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# Lecture 18 - Chapter 2: Resource Theories

edited February 2020

Hi, I'm back from Leiden! I'm ready to rock and roll! Are you ready?

I could spend all day talking about the workshop on applied category theory and what we did there. We figured out the math of how cells "couple" chemical reactions, we learned an amazing connection between "lenses" in bidirectional programming and supervised learning in artificial intelligence, we launched a new journal... and much, much more. My brain is still recovering. But right now it's time to teach you some cool stuff!

Chapter 2 is about "resource theories". Resource theories help us answer questions like this:

1. Given what I have, is it possible to get what I want?
2. Given what I have, how much will it cost to get what I want?
3. Given what I have, how long will it take to get what I want?
4. Given what I have, what is the set of ways to get what I want?

In each of these questions we are given some "inputs" $$x$$ and we want some "outputs" $$y$$. In the first question we want a yes-or-no answer: can we get from $$x$$ to $$y$$. In the second and third questions we want a number for an answer. In the fourth question we want a set for an answer. We'll eventually get a unified framework for answering questions of all these kinds.

For example, we could be trying to make a lemon meringue pie. Have you ever made one? Not me! But here's a way to make one... at least before you put it into the oven: This kind of picture often called a "string diagram". Fong and Spivak call it a "wiring diagram", so we'll do that.

We can use this wiring diagram in many ways. The simplest way is to answer the first question in our list! Yes, it is possible to make an unbaked lemon meringue pie if you have a crust, a lemon, some butter, some sugar, and egg, and some more sugar.

(Of course you'll also need some kitchen tools. These could be included in a more detailed string diagram. For now I'm trying to simplify things. That's also why I'm not saying how much butter and sugar we need.)

Another thing we can do with this wiring diagram is connect it to another wiring diagram and build a bigger wiring diagram, like this: Now we are baking our pie! For this we need an oven.

If all we care about is the first question in our list - is it possible to go from some inputs to some outputs - wiring diagrams are overkill. For this question, we could summarize our first wiring diagram as an inequality:

$$\text{unbaked pie} \le \text{crust} + \text{lemon} + \text{butter} + \text{sugar} + \text{egg} + \text{sugar}$$ where we use $$x \le y$$ to mean "we can get $$x$$ from $$y$$". Similarly, we could summarize the baking process by

$$\text{baked pie} \le \text{unbaked pie} + \text{oven}$$ We can manipulate these inequalities in familiar ways to get

$$\text{baked pie} \le \text{crust} + \text{lemon} + \text{butter} + \text{sugar} + \text{egg} + \text{sugar} + \text{oven}$$ But what are these "familiar ways", exactly?

For starters, we are using the rules for a preorder, which say that

$$x \le x$$ and

$$x \le y \textrm{ and } y \le z \textrm{ imply } x \le z .$$ That's one reason we went through Chapter 1: to understand this stuff, you need to understand posets! But we are also using some rules about the symbol $$+$$. We're not doing ordinary addition of numbers here. So, what does this $$+$$ symbol really mean? More precisely: what rules does should it obey?

This question is one of the big topics of Chapter 2. If you want to peek ahead at the answer, read Section 2.2 of our textbook.

But first I need to say more about why resource theories are important. We do not really need them to make lemon meringue pies! But we do need them to deeply understand chemistry, thermodynamics, operations research, project management, information theory and the study of quantum entanglement... and how these subjects fit together! So, I want to say a bit about that.

If you want to learn more about resource theories, besides our textbook and my lectures I recommend these papers:

The first is more elementary, so start there! Luckily Tobias Fritz is my friend, and he's attending this course - so once we get going, we can ask him some questions!

To read other lectures go here.

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1.

We figured out the math of how cells "couple" chemical reactions

Sounds intriguing, where to learn the details? :)

Comment Source:>We figured out the math of how cells "couple" chemical reactions Sounds intriguing, where to learn the details? :)
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2.

Igor - we'll write a paper this summer and put it on the arXiv. I'll announce it here and on the Azimuth blog.

Comment Source:Igor - we'll write a paper this summer and put it on the arXiv. I'll announce it here and on the [Azimuth blog](https://johncarlosbaez.wordpress.com/).
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3.
edited May 2018

John, do you agree with the book's assertion that in economics liabilities created during production can just be ignored?

I find such an assertion very odd for this course, especially since the accumulation of industrial $$\text{CO}_2$$ (and other industrial pollutants) is a consequence of this attitude.

Comment Source:John, do you agree with the book's assertion that in economics liabilities created during production can just be ignored? I find such an assertion very odd for this course, especially since the accumulation of industrial \$$\text{CO}_2\$$ (and other industrial pollutants) is a consequence of this attitude.
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4.
edited May 2018

Keith - where does it say that, exactly?

In mathematics, the phrase "can be ignored" usually means "we're gonna ignore it for now because things will get more complicated if we don't, and we're trying to keep the math simple at first." That's completely different from "we're gonna ignore it because it doesn't matter".

Comment Source:Keith - where does it say that, exactly? In mathematics, the phrase "can be ignored" usually means "we're gonna ignore it for now because things will get more complicated if we don't, and we're trying to keep the math simple at first." That's completely different from "we're gonna ignore it because it doesn't matter". 
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5.

“Free disposal” is indeed a very common assumption in economic theory. Sometimes this is just done in the sense described by John (a simplification adopted because the model wants to focus on other issues); other times because it is considered realistic (indeed we have a lot of CO2 in the atmosphere precisely because it is/was free to dispose of it, i.e., not subject to disposal charges for the producer). And sometimes, of course, it is not assumed at all, e.g., when the point of the model is precisely to deal with a pollution externality.

Comment Source:“Free disposal” is indeed a very common assumption in economic theory. Sometimes this is just done in the sense described by John (a simplification adopted because the model wants to focus on other issues); other times because it is considered realistic (indeed we have a lot of CO2 in the atmosphere precisely because it is/was free to dispose of it, i.e., not subject to disposal charges for the producer). And sometimes, of course, it is not assumed at all, e.g., when the point of the model is precisely to deal with a pollution externality.
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6.

The problem is though, is that most people will try to get away with 'ignoring the complicated bits' for as long as possible, especially if for instance, the thing in question happens to be an economic liability that everyone wants to play hot-potato with.

Taking to the extreme, since under our current economic system labor costs are produced after the fact an employee has done labor, such an assertion that under production, 'you can throw anything you want away, and it disappears from view.' amounts to, an employer can 'throw away' their liabilities towards their employees and hold on to the money that was otherwise going to be paychecks.

A much much better assertion would be, when thinking about production,

'you can ignore anything you want away, and it temporarily disappears from view.'

To say otherwise would destroy linearity in economics.

(As an aside: if commodities can't technically be duplicated or destroyed, only converted into other commodities, these conditions imply that economics also follows a no-cloning and no-deleting theorems, much like quantum mechanics. Indeed, the quantum no-cloning and quantum no-deleting theorems are required to uphold the linearity of QM).

Comment Source:The problem is though, is that most people will try to get away with 'ignoring the complicated bits' for as long as possible, especially if for instance, the thing in question happens to be an economic liability that everyone wants to play hot-potato with. Taking to the extreme, since under our current economic system labor costs are produced *after the fact* an employee has done labor, such an assertion that under production, '*you can throw anything you want away, and it disappears from view.*' amounts to, an employer can '*throw away*' their liabilities towards their employees and hold on to the money that was otherwise going to be paychecks. A much much better assertion would be, when thinking about production, '*you can **ignore** anything you want away, and it **temporarily** disappears from view.*' To say otherwise would destroy linearity in economics. (As an aside: if commodities can't technically be duplicated or destroyed, only converted into other commodities, these conditions imply that economics also follows a no-cloning and no-deleting theorems, much like quantum mechanics. Indeed, the quantum no-cloning and quantum no-deleting theorems are required to uphold the linearity of QM).
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7.

@Keith: just to avoid misunderstanding, the "free disposal" that is a common assumption in economics is the assumption that "commodities" (or byproducts) can be destroyed (i.e., put out of the model's view) at will; there is no common assumptions that liabilities (i.e., things someone is legally liable for) can be freely disposed of. On the contrary, unless one is explicitly modeling bankruptcy constraints, it is generally assumed that debts will be repaid in full.

1) standard (a.k.a. "neoclassical", even though I dislike the term) economics is really not into "ignoring" things, be it permanently or, worse, temporarily. Model outputs are typically Nash equilibria of games with perfect recall. There are some theoretical models with imperfect recall, but I cannot recall seeing them in applied work.

2) there are plenty of non-linearities in economic models, so I would not worry much about destroying linearity (of course, there are also a lot of linear models and linear approximations of non-linear models, but that is not just in economics). Perhaps the only linearity that is really hard for standard economics to give up is the one embedded in Expected Utility Theory, i.e., the fact that (under the standard assumptions) agents (can be taken to) evaluate "lotteries" using functions that are linear in the probabilities of the various possible outcomes (though typically non-linear in the evaluation of the certain outcomes). Much of behavioral economics was born out of attempts to relax these assumptions and use more general evaluation functions.

Comment Source:@Keith: just to avoid misunderstanding, the "free disposal" that is a common assumption in economics is the assumption that "commodities" (or byproducts) can be destroyed (i.e., put out of the model's view) at will; there is no common assumptions that liabilities (i.e., things someone is legally liable for) can be freely disposed of. On the contrary, unless one is explicitly modeling bankruptcy constraints, it is generally assumed that debts will be repaid in full. A couple more comments: 1) standard (a.k.a. "neoclassical", even though I dislike the term) economics is really not into "ignoring" things, be it permanently or, worse, temporarily. Model outputs are typically Nash equilibria of games with perfect recall. There are some theoretical models with imperfect recall, but I cannot recall seeing them in applied work. 2) there are plenty of non-linearities in economic models, so I would not worry much about destroying linearity (of course, there are also a lot of linear models and linear approximations of non-linear models, but that is not just in economics). Perhaps the only linearity that is really hard for standard economics to give up is the one embedded in Expected Utility Theory, i.e., the fact that (under the standard assumptions) agents (can be taken to) evaluate "lotteries" using functions that are linear in the probabilities of the various possible outcomes (though typically non-linear in the evaluation of the certain outcomes). Much of behavioral economics was born out of attempts to relax these assumptions and use more general evaluation functions.
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8.
edited May 2018

@Valter

My point is that the byproducts of production are liabilities, and as such, they cannot be "freely ignored."

Just as it's your responsibility and no one else's to clean up after yourself when you make a mess (the byproducts of whatever you're doing), it is the entrepreneurs' responsibility to deal with the byproducts of products. In fact, as Prof. Ellerman will happily point out, under our economic system, since the entrepreneurs' have 100% of the discretionary control rights over production, they are only entitled to the finished product once they assume 100% of the responsibility over production, and therefore have swallowed 100% of the liabilities of production (including those byproducts that are produced) as well as own or have direct contractual control over 100% of the assets of production.

An entrepreneur may wish to ignore byproducts of production and 'freely dispose of it' (for instance, hot coals fall off a working train and catch fire to a farmer's field), but the courts would not be swayed by the standard economic appeal that the entrepreneur may 'freely dispose' the byproducts of production and will assign the liability to the entrepreneur.

Comment Source:@Valter My point is that the byproducts of production *are* liabilities, and as such, they cannot be "freely ignored." Just as it's your responsibility and no one else's to clean up after yourself when you make a mess (the byproducts of whatever you're doing), it is the entrepreneurs' responsibility to deal with the byproducts of products. In fact, as Prof. Ellerman will happily point out, under our economic system, since the entrepreneurs' have 100% of the discretionary control rights over production, they are only entitled to the finished product once they assume 100% of the responsibility over production, and therefore have *swallowed* 100% of the liabilities of production (including those byproducts that are produced) as well as own or have direct contractual control over 100% of the assets of production. An entrepreneur may wish to ignore byproducts of production and 'freely dispose of it' (for instance, hot coals fall off a working train and catch fire to a farmer's field), but the courts would not be swayed by the standard economic appeal that the entrepreneur may 'freely dispose' the byproducts of production and will assign the liability to the entrepreneur. 
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9.

Igor - we'll write a paper this summer and put it on the arXiv. I'll announce it here and on the Azimuth blog.

John - cool, looking forward for it.

Comment Source:>Igor - we'll write a paper this summer and put it on the arXiv. I'll announce it here and on the Azimuth blog. John - cool, looking forward for it.
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10.

I have a question about the translation between the wiring diagram and the poset interpretation. John, you wrote:

we use $$x \leq y$$ to mean "given $$x$$, we can get $$y$$"

In the case of the lemon meringue pie, I want "lemon $$\leq$$ lemon filling". However, it seems not that case just "given lemon, we can lemon filling" because we need more ingredients to make the filling. For me, it would be clearer to say: we use $$x \leq y$$ to mean "we need $$x$$, to get $$y$$". Would this still be an accurate interpretation?

Comment Source:I have a question about the translation between the wiring diagram and the poset interpretation. John, you wrote: > we use \$$x \leq y\$$ to mean "given \$$x\$$, we can get \$$y\$$" In the case of the lemon meringue pie, I want "lemon \$$\leq\$$ lemon filling". However, it seems not that case just "given lemon, we can lemon filling" because we need more ingredients to make the filling. For me, it would be clearer to say: we use \$$x \leq y\$$ to mean "we need \$$x\$$, to get \$$y\$$". Would this still be an accurate interpretation?
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11.

Consider the example of getting rid of commodities. "we need x, to get nothing" is not true.

Comment Source:Consider the example of getting rid of commodities. "we need x, to get nothing" is not true.
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12.

@Keith : we are not disagreeing. Of course, there is no reason to ignore relevant by-products other than in the "simplifying assumption" mode. In fact, standard economic theory says that efficiency requires "externalities" (e.g., pollution) to be properly priced or otherwise constrained - see the debates on carbon taxes vs carbon quotas (in models for "normative analysis", i.e., those used to discuss "what should be"). Still, if externalities are not priced and firms can in fact pollute without cost to themselves, then you need to make that assumption to describe their behaviour (in models for "positive analysis", i.e., those used to discuss "what is"). As I wrote earlier, that is how we can explain why we've got so much CO2 in the air: its disposal was free! Different models serve different purposes.

Comment Source:@Keith : we are not disagreeing. Of course, there is no reason to ignore relevant by-products other than in the "simplifying assumption" mode. In fact, standard economic theory says that efficiency requires "externalities" (e.g., pollution) to be properly priced or otherwise constrained - see the debates on carbon taxes vs carbon quotas (in models for "normative analysis", i.e., those used to discuss "what should be"). Still, if externalities are not priced and firms can in fact pollute without cost to themselves, then you need to make that assumption to describe their behaviour (in models for "positive analysis", i.e., those used to discuss "what is"). As I wrote earlier, that is how we can explain why we've got so much CO2 in the air: its disposal was free! Different models serve different purposes.
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13.

@SophieLibkind : I think you are considering a different theory, i.e., one based on a different preorder: John would not write "lemon ≤ lemon filling", but "lemon+butter+sugar+yolk ≤ lemon filling" (unless he decided to abstract from the need of the latter three ingredients in the same way as he is abstracting from the need of using a bowl and other tools).

I suspect that the reason for preferring a theory where "≤" is used for giving information on the resources that are sufficient for the output is more manageable than one in which it is used to give information on merely necessary resources is that it "composes better": if we wrote "lemon+butter ≤ lemon filling", "sugar+yolk ≤ lemon filling", and had one dose each of lemon, butter, sugar and yolk, the formalism would not be able to tell us that we get exactly one dose of lemon filling, rather than two or an undetermined number.

Comment Source:@SophieLibkind : I think you are considering a different theory, i.e., one based on a different preorder: John would not write "lemon ≤ lemon filling", but "lemon+butter+sugar+yolk ≤ lemon filling" (unless he decided to abstract from the need of the latter three ingredients in the same way as he is abstracting from the need of using a bowl and other tools). I suspect that the reason for preferring a theory where "≤" is used for giving information on the resources that are sufficient for the output is more manageable than one in which it is used to give information on merely necessary resources is that it "composes better": if we wrote "lemon+butter ≤ lemon filling", "sugar+yolk ≤ lemon filling", and had one dose each of lemon, butter, sugar and yolk, the formalism would not be able to tell us that we get exactly one dose of lemon filling, rather than two or an undetermined number. 
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14.

Consider not only externalities but also ecosystemic limits. How would you express those in resource theories?

Like, the Oglalla Aquifer has x amount of water that regenerates with y rate. How much water can be pumped out per year for human uses, e.g. irrigated farming?

Comment Source:Consider not only externalities but also ecosystemic limits. How would you express those in resource theories? Like, the Oglalla Aquifer has x amount of water that regenerates with y rate. How much water can be pumped out per year for human uses, e.g. irrigated farming?
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15.

It strikes me that if we can "throw stuff away" then it follows that we always have $$x + y \leq x$$

This makes the mysterious operation $$+$$ start to look a lot like the meet operation $$\wedge$$

Comment Source:It strikes me that if we can "throw stuff away" then it follows that we always have \$$x + y \leq x\$$ This makes the mysterious operation \$$+\$$ start to look a lot like the meet operation \$$\wedge\$$
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16.
edited May 2018

Like, the Oglalla Aquifer has x amount of water that regenerates with y rate. How much water can be pumped out per year for human uses, e.g. irrigated farming?

I don't think this is covered in the book.

For a simple model like this, you use linear programming. There are a number of elegant theorems regarding linear programming, in particular duality theorems.

I couldn't find much investigation into the connection of category theory to linear programming. There's Chavez (1989) who develops the category of $$LP^\ast$$. There's also Hochstattler et al. (1999).

Linear programming is a special class of convex optimization problems. There is bleeding edge research by Conal Elliot (2018) and Brendan Fong (2017) on connecting gradient descent techniques for solving these to category theory.

Comment Source:[Bob Haugen #14](https://forum.azimuthproject.org/discussion/comment/17809/#Comment_17809): > Like, the Oglalla Aquifer has x amount of water that regenerates with y rate. How much water can be pumped out per year for human uses, e.g. irrigated farming? I don't think this is covered in the book. For a simple model like this, you use [linear programming](https://en.wikipedia.org/wiki/Linear_programming). There are a number of elegant theorems regarding linear programming, in particular [duality theorems](http://web.mit.edu/15.053/www/AMP-Chapter-04.pdf). I couldn't find much investigation into the connection of category theory to linear programming. There's [Chavez (1989)](https://www.sciencedirect.com/science/article/pii/0097316589900319) who develops the category of \$$LP^\ast\$$. There's also [Hochstattler et al. (1999)](http://www.kurims.kyoto-u.ac.jp/EMIS/journals/CMUC/pdf/cmuc9903/honeset.pdf). Linear programming is a special class of convex optimization problems. There is bleeding edge research by [Conal Elliot (2018)](https://arxiv.org/abs/1804.00746) and [Brendan Fong (2017)](https://www.researchgate.net/publication/321347450_Backprop_as_Functor_A_compositional_perspective_on_supervised_learning) on connecting gradient descent techniques for solving these to category theory.
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17.

Matthew Doty #16

For a simple model like this, you use linear programming.

Or maybe constraint programming? Lots of limits like that, I only mentioned one as an example. But categories would seem to fit in somewhere...

Comment Source:[Matthew Doty #16](https://forum.azimuthproject.org/discussion/comment/17811/#Comment_17811) > For a simple model like this, you use linear programming. Or maybe constraint programming? Lots of limits like that, I only mentioned one as an example. But categories would seem to fit in somewhere...
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18.
edited May 2018

Keith wrote:

The problem is though, is that most people will try to get away with 'ignoring the complicated bits' for as long as possible, especially if for instance, the thing in question happens to be an economic liability that everyone wants to play hot-potato with.

Sure: in the real world, economics is often a way for people to justify doing what they already wanted to do. But there's nothing about the math of "resource theories" that forces us to ignore the complicated bits.

By the way, you never answered my question about where the book says "in economics liabilities created during production can just be ignored". I would find it shocking if Brendan and David were making serious pronouncements about economics! If they did, I'll tell them to fix it. I can more easily imagine that they were trying to keep their examples very simple.

Comment Source:Keith wrote: > The problem is though, is that most people will try to get away with 'ignoring the complicated bits' for as long as possible, especially if for instance, the thing in question happens to be an economic liability that everyone wants to play hot-potato with. Sure: in the real world, economics is often a way for people to justify doing what they already wanted to do. But there's nothing about the math of "resource theories" that forces us to ignore the complicated bits. By the way, you never answered my question about where the book says "in economics liabilities created during production can just be ignored". I would find it shocking if Brendan and David were making serious pronouncements about economics! If they did, I'll tell them to fix it. I can more easily imagine that they were trying to keep their examples very simple.
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19.
edited May 2018

Perhaps Keith was referring to p. 50?

"The basic idea in manufacturing is exactly the same as that for chemistry, except there is an important assumption we can make in manufacturing that does not hold for chemical reactions:

 You can throw anything you want away, and it disappears from view.


This simple assumption has caused the world some significant problems, but it is still in effect. In our meringue pie example, we can ask: “what happened to the egg shell, or the paper surrounding the stick of butter”? The answer is they were thrown away"

P.S. I just learned that, apparently, pressing Tab posts the comment!

Comment Source:Perhaps Keith was referring to p. 50? > "The basic idea in manufacturing is exactly the same as that for chemistry, except there is an important assumption we can make in manufacturing that does not hold for chemical reactions: > You can throw anything you want away, and it disappears from view. > This simple assumption has caused the world some significant problems, but it is still in effect. In our meringue pie example, we can ask: “what happened to the egg shell, or the paper surrounding the stick of butter”? The answer is they were thrown away" P.S. I just learned that, apparently, pressing Tab posts the comment! 
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20.

Valter - thanks for the reference. So, they didn't fail to point out the problem with this attitude; they're just noting that it's "still in effect".

Mathematically this assumption that you can freely discard anything you want is described by saying "the unit for the tensor product is terminal". We'll get to that in more detail later, I hope!

Comment Source:Valter - thanks for the reference. So, they didn't fail to point out the _problem_ with this attitude; they're just noting that it's "still in effect". Mathematically this assumption that you can freely discard anything you want is described by saying "the unit for the tensor product is terminal". We'll get to that in more detail later, I hope!
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21.
edited May 2018

Matthew - I fixed a typo in your last comment: you kinda mixed up my students Brendan and Brandon. It's confusing, I know! They even wrote a paper together once.

Comment Source:Matthew - I fixed a typo in your last comment: you kinda mixed up my students Brendan and Brandon. It's confusing, I know! They even wrote a paper together once.
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22.

After watching a Bob Coecke video, I see what the authors are trying to say. They want to treat production as a physical, causal theory.

That is to say, we ignore trash in the same way we ignore what's going on when thinking about how production of goods and services is organized.

Dually, if you're a civil engineer, you could ignore the productive bits and only focus on the trash, sewage, etc.

Comment Source:After watching a Bob Coecke video, I see what the authors are trying to say. They want to treat production as a physical, *causal* theory. That is to say, we ignore trash in the same way we ignore what's going on when thinking about how production of goods and services is organized. Dually, if you're a civil engineer, you could ignore the productive bits and only focus on the trash, sewage, etc.
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23.

After watching a Bob Coecke video, I see what the authors are trying to say.

Which video was that?

Is that also where you ignore the reproduction of the production workers? (Hi, Mom...)

Comment Source:> After watching a Bob Coecke video, I see what the authors are trying to say. Which video was that? Is that also where you ignore the reproduction of the production workers? (Hi, Mom...)
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24.
edited May 2018

ACT2018: Bob Coecke and Aleks Kissinger — Causality

If you look closely, you can see John and his wooly hair.

I don't know what you mean by 'ignore the reproduction of the production worker.'

Comment Source:[ACT2018: Bob Coecke and Aleks Kissinger — Causality](https://www.youtube.com/watch?v=JFQQgpqMWSo) If you look closely, you can see John and his wooly hair. I don't know what you mean by 'ignore the reproduction of the production worker.'
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25.

I don't know what you mean by 'ignore the reproduction of the production worker.'

To begin with, I meant the collective, rhetorical, "you", not you personally. Sorry, that was sloppily written and probably confusing.

But I was reading a bunch of statements in this thread about economics ignoring this or that aspect of economics, and in this statement,

we ignore what's going on when thinking about how production of goods and services is organized.

...the first aspect of ignoring what is going on that came to my mind, is that people work in those organizations, and were born and raised and fed etc in families, etc etc. And the work in the family is not paid, thus ignored or more accurately taken advantage of in a lot of economic analyses.

It's difficult to include everything and so I understand narrowing one's focus.

Comment Source:Thanks for the video link. >I don't know what you mean by 'ignore the reproduction of the production worker.' To begin with, I meant the collective, rhetorical, "you", not you personally. Sorry, that was sloppily written and probably confusing. But I was reading a bunch of statements in this thread about economics ignoring this or that aspect of economics, and in this statement, > we ignore what's going on when thinking about how production of goods and services is organized. ...the first aspect of ignoring what is going on that came to my mind, is that people work in those organizations, and were born and raised and fed etc in families, etc etc. And the work in the family is not paid, thus ignored or more accurately taken advantage of in a lot of economic analyses. It's difficult to include everything and so I understand narrowing one's focus.
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edited May 2018

I have a question about applying posets into resource theories.

How would you translate the reflexive property $$x \leq x$$ in a resource theoretic way?

The natural way for me to read it is "what goes in must come out" and ties in nicely with the impossibility of 100% yield in reality. Even processes that seemingly have 100% yield are microscopically less than 100% due to entropy. In the lemon pie example, even if you think you use all the butter you can really never use all the butter since there will be remnants left on everything you use to touch it which ends up going out as waste.

Is this a valid line of reasoning? If it is, then when people assume 100% yield like when you ignore by products, this would no longer be a preorder since reflexivity is disobeyed?

Comment Source:I have a question about applying posets into resource theories. How would you translate the reflexive property \$$x \leq x\$$ in a resource theoretic way? The natural way for me to read it is "what goes in must come out" and ties in nicely with the impossibility of 100% yield in reality. Even processes that seemingly have 100% yield are microscopically less than 100% due to entropy. In the lemon pie example, even if you think you use all the butter you can really never use all the butter since there will be remnants left on everything you use to touch it which ends up going out as waste. Is this a valid line of reasoning? If it is, then when people assume 100% yield like when you ignore by products, this would no longer be a preorder since reflexivity is disobeyed? 
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27.

Michael wrote:

How would you translate the reflexive property $$x \leq x$$ in a resource theoretic way?

To answer this question, you have to 1) decide on your interpretation of $$x \le y$$ and then 2) see what this says in the special case where $$x$$ is $$y$$. (In other words, don't tackle $$x \le x$$ head on; think of it as a special case of something more fundamental.)

1) In resource theories $$x \le y$$ often means "if you have $$y$$ you can use it to make $$x$$". For example, if you have $50 you can use it to make$10 (just give away $40), so$40 $$\le$$ $50. 2) Given this interpretation, $$x \le x$$ means "if you have $$x$$ you can use it to make $$x$$". And this is always true: you just do nothing. [...] when people assume 100% yield like when you ignore by products, this would no longer be a preorder since reflexivity is disobeyed? That's not right if we use my interpretation of $$x \le y$$. I have a feeling that when you are writing $$x \le y$$ you are thinking $$x \lt y$$. Comment Source:Michael wrote: > How would you translate the reflexive property \$$x \leq x\$$ in a resource theoretic way? To answer this question, you have to 1) decide on your interpretation of \$$x \le y\$$ and then 2) see what this says in the special case where \$$x\$$ is \$$y\$$. (In other words, don't tackle \$$x \le x\$$ head on; think of it as a special case of something more fundamental.) 1) In resource theories \$$x \le y\$$ often means "if you have \$$y\$$ you can use it to make \$$x\$$". For example, if you have$50 you can use it to make $10 (just give away$40), so $40 \$$\le\$$$50. 2) Given this interpretation, \$$x \le x\$$ means "if you have \$$x\$$ you can use it to make \$$x\$$". And this is always true: you just do nothing. > [...] when people assume 100% yield like when you ignore by products, this would no longer be a preorder since reflexivity is disobeyed? That's not right if we use my interpretation of \$$x \le y\$$. I have a feeling that when you are writing \$$x \le y\$$ you are thinking \$$x \lt y\$$. 
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28.
edited May 2018

So in a nutshell, $$x \leq y$$ where $$x$$ and $$y$$ are resources is a sort of dependency relationships between resources? That is to say, we care only about if there exists a "way" to "get" from one resource to another.

Comment Source:So in a nutshell, \$$x \leq y \$$ where \$$x \$$ and \$$y \$$ are resources is a sort of *dependency relationships* between resources? That is to say, we care only about if *there exists* a "way" to "get" from one resource to another.
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29.

Also, to get back into the spirit of things:

Puzzle KEP: Give examples of some resource theories.

For instance, academic papers that reference older academic papers are a resource theory.

Comment Source:Also, to get back into the spirit of things: **Puzzle KEP:** Give examples of some resource theories. For instance, academic papers that reference older academic papers are a resource theory. 
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30.

Keith: yes, in resource theory we say $$x \le y$$ when given $$y$$, there exists a way to get $$x$$.

In fact Fong and Spivak write $$y \to x$$ for this, which is perhaps slightly clearer. But it's just a different notation for the same thing.

Comment Source:Keith: yes, in resource theory we say \$$x \le y\$$ when given \$$y\$$, there exists a way to get \$$x\$$. In fact Fong and Spivak write \$$y \to x \$$ for this, which is perhaps slightly clearer. But it's just a different notation for the same thing. 
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31.
edited May 2018

The CNO cycle is a resource network. It's one of the many ways the universe breeds new elements from hydrogen: This is supposed to answer puzzle KEP 1, but I'm too shy to call this network a theory.

I've been researching this stuff in an attempt to use category theory to write fiction. Yes, I mean what I just said.

Comment Source:The CNO cycle is a resource network. It's one of the many ways the universe breeds new elements from hydrogen: <img src="https://upload.wikimedia.org/wikipedia/commons/thumb/2/21/CNO_Cycle.svg/768px-CNO_Cycle.svg.png"> This is supposed to answer puzzle KEP 1, but I'm too shy to call this network a theory. I've been researching this stuff in an attempt to use category theory to write fiction. Yes, I mean what I just said.
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32.
edited May 2018

John Baez wrote :

In resource theories $$x \le y$$ often means "if you have $$y$$ you can use it to make $$x$$".

The aha moment LOL. So we are only looking at the possibility of production and not the actual process. I usually deal with actual processes so when I see flow diagrams I automatically start calculating yields which caused the confusion.

I have a feeling that when you are writing $$x \le y$$ you are thinking $$x \lt y$$.

I am not sure what exactly your interpretation of $$x \lt y$$ is but you are probably right? LOL Following the flow, I would guess this would be the conversion of the actual process? The "or" in "less than or equal to" makes it into a possibility of production. But with just "less than" the conversion has made a choice of outcome?

So if $$a+b \leq c$$ where a,b,c are resources is a process that is possible, then the actual reaction would be $$a+b \lt a+b+c$$?

Comment Source:John Baez wrote : > In resource theories \$$x \le y\$$ often means "if you have \$$y\$$ you can use it to make \$$x\$$". The aha moment LOL. So we are only looking at the possibility of production and not the actual process. I usually deal with actual processes so when I see flow diagrams I automatically start calculating yields which caused the confusion. > I have a feeling that when you are writing \$$x \le y\$$ you are thinking \$$x \lt y\$$. I am not sure what exactly your interpretation of \$$x \lt y\$$ is but you are probably right? LOL Following the flow, I would guess this would be the conversion of the actual process? The "or" in "less than or equal to" makes it into a possibility of production. But with just "less than" the conversion has made a choice of outcome? So if \$$a+b \leq c \$$ where a,b,c are resources is a process that is possible, then the actual reaction would be \$$a+b \lt a+b+c\$$? 
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33.

Michael wrote:

So we are only looking at the possibility of production and not the actual process.

If we're taking option 1, then yes. If we're taking option 4, then no:

1. Given what I have, is it possible to get what I want?
2. Given what I have, how much will it cost to get what I want?
3. Given what I have, how long will it take to get what I want?
4. Given what I have, what is the set of ways to get what I want?

This is the difference between preorders and categories. In a preorder we only ask is it possible to go from $$x$$ to $$y$$, and if so we write $$y \le x$$ (or maybe $$x \le y$$ if we change our conventions about the meaning of $$\le$$. In a category we discuss the set of ways to go from $$x$$ to $$y$$; each way is called a morphism $$f : x \to y$$.

Fong and Spivak spend chapters 1 and 2 on preorders, and then move to categories.

Comment Source:Michael wrote: > So we are only looking at the possibility of production and not the actual process. If we're taking option 1, then yes. If we're taking option 4, then no: 1. Given what I have, _is it possible_ to get what I want? 2. Given what I have, _how much will it cost_ to get what I want? 3. Given what I have, _how long will it take_ to get what I want? 4. Given what I have, _what is the set of ways_ to get what I want? This is the difference between preorders and categories. In a preorder we only ask _is it possible_ to go from \$$x\$$ to \$$y\$$, and if so we write \$$y \le x\$$ (or maybe \$$x \le y\$$ if we change our conventions about the meaning of \$$\le\$$. In a category we discuss the _set of ways_ to go from \$$x\$$ to \$$y\$$; each way is called a **morphism** \$$f : x \to y\$$. Fong and Spivak spend chapters 1 and 2 on preorders, and then move to categories.
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34.
edited May 2018

Michael wrote:

I am not sure what exactly your interpretation of $$x<y$$ is

$$x < y$$ is always defined to mean $$x \le y$$ but not $$x = y$$. So, once you know what $$x \le y$$ means, you know what $$x < y$$ means.

Comment Source:Michael wrote: > I am not sure what exactly your interpretation of \$$x<y\$$ is \$$x < y\$$ is always defined to mean \$$x \le y\$$ _but not_ \$$x = y\$$. So, once you know what \$$x \le y\$$ means, you know what \$$x < y\$$ means. 
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35.

John wrote:

This is the difference between preorders and categories.

Got it. I will wait for the story to unfold.

Comment Source:John wrote: >This is the difference between preorders and categories. Got it. I will wait for the story to unfold. 
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36.

It's great to see that the lectures on Chapter 2 have started! Since David Spivak was here until this morning and we've been hard at work, I'm only getting around to reading this stuff now. I'm in the process of reading everything and will check in on the forum on a regular basis. It's exciting to see this happening here!

Comment Source:It's great to see that the lectures on Chapter 2 have started! Since David Spivak was here until this morning and we've been hard at work, I'm only getting around to reading this stuff now. I'm in the process of reading everything and will check in on the forum on a regular basis. It's exciting to see this happening here!
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37.
edited May 2018

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).

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, which Bob Coecke likes to call a causal theory.

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. 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.

Comment Source: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). --- 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.
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38.
edited May 2018

Hi, Tobias! You wrote:

Now I'm confused about our notational convention for $$\leq$$.

Me too! 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, 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.

Comment Source:Hi, Tobias! You wrote: > Now I'm confused about our notational convention for \$$\leq\$$. Me too! <img src = "http://math.ucr.edu/home/baez/emoticons/tongue2.gif"> 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.
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39.
edited May 2018

Tobias Fritz wrote:

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?

This is a very interesting way to express the connection between yield and economies of scale.

In chemistry, there are two kinds of yield, actual and theoretical. Theoretical yield assumes 100% conversion of the limiting reagent whereas actual yield is smaller to which the degree depends on equilibrium constants, losses, production of by-products, etc. So if I translate my definition of yield to orders, theoretical yield will be more like a poset where you don't have any cycles and reactions are one way tickets ($$a+b \rightarrow c$$)and actual yield will be more like a preorder where cycles are allowed and reactions or processes can go forward and backwards ($$a+b \leftrightharpoons c$$ which leads to the end result at equilibrium $$a+b \rightarrow a+b+c$$).

For example, let's say you have the reaction $$2H_2 + 1O_2 \rightarrow 2H_{2}O$$. Theoretically, 100% conversion then you will get $$2H_{2}O$$. But in reality 100% conversion is never observed in chemistry (It'd be interesting to make a list of real processes that do undergo 100% conversion). Nature forces equilibrium and so if you let this reaction go to equilibrium, you will end up with $$a H_2, b O_2, c H_{2}O$$ where a, b, c are dependent on thermodynamic conditions. You will always have some reagent leftover at the end even if its a microscopic amount.

Now this is speculation but if you make this into a batch process, you will get an accumulation of reagent after each cycle which can be recycled to make more product the following round. And like you said this will reduce the amount of reagent needed to make the same amount of product in subsequent cycles.

I am probably missing out on a lot the detail behind all of this and hope to learn more as the the story progresses :)

Comment Source:Tobias Fritz wrote: >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? This is a very interesting way to express the connection between yield and economies of scale. In chemistry, there are two kinds of [yield](https://en.wikipedia.org/wiki/Yield_(chemistry)), actual and theoretical. Theoretical yield assumes 100% conversion of the limiting reagent whereas actual yield is smaller to which the degree depends on equilibrium constants, losses, production of by-products, etc. So if I translate my definition of yield to orders, theoretical yield will be more like a poset where you don't have any cycles and reactions are one way tickets (\$$a+b \rightarrow c\$$)and actual yield will be more like a preorder where cycles are allowed and reactions or processes can go forward and backwards (\$$a+b \leftrightharpoons c\$$ which leads to the end result at equilibrium \$$a+b \rightarrow a+b+c\$$). For example, let's say you have the reaction \$$2H_2 + 1O_2 \rightarrow 2H_{2}O\$$. Theoretically, 100% conversion then you will get \$$2H_{2}O\$$. But in reality 100% conversion is never observed in chemistry (It'd be interesting to make a list of real processes that do undergo 100% conversion). Nature forces equilibrium and so if you let this reaction go to equilibrium, you will end up with \$$a H_2, b O_2, c H_{2}O\$$ where a, b, c are dependent on thermodynamic conditions. You will always have some reagent leftover at the end even if its a microscopic amount. Now this is speculation but if you make this into a batch process, you will get an accumulation of reagent after each cycle which can be recycled to make more product the following round. And like you said this will reduce the amount of reagent needed to make the same amount of product in subsequent cycles. I am probably missing out on a lot the detail behind all of this and hope to learn more as the the story progresses :)
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40.
edited May 2018

@Tobias and all others following this discussion, if we approach a general welfare as a "product", the maximum on economies of scale would occur for the maximum on the amount of users of this "product", ok? In your opinion, does this argumentation holds on the frame of a global equality promotion defense? What are its strong and weak features? Some initial expansion would trigger a positive feedback loop possible... In this scenario is implicit some limiting resources on the "Earth System". Best

Comment Source:@Tobias and all others following this discussion, if we approach a general welfare as a "product", the maximum on economies of scale would occur for the maximum on the amount of users of this "product", ok? In your opinion, does this argumentation holds on the frame of a global equality promotion defense? What are its strong and weak features? Some initial expansion would trigger a positive feedback loop possible... In this scenario is implicit some limiting resources on the "Earth System". Best
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41.

The difference between theoretical and actual yield, as discussed by Michael, is a very interesting issue. The good news is that we can formalize them both as resource theories! More precisely, the particular resource theory depends on which transformations you consider feasible. And the latter depends on whether you want to analyze an idealized or a realistic setup, and which realistic setup.

For example, if you do industrial chemistry, then indeed the transformation $$2H_2 + 1O_2 \rightarrow 2H_{2}O$$ should not be considered feasible. In other words, we would have a monoidal poset with $$2H_2 + 1O_2 \not\leq 2H_{2}O$$ -- although theoretical chemistry would tell us that such a transformation ought to be possible. So what we may be able to get in practice is perhaps something like $$6H_2 + 3O_2 \not\leq 4H_{2}O$$, which is less efficient. But I agree with Michael that the leftover reagent accumulates, so that we might also have $$2H_2 + 1O_2 + \mathrm{leftover} \leq 2H_{2}O + \mathrm{leftover}$$. In resource-theoretic parlance, $$\mathrm{leftover}$$ is a catalyst.

@Pierre: I personally haven't thought about it enough to comment on the consequences for economic policy. There are certainly many factors other than maximizing yield to take into account. For example, we may also want a robust economy and welfare system, which could be an argument for making it regional rather than global. So I'm not sure, but it's a great question, so perhaps others can say more.

Comment Source:The difference between theoretical and actual yield, as discussed by Michael, is a very interesting issue. The good news is that we can formalize them *both* as resource theories! More precisely, the particular resource theory depends on which transformations you consider feasible. And the latter depends on whether you want to analyze an idealized or a realistic setup, and *which* realistic setup. For example, if you do industrial chemistry, then indeed the transformation \$$2H_2 + 1O_2 \rightarrow 2H_{2}O\$$ should not be considered feasible. In other words, we would have a monoidal poset with \$$2H_2 + 1O_2 \not\leq 2H_{2}O\$$ -- although theoretical chemistry would tell us that such a transformation ought to be possible. So what we may be able to get in practice is perhaps something like \$$6H_2 + 3O_2 \not\leq 4H_{2}O\$$, which is less efficient. But I agree with Michael that the leftover reagent accumulates, so that we might also have \$$2H_2 + 1O_2 + \mathrm{leftover} \leq 2H_{2}O + \mathrm{leftover} \$$. In resource-theoretic parlance, \$$\mathrm{leftover}\$$ is a *catalyst*. @Pierre: I personally haven't thought about it enough to comment on the consequences for economic policy. There are certainly many factors other than maximizing yield to take into account. For example, we may also want a *robust* economy and welfare system, which could be an argument for making it regional rather than global. So I'm not sure, but it's a great question, so perhaps others can say more.
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42.
edited May 2018

One nice feature of resource theories is that they enable us to measure things like "general welfare" or "well-being" using preorders that are subtler than the set of real numbers. I keep thinking a lot of problems with our society are due to "flattening" preorders to total orders that can be embedded in the real numbers, in order to give everything a worth in dollars. When we do this, everything becomes interconvertible with everything else. This is very convenient but I think it's wrong, both morally and conceptually. Will future societies laugh at us for having money that's 1-dimensional?

I haven't developed these thoughts very much; it's hard for me to know exactly what to do with them. But both von Neumann–Morgenstern and Tobias have nice theorems about when preorders can be embedded in the real numbers, so I may talk about those in a future lecture. This lets us see which of the hypotheses of these theorems don't apply in real life.

Comment Source:One nice feature of resource theories is that they enable us to measure things like "general welfare" or "well-being" using preorders that are subtler than the set of real numbers. I keep thinking a lot of problems with our society are due to "flattening" preorders to total orders that can be embedded in the real numbers, in order to give everything a worth in dollars. When we do this, everything becomes interconvertible with everything else. This is very convenient but I think it's _wrong_, both morally and conceptually. Will future societies laugh at us for having money that's 1-dimensional? I haven't developed these thoughts very much; it's hard for me to know exactly what to do with them. But both von Neumann&ndash;Morgenstern and Tobias have nice theorems about when preorders can be embedded in the real numbers, so I may talk about those in a future lecture. This lets us see which of the hypotheses of these theorems don't apply in real life.
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43.

Re limiting resources, multiple constraints, thresholds, allocations, carrying capacity: here are some people working on all that: https://medium.com/@ralphthurm/what-are-thresholds-allocations-and-why-are-they-necessary-for-sustainable-system-value-fe127483c407

I am not advocating their approach, just got the article from some colleagues on a related project, don't know the people, and think "capitals" is a weird word for resources, but they do seem to be tackling some real problems that are related to resource theories.

Comment Source:Re limiting resources, multiple constraints, thresholds, allocations, carrying capacity: here are some people working on all that: https://medium.com/@ralphthurm/what-are-thresholds-allocations-and-why-are-they-necessary-for-sustainable-system-value-fe127483c407 I am not advocating their approach, just got the article from some colleagues on a related project, don't know the people, and think "capitals" is a weird word for resources, but they do seem to be tackling some real problems that are related to resource theories. 
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44.

Since we're in an economics mood, can resource theories be used to keep track who appropriates and who expropriates assets and liabilities?

That is to say, can we view David P. Ellerman's work Hume implies Lock in terms of resource theories?

Also, since there is enough economics discussion going on, perhaps an economics discussion group is in order.

Comment Source:Since we're in an economics mood, can resource theories be used to keep track who appropriates and who expropriates assets and liabilities? That is to say, can we view David P. Ellerman's work [Hume implies Lock](http://www.ellerman.org/wp-content/uploads/2012/12/Hume-implies-Locke.pdf) in terms of resource theories? Also, since there is enough economics discussion going on, perhaps an economics discussion group is in order.
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45.

Hmm, I wish I understood the Hume implies Locke paper!

My first reaction is to say that there are probably better ways of keeping track of ownership than resource theories, such as distributed consensus algorithms. Resource theories are good at expressing which things are possible and which ones are impossible for a single agent, which may be an individual, an entire society, or whatever else.

However, it is possible to develop multi-agent versions of resource theories, where each agent's (cap-)abilities are described by their own order. For example, we may have $$\textrm{flint} + \textrm{wood} \leq_A \textrm{flint} + \textrm{fire}$$ for agent $$A$$ who knows how to start a fire, but $$\textrm{flint} + \textrm{wood} \not\leq_B \textrm{flint} + \textrm{fire}$$ for agent $$B$$ who doesn't. This is secretly a special case of enriched category theory. I don't know if there is any relation to ownership. I'll ponder it some more and will reply here if I can think of more to say.

Comment Source:Hmm, I wish I understood the Hume implies Locke paper! My first reaction is to say that there are probably better ways of keeping track of ownership than resource theories, such as distributed consensus algorithms. Resource theories are good at expressing which things are possible and which ones are impossible for a single agent, which may be an individual, an entire society, or whatever else. However, it *is* possible to develop multi-agent versions of resource theories, where each agent's (cap-)abilities are described by their own order. For example, we may have \$$\textrm{flint} + \textrm{wood} \leq_A \textrm{flint} + \textrm{fire} \$$ for agent \$$A\$$ who knows how to start a fire, but \$$\textrm{flint} + \textrm{wood} \not\leq_B \textrm{flint} + \textrm{fire} \$$ for agent \$$B\$$ who doesn't. This is secretly a special case of [enriched category theory](https://en.wikipedia.org/wiki/Enriched_category). I don't know if there is any relation to ownership. I'll ponder it some more and will reply here if I can think of more to say.
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46.

Tobias, I am madly interested in topics like this, about capabilities of interacting agents. If you find any other leads, I would love to hear about them!

Comment Source:Tobias, I am madly interested in topics like this, about capabilities of interacting agents. If you find any other leads, I would love to hear about them!
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47.

We are also madly interested in topics like these and are actively working on them in that link as well as others.

Comment Source:[We](https://www.valueflo.ws/) are also madly interested in topics like these and are actively working on them in that link as well as [others](https://www.loomio.org/d/3wDCtkoG/structuring-the-oae-around-agents).
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48.

Typo: in sugar + over should be oven

Comment Source:Typo: in sugar + over should be oven
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49.
edited May 2018

Very interesting! I've been reading through Bob's links in order to get an idea of what those things are about. Especially the Valueflows seems close to resource theories in spirit, and I can imagine resource theory to play a part in such a framework, probably together with other components like some kind of powerful higher-order process calculus. However, it's very easy to promise too much of my own toy when I don't understand others' toys very well! One problem is that the multi-agent generalization that I described above does not take into account any potential interaction of the agents; it merely describes the capabilities of the two agents separately. But nobody has really yet worked out the details of this, so there might be much more to say.

Comment Source:Very interesting! I've been reading through Bob's links in order to get an idea of what those things are about. Especially the Valueflows seems close to resource theories in spirit, and I can imagine resource theory to play a part in such a framework, probably together with other components like some kind of powerful higher-order [process calculus](https://johncarlosbaez.wordpress.com/2018/05/12/rchain/). However, it's very easy to promise too much of my own toy when I don't understand others' toys very well! One problem is that the multi-agent generalization that I described [above](https://forum.azimuthproject.org/discussion/comment/17995/#Comment_17995) does not take into account any potential interaction of the agents; it merely describes the capabilities of the two agents separately. But nobody has really yet worked out the details of this, so there might be much more to say.
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50.
edited May 2018

Tobias

I can imagine resource theory to play a part in such a framework, probably together with other components like some kind of powerful higher-order process calculus

I can, too. That's what brought me to this class.

What I am missing in general in these resource flow notations is the representation of the method/process of getting from one resource to another $$x \leq y$$ - like, how do you get y from x?
Comment Source:[Tobias](https://forum.azimuthproject.org/discussion/comment/18021/#Comment_18021) > I can imagine resource theory to play a part in such a framework, probably together with other components like some kind of powerful higher-order process calculus I can, too. That's what brought me to this class. Your [comment upthread](https://forum.azimuthproject.org/discussion/comment/17995/#Comment_17995) said, in part > it is possible to develop multi-agent versions of resource theories, where each agent's (cap-)abilities are described by their own order. What I am missing in general in these resource flow notations is the representation of the method/process of getting from one resource to another \$$x \leq y\$$ - like, how do you get y from x? (Please forgive me if it should have been obvious.) I think the agent capabilities could be described pretty concisely by the processes they can execute. For example, https://en.wikipedia.org/wiki/List_of_manufacturing_processes