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# Eugenia Cheng on "Category theory in life"

edited May 2018

Ross Street. a famous category theorist at Macquarie University near Sydney, sent out an email saying just "Eugenia's in town!", with this photo from his cell phone:

Eugenia Cheng did her Ph.D. with Martin Hyland in Cambridge, developing and fixing some ideas James Dolan and I had about $$n$$-categories. She's become a quite famous explainer of mathematics. She's giving a public talk here, explaining that category theory is "the logical study of the logical study of how logical things work". On May 28th she's giving a talk in Brisbane:

Abstract. Category theory can be thought of as being “very abstract algebra”. It is thought of as “too abstract” by some people, and as “abstract nonsense” by some others. In this talk I will show that while it is abstract, it is far from being nonsense. I will argue that the abstraction has a purpose and that broad applicability is one of the powerful consequences. To demonstrate this, I will show how I apply concepts of category theory to important questions of life such as prejudice, privilege, blame and responsibility. I will introduce the category theory concepts from scratch so no prior knowledge is needed. These concepts will include objects and morphisms, isomorphisms and universal properties.

Here's a talk she gave at "Lambda World" last year:

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edited May 2018

I really liked that video, so much so I spread it around on my social media.

Comment Source:I really liked that video, so much so I spread it around on my social media. 
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edited May 2018

I was there, it was a fantastic talk! Dr Cheng is a wonderful communicator of category theoretic ideas, and just as kind and funny in person. I wish I'd known Ross Street was there, it would have been nice to meet him too!

Comment Source:I was there, it was a fantastic talk! Dr Cheng is a wonderful communicator of category theoretic ideas, and just as kind and funny in person. I wish I'd known Ross Street was there, it would have been nice to meet him too!
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Ken - wow, you were there! I know Eugenia Cheng and Ross Street quite well, since I spent a sabbatical in Macquarie talking to Street, and also visited Cambridge while Eugenia was a grad student there doing her thesis on some math I'd helped develop. They're both great people. Eugenia is also a concert pianist!

Comment Source:Ken - wow, you were there! I know Eugenia Cheng and Ross Street quite well, since I spent a sabbatical in Macquarie talking to Street, and also visited Cambridge while Eugenia was a grad student there doing her thesis on some math I'd helped develop. They're both great people. Eugenia is also a concert pianist!
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Do you think we could get Eugenia to come by and say "Hi"? :)

Comment Source:Do you think we could get Eugenia to come by and say "Hi"? :)
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edited May 2018

She's pretty damned busy these days, since she gave up tenure at the University of Sheffield in order to make her living as a math popularizer and author. She also teaches at the Art Institute of Chicago, but she's running around giving lectures and book tours all the time. So I don't bother her much, though when we last met - in Warsaw, at Higher Dimensional Rewriting and Applications 2015, where we were both speakers - we spent a few days talking a lot and catching up.

Comment Source:She's pretty damned busy these days, since she gave up tenure at the University of Sheffield in order to make her living as a math popularizer and author. She also teaches at the Art Institute of Chicago, but she's running around giving lectures and book tours all the time. So I don't bother her much, though when we last met - in Warsaw, at [Higher Dimensional Rewriting and Applications 2015](http://hdra.gforge.inria.fr/2015/), where we were both speakers - we spent a few days talking a lot and catching up.
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edited June 2018
Comment Source: * [The art of logic: how to think like a mathematician](http://www.rigb.org/whats-on/events-2018/july/public-the-art-of-logic?cm_ven=RIGB&cm_cat=14+subscriber+NL&cm_pla=All+Subscribers&cm_ite=http%3a%2f%2fwww.rigb.org%2fwhats-on%2fevents-2018%2fjuly%2fpublic-the-art-of-logic&cm_lm=003D000002M5ObvIAF&cm_ainfo=&&&&&)
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That's almost exactly the kind of thing I would have liked to do---something Very applied to all kinds of things (as is done to some extent by people at SFI , as well math sociologists, psychologists, economists, econophysicists etc.)

I probably was not PhD level material (though i passed a couple grad school classes as an undergrad) but if you are lucky (unlike me--too stupid to even use what luck i have--win the lottery and either have your jackpot stolen or squander it since that is what your peers tell you to do even if you'd prefer to do something else ) there is a place for people not doing research and college teaching. (You can teach at lower levels which i did--but i had same issues as in academia --personal conflicts, narrow focus, biases and prejudice---so i quit. In age of internet you can go independent and get everything online (except personal and financial support. I'm no Ramanujan but he's an example of what can happen with no support. Julian Barbour managed to find a side gig and be independent. ).

Most of what i could get when involved with academia was stuff like modeling conformations of DNA , and stuff related to biotech. Thats fine and itslef can be interesting and sometimes useful (but in this age of 'Big Pharma' and the opiate epidemic , and obesity fueled by junk food, you can question exactly how valuable biomedical research focusing on genes and biomolecules is---studying prejudice, blame , responsibility and priviledge may be more socially valuable, along with environment, behavioral economics (alot of which now is applied by corporations for marketing) , inequality, intolerance, etc.

In my academic world these topics were basically taboo----people are too busy saving the world to deal with 'victimology'---those are the whiners and complainers we are trying to help through our research. They say 'We Gotta save the (or at least my world)---publish or perish'. 'All boats will rise , and when they do we will start to feed, medicate , educate and find jobs for the whining starving babies, who will then be 100 years old if still alive. 'all happy together, so goes the weather, forever'.

I wonder if the application of category theory to social issues or problems leads to anything new and/or applicable. Maybe all you get is an elegant, aesthetically attractive reformulation of things like 'theory of war, class struggle and conflict ' (eg Louis Fry Richardson, Goodwin --both pioneers of nonlinear dynamics) written in math language as opposed to words.

I see this is from Chicago. My area (Wash DC) has a few gun shootings and homicides every week, Baltimore has maybe 1 /day, and south side chicago has more than that. I bet one could categorize this issue. The set of people who go to U Chicago is not the same as the set who live a few blocks from there.

Comment Source:That's almost exactly the kind of thing I would have liked to do---something Very applied to all kinds of things (as is done to some extent by people at SFI , as well math sociologists, psychologists, economists, econophysicists etc.) I probably was not PhD level material (though i passed a couple grad school classes as an undergrad) but if you are lucky (unlike me--too stupid to even use what luck i have--win the lottery and either have your jackpot stolen or squander it since that is what your peers tell you to do even if you'd prefer to do something else ) there is a place for people not doing research and college teaching. (You can teach at lower levels which i did--but i had same issues as in academia --personal conflicts, narrow focus, biases and prejudice---so i quit. In age of internet you can go independent and get everything online (except personal and financial support. I'm no Ramanujan but he's an example of what can happen with no support. Julian Barbour managed to find a side gig and be independent. ). Most of what i could get when involved with academia was stuff like modeling conformations of DNA , and stuff related to biotech. Thats fine and itslef can be interesting and sometimes useful (but in this age of 'Big Pharma' and the opiate epidemic , and obesity fueled by junk food, you can question exactly how valuable biomedical research focusing on genes and biomolecules is---studying prejudice, blame , responsibility and priviledge may be more socially valuable, along with environment, behavioral economics (alot of which now is applied by corporations for marketing) , inequality, intolerance, etc. In my academic world these topics were basically taboo----people are too busy saving the world to deal with 'victimology'---those are the whiners and complainers we are trying to help through our research. They say 'We Gotta save the (or at least my world)---publish or perish'. 'All boats will rise , and when they do we will start to feed, medicate , educate and find jobs for the whining starving babies, who will then be 100 years old if still alive. 'all happy together, so goes the weather, forever'. I wonder if the application of category theory to social issues or problems leads to anything new and/or applicable. Maybe all you get is an elegant, aesthetically attractive reformulation of things like 'theory of war, class struggle and conflict ' (eg Louis Fry Richardson, Goodwin --both pioneers of nonlinear dynamics) written in math language as opposed to words. I see this is from Chicago. My area (Wash DC) has a few gun shootings and homicides every week, Baltimore has maybe 1 /day, and south side chicago has more than that. I bet one could categorize this issue. The set of people who go to U Chicago is not the same as the set who live a few blocks from there. 
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edited June 2018

Mart Malakoff wrote:

Most of what i could get when involved with academia was stuff like modeling conformations of DNA , and stuff related to biotech. Thats fine and itslef can be interesting and sometimes useful (but in this age of 'Big Pharma' and the opiate epidemic , and obesity fueled by junk food, you can question exactly how valuable biomedical research focusing on genes and biomolecules is---studying prejudice, blame , responsibility and priviledge may be more socially valuable, along with environment, behavioral economics (alot of which now is applied by corporations for marketing) , inequality, intolerance, etc.

I think a bit part of why people don't try to model prejudice, blame, responsibility and privilege. If it doesn't involve numbers, too many people it's un-modelable. To many many people, math is strictly about numbers.

It's interesting though because things like assigning responsibility to the appropriate parties are equivalent to assigning property under the labor theory of property, or as one (and perhaps the only) economist (David P. Ellerman) points this out clearly,

In the context of assigning property rights and obligations, the juridical principle of imputation is expressed as the labor theory of property which holds that people should appropriate the (positive and negative) fruits of their labor. Since, in the economic context, intentional human actions are called “labor”, we can express the equivalence as:

The Juridical Principle of Imputation: People should have the legal responsibility for the positive and negative results of their intentional actions.

The Labor Theory of Property: People should legally appropriate the positive and negative fruits of their labor.

In other words, the juridical principle of imputation is the labor theory of property applied in the context of civil and criminal trials, and the labor theory of property is the juridical principle applied in the context of property appropriation.

Some individuals, such as infants or the insane, are not capable of de facto responsible actions.

The statement that an individual is zurechnungfähig (“responsible”) means that a sanction can be inflicted upon him if he commits a delict. The statement that an individual is unzurechnungsfähig (“irresponsible”)– because, for instance, he is a child or insane–means that a sanction cannot be inflicted upon him if he commits a delict. … The idea of imputation (Zurechnung) as the specific connection of the delict with the sanction is implied in the juristic judgment that an individual is, or is not, legally responsible (zurechnungsfähig) for his behavior. [Kelsen 1985, 364]

Regardless of their causal efficacy, things are, a fortiori, unzurechnungsfähig.

De facto responsibility is not a normative notion; it is a descriptive factual notion. The juridical principle of imputation is a normative principle which states that legal or de jure responsibility should be assigned in accordance with de facto responsibility. In the jury system, the jury is assigned the factual question of “officially” determining whether or not the accused was de facto responsible for the deed as charged. If “Guilty” then legal responsibility is imputed accordingly.

Economics is always on “jury duty” to determine “the facts” about human activities. These are not value judgments (where social scientists have no particular expertise). The economist-as-juror is only required to make factual descriptive judgments about de facto responsibility. In this chapter we are not concerned with the normative principle of juridical imputation (i.e., the labor theory of property applied in the courtroom), only the descriptive question of responsibility. The normative and descriptive questions should be kept conceptually distinct. That separation is difficult since, given the juridical principle, de facto responsibility implies de jure responsibility.

In a given productive enterprise, the descriptive question asks what set of people are de facto responsible for producing the product by using up the various inputs? The economist-as-juror faces that question. The marginal productivity of tools (machine tools or burglary tools) is not relevant to this factual question of responsibility either inside or outside the courtroom. Only human actions can be responsible; the services provided by things cannot be responsible (no matter how causally efficacious). The original question includes the question of who is responsible for using up those casually efficacious or productive services of the tools .

The question of de facto responsibility, whether posed in a courtroom or outside, presupposes the understanding that persons act and things don’t. Yet it is precisely the presupposition that is “overlooked” in economic theory which treats both the services of human beings and the services of capital and land symmetrically as “input services.” Economists choose to limit their description of the human activity of production to an engineering description of the causally efficacy of the various types of input services. The unique responsible agency of human activities is not acknowledged.

Comment Source:Mart Malakoff wrote: >Most of what i could get when involved with academia was stuff like modeling conformations of DNA , and stuff related to biotech. Thats fine and itslef can be interesting and sometimes useful (but in this age of 'Big Pharma' and the opiate epidemic , and obesity fueled by junk food, you can question exactly how valuable biomedical research focusing on genes and biomolecules is---studying prejudice, blame , responsibility and priviledge may be more socially valuable, along with environment, behavioral economics (alot of which now is applied by corporations for marketing) , inequality, intolerance, etc. I think a bit part of why people don't try to model prejudice, blame, responsibility and privilege. If it doesn't involve numbers, too many people it's un-modelable. To many many people, math is strictly about numbers. It's interesting though because things like assigning responsibility to the appropriate parties are equivalent to assigning property under the labor theory of property, or as one (and perhaps the only) economist (David P. Ellerman) points this out clearly, > In the context of assigning property rights and obligations, the juridical principle of imputation is expressed as the labor theory of property which holds that people should appropriate the (positive and negative) fruits of their labor. Since, in the economic context, intentional human actions are called “labor”, we can express the equivalence as: >**The Juridical Principle of Imputation:** People should have the legal responsibility for the positive and negative results of their intentional actions. >**The Labor Theory of Property:** People should legally appropriate the positive and negative fruits of their labor. >In other words, the juridical principle of imputation is the labor theory of property applied in the context of civil and criminal trials, and the labor theory of property is the juridical principle applied in the context of property appropriation. >Some individuals, such as infants or the insane, are not capable of de facto responsible actions. >>The statement that an individual is zurechnungfähig (“responsible”) means that a sanction can be inflicted upon him if he commits a delict. The statement that an individual is unzurechnungsfähig (“irresponsible”)– because, for instance, he is a child or insane–means that a sanction cannot be inflicted upon him if he commits a delict. … The idea of imputation (Zurechnung) as the specific connection of the delict with the sanction is implied in the juristic judgment that an individual is, or is not, legally responsible (zurechnungsfähig) for his behavior. [Kelsen 1985, 364] >Regardless of their causal efficacy, things are, a fortiori, unzurechnungsfähig. >De facto responsibility is not a normative notion; it is a descriptive factual notion. The juridical principle of imputation is a normative principle which states that legal or de jure responsibility should be assigned in accordance with de facto responsibility. In the jury system, the jury is assigned the factual question of “officially” determining whether or not the accused was de facto responsible for the deed as charged. If “Guilty” then legal responsibility is imputed accordingly. >Economics is always on “jury duty” to determine “the facts” about human activities. These are not value judgments (where social scientists have no particular expertise). The economist-as-juror is only required to make factual descriptive judgments about de facto responsibility. In this chapter we are not concerned with the normative principle of juridical imputation (i.e., the labor theory of property applied in the courtroom), only the descriptive question of responsibility. The normative and descriptive questions should be kept conceptually distinct. That separation is difficult since, given the juridical principle, de facto responsibility implies de jure responsibility. >In a given productive enterprise, the descriptive question asks what set of people are de facto responsible for producing the product by using up the various inputs? The economist-as-juror faces that question. The marginal productivity of tools (machine tools or burglary tools) is not relevant to this factual question of responsibility either inside or outside the courtroom. Only human actions can be responsible; the services provided by things cannot be responsible (no matter how causally efficacious). The original question includes the question of who is responsible for using up those casually efficacious or productive services of the tools . >The question of de facto responsibility, whether posed in a courtroom or outside, presupposes the understanding that persons act and things don’t. Yet it is precisely the presupposition that is “overlooked” in economic theory which treats both the services of human beings and the services of capital and land symmetrically as “input services.” Economists choose to limit their description of the human activity of production to an engineering description of the causally efficacy of the various types of input services. The unique responsible agency of human activities is not acknowledged. 
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All this stuff is extremely important, and not very easy to get a firm grip on - at least for me.

Comment Source:All this stuff is extremely important, and not very easy to get a firm grip on - at least for me.
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I don't think most mathematicians, engineers, and scientists would find notions of responsibility to be intuitive at all, as Mr. Ellerman points out,

The natural sciences take no note of responsibility. The notion of responsibility (as opposed to causality) is not a concept of physics and engineering. The difference between the responsible actions of persons and the nonresponsible services of things would not be revealed by a simple engineering description of the causal consequences of the actions/services. Therefore when economists choose to restrict their description of the production process to an engineering production function, they are implicitly or explicitly deciding to ignore the difference between the actions of persons and the services of things.

David Ellerman has a point. In math, physics, and engineering, we don't deal with responsibility, only the causality of inanimate objects.

In fact, earlier in the course when we went over preimages of subsets, and I asked if responsibility could be modeled as a sort of preimage operation on a function from a set of people to a set of outcomes of their actions (given a specific time), and you confirmed that giving semantics to responsibility in terms of a preimage operation is technically possible. To us, it was no big deal and we moved on, but to someone like a lawyer, the idea that math (what most people think of as "just number manipulation"), can be used to give formal semantics to something as abstract as assigning responsibility, must be mindblowing.

Comment Source:I don't think most mathematicians, engineers, and scientists would find notions of responsibility to be intuitive at all, as Mr. Ellerman points out, >The natural sciences take no note of responsibility. The notion of responsibility (as opposed to causality) is not a concept of physics and engineering. The difference between the responsible actions of persons and the nonresponsible services of things would not be revealed by a simple engineering description of the causal consequences of the actions/services. Therefore when economists choose to restrict their description of the production process to an engineering production function, they are implicitly or explicitly deciding to ignore the difference between the actions of persons and the services of things. David Ellerman has a point. In math, physics, and engineering, we don't deal with responsibility, only the causality of inanimate objects. In fact, earlier in the course when we went over preimages of subsets, and I asked if responsibility could be modeled as a sort of preimage operation on a function from a set of **people** to a set of outcomes of their actions (given a specific time), and you confirmed that giving semantics to responsibility in terms of a preimage operation is *technically* possible. To us, it was no big deal and we moved on, but to someone like a lawyer, the idea that math (what most people think of as "just number manipulation"), can be used to give formal semantics to something as abstract as assigning responsibility, must be mindblowing. 
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I wrote a presentation of a model of personal responsibility for social calamities. It gives a way of identifying the people must culpable - the ones who are willing to make sacrifices in order to make the calamity happen and pull it all together - who desire it, recruit others and justify it. The idea is that a calamity requires a dedicated learner to make it happen. The next most culpable are the creative workers, the organizers who do at least one of those three - desire it, recruit others or justify it - typically for ulterior motives. The model came out of a historical study I made. Determining Personal Responsibility for a Social Calamity: The Origins of the Holocaust in Lithuania

Comment Source:I wrote a presentation of a model of personal responsibility for social calamities. It gives a way of identifying the people must culpable - the ones who are willing to make sacrifices in order to make the calamity happen and pull it all together - who desire it, recruit others and justify it. The idea is that a calamity requires a dedicated learner to make it happen. The next most culpable are the creative workers, the organizers who do at least one of those three - desire it, recruit others or justify it - typically for ulterior motives. The model came out of a historical study I made. [Determining Personal Responsibility for a Social Calamity: The Origins of the Holocaust in Lithuania](http://www.ms.lt/sodas/Book/20170708PersonalResponsibility)
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edited June 2018

Well, thinking about it, responsibility is Boolean: either you've acted to bring about a certain consequence (consciously or not) or you strictly aren't.

$\mathrm{Resp}(P,o) = \begin{cases} \texttt{true} & \text{If } P \text{ is a set of people, } o \text{ is a set of outcomes,} \\ & \text{and the set of people } P \text{ caused the outcomes } o.\\ \texttt{false} & \text{Otherwise.} \end{cases}$

or diagrammatically,

$\begin{matrix} \mathrm{People} & \overset{\text{Caused}}\rightarrow & \mathrm{Outcomes}\quad\quad \\ !_\mathrm{People}\downarrow & & \downarrow \mathrm{Resp} \\ \quad\quad\ \ \mathbf{1} & \underset{\texttt{true}}\rightarrow & \mathbf{Bool}\quad\quad\quad \end{matrix}$

However, we could speak of various degrees of responsibility. For instance, keeping all other things equal, it seems reasonable to say people who aren't aware of what they are doing should have less than or an equal amount of responsibility as a person fully aware of what they are doing, but never strictly more responsible.

I believe that could be modeled using the preorder given by the product of the two preorders.

Comment Source:Well, thinking about it, responsibility is Boolean: either you've acted to bring about a certain consequence (consciously or not) or you strictly aren't. \$\mathrm{Resp}(P,o) = \begin{cases} \texttt{true} & \text{If } P \text{ is a set of people, } o \text{ is a set of outcomes,} \\\\ & \text{and the set of people } P \text{ caused the outcomes } o.\\\\ \texttt{false} & \text{Otherwise.} \end{cases} \$ or diagrammatically, \$\begin{matrix} \mathrm{People} & \overset{\text{Caused}}\rightarrow & \mathrm{Outcomes}\quad\quad \\\\ !_\mathrm{People}\downarrow & & \downarrow \mathrm{Resp} \\\\ \quad\quad\ \ \mathbf{1} & \underset{\texttt{true}}\rightarrow & \mathbf{Bool}\quad\quad\quad \end{matrix} \$ However, we could speak of various *degrees* of responsibility. For instance, keeping all other things equal, it seems reasonable to say people who aren't aware of what they are doing should have less than or an equal amount of responsibility as a person fully aware of what they are doing, but never strictly more responsible. I believe that could be modeled using the preorder given by the product of the two preorders.