> 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.
Got lost here. I think I'm officially too dumb for math.
You can think of the "surreal numbers" as being built up step by step. We start out with no numbers at all, and then we repeatedly do a construction that makes some new numbers.
A surreal number is made from two sets of (pre-existing) surreal numbers. We typically call them L and R, for "left" and "right", and sometimes write it as L|R or {L|R} or something like that. The "left" numbers have to be smaller than the "right" numbers. The resulting number will turn out to be, in a certain sense, the "simplest" number in between all the left numbers and all the right numbers.
Now, as I said, we start out with no numbers at all. It might seem like that gives us no way to proceed, but it does: even given no numbers at all, we can still make a set of numbers, namely the empty set! So we can use that for both L and R, getting ∅|∅. Empty sets on both sides. We call this 0, and it will turn out to behave in the way you'd expect the number 0 to behave.
Now we suddenly have another set available, namely {0}, the set containing only zero. Which means that instead of being able to make one number, maybe we can make four: ∅|∅, ∅|{0}, {0}|∅, {0}|{0}. The first of these we already knew about. The last isn't actually admissible -- remember that the "left" numbers have to be smaller than the "right" numbers, which is "vacuously" true when one of those sets is empty (it means "if you have a number x in the left set, and a number y in the right set, then x<y", and if there are no numbers in the left set or no numbers in the right set then that's trivially true) but isn't true when both sets contain 0 because 0<0 is false.
So actually we get two new numbers: ∅|{0} and {0}|∅. The first fits into what OP calls the gap "between nothing and zero". The second first into what OP calls "the gap between zero and nothing". In both cases, "zero" means a number and "nothing" means a space where we don't yet have any numbers.
The number ∅|{0} is called -1 (it has to lie to the left of 0, and there's no constraint on its left, and -1 is "the simplest number less than 0") and the number {0}|∅ is called +1 (it has to lie to the right of 0, and there's no constraint on its right, and +1 is "the simplest number greater than 0").
I should explicitly acknowledge that I haven't defined what "less than" and "greater than" actually mean for these numbers, nor anything else about how they relate to one another that could possibly justify giving these things the specific names 0, -1, and +1. But there are definitions for "less than" and "greater than" and "plus" and "minus" and so forth, and the whole thing does turn out to work very nicely.
Anyway, once we've got these numbers we have eight possible sets that can go on the left or on the right. The requirement for left-things to be smaller than right-things reduces the possibilities somewhat, and the actual new numbers we get next time around are: ∅|{-1}, which turns out to be -2; {-1}|{0} which turns out to be -1/2; {0}|{+1} which turns out to be +1/2; {+1}|∅ which turns out to be +2. We also get some already-existing numbers in new ways; for instance, {-1}|{+1} is actually equal to 0 ("0 is the simplest number between -1 and +1"). Again, I should explicitly acknowlege that I haven't said anything about how you determine when two of these things are actually equal; again, it does all turn out to work properly.
If you keep going with this construction, you produce all the integers, two at a time, and also all the "dyadic rationals", meaning fractions where the denominator is a power of 2. And then, once you've got all those, at the next stage of construction you abruptly get all the real numbers -- e.g., the square root of 2 is L|R where L = {dyadic rational numbers that are negative or have a square smaller than 2} and R = {dyadic rational numbers that are positive and have a square larger than 2} -- and you also get {0,1,2,3,4,...}|∅, conventionally written as a lower-case Greek letter omega, which is an infinite number, larger than all the integers. (And its negation.) And {0}|{1,1/2,1/3,1/4,...} which is an infinitesimal number, positive but smaller than any ratio of positive integers. And you can then proceed further and construct a vast infinitude of numbers, including all the real numbers (which we've already made) and all of the so-called infinite ordinals (which you can kinda think of as being a sort of "infinite positive integer", though there's more to them than that) and much more, all in a system that lets you do arithmetic and suchlike. It's very elegant, if your brain has been twisted into the mathematician-y shape that finds such things elegant.
First: the construction of the real numbers from (traditionally) the rational numbers by means of "Dedekind cuts" (sometimes called "Dedekind sections"). The idea is that if you're trying to build up the machinery of mathematics from scratch, it's not too hard to go step by step from (say) sets to nonnegative integers to integers to rational numbers, but it's harder to get from there to the real numbers, and Dedekind's idea is to say that e.g. the square root of 2 is the way of chopping the rational numbers into "things less than the square root of 2" and "things greater than the square root of 2".
Second: the construction of the ordinal numbers (a sort of generalization of the notion of "nonnegative integer" that allows the numbers to get very infinite) due to von Neumann: you start off saying that zero "is" the empty set, and then you repeatedly say: the next ordinal "is" the set of all the ordinals you've constructed so far. So, e.g., 1 = {0}, and then 2 = {0,1}, etc. -- but once you've constructed all the nonnegative integers you can then look at {0,1,2,...} and that's a new ordinal typically called ω, and then you can take {0,1,2,...,ω} and call it ω+1, and so on and so forth.
Both of these are special cases of what Conway does: Dedekind's is the case where all the numbers are rational numbers and you don't allow either set to be empty, and von Neumann's is where you _require_ the right-hand set to be empty.
There's a further connection, which I believe is how Conway found these things in the first place: if in the definition of surreal numbers you delete the requirement that everything in L has to be less than everything in R, then what you've got is (more or less) the definition of a position in a two-player game. L is the set of positions one player can move to, R is the set of positions the other player can move to. (I say "more or less" because e.g. in many games you're allowed to repeat positions, and games may have complicated winning conditions or involve chance or whatever.) And there's a whole rather nice thing called "combinatorial game theory" that's all about these, and from that perspective numbers are just one particular kind of (position in a) game. (Specifically, a number is a game in which at no point in the subsequent gameplay can it ever make your position better for you to make a move: you'd always rather pass if you could.)
For the infinitesimal number, I think it makes more sense to use {0}|{1,1/2,1/4,1/8,...} since it gets born at the same day as say 1/3. So it is easier to understand how it arises without "waiting" for all reals.
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.
Only as a mental abstraction that's based on our experience/concept of space+time.
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?
Sorry it was confusing.
Edit: the picture is now edited into the article.
Some other comments clarify that "nothing" is more accurately "the empty set". This is helpful because at first I wrongly synonomized "nothing" with zero. But now I get tripped up on the "between" language. Maybe it's a lack of background in sets, but I don't know what "between" implies for an integer (zero) and a set (the empty set).
My favorite intro to surreal numbers is https://www.infinitelymore.xyz/p/surreal-numbers, but it is behind a registration wall.
That's a mistake. They should've written "empty set of surreal numbers" and not "nothing".
It is a constructive theory, like sets/ordinals. For ordinals you can use ∅, { }, ∪ and you construct
∅, {∅}, {{∅},∅}, ... (von Neumann ordinals).
You build up surreal numbers as
{∅|∅}, {∅|{∅|∅}}, {{∅|∅}|∅}
---
Edit: A happy accident: I denoted the "empty set of surreal numbers" with the symbol ∅. It works, and it gives you the surreal numbers. But if you think of ∅ as the empty set in set theory, then the same construction (using an ordered-pair construction) gives you the surreal numbers as sets!
I’m slightly stretching the metaphor here because the number lines gives me enough structure (order expressed visually) and I only deal with at most one set item at a time (since we construct in the order of simplicity and can use the already constructed numbers), so it (IMO) unnecessarily complicates things to even talk about sets when we’re sort of just making cuts on the line. But in either case I don’t see the problem with colloquially saying “nothing” here.
I'm hoping someone develops an interactive tutor that can teach any subject to any depth.
The tutor should optimize its pedagogy. It should use online RL to adapt to a learner's ideal learning style, model what the student understands and to what degree, and understand what the gaps and next steps are.
I'd subscribe in a heartbeat.
- https://github.com/mattpocock/skills/blob/main/skills/produc...
ie. i am an expert at zig, explain this c++ in terms of zig
Universities are great for networking, starting projects with other students (not the ones professors mandate), and learning lab sciences. In research, they're great for institutional knowledge, having a community of peers, getting guidance from research advisors, having real equipment and funding, etc. But there's a great need for AI tools to accelerate learning outside of that setting.
Anecdotally, I'm a working adult. I'm not going to waste time in college again. I need this for me.
> (crucially, “to the left of all” and “to the right of all” also count as “gaps”)
So there are two "nothings" here, left of "all" - i.e. the zero - and right of it.
Though I'm not quite sure how you'd get infinite or irrational numbers by this procedure. Wouldn't you simply get the rational numbers by this?
(unless the "put a number" step is doing more work here than it seems. He doesn't really say which number to put there. In the examples, he mostly did "new number = (left number + right number) / 2", with special cases if any number is "nothing" - but he never actually wrote what the rules are here.)
In each episode they make a major science-fiction style breakthrough and grapple with the consequences without revealing themselves.
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.
“Doing better than a totally useless explanation in fewer characters” is in general impossible, of course, eg if the first explanation has only one character.
> We may say that Cantor was only interested in moving ever rightwards, whereas Dedekind stopped to fill in the gaps, so that R was always empty for Cantor, never empty for Dedekind. It is remarkable that by dropping these restrictions we obtain a theory that is both more general and more easy to work with.
This is precisely the intuition I present to the reader of the article. I am relying on visual aid (concretely, the ordered number line) to imply the machinery explicit in the actual recursive definition. The intended reader of this article is not a mathematician, and I think intuition is vastly more important here.
And I don't think I'm conflating 0 with {0} as you claim. When I say zero is "between nothing and nothing", I mean 0 := {|}. When I say one is "between zero and nothing", I mean 1 := {0|}. When I say 1/2 is "between 0 and 1", I mean 1/2 := {0|1}. And so on. I elide "the simplest number" because I am already going in the order of simplicity. I do not need to explain that alternative spellings like 1/2 = {0.2 | 1} are valid because it is not relevant to establishing the mental model of birthdays.
For the finite cases in my explanation, I do not need to explain that the left and the right parts form sets because I only ever need at most one surreal on either side to define the next generation. I also do not need to state the left/right order condition because it is already visually implied by the picture. For the same reason, I do not need to explicitly quantify over the set of earlier-born surreals, since in these finite cases, if we go birthday by birthday, each next day's surreals are definable via the numbers already constructed by the previous day.
I agree that these finite examples don't spell out how to handle infinitely many bounds at the omega-th day, which is where I believe the illustration embedded below is more helpful. I still think "a gap beyond 0, 1, 2, 3, ... with nothing on the right" is a useful intuition when we get there.
For a more precise but accessible treatment, I think https://www.infinitelymore.xyz/p/surreal-numbers is much clearer than Wikipedia.