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Donald Knuth: Surreal Numbers [video]
- zargay01 10y agoWhat is to this day unknown about surreal numbers? What are the important/fun/etc. conjectures?
- drdeca 10y agoI 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.
- ikeboy 10y agoSome quick googling turns up http://www.sciencedirect.com/science/article/pii/0097316572900635 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 10y agoWrite a comment under the video, the numberphile people will definitely see it and perhaps pass it on
- ikeboy 10y agoDone.
- vanderZwan 10y agoIt's not showing under recent comments for me?
- ikeboy 10y agoThat'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 10y agoHeh. 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.
- eternalban 10y agoSuprised 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. ;)
- IngoBlechschmid 10y agoI 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_combinatorial_game_theory https://en.wikipedia.org/wiki/Surreal_number#Application_to_... for more information on this point.
- eternalban 10y ago> 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 10y agoThose 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.
- decentrality 10y agoI 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 https://youtu.be/mPn2AdMH7UQ?t=5m31s "Because we liked to have an affair in the hotel room." "So that was the plan."
- droopybuns 10y agoWhat is a 3:16 experimenter?
- Kristine1975 10y agoI guess this refers to Donald Knuth's "3:16 Bible Texts Illuminated": http://www-cs-faculty.stanford.edu/~uno/316.html 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 10y agoAnd for thr profane: http://www.larry.denenberg.com/Knuth-3-16/ http://www.larry.denenberg.com/Knuth-3-16/
- spot 10y agonot mentioned in the video: conway got the idea for surreal numbers by playing and studying go.
- xyience 10y agoHaven'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 10y ago{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.
- peter303 10y agoSo 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.
- VodkaHaze 10y agoI imagine it's because the dialogue format flows naturally compared to the traditional monologue paradigm in books.
- pasquinelli 10y agoseems pretty natural to put the questions you have answers to into your reader's head and then answer them.
- cJ0th 10y agoApparently the book is available on archive.org https://archive.org/details/SurrealNumbers 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 10y agoAnyone 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_ 10y agohttps://blog.archive.org/2016/06/02/copyright-offices-proposed-notice-and-staydown-system-would-force-the-internet-archive-and-other-platforms-to-censor-the-web/ https://blog.archive.org/2016/06/02/copyright-offices-propos...
- elasolova 10y agoSo 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 10y agoNo, 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 10y agoThe 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.
- jobigoud 10y agoSeveral 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?
- jarmitage 10y agoThis guy is trying to create a stream of conscious recording Memex: http://quantifiedself.com/2009/09/the-social-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."
- Bartweiss 10y agoI 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.
- bisby 10y agohttps://www.youtube.com/watch?v=1N6cOC2P8fQ 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).
- my_first_acct 10y agoRandom observation: On the bookshelf behind Prof. Knuth in the video, Stephen Wolfram's book "A New Kind of Science" is clearly visible.
- safiume 10y agoThis 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.
- ianopolous 10y agoI wrote a magazine article on surreal numbers: https://ianopolous.github.io/maths/surreal https://ianopolous.github.io/maths/surreal
- Iv 10y agoNeither Firefox nor chromium manage to render your mathematical expressions, which makes the whole thing hard to read.
- ianopolous 10y agoThank you for alerting me to that. Love a backwards compatible web!
- majewsky 10y agoUnless you're writing LaTeX frequently. :)
- ianopolous 10y agoShould hopefully be fixed now. The MathJax was loading over http, not https.
- unusximmortalis 10y agofrom what he's describing professor Donald Knuth had a channeling session when he wrote the theory of surreal numbers :)