numbers.

So for any positive number e > 0, the measure of the rational numbers is less than e, meaning that the measure of the rational numbers is 0.

To turn this into actual probabilities requires a little bit more work, since the measure of the number line, and hence that of the irrational numbers, is infinite, not "100%". But you could look at the probability of selecting a rational number in the interval [0, 1] and use the same reasoning as above to get a probability of 100% for an irrational number and 0% for a rational number.

The other three items listed for Socrates are:

* Knowledge of the universe does not affect our behavior.

* We should pity the perpetrator of injustice, not the victim of injustice.

* No one really knowingly does wrong.

The closest approximation I can find is pp. 292--293. (I don't have a copy and haven't read it in about twenty years, so there may be a better approximation available.)

Hadden and Ellie have a brief discussion about the potential to market dying in space as a "really nifty last indulgence", whereupon Hadden segues to the topic of immortality and says the following:

"Now, I'm not bringing this up so I can boast. I'm bringing it up for a practical reason. If we're figuring out ways to extend our lifespans, think of what those creatures on Vega must have done. They probably are immortal, or close enough. I'm a practical person, and I've thought a lot now about immortality. I've probably thought longer and more seriously about it than anyone else. And I can tell you one thing for sure about immortals: They're very careful. They don't leave things up to chance. They've invested too much effort in becoming immortal. I don't know what they look like, I don't know what they want from you, but if you ever get to see them, this is the only piece of practical advice I have for you: Something you think is dead cinch safe, they'll consider an unacceptable risk. If there's any negotiating you get to do up there, don't forget what I'm telling you."

(Please forgive transcription errors.)

https://books.google.com/books?id=Q6o51-W_z8MC&lpg=PP1&dq=sa...

Many libraries will purge checkout records once a patron returns books, but the increasing reliance on centralized cloud-based ILS vendors is making this harder to do reliably.

I'd appreciate it if you could name the online system your county uses. (Or if you can provide a link.)

Yes. Diaconis, Holmes, and Montgomery demonstrate (but do not provide a full specification for) a coin-flipping machine which will yield heads 100%* of the time (as long as it starts on heads).

"Dynamical bias in the coin toss". https://statweb.stanford.edu/~susan/papers/headswithJ.pdf

* Or at least every time the experiment is performed.

Yup, you never get to \omega that way. Cardinality is a coarser notion than "ordinality". So your example shows that Card(\omega * \omega) = Card(\omega) = \aleph_0, even though \omega * \omega and \omega have different order types.

EDIT: Just wanted to add that an order isomorphism has two requirements:

(1) it needs to be a bijection (so order-isomorphic objects have the same cardinality); and

(2) it needs to preserve all inequalities (so a strict inequality among items in one object turns into a strict inequality in the same direction among the corresponding items in the other object).

Thomas Pynchon's essay does a good job, I think, of explaining what the Luddites really fought against, and he does not, as this article seems to suggest, say a Luddite is "someone who opposes technological progress".

https://archive.nytimes.com/www.nytimes.com/books/97/05/18/r...

> But it's important to remember that the target even of the original assault of 1779, like many machines of the Industrial Revolution, was not a new piece of technology. The stocking-frame had been around since 1589 .... Now, given that kind of time span, it's just not easy to think of Ned Lud as a technophobic crazy.

The pronunciation of PNG is provided in the spec:

> This International Standard specifies a datastream and an associated file format, Portable Network Graphics (PNG, pronounced "ping"), for a lossless, portable, compressed individual computer graphics image transmitted across the Internet.

https://www.w3.org/TR/2003/REC-PNG-20031110/#1Scope

I believe that the spec for MNG also specifies its pronunciation.

According to the abstract at https://arxiv.org/abs/1008.1459 , this document has been revised many times (the current version, from 2015, is v38). Is there a particular reason that v8 was the version selected for submission?

Here is some support for the claim that he sometimes used a capitalized version of his name.

Quoting from http://faculty.gvsu.edu/websterm/cummings/caps2.html :

> As we may have mentioned, due to the kindness of D. Jon Grossman's son, Jerome, we have the complete file of Jon's correspondence with Cummings. On making a preliminary tour through these letters, we found Jon preparing a French edition of his translations of Cummings' poetry, and on 27 February 1951 he wrote to the poet: "are you E.E.Cummings, ee cummings, or what?(so far as the title page is concerned)wd u like title page all in lowercase?"

> The poet replied on 1 March 1951: "E.E.Cummings, unless your printer prefers E. E. Cummings/ titlepage up to you;but may it not be tricksy svp[.]"

> That seems definitive to us: may it not be tricksy!

This puzzle game engine was written by Stephen Lavelle, who may be best known for Stephen's Sausage Roll.

Here's Sokogoban, a cute recent game made with the engine:

https://www.puzzlescript.net/play.html?p=8726ac1f3addfe42cc5...

I recommend clicking the "hack" link at the bottom to see the code for the game.

The article suggests using nonviolent communication to clarify which sort of support is needed:

> Beyond that, ask for what you want, and check with your partner about what they want. If you want your partner’s support about something, let them know whether you’re looking for understanding or advice. Don’t make them guess. Don’t wait to see what they do. Tell them what you want.

I can see that the "Default to emotional support" can seem to suggest that you must provide emotional support whenever asked, but if you can't provide emotional support (whether from low EQ or from being drained), you can't. The suggested strategy of making the request for support unambiguous allows a response along the lines of (but probably not worded this way) "I'd like to provide support, but I can't."

The distinction between "stop giving X" and "stop asking for X" is important — in the latter case, someone is asking for what they need, whether or not the need can be fulfilled, while in the former case, someone is providing X, regardless of whether it is desired.

Is this the paper you have in mind?

* Daniel G. Boyce, Marlon R. Lewis & Boris Worm. Global phytoplankton decline over the past century. Nature 466, 591–596 (2010).

https://www.nature.com/articles/nature09268

I believe I found some references to an attempted bombing shortly after Rudolf Hess's death.

* "Hess Death Sparks Neo-nazi Activity", Jewish Telegraphic Agency, August 24, 1987. https://www.jta.org/1987/08/24/archive/hess-death-sparks-neo... . "In Frankfurt, police arrested two young neo-Nazis who placed a bomb which did not go off in the city's central railway station."

* _Right-Wing Terrorism in the 21st Century: The 'National Socialist Underground' and the History of Terror from the Far-Right in Germany_, Daniel Koehler, Routledge, 2016. https://books.google.com/books?id=1w8xDQAAQBAJ&pg=PT211 "On August 20, 1987, two neo-Nazis were arrested after they had placed materials for a combined explosive and arson device in a locker at Frankfurt Main central station."

The latter account is sourced from p. 125 of

* _Verfassungsschutz 1987_, Bonn: Bundesministerium des Innern, 1988.

I was not able to find volumes of _Verfassungsschutz_ as far back as 1987 online.

I didn't downvote — and talking about downvotes just invites more — but the faux-quote summary you provide in your comment is inaccurate. A more accurate summary, provided in the article, is "Get news. Not too quickly. Avoid social."

The author explicitly rejects the claim you are imputing to him: "You don't have to read a print newspaper to get a better relationship with the news."

The whole article, as I understood it, was about the deeper understanding you can gain about a situation if you take things slow, in batches, instead of seeking to read and post shallow reactions quickly.

The page being quoted points that out:

> Of course, in reality none of this could ever happen. Sorry to break it to you. The truth is, the Pacific Ocean will boil off as the Sun becomes a red giant before you could even take your fifth step in your first trek around the world. Somewhat more of an obstacle, however, is the fact that all the stars in the universe will eventually burn out leaving space a dark, ever-expanding void inhabited by a few scattered elementary particles drifting a tiny fraction of a degree above absolute zero. The exact details are still a bit fuzzy, but according to some reckonings of The Reckoning, all this could happen before you would've had a chance to reduce the vast Pacific by the amount of a few backyard swimming pools.

The link may be dead, but the site still exists. Here's the 1401-digit prime:

https://primes.utm.edu/curios/page.php?number_id=953

CC0 is Creative Commons's license for dedication to the public domain:

https://creativecommons.org/publicdomain/zero/1.0/

You make the paperclips with wire (and your own labor, to start). You start off with 1,000 inches of wire.

Maybe I'm missing something, but the NYT article says they were hauling rocks "to gather samples" (of "fossilized ash deposit" for analysis). The next paragraph explains that they were looking for crystals in the samples because that can help give a timeline for relevant changes in the crystal's environment:

> Ms. Shamloo later analyzed trace crystals in the volcanic leftovers, allowing her to pin down changes before the supervolcano’s eruption. Each crystal once resided within the vast, seething ocean of magma deep underground. As the crystals grew outward, layer upon layer, they recorded changes in temperature, pressure and water content beneath the volcano, much like a set of tree rings.

It's true that this is not sufficient information to allow readers to replicate the analysis at home, but it seems like a good level of introductory detail so people can decide whether they want to follow the link in the NYT article to the volcanology conference.

The term "free-speech zone" is older than that. It goes back at least to 2004, and a similar concept under a different name ("protest zone") is even older.

Example from 2004: https://www.acslaw.org/acsblog/free-speech-zones-balancing-f...

Is that https://www.nytimes.com/2017/08/04/opinion/sunday/motherhood... by any chance?

> Phil. of mind and neuroscience has different subject of study, in most cases

You may find the work of Patricia Churchland an interesting exception.

I believe this exactly is the situation that Damasio describes in the book I mentioned -- his patient "Elliot" had no emotions and couldn't make decisions.

See for instance:

* http://nymag.com/scienceofus/2016/06/how-only-using-logic-de...

* https://intentionalworkplace.com/2012/03/15/how-emotion-shap...

By any chance are you thinking of Antonio Damasio's book _Descartes' Error: Emotion, Reason and the Human Brain_?

Also, this (paywalled, sorry) article seems relevant.

Antoine Bechara, Antonio R. Damasio, Hanna Damasio, Steven W. Anderson, Insensitivity to future consequences following damage to human prefrontal cortex, Cognition, Volume 50, Issues 1–3, April–June 1994, Pages 7-15, ISSN 0010-0277, http://dx.doi.org/10.1016/0010-0277(94)90018-3. (//www.sciencedirect.com/science/article/pii/0010027794900183)

Abstract: Following damage to the ventromedial prefrontal cortex, humans develop a defect in real-life decision-making, which contrasts with otherwise normal intellectual functions. Currently, there is no neuropsychological probe to detect in the laboratory, and the cognitive and neural mechanisms responsible for this defect have resisted explanation. Here, using a novel task which simulates real-life decision-making in the way it factors uncertainty of premises and outcomes, as well as reward and punishment, we find that prefrontal patients, unlike controls, are oblivious to the future consequences of their actions, and seem to be guided by immediate prospects only. This finding offers, for the first time, the possibility of detecting these patients' elusive impairment in the laboratory, measuring it, and investigating its possible causes.

>> Public education is under attack around the world, and in response, student protests have recently been held in Britain, Canada, Chile, Taiwan and elsewhere. >What protests are you referring to? When?

By using the search terms "student protests britain 2012" and varying the third term to "canada", "chile", or "taiwan", I found the following:

https://en.wikipedia.org/wiki/2010_United_Kingdom_student_pr...

https://en.wikipedia.org/wiki/2012_Quebec_student_protests

https://en.wikipedia.org/wiki/2011%E2%80%9313_Chilean_studen...

https://en.wikipedia.org/wiki/Sunflower_Student_Movement

>> The Los Angeles Times reports... >Link please.

Chomsky quotes the following line from the article: "California State University officials announced plans to freeze enrollment at most campuses." By using that string as a search query I found

http://articles.latimes.com/2012/mar/20/local/la-me-cal-stat...

In the United States, that's prohibited (for federal elections, at least) by the 24th Amendment.

https://www.law.cornell.edu/constitution/amendmentxxiv

I can't comment on applications of category theory to computer programming or computer science, but I believe that the book Conceptual Mathematics by Lawvere and Schanuel is a nice introduction to the basic ideas of category theory.

http://www.cambridge.org/catalogue/catalogue.asp?isbn=978052...

I believe this book has been discussed "recently" on HN, but I couldn't find a thread.

Mathematicians write for multiple audiences, but there are two major mathematical audiences for their works: specialists in the same field and researchers in other fields. Phrases like "relatively little mathematical background" normally signal that a work is intended to be accessible to non-specialists, but it's often safe to assume that it is aimed at research mathematicians. I think if a work actually requires relatively little mathematical background, a mathematician is more likely to say something like "no, really, you don't need to know mathematics to understand this!" even when it's not quite true.