Surreal Numbers (writing the first book) - Numberphile
Вставка
- Опубліковано 26 чер 2016
- Donald Knuth coined the term "Surreal Numbers" and wrote the first book about them after lunch with the man who devised them, John Conway.
More links & stuff in full description below ↓↓↓
More from this interview: • Surreal Numbers (Don K...
John Conway interviews: bit.ly/ConwayNumberphile
Surreal Numbers AMAZON BOOK LINK: bit.ly/KnuthSurreal
John Conway AMAZON BOOK LNK: bit.ly/GeniusAtPlayUS
Surreal Number graphic by Lukas Lanky
Knuth Dragon Curve: • Wrong Turn on the Drag...
Support us on Patreon: / numberphile
NUMBERPHILE
Website: www.numberphile.com/
Numberphile on Facebook: / numberphile
Numberphile tweets: / numberphile
Subscribe: bit.ly/Numberphile_Sub
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
Videos by Brady Haran
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanblog.com/
Sign up for (occasional) emails: eepurl.com/YdjL9
Numberphile T-Shirts: teespring.com/stores/numberphile
Other merchandise: store.dftba.com/collections/n... - Наука та технологія
"My wife did visit me a couple times that week. Because we always wondered what it would be like to have an affair in a hotel room." The man, the legend Donald Knuth.
nitowa1 wonder if there was any LaTeX involved in those visits
average numberphile enjoyer
Kids on the playground: "Yeah, well I'm infinity plus one!"
Grade school: "No such thing."
First year calculus: "We have to be careful not to treat infinity as a number, but it's useful for some calculations."
Mathematicians on a napkin over lunch: "The kid is onto something."
lol
You have to know the rules before you can break them.
@@jacobshirley3457 I disagree strongly in the case of mathematics. All you need to do is have respect for the concept of rules and consistency. That is, it's okay to break the rules if you can still find a way for things to be consistent, and once you have done so, you have found something truly marvelous.
and now we know that at least some math is completely absurd and should be discarded. There is no such thing as infinity +1 as by definition infinity is infinitely greater than the largest known number and adding one still yields infinity.
@@kennethgee2004 That's only with cardinal numbers. In ordinals, there is a family of infinities. There literally is infinity + 1. Aleph Null and such. There are also smaller and larger infinities. The whole numbers (1, 2, 3, 4, ... ∞) is a smaller infinity than [0, 1], or the real numbers that are in between 0 and 1 are larger than all the whole numbers, it's a larger infinity. 0, ... 0.00001 ... 0.0001 ... 0.3, ..... 0.6, ... 0.99 ... 0.99999999 ... 0.999999999999999999 ... 1.
Is this the man who invented the arrow notation?
yes
how does one write square roots? or ANY non-[in between |or extended from] function?
Also the man who invented TeX?
Yes, and he also wrote The Art of Computer Programming.
Actually, he’s still writing it.
It's always fun to see a legendary genius speaking as a normal guy in a one-on-one interview.
Wrote it in 6 days and rested on the 7th
Is Donald Kunth god?
He's surreal.
Conway is according to the book
He created Life.
Edit: He *could have* created Life, if he were Conway.
*****
D'oh! That's right.
I'm being oppressed by facts!
No; the rhetoric was successful the first time, why not try it again?
He was just trying to be interesting. IMO, dragging religious connotations into this just fouls it up.
Some quick googling turns up "The theory of well-quasi-ordering: A frequently discovered concept", 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.'
Cool
Name awesome! I hope he got this information
Fifty years behind on his writing? Wow, he procrastinates worse than I do.
I would worship him, but I'm watching UA-cam
+Callum Munro I'll do it after I finished watching UA-cam XD
Because it obviously wasn't obvious?
+Callum Munro I'll worship him tomorrow! Or maybe later, since I have many important UA-cam videos to watch…
+true reality writing ", obviously" IS a valid way to symbolise sarcasm.
Kappa makes as much sense as "obviously" - language is just what it is. I think "obviously" isn't as obvious at marking sarcasm as other methods tho.
BTW: what does kappa actually mean?
"There's no reason for us to think that the universe obeys the laws of real numbers."
This guy is a legend. One of the few names I remember from times when I studied CS in the 90-s. And now for the first time I actually see him talking ;)
I'm starting to come to the conclusion that a large part of Mathematics is really just a branch of Philosophy.
Most systems of thought are defined as being a specific branch of philosophy (e.g Science) . You would be correct
I think along similar lines too. If philosophy is putting forward an argument by making a premise and initial statements and following them to a logical conclusion with words, then maths could well be described as philosophy codified in numbers.
Will Fairweather
I didn't mean philosophy in the classical sense where everything is a philosophy. I meant in the existential "what does it all mean?" kind of philosophy. You knew what I meant. You're just trying to be contrary.
+TheDarxide23 You're asking a worldview question -- this has more to do with starting points (or initial premises) than logical conclusions. To arrive at viable premises, the best tools we have (as far as I know) are to evaluate existing belief systems for (1) correspondence to reality and (2) internal consistency.
The biggest difference between math and philosophy is that philosophers almost always disagree on premises--what we are accepting as true, what we are taking as definitions. Mathematicians rarely disagree on this. They will when something is first introduce, and over the years the best way of thinking about thing will be established. This is a HUGE difference between philosophy and mathematics. Math only need to accept the definitions that makes the math work out the way WE want. We don't actually need it to be "true" in the sense philosophers do. This makes mathematics much less restrictive.
This guy reminds me of my father who passed away last year, meekness & gentleness covering a awe inspiring intellect!
You can tell by the way he talks that this is someone with ALOOOOOOOOOT going on in his mind :O
He's just old. Not a bad thing as much as an inevitable thing, if you live long enough.
@@jacobshirley3457 Try getting old and tell me again that it's not so bad.
@@megamaser that's not what he meant
@@megamaser It's better than the alternative
@@iank2615 only up to a point
So in other words the book was the program Knuth wrote to help him remember (rediscover, via the help of two abstract entities in the universe he created) the proofs Conway provided. Sounds about right.
I can't thank you enough Numberphile for getting Donald tell the story live on camera. This is pure gold, and future generations will thank you for this :).
Man, right off the bat: I love this guy's creativity!
Get Donald Knuth to make a computerphile video.
get donald kunth to make any video
Yeah totally. Could talk about the failure of itanium for example. I think it's something he weighed in on before. Well anything really. He's one of the all time greats for sure.
Get Donald Knuth.
University/Graduate Mathematics : Numberphile :: Art of Programming : (???)
I have wanted this video for ages. Thank you!
watching this was sad he has so much to say but cant say it....
Who should explain it?
stefanozurich Like a stutter? I do the same thing but, for me, it's more like diarrhea of the mouth, constipation of the brain.
frtard Cool, who should explain it though? Who can explain it better?
He tries so hard
#SAVETHENUMBERS
Is this the guy who invented TeX?
yes
Yes, also invented Knuth Up Arrow Notation
@@hamiltonianpathondodecahed5236 This is a difference in opinion, and some people tend to believe mathematics is inventyed, others believe it is discovered. Personally, I tend to be on the more "discovered" side of thinking. However, notation is not a fundamental fact of the universe. There are multiple ways to represent the operations which can be represented in Knuth Up-Arrow Notation, and there is nothing inherent about using arrows to represent them. Your argument is like saying I "discovered" the word "florbifod" despite just making it up just now.
@@hamiltonianpathondodecahed5236 It's a bit of both. We invent axioms/definitions then discover facts about them.
Arnav kumar Sinha Well, notation isn’t discovered since it’s pretty arbitrary if you think about it. It’s invented ti describe concept that are discovered (e.g. hyperoperators). Still, Knuth’s a genius. No doubt about that ;)
I've been wanting you to do a video on surreals since your interviews with Conway. Excellent interview!
Fascinating , immediately purchased this book ! John Conway is a real genius !
I really enjoyed this, seems like such a genuine dude as well. Will have to give his book a read
How simple and humble was JC in referring to his own death.
Now, killed by the virus of COVID19 it seems to me a synchronicity.
Something surreal.
To infinity, and beyond John Conway!
Nino Meloni I didn’t know! I’m so sorry to hear this.
Fake. Comorbidity.
@@nintendo9231889 Monster group
Probably one the best videos by Numberphile.
That feeling, when you get back from a convention and are happy that you got a signature from someone you admire and the internets adds new people to that list immediately...
Man we got to bring 70s math back. Dialogues, games, fantastic stuff
The main application of Conway's version of surreal numbers is in game theory, where the advantage to a player of some well-balanced games is more than 0 but smaller than any positive real number. The books "Winning Ways" by Conway, Guy, and Berlecamp give details.
after watching Numberphile for so many years, it's still blows my mind, finding 'a new topic for me' that you already explored 6 years ago. Feels a bit like a worldline-spacetime-dilation ;-) THANK YOU
Wow, the legend himself!
Thank you, Numberphile :D
Thanks for the TeX, Don!
It seems so much a theoretical and outlandish concept, but actually I just recently used these to simplify an algorithm significantly. Basically I stripped all of border handling from the algorithm by only looking at the numbers between the reals (where no border could be) instead of the reals themselves. It went from about 250 LOC to 30 LOC (plus about 80 for obvious but many-cases back-and-forth transformation to ordinary reals). It is now also much easier to read and understand. It wasn't even a particularly sophisticated algorithm, it just simplified some filter expressions for number data.
I bought On Numbers and Games by Conway several years ago, and I have tried to read it a few times, but always get lost after a few chapters. I'm definitely going to have to find a copy of Knuth's book, I imagine after reading it, On Numbers and Games might finally make sense! This was a fantastic video, both technically informative and a great insight into the creative, artistic side of intellectual exploration that many people don't even know exists. Most people picture intellectual work as very austere and devoid of passion or art, and the truth couldn't be further from that!
When he said that about dropping slithers into holes we did not know that they existed that reminded me of fractals and the problem of measuring the length of the coastline, because there you also zoom closer and closer and find more and more nooks and stuff.
Numbers are fractals in that way.
Same when he talked about creating universes from simple rules. That reminds me of fractal curves.
His choppy speech is driving me insane. I'm trying to listen for his words but it's making me crazy.
I think he is getting old and you slow down a bit and the idea he was trying to explain is a complicated one.
Get over yourself
He sounds like that on 20-year-old lecture videos. Not as pronounced as here, though.
His finished thoughts are great! He just has the habit of speaking before his brain has fully finished his thought.
Pretty sure he is halting a stutter. Pretty well too I would say.
lol this is so brilliant... “the mistakes that I would make, the characters in the story would make those same mistakes”
Bradford Folkens I loved that :)
I'm so glad I ran across this interview. I should read more into the surreal numbers. I've been dissuaded by those stuck in the reals, but now that I'm no longer in school or stuck in endless internet debates, I think I should review the surreals.
That was amazing! Thank you!!
RIP John Conway. Your works shall be remembered!
Sad days
The voice of this man make me cry
Never trust anyone who sounds to confident in himself. Such a person has not, and will never begin to grasp the universe.
Only a very self-confident person would say that.
Really, i didnt finish the video because of that
Bronze To Challenger ?
Why?
he is as pure and this is as pure as art can ever be! he is a true artist.
1:02 - probably it was Pierre Jullien, „Sur la comparaison des types d’ordres dispersés“ from 1967
i actually loved how he speaks about how he created it, it's so cool
Seems like it took a lot of time and effort to edit this video. Thanks for making it flow well Brady!
This was a really fun book! Anyone who’s into recreational mathematics will enjoy it.
Do the series of Millenium Problems.
You mean gender studies? :)
+Gazeth Sonica Huh?
+Gazeth Sonica I meant the Millennium Problems (7) set by Clay Institute of Mathematics. One has been solved.
Not gonna pretend i knew, but in was under the assumption you probably did mean something related to math. :)
+Gazeth Sonica I think you heard of the Riemann Hypothesis. That is one of them, desperately waiting for someone to solve it.
...and on the seventh day he rested..
Donald is pure enthusiasm, intelligence and inspiration.
Beautiful interview.
I feeeeeeeel like they're just making stuff up XD
Math is really the study of systems of rules and their logical consequences. Mathematicians are basically allowed to make up whatever rules they want as long as the rules don't contradict themselves and they have interesting consequences.
I recommend the video "Mathematics: Measuring x Laziness^2" by Zogg from Betelgeuse.
Mathematics is a series of games, laying some groundwork and seeing what structure we can build on top of it. It just so happens that so many of our games become these magnificent descriptions of the real world and enhance our scientific understanding of the universe immensely.
I totally agree with you guys haha I just found it funny
No... Surreal numbers are a byproduct of studying mathematics via model theory. Model theory isn't just some random rules that were created to create the surreal numbers, maths isn't done like that. You don't just decide that you want surreal numbers and find some convenient rules that imply their existence, model theory is a way of studying the fundamental structures of all mathematical universes, and the insights of studying mathematics at this level of generality, among other achievements, uncovered the existence of surreal numbers naturally.
There is a whole area of mathematics, that features in most numberphile video not to do with number theory, done by mathematicians who do indeed just make rules up, but they do so to find interesting and challenging problems, not to create whole new theories of mathematics; mathematical theories come from studying first known 'natural' mathematical objects, then older theories that study general classes of these objects. Theory building is bottom up, not top down. The bottom however, as I've just explained, is not arbitrarily decided upon as you are suggesting.
knuth's stutters make me feel less insecure about my own :) since i stutter on words too because my mouth cant keep up as fast as im thinking
Gave me some perspective on what genius really is. Such an abstract thinker that he can't even keep up with himself sometimes! Great man and a fascinating mind.
first time see him talking, une légende ... emotions ...
It sounds like the Genesis of math.
"In the beginning there was nothing,
therefore there was one thing one void,
Thus there was two things, one and the void, zero."
The book cover is quite pretty!
Though I was also expecting some fancy math.
The very existence of nothing means there exists one thing.
It's beautiful.
The 7 day hotel room story alludes to the bible even further (and for me, much more obviously).
Nothing is not something, but the concept of nothing is not nothing.
This is how numbers are constructed from set theory. For the natural numbers it is called the Zermelo-Fraenkel set, after the guys who worked it out. Surreal numbers take a just slightly different approach, and suddenly ε is a number. I still can't believe it is really a thing. (But surreally it is.)
@@Woodside235 so are you Artrax from Das Dorf?
Am I the only one that caught the "because we had always wondered what it would be like to have an affair in a hotel room" @ 5:30 ?
Being so honest about this subject is really impressive ! Gogo Donald Knuth !
I find that very romantic to say.
I started this video after reading half of Knuths book on Surreal Numbers. It's really fun, funny and more detailed and so far, very easy to me, with little mathematical maturity. Recommended.
I bought the book several years ago. I think it was mentioned in some other book(maybe some psychology book?) i was reading at the time. Very cool book.
I'm amazed. 3 years of being a Numberphile subscriber, and this is the first video that has made absolutely no sense to me. I thought at first that the guy was just crazy, but apparently this is the man who created arrow notation, among other things. So I'm assuming he's just speaking on a higher level of intelligence than I can comprehend.
No, he just doesn't understand his own concept.
Did you check on Wikipedia who the heck is this grandpa? Please do!
@@EGarrett01 its John Conways concept. And to understand it, you need to interprete the sides in the set as options for Alice and Bill playing a combinatorial game
@@gereonlanzerath9286 He's the interviewee, he's presenting the idea. It's 'his own' in this case similar to how a person who burns meat would screw up "his own" steak that he was serving.
@@EGarrett01 ok. Thought you were talking about the concept of surreal numbers
I just invented a number that is bigger than zero but smaller than any surreal number and still it is as big as any infinity and it is also a negative number and if you divide it with zero you get the answer twenty six. I call it Zsrlinmber and to write it you use the letter H
Maan, I was keep getting 42 as the answer referring to an old research paper, idk it was in French, or maybe English, I can't recall. Thanks though
So you literally need to break mathematics in order to obtain the Zsrlinmber. I'm intrigued.
What happens when you divide H by plaid?
Emma May I have a hunch you get somewhere in or near the North Pole. Don't quote me on that though.
If you assume contradictory properties for the number, you can indeed prove anything - even that when you divide the number by 0, you get 26.
Soo cool!! Your books are amazing!!
I love this guy! What a nice story.
HAHAHAHA "...where Ibsen used to live, so I could get some of his VIBES..."
Vibesen.
I did not get the joke earlier, but thanks to you now I do ^_^
10:00 How inspiration works
I own an H. Ipsen poem book, printed in 1875. Some of his work is amazing!
When I was a kid I came up with the idea of 0.0 repeating 1 as the smallest number over 0, a few years ago I discovered from numberphile, that I was actually on to something, that number is an example of a surreal number!
Have we had Don on Computerphile yet?
I did my dissertation for a Masters on this - to this day still don't get it really.
Wow, that last comment from Conway has a new depth to it. RIP.
How wonderful that you got Donald Knuth! Thank you for this.
"Book tells how mathematical research is done"
"He gave me a napkin with some notes defining numbers"
Sounds funny, mathematicians are nice guys (:
8:33 These are smileys from an other planet.
One over a tiny sliver would be a great big sliver. Which makes me smile to put it like that.
Ah good old Don Knuth. Huge person in computer science... I still need to get and read his "Art of Computer Programming"
The concept seems rather similar to the one described in the video about "monads". I wonder what the technical differences are...
I think after watching this video, NJ Wiildberger's would be loosing his mind over surreal numbers lololol
you're still awesome, professor Wildberger ^_^
He wouldn't buy any of this hehe ^__^
He would have few things to say about it i would think.
+Blackadder 1620 he might as well try to refute that surreal numbers do not exist. Honestly, I might have forgotten the whole video if they actually have a formal definition what a surreal number is and if it is a subset of the complex. I admire the construction tho. Very creative.
+mdphdguy1 that I completely agree. I am curious to read his book.
+Johanus Gauss he really makes you think about the basics but, we have algebra and, it's hard to live without it even if some of the foundations aren't as strong as we would like. Give it another 100 years or so, im sure we'll know more.
Hearing Conway talking about his impending death at the end destroyed me.
he is as pure and this is as pure as art csn ever be! he is a true artist.
Um, why don't you just take more of whatever you were on during that week?
5:26 slowly clicks away from the video...
donald is the best! im surely gonna read that book of his, it's not even that long.
One of my favorite videos so far.
Can we get more explanation on what exactly is going on and why this is helpful. It seems like he's just replacing normal numbers with dots and lines and then he starts writing in normal fractional numbers
If you're interested in what's going on, the book "On Numbers and Games" by John Conway is arguably the best book.
It does not only cover the surreal numbers, but also games, which has the surreal numbers as subset (crazy, i know).
I agree with you that they should have gone much further.
He replaced normal numbers with dots and lines to represent how surreal numbers can be used to define any numbering system. While I do agree it's more convenient to use numbers already in place, the book is more of a proof of concept than anything else.
The surreals are my favorite ordered field. They're very interesting, and as evidenced by the disbelief about their existence in the comments here they're highly counterintuitive.
I was taught AP Calculus AB by his son. Needless to say, he was a great teacher.
Lovely. Thank you!
Donald Knuth
Mr Trump! What an opportunity to ask you a question that has troubled me. Some psychologists, although they haven't met with you personally, have published a description of your behavior as that of an "extreme narcissist." This obviously a serious diagnosis. It means that you may be suffering from a near-psychotic condition, which means you may actually be delusional. My question to you is, have you personally ever sought the help of a psychiatrist or psychologist? If so, what was their diagnoses? If you haven't been to a psychiatrist or psychologist, don't you feel it is time for you to consult one? After all, you yourself ought to be sympathetic to the notion that the commander-in-chief of US military should not be led by a madman.
Of course, I realize I speaking to a wall. A narcissist, especially an extreme one, seldom ever acknowledges his faults.
@@tapolna lern sum gramer
Normal people: 70% water
Gamers: 70% salt
Mathematicians: 70% acid
explanation needed
Arnav kumar Sinha
People are ~70% water by volume
Gamers, or any competitive person tend to be upset when they lose, or "salty"
The last line is obvious, so I won't waste time explaining it.
Hope this helps!
Conky im pretty sure he means the drug acid
Conky I would say chemist
Hel lo oh...! dang I feel dense - thanks
Gosh, another gentleman with infectious enthusiasm. Math seems to bring that outta people...
It is fascinating to see how inspired many of these mathematicians are.
This man is "The Man"... As a programmer myself, he is one of my references, i admire him, I respect him, I wish I could have learn with him and had a chance to work with him...
Those generation had great brains, Knuth, Kernighan & Richie, Aho Sethi And Ullman, van Dam, Foley, etc.
When I read some post-2k kids who believe THEY are the digital generation and mock "elders" as if they knew nothing about "the net", it pisses me off. All this brain washing caused by hyped media, who only knows about facebook and twitter, confusing internet and www, etc...
Not trying to be rude, just curious, but does Mr. (Dr?) Knuth have a stutter?
No. Did you not listen?
He wants to speak a lot of things but obviously no one can. So much information wanting to be put out and the mouth can't handle it.
Wow, you registered it? It's a wonder your achievements will never even approach his.
Absolutely brilliant
2:25 There exists an empty, a null set.
2:27 Each number has one after it.
It's the first two Russells' prepositions for his following reasonings in Principia Mathematica.
6:53 Set of all set that does not contain themselves (Russell's paradox).
Gentleman on the recording has recreated a set theory.
7:45 Ok, synthetic, formal languages.
11:15 Tape with whole numbers from a busy beaver POV.
I love the book reviews. He truly gives 1/ω shits. :)
So he at least cares a little bit.
Yup... the littlest bit.
What about cereal numbers? Are there any?
There’s 0 , the cheerio number.
He has the same copy of wolfram's A New Kind of Science on his shelf as I have on my shelf. I like him
what a mind blowing talk
1:42
"That's what almost everybody thought except Kanye"
I soon realised that he didn't say 'Kanye'.
Funnily enough, I remember going into a store during my teen years and asking for the new "Conway West" album 😆
Just wondering if i understand this right, because im a layman.
1=0,99999_
but if you use surreal numbers then X can exist where
1>X>0,99999_
Would that be true?
Yup
Surreals are weird.
Well, I don't think that the notation 0.999... makes any sense here. And if 0.999...=1, then that's it, and there wouldn't be any X between, because then 0.999.. and 1 couldn't be equal anymore.
Zardo Dhieldor
I think that's the reason why X can only exist as a surreal number. What you describe is all still within real numbers.
I'm not saying that with intent of 'correcting' you btw. As i mentioned earlier I'm a layman and quite ignorant of this topic.
Gazeth Sonica
Not really. It just depends on what you mean by 0.999...!
It's rare to get the chance to listen to genius
I love the image of the Research files!
I would be frustrated as hell reading that book. It's Alice and Bob! Not Alice and Bill!
I was just about to say the same thing!