Hi, Tobias! You wrote:
> Now I'm confused about our notational convention for \\(\leq\\).
I'm not really confused, just pulled in different directions by conflicting desires built into the heart of mathematics. I think it's time to talk about that.
> But in the main post, you seem to be writing y≤x for "given y, there exists a way to get x", presumably since it matches up with the standard category theory conventions. So which one are we using?
Starting around [Lecture 20](https://forum.azimuthproject.org/discussion/2081/lecture-20-chapter-2-manufacturing#latest), I got really serious about using the convention that says "10 dollars is \\(\le\\) 20 dollars because given 20 dollars there exists a way to get 20 dollars". This seems so natural that the reverse convention would confuse everyone!
But this convention conflicts with another convention which I'd been using earlier, says "we write \\(x \le y\\) to mean \\(x \to y\\), that is, there exists a way to get from \\(x\\) to \\(y\\)". (Fong and Spivak won't mention categories until Chapter 3, but this means there's a _morphism_ from \\(x\\) to \\(y\\)."
I see that in this lecture I was still using that other convention. I'll edit it to fix that.
This conflict is built into the heart of mathematics, so there's really no way to avoid it, and ultimately everyone needs to understand it and get used to it.
We bumped into it earlier as follows: for any set \\(X\\), the power set \\(P(X)\\) becomes a poset where \\(S \le T\\) means \\(S \subseteq T\\). But in logic, these subsets correspond to propositions, and we say \\(S\\) implies \\(T\\) if \\(S \subseteq T\\). We could write this as \\(S \to T\\), though I've avoided doing that in my lectures.
Then we're in the situation where "from something small, we can get something big".
For example, the \\(\emptyset\\) corresponds to "false", and \\(\emptyset \subseteq S\\) for every \\(S \in P(X)\\). This says that "false implies anything", which most of us are used to. But it also says "from nothing you can get anything" - which sounds very bad if we're talking about resources theories!
There's no contradiction here, just cognitive dissonance. Ultimately one needs to get used to "opposite categories", or at least the *opposite* of a preorder, where we redefine \\(x \le y\\) to mean \\(y \le x\\).
But I will correct my post above to make it match my current conventions.