HN Theater @HNTheaterMonth

Knuth has a nice book on Surreal numbers.

https://www-cs-faculty.stanford.edu/~knuth/sn.html

This interview is excellent https://www.youtube.com/watch?v=mPn2AdMH7UQ

He wrote the book in less than a week, the book was just flowing for days and after that he couldn't write a thing. Then he realized he missed a detail and had to rewrite the whole book.

You can safely ignore it, the document is just a summary of Knuth's book. And it's a good one, I found it very helpful when I was working through the book. I found I had to read the material from several different sources to understand, in addition to working things out on paper/in code.

I'm not religious myself but honestly that seems like a perfectly reasonable motivation. While there are some people who believe the Surreals could lead to a deeper form of analysis, I think the most common reason to learn them is to apprehend something beautiful without any sort of utility. But as a heads up, if that doesn't sit right with you, then Knuth's book is probably not right for you either; it's framed in the style of the old testament. On the first day, Conway created zero, and saw that it was good; things like that.

This may be more to your taste.

https://www.youtube.com/watch?v=mPn2AdMH7UQ

https://www.youtube.com/watch?v=1eAmxgINXrE

https://core.ac.uk/download/pdf/51289919.pdf

⬐ exmadscientist

For what little it's worth, I am not a religious person, but I found Knuth's Surreal Numbers perfectly reasonable, much more on the tongue-in-cheek side than the proselytizing side.

And the surreal numbers are over of my favorite things in mathematics, so I'd say they're worth it :)

⬐ weatherlight

Thanks, this was constructive and informative.

⬐ dvt

> You can safely ignore it, the document is just a summary of Knuth's book.

Knuth is a Lutheran, by the way (uh-oh). He even wrote 3:16 Bible Texts Illuminated[1].

[1] https://www.amazon.com/3-16-Bible-Texts-Illuminated/dp/08957...

⬐ dhosek

Perhaps even more interesting is Knuth’s Things a Computer Scientist Rarely Talks About which goes into a bit more of his religious life and how he applies some mathematical techniques (particularly Monte Carlo–style random selection). The genesis of 3:16 (so to speak) is discussed as well.

http://www.amazon.com/exec/obidos/ASIN/157586326X/donhosek

⬐ globular-toast

Knuth isn't evangelical, though. Like Bach and many others, Knuth's works could be classified as either religious or secular. He doesn't shoehorn religion into every work.

⬐ maxbond

I didn't know that, that's interesting. Thank you.

To clarify, I didn't mean, you can ignore it because it's irrelevant, as much as, you can derive the bulk of the value from the text if you ignore the introduction.

⬐ ikeboy

Some quick googling turns up http://www.sciencedirect.com/science/article/pii/00973165729..., which cites a French paper in footnote 9 as the origin of the name surordinal, almost certainly the one Knuth refers to. Someone want to tell him?

>In 1967, writing in French, Pierre Jullien [9] generalized the concept of an ordinal slightly to what he called a “surordinal.” He announced several results, among which is the statement that the class of “surordinaux” are wqo.

>9. P. JULLIEN, Theorie des relations. - Sur la comparaison des types d’ordres disperses, C. R. Acad. Sci. Paris, Se’r. A 264 (1967), 594-595.

⬐ vanderZwan

Write a comment under the video, the numberphile people will definitely see it and perhaps pass it on

⬐ ikeboy

Done.

⬐ vanderZwan

It's not showing under recent comments for me?

⬐ ikeboy

That's weird, only shows up for me. I just reposted without the link and it does show up in incognito, so the link must have filtered it.

Youtube comments suck, can't even get paragraphs formatted normally.

⬐ lfowles

Heh. I was just following along with some Unreal Engine tutorials yesterday. Unfortunately the engine has changed since the tutorials were uploaded, so some kind souls pasted the new C++ code in the comment field. What. A. Mess.

⬐ cJ0th

Apparently the book is available on archive.org

https://archive.org/details/SurrealNumbers

By the way: does any one know whether all downloads on archive.org can be considered legal? It is amazing how many old books which, however, aren't old enough to be out of copyright can be found on that site.

⬐ ikeboy

Anyone can upload files there, and I've definitely found recent books there. Of course they comply with the DMCA and will take down files.

⬐ f00_

https://blog.archive.org/2016/06/02/copyright-offices-propos...

⬐ bisby

https://www.youtube.com/watch?v=1N6cOC2P8fQ

day9 (an esports personality, with a math degree) tells the story of graham's number and how donald knuth invented up arrow notation for graham's number to be possible.

He tells it quite entertainingly with a great ending. (I was just showing this to a coworker this morning).

⬐ eternalban

Suprised that Donald Knuth is making such a categorical error. Per his description of Conway's construct, these numbers are progressively higher dimensional. In a same manner that we can not faithfully order a mix of Real and Complex numbers -- is 1.000001+i > 1.000001? -- we also can not meaningfully speak of a partial order of a set of numbers that include the form <:>.

p.s. Re. "worked for 6 days and rested on the sevent" and "J.H.W.H." -- woah there cowboys. Get ye down to earth. ;)

⬐ etatoby

> we can not faithfully order a mix of Real and Complex numbers

A mix of real and complex numbers is just a set of complex numbers, because every real number is a complex number. And complex numbers do not possess a total ordering: https://proofwiki.org/wiki/Complex_Numbers_cannot_be_Totally...

⬐ JadeNB

> complex numbers do not possess a total ordering

The set of complex numbers does not possess a total ordering that respects the algebraic operations. The set of complex numbers is in bijection (in many, meaningless, ways) with the set of real numbers, and so certainly admits a (indeed many, meaningless) total ordering.

⬐ vanderZwan

Not a mathematician, but Wikipedia says the following:

> In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including a total order ≤ and the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field.

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

⬐ dnautics

" we also can not meaningfully speak of a partial order of a set of numbers "

Actually surreal numbers have a well-defined partial order despite the dimensionality concerns; keep in mind surreal numbers are a class, not a set. (They contain as a strict subclass the transfinite ordinals, so it must be a class).

Moreover I think the form <:> is best considered an operator (or generator) than a cartesian product. You can put a set of surreals on one side of <:> and out pops another surreal.

⬐ IngoBlechschmid

I can't watch the video right now. But I can confirm that one can define what it means for a surreal number to be smaller than another surreal number in a meaningful way: So that the surreal numbers are totally ordered (for any two surreal numbers a and b, either a < b or a = b or a > b) and that the usual expected rules for transforming inequalities hold (for instance that the "<" sign reverses when multiplying with a negative number).

This total order is also very useful. Namely one can associate surreal numbers to positions of a large class of two-player games. The sign of the associated surreal number (whether it's positive, zero, or negative) then tells you which player possesses a winning strategy. See https://en.wikipedia.org/wiki/Surreal_number#Application_to_... for more information on this point.

⬐ eternalban

> for a surreal number to be smaller than another surreal number in a meaningful way

By definition! He was discussing ordering a mix e.g. π-sn, π, π+sn, π+smallestRealNumber.

⬐ jerf

Those words are translations into conventional words, which are fairly real-number-oriented. They aren't really quite accurate. (It's fair to say that English words aren't really real numbers, but it's also fair to say that's what most mathematically-educated people would interpret it as.)

Look, taking on one or the other of Conway or Knuth on a topic like this would be a sobering prospect, but both of them at once? Maybe you should be phrasing your "objections" in the form of questions.

⬐ eternalban

> questions.

How can one meaningfully speak of comparative measure between constructs that occupy distinct dimensional spaces. Question mark. (point well taken! :)

⬐ JadeNB

> How can one meaningfully speak of comparative measure between constructs that occupy distinct dimensional spaces?

Because there is only a colloquial, not a mathematical, relationship between the notion of order, and most notions of dimension. An 'order' is any binary relation obeying certain properties (antisymmetry, transitivity, and, depending on whether you want a strict order, possibly irreflexivity); one can put an order on a set without any concern about whether it obeys your or anyone's intuition, as long as it satisfies those properties.

⬐ eternalban

Thank you. So it is correct to think that such orderings are ~toplogical in contrast to the our ~geometric intuition?

⬐ JadeNB

This sounds a little snarky, but I mean it sincerely.

Unfortunately, I'm not sure how to define the notions of "approximate topological-ness" or "approximate geometric-ness" of an order in any rigorous way, so I am reluctant to answer the question. I would agree that trying to fit the order on the surreal numbers into an existing geometric, or even probably topological, intuition is likely to fail. For me, the order is in some sense a combinatorial construction; but I don't know any way to make that rigorous other than that the definition is game-theoretic, and I think of game theory as a particular kind of combinatorics (as probably no specialist in either field ever would).

⬐ JadeNB

> Look, taking on one or the other of Conway or Knuth on a topic like this would be a sobering prospect, but both of them at once? Maybe you should be phrasing your "objections" in the form of questions.

By the way, after having noticed that the conversation in https://news.ycombinator.com/item?id=11952290 tended to involve frequent (what seemed to me to be) appeals to authority, which I think do not belong in mathematical discussion, I wanted to thank you for your measured tone: not "don't question these lofty personalities", but rather explicitly "do question them" (rather than assuming out of hand that an apparent objection is an actual one).

⬐ None

None

⬐ my_first_acct

Random observation: On the bookshelf behind Prof. Knuth in the video, Stephen Wolfram's book "A New Kind of Science" is clearly visible.

⬐ elasolova

So am I getting this wrong? Isn't this exactly how a float described in bits? If you had infinite memory then this would possibly be implemented with floats.

⬐ dnautics

No, floats have explicit exponent and fraction parts. For the "finite-ish" parts of the knuth construction, the maximum size depends on the "generation" of the construct. A "knuth"-string float would have a variable length, depending on the amount of precision necessary to express the value you seek.

⬐ gjkood

The only words that come to my mind when I hear the name Donald Knuth are "Never was so much owed by so many to so few".

The original context is different but I wonder if we can ever compute all the wealth and impact that has been created by the practitioners of his art.

⬐ peter303

So Knuth wrote up treatise as a dialog between students. (I havent read it.) Hofstader uses mathe-philosophical dialogs in Goedel Escher and Bach. The Greek philosophers like Plato and Zeno often used such dialogs. Then too Hofstader wrote his book in 1976 as a postdoc at Stanford, two years after Stanford prof Knuth published his Surreal book. Maybe there is an influence.

⬐ pasquinelli

seems pretty natural to put the questions you have answers to into your reader's head and then answer them.

⬐ VodkaHaze

I imagine it's because the dialogue format flows naturally compared to the traditional monologue paradigm in books.

⬐ zargay01

What is to this day unknown about surreal numbers?

What are the important/fun/etc. conjectures?

⬐ drdeca

I believe that there has been some work trying to define integrations and derivatives for function No->No , and that while there has been some good progress, there are still significant problems with at least one of the two.

⬐ safiume

This is the kind of book I'd imagine is on a reading list for young adults and precocious children who want to bend their mind. I'm going to assume it's being taught at Altschool and Ad Astra. It's a total gem of a book. I recommend it for those that enjoy math as well as those that dislike math at any level.

⬐ spot

not mentioned in the video: conway got the idea for surreal numbers by playing and studying go.

⬐ xyience

Haven't watched the video yet, but then Conway and others formalized a bunch of theory for 2-player combinatorial games like Go, using surreal numbers for game states. (e.g. the number * = {0|0}, or, whoever moves first wins.)

⬐ drdeca

{0|0} = * isn't strictly speaking a surreal number. It is a member of a class that the surreal numbers are a subclass of, called the class of Games.

The definition of a surreal number requires that everything on the left set be less than everything in the right set, but 0 is not less than 0 , so {0|0} does not name a surreal number.

Games do not have this restriction.

The surreal numbers form a field (except that it has a proper class of elements instead of a set of elements) , while the Games do not form a field. The Games do however form a group under addition iirc. Yes, that seems true to me.

The Games represent all perfect information 2 player games with what is iirc called the normal play condition (i.e. you lose iff it is your turn and you have no valid moves), and are such that there is no sequence of moves by the 2 players such that the game never ends.

(Chess for example, is almost a Game , except that it is possible to reach a draw in chess, and the win conditions are somewhat different.

Tic-Tac-Toe with the change that, instead of winning when you have 3 in a row, you win when it is the other player's turn and they have no valid moves, and you cannot move if the other player has 3 in a row, is a Game in this sense.)

Note: I think that Games is often capitalized because of it being a proper class, but I am not sure.

⬐ ianopolous

I wrote a magazine article on surreal numbers:

https://ianopolous.github.io/maths/surreal

⬐ Iv

Neither Firefox nor chromium manage to render your mathematical expressions, which makes the whole thing hard to read.

⬐ ianopolous

Should hopefully be fixed now. The MathJax was loading over http, not https.

⬐ ianopolous

Thank you for alerting me to that. Love a backwards compatible web!

⬐ majewsky

Unless you're writing LaTeX frequently. :)

⬐ unusximmortalis

from what he's describing professor Donald Knuth had a channeling session when he wrote the theory of surreal numbers :)

⬐ decentrality

I love how a famous forefather ( turned 3:16 experimenter and muser on metaphysics, etc ) suddenly throws in how he wanted to role play cheating in a hotel during the course of research in Oslo, and that was his wife's expectation if he was going to squeeze composing another book into their marriage.

It was so nonchalant it was just thrown in as if he was describing the napkin with the theory on it:

https://youtu.be/mPn2AdMH7UQ?t=5m31s

"Because we liked to have an affair in the hotel room."

"So that was the plan."

⬐ droopybuns

What is a 3:16 experimenter?

⬐ Kristine1975

I guess this refers to Donald Knuth's "3:16 Bible Texts Illuminated": http://www-cs-faculty.stanford.edu/~uno/316.html >The text found in chapter 3, verse 16, of most books in the Bible is a typical verse with no special distinction. But when Knuth examined what leading scholars throughout the centuries have written about those verses, he found that there is a fascinating story to be learned in every case, full of historical and spiritual insights. This book presents jargon-free introductions to each book of the Bible and in-depth analyses of what people from many different religious persuasions have said about the texts found in chapter 3, verse 16, together with 60 original illustrations by many of the world's leading calligraphers. >The result is a grand tour of the Bible -- from Genesis 3:16 to Revelation 3:16 -- a treat for the mind, the eyes, and the spirit.

⬐ tamana

And for thr profane:

http://www.larry.denenberg.com/Knuth-3-16/

⬐ jobigoud

Several times in the video he mentions loosing a piece of inspiration because it's flowing too fast for writing it down.

I wonder how many ideas and insights have been lost throughout history to this phenomenon.

Sometimes you crack a problem from an unexpected angle and an entire forest of insights fall down onto you. You take a minute to type the first one but already some of the others are lost forever. Plus you broke out from the free flowing zone.

Will we ever come up with a tool to fix this tragedy?

⬐ AYBABTME

I've been thinking a lot about this problem. I have at least an analogy for it.

Imagine that you have a high resolution, high fps camera. To record its feed requires a memory card that is capable of handling the data rate that the camera produces. Otherwise, the events captured by the camera are lost forever as they are flushed away by the forthgoing frames. Essentially, the problem boils down to the ability to encode the information at least as quickly as the lifetime of the information.

I believe the same thing happens in thought. We have a state of mind, an idea, that we want to share. Because we can't simply transfer whole brain states between each of us, we first need to serialize our ideas in writing or speech. That process is lossy, and possibly lengthy (or at least longer than your train of thought). In the mean time, a variety of new thoughts come to your mind, flushing the original idea.

So following this analogy, a solution to the problem of lost ideas would be to have higher bandwidth means of expressing and recording ideas. Personally this problem annoys me quite a lot in a variety of incarnations, and I think advances in Human Computer Interactions will directly address this.

For instance, consider a program. Sometimes I have a full blown picture of the program I want to build in mind. Then along the many hours it takes to serialize this into a programming language, I battle myself, attempting to keep the idea fresh. There many steps along the way that contribute to the serialization cost:

    - The abstractions of the programming language.
    - The step from "programming language" to "abstract syntax tree".
    - The letters I have to type.
    - The movement of my fingers on the keyboard.
    - The translation of my brain from program construct, to
      structure, to sentences, to words, to tokens, to letters,
      to movement of my fingers, to keypresses.

There's many steps in this process that could potentially be removed to reduce the cost of serialization.

   * * *

So yes, I think this is a real problem, and I think that it can be addressed with better tools and techniques.

⬐ amasad

This seems like it could easily be confirmation bias. When you have an idea that comes as a fast-flowing "inspiration" and you manage to write it down, and it turns out to be relevant, it registers in your mind as a close-call. However, you'd also have lots of those fast-flowing ideas that are actually not that great and are just ignored.

⬐ Bartweiss

I know of at least one author (Roald Dahl?) who swears that he lost several novel-quality ideas to this pattern. The sort of thing where you're lying awake at night and go "that would be a great idea for a book, I should write it down in the morning!" He also recounts saving one published story he came up with in the car: he pulled over and scribbled it in the dust on the rear windshield.

Among authors alone, I would imagine we've lost a great many brilliant books because brainstorming is tricky and variable. Some days you get nothing, some days you get a dozen ideas and only one makes it out to the rest of the world.

I can't think of a good answer. Meditation and memory training might help, but are not silver bullets. Typing tends to be flow breaking (though I'm sure smartphones have improved matters) and vocalization is slow and awkward. It's hard to imagine anything short of EEG-translation removing the I/O limits on the mind.

⬐ sampl

> ...loosing a piece of inspiration because it's flowing too fast for writing it down

I've been using the voice recorder on my phone to capture inspiration for years now (plus journals, ambient recordings, etc). I've got 1,700 recordings now, and whenever I dive deep into a new project I listen through the relevant ones for help.

⬐ redler

If it's been years, how do you know which of the 1700 are the relevant ones? Are you tagging or transcribing them? Seems like this would be a good use case for even moderately accurate transcription of the audio files by software -- the way even moderately accurate OCR of a bag of scanned PDFs can provide the fodder for search tools like Spotlight to interactively surface approximately useful results.

⬐ sampl

Yes, I usually tag them when I'm done. Would love to have some good transcribing software for deeper search though.

⬐ jarmitage

This guy is trying to create a stream of conscious recording Memex:

http://quantifiedself.com/2009/09/the-social-memex/

----

Relevant excerpt from The Mars Trilogy by Kim Stanley Robinson:

"Actually, now that he thought of it, losing his train of thought had been happening a lot. Or so he seemed to remember. It was an odd problem that way. But certainly he had been aware of losing trains of thought, which seemed, in their blank aftermath, to have been good thoughts. He had even tried to talk into his wrist pad when such an accelerated burst of thinking began, when he felt that sense of several strands braiding together to make something new. But the act of talking stopped the mentation. He was not a verbal thinker, it seemed; it was a matter of images, sometimes in the language of math, sometimes in some kind of inchoate flow that he could not characterise. So talking stopped it. Or else the lost thoughts were much less impressive than they had felt; for the wrist recordings had only a few phrases, hesitant, disconnected, and most of all slow – they were nothing like the thoughts he had hoped to record, which, especially in this particular state, were just the reverse – fast, coherent, effortless – the free play of the mind. That process could not be captured; and it struck Sax forcibly how little of anyone’s thinking was ever recorded or remembered or conveyed in any way to others – the stream of one’s consciousness never shared except in thimblefuls, even by the most prolific mathematician, the most diligent diarist."