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Modeling the Composition of Perspectives

I'm interested to try to use category theory to model the composition of perspectives. I wonder to what extent that's already been done. Somewhat related is Gilles Fauconnier's important work on Mental Spaces.

For example, consider a lost child in an airport. An oblivious child doesn't even realize that they are lost. A foolish child goes looking for their parents. A wise child realizes, "I am the child and they are the parent. They are supposed to be looking for me!" And so the wise child goes to wherever their parents will most easily find them.

The intelligence of the children is given by the length of composition:

  • The oblivious child entertains: The child's view of the parent's view of the child's view. ("My parents hear me.")

  • The foolish child entertains: The child's view of the parent's view of the child's view of the parent's view. ("My parents need me to understand that they have lost me.")

  • The wise child entertains: The child's view of the parent's view of the child's view of the parent's view of the child's view. ("My parents should be concerned that I am wondering if my parents know what I am going through.")

The wise child's chain of perspectives is extraordinary in that it allows the child and the parents to coordinate their actions (the child will go where the parents will most easily find them) without any communication but simply the knowledge of asymmetry (who is the parent, who is the child, and who should be looking for whom). For me, personally, I believe it's a chain of the maximal length that I am capable of.

I'm not sure how to define "perspective" although I have thought a lot about perspectives. For me, a synonym would be "point of view". I suppose that it relates to the knowledge that is available in a given circumstance. As such, it may also depend on the observer taking up the perspective.

It's also important for me to be able to model "stepping-into" a perspective and "stepping-out-of" any perspective. Stepping-into a perspective means that I am engrossed in a perspective and no longer cognizant that I happen to be in a particular perspective. Stepping-out means that I am not entertaining any perspective but cognizant of the different perspectives that I may take up. We are subjective when we are stepped-in and objective when we are stepped-out. In a healthy discourse, participants alternate unpredictably as to who is stepped-in and who is stepped-out, and there is even a sense of a fluttering spirit over the discourse. I'm wondering if there is an adjoint type of relationship as in "there exists a perspective in which we know..." and "for all perspectives we know..."

It's not clear if composition of perspectives is associative. I'm not sure if category theory is ultimately the right mathematical framework but I'm curious to try. Success would be great and failure might help discover a more appropriate framework.

Comments

  • 1.

    I suppose that the point of this "algebra of perspectives" is to distinguish between our level of consciousness (the syntax of the perspectives, which is described by the algebra of perspectives) and our ability to unconsciously know or intuit (which is the semantics which a perspective avails, its inner knowledge). Thus two people or two beings may differ in terms of what they can intuit, but we can still discuss their consciousness with regard to that. The algebra of perspectives describes how we can empathize with others even though we don't have direct access to their direct experience. Category theory may give a lot of clues about the duality between the syntax (the external relatlonships which the algebra of perspectives expresses and we can be conscious of) and the semantics (the internal knowledge that structures one's direct experience of a perspective). This is to say that we can nontrivially empathize with each other.

    Comment Source:I suppose that the point of this "algebra of perspectives" is to distinguish between our level of consciousness (the syntax of the perspectives, which is described by the algebra of perspectives) and our ability to unconsciously know or intuit (which is the semantics which a perspective avails, its inner knowledge). Thus two people or two beings may differ in terms of what they can intuit, but we can still discuss their consciousness with regard to that. The algebra of perspectives describes how we can empathize with others even though we don't have direct access to their direct experience. Category theory may give a lot of clues about the duality between the syntax (the external relatlonships which the algebra of perspectives expresses and we can be conscious of) and the semantics (the internal knowledge that structures one's direct experience of a perspective). This is to say that we can nontrivially empathize with each other.
  • 2.

    This is great! I always thought the Yoneda Lemma was saying something deep about the nature of perspective...

    There's a notion of an 'algebra of perspectives' in the topos theoretic approach to quantum theory that perhaps could be generalized. It has more to do with the notion of a collection of possible measurements on a quantum system (and how they interrelate in a non-trivial way)... but it could be a good starting point.

    Related to Fauconnier is the rich body of category theoretic ideas from late computer scientist Joseph Goguen. His essay 'What is a Concept?' https://cseweb.ucsd.edu/~goguen/pps/iccs05.pdf uses category theory to unify the symbolic mental spaces of Fauconnier and the geometric conceptual spaces of Gardenfors.

    Comment Source:This is great! I always thought the Yoneda Lemma was saying something deep about the nature of perspective... There's a notion of an 'algebra of perspectives' in the topos theoretic approach to quantum theory that perhaps could be generalized. It has more to do with the notion of a collection of possible measurements on a quantum system (and how they interrelate in a non-trivial way)... but it could be a good starting point. Related to Fauconnier is the rich body of category theoretic ideas from late computer scientist Joseph Goguen. His essay 'What is a Concept?' https://cseweb.ucsd.edu/~goguen/pps/iccs05.pdf uses category theory to unify the symbolic mental spaces of Fauconnier and the geometric conceptual spaces of Gardenfors.
  • 3.
    edited May 12

    I don't think we even need a full topos. What it sounds like what this view of perspectives is being described are over- and under categories of a particular person in a sort of category of people, maybe call it Society, where the objects are persons and the morphisms are social relationships between individuals.

    For instance, taking a person in our category, $$ \{\mathrm{Keith E. Peterson}\} \in Ob(\mathbf{Society}), $$ we can take that person's over category, $$ \mathbf{Society} / \{\mathrm{Keith E. Peterson}\} $$ where the objects are all social relationships that end with that person,

    $$ r \in \mathrm{Arr}(\mathbf{Society}) \ \mathrm{such} \ \mathrm{that} \ \mathrm{cod} (r)={\mathrm{Keith E. Peterson}} $$ and morphisms are social relations between people,

    $$ X,Y \in Ob(\mathbf{Society}), f \in Arr(\mathbf{Society}), \ X \overset{f}{\longrightarrow}Y $$ such that

    $$ r: X \rightarrow {\mathrm{Keith E. Peterson}}, \ r': Y \rightarrow {\mathrm{Keith E. Peterson}}, \ \mathrm{and } : r' \circ f = r. $$ If you think about it, this models how a person can view society from all the social relationships in towards themselves.

    The under category construction where a person views society based on social relationships out from themselves proceeds dually.

    Comment Source:I don't think we even need a full topos. What it sounds like what this view of perspectives is being described are over- and under categories of a particular person in a sort of category of people, maybe call it **Society**, where the objects are persons and the morphisms are social relationships between individuals. For instance, taking a person in our category, $$ \{\mathrm{Keith E. Peterson}\} \in Ob(\mathbf{Society}), $$ we can take that person's over category, $$ \mathbf{Society} / \{\mathrm{Keith E. Peterson}\} $$ where the objects are all social relationships that end with that person, $$ r \in \mathrm{Arr}(\mathbf{Society}) \ \mathrm{such} \ \mathrm{that} \ \mathrm{cod} (r)=\{\mathrm{Keith E. Peterson}\} $$ and morphisms are social relations between people, $$ X,Y \in Ob(\mathbf{Society}), f \in Arr(\mathbf{Society}), \\ X \overset{f}{\longrightarrow}Y $$ such that $$ r: X \rightarrow \{\mathrm{Keith E. Peterson}\}, \\ r': Y \rightarrow \{\mathrm{Keith E. Peterson}\}, \\ \mathrm{and } \: r' \circ f = r. $$ If you think about it, this models how a person can view society from all the social relationships *in* towards themselves. The under category construction where a person views society based on social relationships *out* from themselves proceeds dually.
  • 4.
    edited May 12

    Van, Keith, Thank you for responding!

    Keith, I wasn't aware of over categories (slice categories) and under categories (coslice categories). Your example helps me think about that.

    I suppose, though, that there has to be more to it. The perspectives are like "windows", so that we can have "windows upon windows". A person may look through one particular window, or a person may stand back and see a set of windows. A person may look through one window onto a further window. That further window may be clear or opaque.

    I'm not even sure if the windows are objects or arrows or perhaps both. There may be more than one category involved.

    Comment Source:Van, Keith, Thank you for responding! Keith, I wasn't aware of [over categories](https://ncatlab.org/nlab/show/over+category) (slice categories) and [under categories](https://ncatlab.org/nlab/show/under+category) (coslice categories). Your example helps me think about that. I suppose, though, that there has to be more to it. The perspectives are like "windows", so that we can have "windows upon windows". A person may look through one particular window, or a person may stand back and see a set of windows. A person may look through one window onto a further window. That further window may be clear or opaque. I'm not even sure if the windows are objects or arrows or perhaps both. There may be more than one category involved.
  • 5.

    Van, That's a great article for me to read! It will take some time. Key ideas seem to be "frames" and "institutions".

    Actually, Joseph Goguen is the person who first encouraged me to study category theory! He was a very nice man and a daringly brilliant man and I always feel sad to know that he passed away.

    I got my Ph.D. in Math at UCSD in 1993 but he wasn't there yet. I moved to Lithuania, where, in trying to make living, I set up Minciu Sodas, "The Orchard of Thoughts", an online laboratory for serving independent thinkers around the world. A first project that I pursued was an import/export standard for tools for organizing thoughts such as TheBrain and MindManager (they both provided some funding). As part of that, I wrote a paper, Organizing Thoughts into Sequences, Hierarchies and Networks which argues that we never visualize these structures in isolation but rather we always have one structure restructure another structure, yielding six different visualizations. More recently, I've argued this yields a taxonomy for paradoxes. I'll be speaking about that in July at the Congress of Universal Logic: Visualization as Restructuring and thus a Source of Logical Paradox. I was hanging out at UCSD for several months around 1999 and I was curious who might be interested. I think that's when and certainly how I got to know him.

    He told me about institutions. But I couldn't understand category theory. I do a lot of vague and abstract thinking but it's always very purposeful and I couldn't intuit the purpose of category theory. Although I understood that it could well be relevant for this "algebra of perspectives". So I tried a second time in 2012. I decided to learn some algebraic topology so that the category theory would be more concrete. I watched through all of Norman Wildberger's video lectures on algebraic topology. I very much enjoy his teaching. I was living in Pilsen, Chicago at the time, and so I went to a seminar at the University of Illinois where I got to know Louis Kauffman, another friendly and wonderfully creative brilliant thinker. I got to tell him about my systematization of the ways of figuring things out in mathematics, and also he and his wife Diane came to see a couple of my art shows on my philosophy. Finally, in 2016, I tried for a third time to understand category theory, and now I feel like I'm getting the hang of it, but certainly the victories are hard fought. I'm very grateful for this class, where finally I can make sense of adjoints as "approximate inverses" from above and below, as with the least upper bound and the greatest lower bound.

    At the time, I found algebraic semiotics to be for me the most useful part of Joseph Goguen's work. He invited me to talk to his class about the Algebra of Copyright, where I showed the significance of relating parsers that act on different levels. I suppose that may be quite related to this algebra of perspectives.

    I will share a couple of relevant letters that Joseph Goguen wrote at my online lab. Also, I'm honored that he listed me and Minciu Sodas on his What's Cool page. And in particular, to be mentioned in his and Ryoko's 2005 summer account, where they enjoyed some time in Vienna with Franz Nahrada, the leader of the Global Villages working group at Minciu Sodas.

    Comment Source:Van, That's a great article for me to read! It will take some time. Key ideas seem to be "frames" and ["institutions"](https://cseweb.ucsd.edu/~goguen/projs/inst.html). Actually, Joseph Goguen is the person who first encouraged me to study category theory! He was a very nice man and a daringly brilliant man and I always feel sad to know that he passed away. I got my Ph.D. in Math at UCSD in 1993 but he wasn't there yet. I moved to Lithuania, where, in trying to make living, I set up [Minciu Sodas, "The Orchard of Thoughts"](http://www.ms.lt/sodas/Book/20170506PostTruth), an online laboratory for serving independent thinkers around the world. A first project that I pursued was an import/export standard for tools for organizing thoughts such as TheBrain and MindManager (they both provided some funding). As part of that, I wrote a paper, [Organizing Thoughts into Sequences, Hierarchies and Networks](http://www.ms.lt/papers/organizingthoughts.html) which argues that we never visualize these structures in isolation but rather we always have one structure restructure another structure, yielding six different visualizations. More recently, I've argued this yields a taxonomy for paradoxes. I'll be speaking about that in July at the Congress of Universal Logic: [Visualization as Restructuring and thus a Source of Logical Paradox](http://www.ms.lt/sodas/Boox/20180621VisualizationAsRestructuring). I was hanging out at UCSD for several months around 1999 and I was curious who might be interested. I think that's when and certainly how I got to know him. He told me about institutions. But I couldn't understand category theory. I do a lot of vague and abstract thinking but it's always very purposeful and I couldn't intuit the purpose of category theory. Although I understood that it could well be relevant for this "algebra of perspectives". So I tried a second time in 2012. I decided to learn some algebraic topology so that the category theory would be more concrete. I watched through all of [Norman Wildberger's video lectures on algebraic topology](https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8). I very much enjoy his teaching. I was living in Pilsen, Chicago at the time, and so I went to a seminar at the University of Illinois where I got to know [Louis Kauffman](http://homepages.math.uic.edu/~kauffman/), another friendly and wonderfully creative brilliant thinker. I got to tell him about [my systematization of the ways of figuring things out in mathematics](http://www.ms.lt/sodas/Book/DiscoveryInMathematics), and also he and his wife Diane came to see a couple of my art shows on my philosophy. Finally, in 2016, I tried for a third time to understand category theory, and now I feel like I'm getting the hang of it, but certainly the victories are hard fought. I'm very grateful for this class, where finally I can make sense of adjoints as "approximate inverses" from above and below, as with the least upper bound and the greatest lower bound. At the time, I found algebraic semiotics to be for me the most useful part of Joseph Goguen's work. He invited me to talk to his class about the [Algebra of Copyright](http://www.ms.lt/papers/2003-The-Algebra-Of-Copyright.pdf), where I showed the significance of relating parsers that act on different levels. I suppose that may be quite related to this algebra of perspectives. I will share a couple of relevant letters that Joseph Goguen wrote at my online lab. Also, I'm honored that he listed me and Minciu Sodas on his [What's Cool](https://cseweb.ucsd.edu/~goguen/cool.html) page. And in particular, to be mentioned in [his and Ryoko's 2005 summer account](http://cseweb.ucsd.edu/~goguen/trips/germanic05.html), where they enjoyed some time in Vienna with Franz Nahrada, the leader of the Global Villages working group at Minciu Sodas.
  • 6.

    Andrius, 2005.08.22, Living by Truth working group.

    I have been thinking about an "algebra of views". The basic idea is that "truth" has a different nature depending on the chain of views involved. From one view, truth is soft, in that all things are true. But as we look through a series of views, and if they become opaque, then truth grows hard, so that we have statements, and they may be false, and ultimately they are either true or false.

    I have thought about two domains where I will try to apply my thinking. One is "moral responsibility". The way that I interact morally with other people depends on my understanding of them. So, for example, if a client is a stranger and asks me to do some work and I don't understand what it will be used for, then I will want to find out more about the purposes because I want to know that the client can and will take moral responsibility. Whereas if a client is somebody who I know to operate under some kind of morality, then I may be willing to do things I don't understand if I feel that my client can and will take the moral responsibility on our behalf.

    Another domain is "humor". Yesterday I watched on television an adaptation of Oscar Wilde's "The Importance of Being Earnest". I found it quite funny, and much of the humor arose from who knowing what. The main character was living under two names, as two brothers - the responsible brother John at his country estate, and the rascalous brother Ernest in London. The ignorance of the truth in most people's minds set up a lot of funny situations. So it would be interesting to see, more generally, in what sense humor depends on opaqueness and transparency. In a sense, everything opaque is just a temporary state.

    Joseph Goguen, 2005.08.22, Living by Truth

    dear andrius,

    im thinking that your algebra ov views could be a monoid, i.e., an associative binary operation with identity. for example, A's view of B's view would be "A of B" where "of" is the binary operation. perhaps the identity I is your "God"? since I of A = A and A of I = A. it is interesting to think about when relative idempotence holds, i.e., A of A of B = A of B, and when it doesnt - "Alex thinks his view of Betty is wrong"). relative commutativity is also interesting. note that all his is similar to Levi-Strauss' work on kinship role in various cultures, e.g., brother of father = uncle (though i think he used semigroups since the identity doesnt seem to make much sense here); you can easily find stuff on this with google.

    hope this helps,

    joseph

    Comment Source:[Andrius, 2005.08.22, Living by Truth working group.](https://groups.yahoo.com/neo/groups/livingbytruth/conversations/messages/493) I have been thinking about an "algebra of views". The basic idea is that "truth" has a different nature depending on the chain of views involved. From one view, truth is soft, in that all things are true. But as we look through a series of views, and if they become opaque, then truth grows hard, so that we have statements, and they may be false, and ultimately they are either true or false. I have thought about two domains where I will try to apply my thinking. One is "moral responsibility". The way that I interact morally with other people depends on my understanding of them. So, for example, if a client is a stranger and asks me to do some work and I don't understand what it will be used for, then I will want to find out more about the purposes because I want to know that the client can and will take moral responsibility. Whereas if a client is somebody who I know to operate under some kind of morality, then I may be willing to do things I don't understand if I feel that my client can and will take the moral responsibility on our behalf. Another domain is "humor". Yesterday I watched on television an adaptation of Oscar Wilde's "The Importance of Being Earnest". I found it quite funny, and much of the humor arose from who knowing what. The main character was living under two names, as two brothers - the responsible brother John at his country estate, and the rascalous brother Ernest in London. The ignorance of the truth in most people's minds set up a lot of funny situations. So it would be interesting to see, more generally, in what sense humor depends on opaqueness and transparency. In a sense, everything opaque is just a temporary state. [Joseph Goguen, 2005.08.22, Living by Truth](https://groups.yahoo.com/neo/groups/livingbytruth/conversations/messages/494) dear andrius, im thinking that your algebra ov views could be a monoid, i.e., an associative binary operation with identity. for example, A's view of B's view would be "A of B" where "of" is the binary operation. perhaps the identity I is your "God"? since I of A = A and A of I = A. it is interesting to think about when relative idempotence holds, i.e., A of A of B = A of B, and when it doesnt - "Alex thinks his view of Betty is wrong"). relative commutativity is also interesting. note that all his is similar to Levi-Strauss' work on kinship role in various cultures, e.g., brother of father = uncle (though i think he used semigroups since the identity doesnt seem to make much sense here); you can easily find stuff on this with google. hope this helps, joseph
  • 7.

    "Views" can be modeled by a comonad called the state comonad.

    Indeed, the coalgebras of this comonad are called lenses, which can get a particular view and set (or update) a view to a new value.

    See for instance: Lenses are the coalgebras for the costate comonad

    Comment Source:"Views" can be modeled by a comonad called the [state comonad](https://ncatlab.org/nlab/show/store+comonad). Indeed, the coalgebras of this comonad are called lenses, which can **get** a particular *view* and **set** (or update) a *view* to a new value. See for instance: [Lenses are the coalgebras for the costate comonad](https://patternsinfp.wordpress.com/2011/01/31/lenses-are-the-coalgebras-for-the-costate-comonad/)
  • 8.
    edited May 13

    Keith, Thank you! That seems very relevant for me to learn about.

    Is it called the costate comonad or the state comonad? I couldn't find the latter.

    It's interesting for me to read what Bartosz Milewski writes about lenses. I very much enjoy his videos on category theory. He mentions the Yoneda lemma. I've been trying to understand and explain that in terms of four levels of knowledge: Whether, What, How and Why. For example, for us to know Why there is a bottle, we would ultimately have to know all of its relationships with absolute everything else.

    The word "lens" captures the "windows" aspect and the composition of windows. I'm also trying to be able to model the distinction between views and perspectives, and ultimately, concepts, positions, principles, and all that has to do with knowing. An important distinction is between what we unconsciously know (the semantics, as with neural-network-thinking) and what consciously don't know (the syntax, as with logic). So, for example, Alpha Zero accumulates intuition about chess by playing millions of games against itself, but it doesn't develop any principles that could help us explain why it wins. It's simply tuning itself. It's just a chess muscle, like the retina is an organ for seeing. We see with the retina but we have no idea why we see what we see. Whereas Stockfish is a chess program that implements the human principles developed over more than a thousand years of civilization. So it is straightforward to pick apart and understand why Stockfish chooses the moves that it does. Logic is syntactic in that it doesn't care about the actual content, just the form. Logic thus deals with what we don't know. Whereas semantics is knowledge about the content, built up as associations. A concept is like a bubble that separates the syntactic outside (where we can relate bubbles or nodes with sequences, hierarchies and networks, which we can visualize) and the semantic inside (which is like a database record of fields which we typically don't, as such, visualize). That's all just my thinking that I'm developing. But the point I want to make is that a view is what we actually see (or think or know) whereas a perspective is what we can potentially see or think or know from our particular circumstances.

    I've been thinking that it might be best to say that the composition is not associative. For example, I doubt that "Andrius's view of (Rima's view of Jonas's view)" is the same as "(Andrius's view of Rima's view) of Jonas's view)". One issue is where to locate and thus define all of these primitive and composite views? Let's imagine there is some absolute framework, Absolute. Then Absolute's view of Andrius's view is not the same as Andrius's view of Andrius's view. But the Absolute's view of Andrius's view is probably a greater knowledge than simply Andrius's view. So we need a framework that just takes Andrius's view as he does, which would be Empathy's view. Thus Empathy's view of Andrius's view equals Andrius's view, let's say. Anyways, the problem is that Andrius may have false knowledge, which is what makes it interesting to model the algebra of perspectives. Andrius may think that Rima doesn't know Jonas but perhaps she actually knows him very well. Now what do I mean by "Andrius's view of Rima's view"? I mean that Andrius has a working model of Rima's mind, her circumstances and her attention. And that model may yield conclusions about her view of Jonas's view, for example, that she doesn't even know Jonas. Here there's a question of whether ownership is relevant, that is, could she have a view about his view nevertheless? Let's say we're talking about Jonas's views on smoking. But still, I may think that Rima doesn't even know what smoking is, but yet she may very well know what smoking is.

    I will rephrase in terms of working models. Is "Andrius's working model of (Rima's working model of Jonas's working model)" the same as "(Andrius's working model of Rima's working model) of Jonas's working model)"? Thus stated the ownership need not be relevant. The most important point is that a working model doesn't have to be consistent! But it can be partially consistent. Thus the order of composition can yield different results regarding particular views on smoking.

    In particular, I think that "my view of (my view of my view)" need not equal "(my view of my view) of my view". This is all to say that there are real challenges in modeling this, but I think that it is highly meaningful. Probably our most highly developed intuitions in life address these challenges. So it's interesting to try to formulate them in category theory and related mathematics.

    Keith, the category theory that you point me to seems relevant because it may deal with modeling problems with inconsistent knowledge. I wonder where I could learn more about "the update problem"?

    Comment Source:Keith, Thank you! That seems very relevant for me to learn about. Is it called the costate comonad or the state comonad? I couldn't find the latter. It's interesting for me to read [what Bartosz Milewski writes about lenses](https://bartoszmilewski.com/2013/10/08/lenses-stores-and-yoneda/). I very much enjoy [his videos on category theory](https://www.youtube.com/user/DrBartosz/playlists). He mentions the Yoneda lemma. I've been trying to understand and explain that in terms of four levels of knowledge: Whether, What, How and Why. For example, for us to know Why there is a bottle, we would ultimately have to know all of its relationships with absolute everything else. The word "lens" captures the "windows" aspect and the composition of windows. I'm also trying to be able to model the distinction between views and perspectives, and ultimately, concepts, positions, principles, and all that has to do with knowing. An important distinction is between what we unconsciously know (the semantics, as with neural-network-thinking) and what consciously don't know (the syntax, as with logic). So, for example, Alpha Zero accumulates intuition about chess by playing millions of games against itself, but it doesn't develop any principles that could help us explain why it wins. It's simply tuning itself. It's just a chess muscle, like the retina is an organ for seeing. We see with the retina but we have no idea why we see what we see. Whereas Stockfish is a chess program that implements the human principles developed over more than a thousand years of civilization. So it is straightforward to pick apart and understand why Stockfish chooses the moves that it does. Logic is syntactic in that it doesn't care about the actual content, just the form. Logic thus deals with what we don't know. Whereas semantics is knowledge about the content, built up as associations. A concept is like a bubble that separates the syntactic outside (where we can relate bubbles or nodes with sequences, hierarchies and networks, which we can visualize) and the semantic inside (which is like a database record of fields which we typically don't, as such, visualize). That's all just my thinking that I'm developing. But the point I want to make is that a view is what we actually see (or think or know) whereas a perspective is what we can potentially see or think or know from our particular circumstances. I've been thinking that it might be best to say that the composition is not associative. For example, I doubt that "Andrius's view of (Rima's view of Jonas's view)" is the same as "(Andrius's view of Rima's view) of Jonas's view)". One issue is where to locate and thus define all of these primitive and composite views? Let's imagine there is some absolute framework, Absolute. Then Absolute's view of Andrius's view is not the same as Andrius's view of Andrius's view. But the Absolute's view of Andrius's view is probably a greater knowledge than simply Andrius's view. So we need a framework that just takes Andrius's view as he does, which would be Empathy's view. Thus Empathy's view of Andrius's view equals Andrius's view, let's say. Anyways, the problem is that Andrius may have false knowledge, which is what makes it interesting to model the algebra of perspectives. Andrius may think that Rima doesn't know Jonas but perhaps she actually knows him very well. Now what do I mean by "Andrius's view of Rima's view"? I mean that Andrius has a working model of Rima's mind, her circumstances and her attention. And that model may yield conclusions about her view of Jonas's view, for example, that she doesn't even know Jonas. Here there's a question of whether ownership is relevant, that is, could she have a view about his view nevertheless? Let's say we're talking about Jonas's views on smoking. But still, I may think that Rima doesn't even know what smoking is, but yet she may very well know what smoking is. I will rephrase in terms of working models. Is "Andrius's working model of (Rima's working model of Jonas's working model)" the same as "(Andrius's working model of Rima's working model) of Jonas's working model)"? Thus stated the ownership need not be relevant. The most important point is that a working model doesn't have to be consistent! But it can be partially consistent. Thus the order of composition can yield different results regarding particular views on smoking. In particular, I think that "my view of (my view of my view)" need not equal "(my view of my view) of my view". This is all to say that there are real challenges in modeling this, but I think that it is highly meaningful. Probably our most highly developed intuitions in life address these challenges. So it's interesting to try to formulate them in category theory and related mathematics. Keith, the category theory that you point me to seems relevant because it may deal with modeling problems with inconsistent knowledge. I wonder where I could learn more about "the update problem"?
  • 9.

    Keith, I looked around a bit about lenses. I found Bartosz Milewski's video about lenses. I will need to learn about comonads, thus monads, thus become truly adept at adjoint functors, which is a good reason to be here.

    Functional Lenses, How Do They Work by Drew Tipson is a helpful article for building my intuition, along with his article, Functional Programming is for Dummies. So I'm getting an inkling of what it means to modify an object by being able to get some particular information inside of it and perhaps set that to a new value and retrieve the entire object.

    It reminds me a little of what I mean by "knowing everything", which is to have a perspective upon the big picture in its entirety, but then (being mentally shortsighted), being able to crawl out in any direction to the matter of interest, inspect it up close, and then be able to crawl back to where you started.

    Comment Source:Keith, I looked around a bit about lenses. I found [Bartosz Milewski's video about lenses](https://www.youtube.com/watch?v=9_iYlp8smc8&t=552s). I will need to learn about comonads, thus [monads](https://en.wikipedia.org/wiki/Monad_(category_theory)), thus become truly adept at adjoint functors, which is a good reason to be here. [Functional Lenses, How Do They Work by Drew Tipson](https://medium.com/@dtipson/functional-lenses-d1aba9e52254) is a helpful article for building my intuition, along with his article, [Functional Programming is for Dummies](https://medium.com/@dtipson/functional-programming-is-for-dummies-fa130a629250). So I'm getting an inkling of what it means to modify an object by being able to get some particular information inside of it and perhaps set that to a new value and retrieve the entire object. It reminds me a little of what I mean by "knowing everything", which is to have a perspective upon the big picture in its entirety, but then (being mentally shortsighted), being able to crawl out in any direction to the matter of interest, inspect it up close, and then be able to crawl back to where you started.
  • 10.
    edited May 20

    Van, I found an old letter by Joseph Goguen from May 9, 2005, about "What is knowing?". He mentions his paper that you mention, What is a Concept?

    Here's the introduction:

    My first step in answering the question "What is knowing?" would be to break it into two parts: "What is a concept?" and "What is truth?" since true concepts will be knowledge.

    I would also like to "de-reify" the question, since i think the processes of knowing are more fundamental than the results. So we should ask about processes of conceptualization, and of reasoning, while still noting that a great deal can be learned from looking at the reified notions of concept and truth.

    As you say in your analysis of "everything", knowledge is relative, and hence always uncertain, perhaps even contradictory; it is also uncertain to varying degrees.

    As noted long ago by Charles Sanders Peirce, the problems of relativity can be overcome to some extent by making the truth of what concepts refer to relative to context, in a very broad sense of context that includes the "knower" and his/her point of view, background knowledge, perceptions, etc., as well as what is in the world.

    So now we want to look at concepts and how they refer in variable contexts, and how we can reason with concepts in a way that allows the result to truthfully refer, not forgetting that concepts can of course refer to other concepts as well as to percepts.

    It does not seem to be as well known as it should be that there is a great deal of recent research on concepts, how they refer, and how we reason with them. This work has been done under labels that include cognitive linguistics, cognitive semantics, semiotics, and experimental psychology.

    It is also not very well known that a theory of logics has recently been developed, that includes a notion of satisfaction of a sentence by a model that can depend on context. It also includes as special cases all the classical logics (first order, modal, higher order, and so on), and even has generalizations of much of classical model theory (Craig interpolation, Beth definability, and so on). Moreover, it has had applications to mainstream programming languages (C++, Ada, ML) as well as to many specification languages, to database systems, ontologies (in the sense of the semantic web), and more, but for the purpose of this note, especially concepts. (Also the truth values of the satisfaction relation can be fuzzy.) This theory is known as the "theory of institutions."

    Id like to review just a little of work in these two areas, concepts and logics, give some references, and then some conclusions.

    also, from his conclusion...

    As far as i can tell, all of this is subsumed by the theory of institutions, and two recent papers of mine draw out many of the connections. These are What is a concept (written for the 2005 Conceptual Structures meeting) and Information Integration in Institutions (written for a memorial volume for Jon Barwise). ... much more in more technical papers listed on the institutions homepage ... especially the recent paper "What is a logic?" Applications to databases can be found in the paper "Data, Schema and Ontology Integration", also linked from the institutions homepage.

    The time seems to have come when the technology that "we" (meaning, Western society in general but also Minciu Sodas) are developing, and the applications that "we" have in mind for it, requires a more sophisticated understanding of knowledge, concepts, and logic than has previously been available. Such an understanding is developing rapidly on a number of fronts, overthrowing millenia of philosophical prejudices in favor of results that have an empirical basis either in laboratory experiments or in working prototype computer based systems. Database integration, robotics, the semantic web vision of Berners-Lee, the cognitive linguistics of Lakoff, Fauconnier, Turner, the multi-logic specification languages CafeOBJ and CASL (by Futatsugi, and by a European collective called CoFI), and so on, are all parts of this. One can also see it in the practice of contemporary artists (such as Bill Viola) and musicians (such as Beck), and many many others.

    I have taken on the crazy task of trying to formalize all this, and also developing some prototype systems that implement the aspects of the formalizations, such as an interactive poetry generation system, a blending algorithm, a database schema matching system, and an algebraic specification language. Obviously this is just a small part of a much bigger movement, but i think we can say that the goal of everyone involved in this large and diffuse area is to answer the questions raised by Ibrahim and Andrius, and i think we can say that we are getting answers, though slowly and often with considerable technical difficulty, in spite of which, the field as a whole seems to be moving very fast, at least to those who try to keep up with all (or a large part) of it. (Unfortunately there is not even an accepted name for the whole field!)

    I have also been trying to relate my ideas about concepts and logic to consciousness, and in particular, to qualia, which i define as segments of perception that are perceived as wholes (though they may still be seen to have parts), and i have worked on applications to free jazz improvisation.

    So I'll try to understanding something about his institutions. It seems that it is one of three approaches in universal logic, the other two being categorical model theory (a topological/categorical approach based on sketches), and Jon Barwise's abstract model theory.

    But I should certainly focus on my own approach which emphasizes perspectives instead of concepts. One advantage is that perspectives can be composed. Perspectives are like filters, criteria or lenses in that you don't need to define What they see, but rather How. I will think about how I imagine perspectives and concepts are related.

    Comment Source:Van, I found an old letter by [Joseph Goguen from May 9, 2005, about "What is knowing?"](https://groups.yahoo.com/neo/groups/livingbytruth/conversations/messages/408). He mentions his paper that you mention, What is a Concept? Here's the introduction: > My first step in answering the question "What is knowing?" would be to break it into two parts: "What is a concept?" and "What is truth?" since true concepts will be knowledge. > I would also like to "de-reify" the question, since i think the processes of knowing are more fundamental than the results. So we should ask about processes of conceptualization, and of reasoning, while still noting that a great deal can be learned from looking at the reified notions of concept and truth. > As you say in your analysis of "everything", knowledge is relative, and hence always uncertain, perhaps even contradictory; it is also uncertain to varying degrees. > As noted long ago by Charles Sanders Peirce, the problems of relativity can be overcome to some extent by making the truth of what concepts refer to relative to context, in a very broad sense of context that includes the "knower" and his/her point of view, background knowledge, perceptions, etc., as well as what is in the world. > So now we want to look at concepts and how they refer in variable contexts, and how we can reason with concepts in a way that allows the result to truthfully refer, not forgetting that concepts can of course refer to other concepts as well as to percepts. > It does not seem to be as well known as it should be that there is a great deal of recent research on concepts, how they refer, and how we reason with them. This work has been done under labels that include cognitive linguistics, cognitive semantics, semiotics, and experimental psychology. > It is also not very well known that a theory of logics has recently been developed, that includes a notion of satisfaction of a sentence by a model that can depend on context. It also includes as special cases all the classical logics (first order, modal, higher order, and so on), and even has generalizations of much of classical model theory (Craig interpolation, Beth definability, and so on). Moreover, it has had applications to mainstream programming languages (C++, Ada, ML) as well as to many specification languages, to database systems, ontologies (in the sense of the semantic web), and more, but for the purpose of this note, especially concepts. (Also the truth values of the satisfaction relation can be fuzzy.) This theory is known as the "theory of institutions." > Id like to review just a little of work in these two areas, concepts and logics, give some references, and then some conclusions. also, from his conclusion... > As far as i can tell, all of this is subsumed by the theory of institutions, and two recent papers of mine draw out many of the connections. These are [What is a concept](http://www.cs.ucsd.edu/~goguen/pps/iccs05.pdf) (written for the 2005 Conceptual Structures meeting) and [Information Integration in Institutions](http://www.cs.ucsd.edu/~goguen/pps/ifi04.pdf) (written for a memorial volume for Jon Barwise). ... much more in more technical papers listed on [the institutions homepage](http://www.cs.ucsd.edu/~goguen/projs/inst.html) ... especially the recent paper "What is a logic?" Applications to databases can be found in the paper "Data, Schema and Ontology Integration", also linked from the institutions homepage. > The time seems to have come when the technology that "we" (meaning, Western society in general but also Minciu Sodas) are developing, and the applications that "we" have in mind for it, requires a more sophisticated understanding of knowledge, concepts, and logic than has previously been available. Such an understanding is developing rapidly on a number of fronts, overthrowing millenia of philosophical prejudices in favor of results that have an empirical basis either in laboratory experiments or in working prototype computer based systems. Database integration, robotics, the semantic web vision of Berners-Lee, the cognitive linguistics of Lakoff, Fauconnier, Turner, the multi-logic specification languages CafeOBJ and CASL (by Futatsugi, and by a European collective called CoFI), and so on, are all parts of this. One can also see it in the practice of contemporary artists (such as Bill Viola) and musicians (such as Beck), and many many others. > I have taken on the crazy task of trying to formalize all this, and also developing some prototype systems that implement the aspects of the formalizations, such as an interactive poetry generation system, a blending algorithm, a database schema matching system, and an algebraic specification language. Obviously this is just a small part of a much bigger movement, but i think we can say that the goal of everyone involved in this large and diffuse area is to answer the questions raised by Ibrahim and Andrius, and i think we can say that we are getting answers, though slowly and often with considerable technical difficulty, in spite of which, the field as a whole seems to be moving very fast, at least to those who try to keep up with all (or a large part) of it. (Unfortunately there is not even an accepted name for the whole field!) > I have also been trying to relate my ideas about concepts and logic to consciousness, and in particular, to [qualia](http://www.cs.ucsd.edu/~goguen/projs/qualia.html), which i define as segments of perception that are perceived as wholes (though they may still be seen to have parts), and i have worked on applications to free [jazz improvisation](http://www.cs.ucsd.edu/~goguen/projs/arts.html). So I'll try to understanding something about his institutions. It seems that it is one of three approaches in [universal logic](https://en.wikipedia.org/wiki/Universal_logic), the other two being categorical model theory (a topological/categorical approach based on [sketches](https://en.wikipedia.org/wiki/Sketch_(mathematics))), and Jon Barwise's [abstract model theory](https://en.wikipedia.org/wiki/Abstract_model_theory). But I should certainly focus on my own approach which emphasizes perspectives instead of concepts. One advantage is that perspectives can be composed. Perspectives are like filters, criteria or lenses in that you don't need to define What they see, but rather How. I will think about how I imagine perspectives and concepts are related.
  • 11.

    Andrius, Thanks for sharing these letters. They are quite invaluable, to say the least! I came to know of his work through the theoretical biologist Francisco Varela. I believe they coedited the Journal of Consciousness together for a time. Varela's work has had a significant influence on my own development. I wonder if you met him as well, considering he also did some work with Louis Kaufmann.

    Here are a couple of papers I've been reading in light of what you've said:

    A interesting thread running through these papers - and pretty much all of Varela's subsequent work - is the interplay between the capacity of an observer to make distinctions in an environment, and the autonomy and self-referential capacity of that same observer. Somehow a perspective is 'the distinctions one makes as consequence of their autonomy'.

    It would be nice to able to say something like 'perspectives are autonomous organizations'. Varela and Goguen use adjoint functors and the idea of complementarity to express this idea.

    Maybe Goguen's Theory of Institutions is a way to do this: the manner in which each one of us integrates information is representative of how we'll deal/interpret/make sense of new information. Somewhat related may be Integrated Information Theory , which is being used to describe causal structure in biological systems .

    Comment Source:Andrius, Thanks for sharing these letters. They are quite invaluable, to say the least! I came to know of his work through the theoretical biologist Francisco Varela. I believe they coedited the Journal of Consciousness together for a time. Varela's work has had a significant influence on my own development. I wonder if you met him as well, considering he also did some work with Louis Kaufmann. Here are a couple of papers I've been reading in light of what you've said: - <a href="https://www.tandfonline.com/doi/abs/10.1080/03081077908960886"> Systems and Distinctions: Duality and Complementarity </a> Joseph Goguen and Francisco Varela - <a href="http://www.univie.ac.at/constructivism/journal/articles/13/1/011.kauffman.pdf" >The Mathematics of Francisco Varela </a> Louis Kauffmann A interesting thread running through these papers - and pretty much all of Varela's subsequent work - is the interplay between the capacity of an observer to make distinctions in an environment, and the autonomy and self-referential capacity of that same observer. Somehow a perspective is 'the distinctions one makes as consequence of their autonomy'. It would be nice to able to say something like 'perspectives are autonomous organizations'. Varela and Goguen use adjoint functors and the idea of complementarity to express this idea. Maybe Goguen's Theory of Institutions is a way to do this: the manner in which each one of us integrates information is representative of how we'll deal/interpret/make sense of new information. Somewhat related may be <a href="http://integratedinformationtheory.org/"> Integrated Information Theory </a>, which is being used to <a href="https://arxiv.org/abs/1708.07880"> describe causal structure in biological systems </a>.
  • 12.

    Van, Thank you very much for your ideas and links! I only know of Maturana and Varela from a book by Robert Axelrod on complex adaptive systems. I'm not able to access the Goguen and Varela paper, but it looks interesting. There is a lot to study and consider, but I will try to think and write especially about my own work.

    Louis Kauffman is very interested in Spencer Brown's Laws of Form, which is based on the act of distinction (making a mark) as a fundamental primitive which generates all structure.

    I have likewise taken on the question of how to generate all structure? I consider what I could imagine might motivate a primordial God who is prior to time, space, being, meaning, logic, etc. For such a God, being and nonbeing are the same, simply words. But would God exist even if God didn't exist? I think this is the motivating question. So an unconditional God removes himself to show that God would arise even in conditions. I explore this is in my talk, God's Question: Is God Necessary?, which I'll be presenting this August at the World Congress of Philosophy in Beijing. Basically, what I mean by God could be thought of as a state of contradiction. The idea is to explore how a state of contradiction can give rise to a state of noncontradiction, which is to say, to a logic. I proposed a talk on that, Divisions of Everything: Cognitive Frameworks Which Ground Contradiction and Noncontradiction. The basic structures divide everything into two perspectives for issues of being, three perspectives for participation, and four perspectives for knowledge. Here's a talk which relates that to consciousness: Consciousness as the Social Awareness Schema of a Disembodying Mind I also note the curious way that evolution of the central nervous system tends towards replacing the world with ever more abstract models, focusing resources on what we don't know rather than what we know.

    As regards perspectives, how do they arise? The crucial activity of God is that, in order to remove himself, God goes beyond himself into himself and thus create his self (or makes his mark, makes a distinction between what is "beyond system" and what is "within system"). If God is to explore his nonbeing, then God has to remove himself. But where does God go? There is only God. So God divides into two possibilities: A) If God exists, then God exists (as in the spiritual world), B) If God does not exist, then yet God still exists (which would model the physical world, for example, its tendency towards total abstraction). The only place that God has to go is his own self, but yet as his "self", which thus arises. Thus he goes beyond himself into his self, thereby creating his self, which is Everything. Everything is the structure of God, and God is the spirit of Everything.

    This contradictory process of "going beyond oneself" is the basis for perspective. I think the difference here is that perspective arises when we already have our self. Thus we are God who arises in conditions and wonders, Am I God? Perspective is the way that we go beyond ourselves, outside of system to what is beyond system, outside of what we know to what we don't know. Then perspective rests on a crucial sense of self. Without self, there is no perspective. And perspective is an act of spirit, which is to say, it is fundamentally self-contradictory because it is rooted in what is prior to system. I suppose this is my starting point.

    Comment Source:Van, Thank you very much for your ideas and links! I only know of Maturana and Varela from a book by Robert Axelrod on complex adaptive systems. I'm not able to access the Goguen and Varela paper, but it looks interesting. There is a lot to study and consider, but I will try to think and write especially about my own work. Louis Kauffman is very interested in Spencer Brown's Laws of Form, which is based on the act of distinction (making a mark) as a fundamental primitive which generates all structure. I have likewise taken on the question of how to generate all structure? I consider what I could imagine might motivate a primordial God who is prior to time, space, being, meaning, logic, etc. For such a God, being and nonbeing are the same, simply words. But would God exist even if God didn't exist? I think this is the motivating question. So an unconditional God removes himself to show that God would arise even in conditions. I explore this is in my talk, [God's Question: Is God Necessary?](http://www.ms.lt/sodas/Book/GodsQuestion), which I'll be presenting this August at the World Congress of Philosophy in Beijing. Basically, what I mean by God could be thought of as a state of contradiction. The idea is to explore how a state of contradiction can give rise to a state of noncontradiction, which is to say, to a logic. I proposed a talk on that, [Divisions of Everything: Cognitive Frameworks Which Ground Contradiction and Noncontradiction](http://www.ms.lt/sodas/Book/20180616DivisionsOfEverything). The basic structures divide everything into two perspectives for issues of being, three perspectives for participation, and four perspectives for knowledge. Here's a talk which relates that to consciousness: [Consciousness as the Social Awareness Schema of a Disembodying Mind](http://www.ms.lt/sodas/Book/20171011DisembodyingMind) I also note the curious way that evolution of the central nervous system tends towards replacing the world with ever more abstract models, focusing resources on what we don't know rather than what we know. As regards perspectives, how do they arise? The crucial activity of God is that, in order to remove himself, God goes beyond himself into himself and thus create his self (or makes his mark, makes a distinction between what is "beyond system" and what is "within system"). If God is to explore his nonbeing, then God has to remove himself. But where does God go? There is only God. So God divides into two possibilities: A) If God exists, then God exists (as in the spiritual world), B) If God does not exist, then yet God still exists (which would model the physical world, for example, its tendency towards total abstraction). The only place that God has to go is his own self, but yet as his "self", which thus arises. Thus he goes beyond himself into his self, thereby creating his self, which is Everything. Everything is the structure of God, and God is the spirit of Everything. This contradictory process of "going beyond oneself" is the basis for perspective. I think the difference here is that perspective arises when we already have our self. Thus we are God who arises in conditions and wonders, Am I God? Perspective is the way that we go beyond ourselves, outside of system to what is beyond system, outside of what we know to what we don't know. Then perspective rests on a crucial sense of self. Without self, there is no perspective. And perspective is an act of spirit, which is to say, it is fundamentally self-contradictory because it is rooted in what is prior to system. I suppose this is my starting point.
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