
Donald Knuth: Surreal Numbers [video] - Jerry2
https://www.youtube.com/watch?v=mPn2AdMH7UQ
======
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?

~~~
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.

------
ikeboy
Some quick googling turns up
[http://www.sciencedirect.com/science/article/pii/00973165729...](http://www.sciencedirect.com/science/article/pii/0097316572900635),
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](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.

~~~
f00_
[https://blog.archive.org/2016/06/02/copyright-offices-
propos...](https://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/)

------
bisby
[https://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).

------
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. ;)

~~~
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_...](https://en.wikipedia.org/wiki/Surreal_number#Application_to_combinatorial_game_theory)
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).

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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.

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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.

~~~
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](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.

~~~
majewsky
Unless you're writing LaTeX frequently. :)

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

