Subtitle: Existential derivatives, and anti-derivatives.

Two definitions to start:

  • A natural number is a whole positive integer. 1, 2, 598, one hundred million and one, etc. Sometimes people include zero in that list, sometimes they don't.

  • A decimal is any number in between. 0.483694586238, or 0.33333333 (repeating), for example.


God advanced calculus was hard for me. (This was during my first year and a half at IIT, which were my only years at IIT.) I imagine now that it was autistic overwhelm limiting my already-limited working memory, and when you're trying to learn a new system without having any ability to take it in as new physical metaphor, ... well, it makes sense, from that perspective.

(My professor for those courses was kind. I'm grateful for that.)

An infinite decimal gives you infinitely accurate precision for a point on the line of all real numbers. That's one thing. The existence of an infinite set of natural numbers doesn't give you exactly that (because there's an interval of size 1 between each number), but your possession of the infinite set of natural numbers gives you just as much possibility for expression and specificity as does possession of infinite decimals.

Two asides, before continuing:

  • First, I got halfway through the above paragraph using the indefinite pronoun, as in "an infinite decimal gives one" as opposed to "gives you", and "but one's possession" as opposed to "but your possession". Mid-paragraph, I realized the ambiguity of "one" in a paragraph that talks about numbers, and even particularly mentioning 1-sized intervals. I decided to swap out "one" for "you" and inadvertently stumbled into the best metasemantic-slash-existential pun I've ever found, lol.

  • And second, though infinite decimals and infinite natural numbers are both infinite, I find myself thinking of decimals as not existing prior to addressing them, whereas I think of all natural numbers as definitely existing even if I haven't gotten to them yet. This is important!

A constant goes away when you take the derivative of an expression. That interval of size 1 (between natural numbers) collapses. That's how we jump the gap, in a style that feels distinctly Gödelian (will get to that in a sec), between the infinite addressability of our natural number world and the infinite addressability of our decimal world.

I wonder if I tend to live in natural number world most of the time. It'd make sense; it was a while before human society came up with the concept of zero, let alone negative numbers or decimals. I can count the number of people and places and things around me, and there are gaps between them, things are definitely separated. This is fairly analogous to the series of natural numbers: I can count them, and there are gaps clearly separating them. I wonder if this is why I feel like all natural numbers exist, even if I haven't thought to go looking for a particular number. Just because I haven't been to a specific natural number before, doesn't mean it doesn't exist, right? If you'd never met me, I'd still exist, right? ("Right???", he asked, suddenly terrified. lol)

If I live in natural-number-world, if I feel like I exist here, and if it feels to me like the whole of this world must continue to exist out into infinity, continuously and without any gaps of non-existence, then what do I make of decimal-world? It's a space I can imagine but that I do not live in. It's hard for me to feel my way in that space, feel how I fit in. It's easy to count myself (or maybe account for myself) in natural-number world:

  • 1 exists (the number)

  • one exists (the indefinite article)

  • I exist (the identity)

  • I am one

  • I am 1

Cool. I get that. But I don't know where I would fit in decimal-world. Perhaps because of that, I can't conjure up the physical experience of moving from one infinite decimal to the next. What would that even mean? I know what it means to move from 1 to 2, I can kinda even talk about how it physically feels, but how do you talk about a single infinite decimal, and the single infinite decimal adjacent to it? (Picture pi, an infinite decimal. What's right next to it?) And yet it is conceivable. I can define those terms with rigor, and there are maths that work in this world. But because I can't feel-know it, I gotta trust what the math is telling me. This is hard for me. I (Isaac) move through this world exclusively by physical intuition. Because my physical intuition doesn't work for me in this other concept-space, I am limited to standing at the window (or at the microscope), peering through. To decimal-world. That's what we're talking about. (Or is it????)

But I can imagine living in decimal-world, by projecting my experience of whole-number-world onto it. And it turns out that kind of thing works - if one (lol) defines a logically complete system, one can use that system to construct a statement like "this statement cannot be expressed using the logically complete system that this statement is expressed with". Because that statement was constructed using a logically complete system, the statement must be true. And yet, on its face, it seems like a contradiction. In investigating why it works, we (Gödel, originally) discover that a logically complete system can be used to define the terms of a new system, and crucially we find that the new system is also therefore and necessarily logically complete. It's like a natural-number-world citizen declaring "this problem can't be solved with natural numbers", and then using that as a springboard into decimal-world, where the problem is solvable, realizing that we only got there by building an interface to decimal-world using the tools we found in natural-number-world.

I mentioned earlier that it feels to me (using my physical intuition) that all things around me exist, even if I haven't personally encountered them yet, and I connected this to the idea that I am a born-and-raised resident of natural-number-world. I mentioned that I have a hard time thinking of all infinite decimals as existing, in contrast.

If we built that interface from natural-number-world to decimal-world, were I to use it and project myself into decimal-world, would I find myself thinking that all infinite decimals around me exist, even if I haven't met them yet?

On the other hand, what if this isn't natural-number world? I can take one thing, and split it in half, and the laws of reality are not violated. We think of splitting 1 apple into 2 halves, but we could also look at this through the lens of apple counting, whereupon we'd start with 0 (zero) apples, and then we'd have 0.5 apples, and then we'd have 1.0 apples. We might as well be in decimal-world.

If we're in decimal-world, what does whole-number-world look like? This thing should work in reverse, right? Never mind what's down there, what's up there? And if I were up there looking down here, would I still have a hard time thinking of everything down here as existing before I got to it?

Once more: Obviously I can come up with any infinite decimal (0.25983698273!) and say it exists (it exists!), and I could do that an infinite number of times (but I won't!). Logic tells me this is tantamount to demonstrating that all infinite decimals exist, but it still doesn't feel like they do, you know? What good is it to say they all exist, anyway? We don't have any actual need for decimal-world to be real to us in that way; decimal math is useful for us as a concept, without having to teach our physical intuition to think of it as a validly extant playing field.

That's my perspective from whole-number world, looking onto decimal-world. But if this works in reverse (and I feel that it must), then (1) what the hell does this world look like from further up, and (2) what does that world up there look like? If I'm here in decimal-world, and you're up in natural-number-world, ... what does that look like? And, you know, can we talk?

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