On numbers and games

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1. Smaug123Sep 2026
   (Apparently I was extremely unclear with this text. For clarity: if you want to actually understand surreal numbers, go and read On Numbers and Games, by Conway, which is a delightful book; or get an LLM to talk you through Wikipedia. Original text follows.)

   It’s a terrible explanation. A surreal number is defined as a pair of sets of surreal numbers (where you fiddle around the recursion in that definition by defining them in waves, so strictly speaking you’re defining “the surreal numbers born at time T” for each individual T given access to the surreal numbers born at all earlier times, and then you “take the union across all times”, scare quotes because there are too many times for this to result in a set). Zero is a surreal number but the LLM is using the word “zero” to mean “the set containing just the surreal number 0”; “nothing” here is the LLM’s obtuse word for the empty set. Wikipedia may actually be easier to follow.

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2. oerstedDec 2023
   Could someone recommend a good maths research book that covers the whole messy discovery process, rather than just the end result?

   I don't mind in which field, and it's fine if it's quite formal (not looking for pop science). But it would be good if it is relatively self-contained (doesn't require too much prior knowledge, it establishes some foundations and builds from them).

   I'm a fan of On Numbers and Games by Conway for instance, and the more literary Surreal Numbers by Knuth (written earlier but inspired by Conway's research before publication).

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3. shagieJan 2023
   If this interests you, the series "Winning Ways for Your Mathematical Plays" is something to add to your bookshelf. And if you ever see "On Numbers and Games" in a used book store (the $200 price tag on Amazon is a bit hard to swallow), its another one to get.
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4. gjm11Feb 2022
   My brain is absolutely hopeless at top-10-like things. But here are a few gems from some of those categories.

   Mathematics: John Conway's "On Numbers and Games". (If you've run across the term "surreal numbers", this is the original source. Absolutely gorgeous, but heavy going if you aren't actually a mathematician.) David McKay's "Information theory, inference, and learning algorithms". (What it says on the tin. Don't expect this to be a practical guide to training neural networks or anything like that; it is a mathematics book.)

   Philosophy: Derek Parfit's "Reasons and Persons". (Contains some exceptionally clear thinking on thorny questions about personal identity and the like.) Gary Drescher's "Good and Real". (Drescher is primarily a computer-science-and-AI guy rather than a philosopher. He addresses a bunch of hairy philosophical questions from that perspective and I personally like it a lot. I confidently predict that some people will hate it.)

   Science: Gnaedig, Honyek & Riley's "200 puzzling physics problems". (Just what it says. Roughly undergraduate-physics level, some easier, some harder.) Steven Vogel's "Life in moving fluids". (Explores, e.g., the very considerable differences in what living in water means for a bacterium and for a whale on account of their different size. Fairly technical; he has other lighter books.)

   Computing: Donald Knuth's "TeX: the program". (A single volume of literate code, constituting the whole of his TeX typesetting system. The style is pretty old-fashioned but he's a genius.) Steven Skiena's "The algorithm design manual". (This would not be my recommendation if you want just one algorithms book; that should be Cormen, Rivest, Leiserson & Stein. But it's an excellent complement to a more standard sort of algorithms book.)

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5. ColinWrightJun 2021
   >> 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.

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6. User23Apr 2020
   On Numbers and Games is one of the most entertaining mathematical books I've ever read. We've lost a genius in the true sense of the word.
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