Thread

1. howunfortunate · 1789743040 · focus · HN ↗

   > On the second day, there are two gaps: “between nothing and zero” and “between zero and nothing”. Two numbers spawn in those two gaps. Call them –1 and 1.

   Got lost here. I think I'm officially too dumb for math.

   1. dubcanada · 1789743367 · focus · HN ↗

      I think it's just we don't have a way to say/write these numbers. So you make up a way to write them (-1 and 1) and continue.

      The numbers don't matter and you could replace -1 and 1 with anything. It's just easier to begin your new fake number at - 1 and 1. Because position does matter.

      1. xg15 · 1789744537 · focus · HN ↗

         There has to be some procedure how to come up with "new" numbers though, if you want to have more in the end than just a fancy binary tree - in particular if you want to map your "fake numbers" to the reals, infinity, etc.

         1. danabramov · 1789745834 · focus · HN ↗

            This procedure is enough. If you define addition and other operations in a certain way (as Conway did), it turns out that on the omega-th day (i.e. after initial infinite steps), all reals will be born.
      2. skeledrew · 1789744554 · focus · HN ↗

         > position does matter

         Only as a mental abstraction that's based on our experience/concept of space+time.

      3. howunfortunate · 1789745630 · focus · HN ↗

         This is very helpful!

         Basically what I take away is that we're inventing a new number system from scratch. So we're not "proving" that 1 is a number between 0 and the empty set. We're defining it as such, and it just so happens that a number system defined this way works out in convergent ways with other mathematics.

         Is that roughly right?

         1. danabramov · 1789745789 · focus · HN ↗

            Exactly.
      4. [deleted] · 1789746698 · focus · HN ↗

         [deleted]

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