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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 meani

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_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 ago
  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 10y ago
    > questions. How can one meaningfully speak of comparative measure between constructs that occupy distinct dimensional spaces. Question mark. (point well taken! :)
    - JadeNB 10y ago
      > 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 10y ago
        Thank you. So it is correct to think that such orderings are ~toplogical in contrast to the our ~geometric intuition?
        JadeNB 10y ago
        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).
  - JadeNB 10y ago
    > 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. By the way, after having noticed that the conversation in https://news.ycombinator.com/item?id=11952290 https://news.ycombinator.com/item?id=11952290 tended to involve frequent (what seemed to me to be) appeals to authority, which I think do not belong in mathematical discussion, I wanted to thank you for your measured tone: not "don't question these lofty personalities", but rather explicitly "do question them" (rather than assuming out of hand that an apparent objection is an actual one).
deleted 10y ago
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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 ago
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 10y ago
  > questions. How can one meaningfully speak of comparative measure between constructs that occupy distinct dimensional spaces. Question mark. (point well taken! :)
  - JadeNB 10y ago
    > 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 10y ago
      Thank you. So it is correct to think that such orderings are ~toplogical in contrast to the our ~geometric intuition?
      - JadeNB 10y ago
        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).
- JadeNB 10y ago
  > 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. By the way, after having noticed that the conversation in https://news.ycombinator.com/item?id=11952290 https://news.ycombinator.com/item?id=11952290 tended to involve frequent (what seemed to me to be) appeals to authority, which I think do not belong in mathematical discussion, I wanted to thank you for your measured tone: not "don't question these lofty personalities", but rather explicitly "do question them" (rather than assuming out of hand that an apparent objection is an actual one).

deleted 10y ago

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