Sophie Libkind wrote:

> For me, it would be clearer to say: we use x≤y to mean "we need x, to get y".

That's not exactly what it means, since there may be many other \\(x'\\) with \\(x'\leq y\\). For example, in the resource theory that describes my baking, it is true both that \\(\textrm{egg white} + \textrm{sugar} \leq \textrm{meringue}\\) and that \\( $10 \leq \textrm{meringue}\\), since I can just go to the store and *buy* some meringue in case that I'm too lazy to make it myself (as often happens in practice).

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Ammar Husain wrote:

> "we need x, to get nothing" is not true.

Right. The correct interpretation of \\(x\leq 0\\) is "given \\(x\\), we can use it to obtain nothing", or equivalently we can discard \\(x\\) whenever we have it. As Keith says, at the level of categories rather than posets, this corresponds to the unit object [being terminal](https://golem.ph.utexas.edu/category/2016/08/monoidal_categories_with_proje.html), which Bob Coecke likes to call a [causal theory](https://arxiv.org/abs/1510.05468).

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John Baez wrote:

> in resource theory we say x≤y when given y, there exists a way to get x.

Now I'm confused about our notational convention for \\(\leq\\). That could equivalently be written as \\(y\geq x\\), for which the "given\\(y\\), there exists a way to get \\(x\\)" is my personally preferred way of reading it, since it's consistent with saying that "\\(y\\) is at least as good as \\(x\\)", which places \\(y\\) above \\(x\\). But in the main post, you seem to be writing \\(y\leq 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?

One of the nice things about the math is that it doesn't matter how we interpret it: all our theorems will be true regardless, and we can also come up with other interpretations, like "I prefer \\(y\\) over \\(x\\)" [as in economics](https://en.wikipedia.org/wiki/Preference_(economics)). But it's still important to have a convention in order to talk about things and in order to guide mathematicians in formulating and proving theorems and coming up with good definitions.

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Michael Hong wrote:

> I automatically start calculating yields

You can still do this in a resource theory, and (for me) this is one of the most interesting aspects of the story! For example, we may have \\(5\, \textrm{eggs} + 3\, \textrm{lemons} \leq 1\, \textrm{pie}\\); let me ignore the other ingredients and the fact that such a pie would end up a bit sour for simplicity. Here, \\(3\, \textrm{lemons}\\) is just shorthand for \\(\textrm{lemon} + \textrm{lemon} + \textrm{lemon}\\). The rules of the game imply that we can add this inequality to itself, resulting in \\(10\, \textrm{eggs} + 6\, \textrm{lemons} \leq 2\, \textrm{pies}\\). But perhaps in making one pie, we only make partial use of the third lemon. So when we make two pies, we may be able to get by with one lemon less, resulting in \\(10\, \textrm{eggs} + 5\, \textrm{lemons} \leq 2\, \textrm{pies}\\). This is called *economy of scale*. We can now ask what happens when we try to make *many* pies---then how many eggs and lemons do we need per pie? This would be the yield, right? Now we're in the territory of linear programming, as others have already pointed out, or more generally linear optimization over a convex set.

> For me, it would be clearer to say: we use x≤y to mean "we need x, to get y".

That's not exactly what it means, since there may be many other \\(x'\\) with \\(x'\leq y\\). For example, in the resource theory that describes my baking, it is true both that \\(\textrm{egg white} + \textrm{sugar} \leq \textrm{meringue}\\) and that \\( $10 \leq \textrm{meringue}\\), since I can just go to the store and *buy* some meringue in case that I'm too lazy to make it myself (as often happens in practice).

---

Ammar Husain wrote:

> "we need x, to get nothing" is not true.

Right. The correct interpretation of \\(x\leq 0\\) is "given \\(x\\), we can use it to obtain nothing", or equivalently we can discard \\(x\\) whenever we have it. As Keith says, at the level of categories rather than posets, this corresponds to the unit object [being terminal](https://golem.ph.utexas.edu/category/2016/08/monoidal_categories_with_proje.html), which Bob Coecke likes to call a [causal theory](https://arxiv.org/abs/1510.05468).

---

John Baez wrote:

> in resource theory we say x≤y when given y, there exists a way to get x.

Now I'm confused about our notational convention for \\(\leq\\). That could equivalently be written as \\(y\geq x\\), for which the "given\\(y\\), there exists a way to get \\(x\\)" is my personally preferred way of reading it, since it's consistent with saying that "\\(y\\) is at least as good as \\(x\\)", which places \\(y\\) above \\(x\\). But in the main post, you seem to be writing \\(y\leq 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?

One of the nice things about the math is that it doesn't matter how we interpret it: all our theorems will be true regardless, and we can also come up with other interpretations, like "I prefer \\(y\\) over \\(x\\)" [as in economics](https://en.wikipedia.org/wiki/Preference_(economics)). But it's still important to have a convention in order to talk about things and in order to guide mathematicians in formulating and proving theorems and coming up with good definitions.

---

Michael Hong wrote:

> I automatically start calculating yields

You can still do this in a resource theory, and (for me) this is one of the most interesting aspects of the story! For example, we may have \\(5\, \textrm{eggs} + 3\, \textrm{lemons} \leq 1\, \textrm{pie}\\); let me ignore the other ingredients and the fact that such a pie would end up a bit sour for simplicity. Here, \\(3\, \textrm{lemons}\\) is just shorthand for \\(\textrm{lemon} + \textrm{lemon} + \textrm{lemon}\\). The rules of the game imply that we can add this inequality to itself, resulting in \\(10\, \textrm{eggs} + 6\, \textrm{lemons} \leq 2\, \textrm{pies}\\). But perhaps in making one pie, we only make partial use of the third lemon. So when we make two pies, we may be able to get by with one lemon less, resulting in \\(10\, \textrm{eggs} + 5\, \textrm{lemons} \leq 2\, \textrm{pies}\\). This is called *economy of scale*. We can now ask what happens when we try to make *many* pies---then how many eggs and lemons do we need per pie? This would be the yield, right? Now we're in the territory of linear programming, as others have already pointed out, or more generally linear optimization over a convex set.