Surreal Numbers – An Introduction (2019) [pdf] | Hacker News Reader

Surreal Numbers – An Introduction (2019) [pdf]

tondering.dk

110 pointsColinWright73 comments

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.

Here's an intro to surreal numbers I wrote a while ago: https://ianopolous.peergos.me/maths/surreal

This is a wonderful intro! Thanks for sharing..

Thanks! Glad you like it. I wrote it as an undergrad for the Oxford maths magazine at the time, Eureka. I'm still blown away by how beautiful Surreal numbers are.

I thought there had been more threads on this, but found only one:

Donald Knuth: Surreal Numbers [video] - https://news.ycombinator.com/item?id=11987415 - June 2016 (54 comments)

Do you literally just do this deduplication with your brain alone? Or do you have some kind of clever indexing scheme and, if the latter, have you ever written it up? Because that would be interesting to read about.

It is all explained at https://news.ycombinator.com/item?id=27284079 and the links back from there.

One helpful intuition for the surreals - the standard numbers are basically mapped to a ruler where the 'gap' between any two consecutive numbers (eg, 4 and 5) is intuitively equivalent to any other (eg, the gap between 0 and 1).

The concept of distance is abandoned in the construction of the surreals so it no longer makes sense to talk about the gap between 1 and 2 being the same as the gap between 2 and 3.

It is a very pure conception of numbers and it is quite pretty that it contains everything (integers, reals, infinitesimals, infinities, concept of infinity + 1, etc.

I was introduced to this concept through the Numberphile video and thought it was more about having fun with symbols / notation than contributing anything mathematically new or useful. I particularly object to the playing with infinities and infinitesimals as though they were numbers. As the pdf says on page 41:

> "According to theorem 5, {Z | } is a number that is greater than all integers. Therefore, its value is infinity!"

That makes no sense, nor does much of the exploration from this point. Other than playing with symbols, I'm not sure it has any value by itself.

That said, "playing with symbols" can still be of great value as how we symbolize and notate things can change our understanding of things.

When we analyse games, especially combinatorial games, we find that for a given game we can give positions "values". Then there's a concept of "Adding" game positions, and the value of the sum of positions is the sum of the values of the positions. So using "P" for position, and V(P) for the value of a position, we have V(P1+P2) = V(P1)+V(P2). (The additions here are on different types.)

So things work nicely, and it feels natural.

Then we find that the integers can be embedded naturally into the values we get. So the values from positions form an algebraic structure that extends the integers.

Then we find that there are game positions whose values are larger than all the integers. The obvious interpretation of this is that they are infinite, but are all part of a larger structure, one where addition and subtraction of these things still makes sense, and are still well defined.

When you say "That makes no sense", you're thinking of this in the limited space of counting, and ordinary numbers. But there are wider contexts where "simple counting" isn't enough, and in those cases it makes perfect sense, and is useful.

> "Then we find that there are game positions whose values are larger than all the integers. The obvious interpretation of this is that they are infinite, but are all part of a larger structure, one where addition and subtraction of these things still makes sense, and are still well defined."

But isn't this really more about notation than having "values" that are necessarily "larger than all integers"? That is, isn't this about needing a convenient way to notate something when we're already using integers for something else?

And/or do you know of any books or resources that go into more detail about this?

(As far as I can tell, the idea of surreal numbers still imagines a one-dimensional layout through which numbers are ordered. It's the adding of "infinity" and "1/infinity" to this line to which I object on nonsense grounds; using a non-integer symbol to represent an extension of some sort (non-spatially related) for some context still makes sense, I just don't think it implies a value "greater than all integers".)

>> Then we find that there are game positions whose values are larger than all the integers ...

> But isn't this really more about notation than having "values" that are necessarily "larger than all integers"?

We have a structure which is "values for games". That structure has addition and subtraction, "greater than" and "less than", and they behave a lot like numbers. We find that the integers embed naturally into this structure, so it's reasonable to think of this structure as an extension of the integers.

There are things in that structure that are greater than all the integers.

Call them what you like, the people who work with these things find it natural to refer to them as "infinities". I feel that your objections are arising from your desire to think of these things in the existing ways you think of the integers, but these things are not the integers, they are a structure in which the integers naturally live as a sub-structure. Not all of your intuitions will be helpful (although some will, and they are certainly understandable).

The classic works here are "On Numbers and Games", or "Winning Ways". The freely available on-line resources are scattered, non-comprehensive, and occasionally bordering on wrong, but they do exist, and they are free. Magic phrases to search for are things like "nimbers", "Combinatorial Game Theory", and "Sprague-Grundy Theorem". You might like to read the related thing that I wrote:

https://www.solipsys.co.uk/new/TheIndependenceGame.html?UF03...

That deals only with the most elementary aspects, specifically Nimbers, but if nothing else, you might like to skip to the references section of that for other links.

Looks interesting, will check it out! Thanks for the dialogue. :)

You're welcome ... your questions and observations are absolutely reasonable, so it's nice to have the opportunity to offer an alternate point of view.

Cheers!

Wikipedia also has a nice overview:

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

2019

Added above. Thanks!

Don't get weirded out by somebody thanking God for something. People thank God for many things, for instance thank God it's Friday.

I'm sure that your comment was well-intentioned, but it's best to leave provocations (such as religious provocations) in place rather than opening a new loop in the threads.