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Here's a nice quote.
Philosophy will not regain its proper status until the gradual
elaboration of categorial schemes, definitely stated at each stage of
progress, is recognized as its proper objective. There may be rival
schemes, inconsistent among themselves; each with its own merits
and its own failures. It will then be the purpose of research to
conciliate the differences. Metaphysical categories are not
dogmatic statements of the obvious; they are tentative
formulations of the ultimate generalities.
(Whitehead l978:8)
... from this article https://tidsskrift.dk/signs/article/download/26864/23624
Søren Brier
Cybersemiotics: A New Foundation for Transdisciplinary Theory of Information, Cognition, Meaning, Communication and Consciousness
More from Whitehead: “It is rigid dogma that destroys truth; and, please notice, my emphasis is not on the dogma, but on the rigidity.”
Would something like ‘Structuralism, semiotics, cybernetics and CT’ be a better stream name?
@janet singer i like it very much, but i think shorter is better. from the little zulip docs i have read so far, it said somewhere "after a while, you'll pick the right 2-3 word length stream titles". It looks like the medium has its own idea of structuralism.
ok, we take that accident + constraint as an opportunity :slight_smile:
So many issues with naming/branding in these fields that end up raising new obstacles – that could be a stream in its own right
Let me try to sketch a connection between exchange of signs - codes and CT now.
What would the ability to encode a process interpret correspond to, in CT? enriched categories/n-category theory?
Does this exist in semiotics? Some form of "reflection"
From programming languages, I would reach for "staged computation" which can be shown to correspond to modal operators in logic.
We would want to proof that the setup above is a category. What is it that composes? The processes themselves, but what happens to their staged interactions when they compose?
Have you seen Leandro’s categorical semiotics?
https://arxiv.org/abs/1604.02790
@Burak Emir The processes are what is composed. That means these are the arrows in the category. What are the objects?
Thanks @Johannes Drever - I guess that makes the possible interactions between processes good candidates for arrows. Staged interactions could be connected via arrows that compose on higher levels.
In the paper "A Compositional Framework for Markov Processes" the morphisms of the category are Markov processes. Do you mean something along these lines?
Burak Emir said:
Let me try to sketch a connection between exchange of signs - codes and CT now.
- We imagine a set of processes that communicate with each other (a concurrent system, immediately symmetric monoidal categories)
- Processes can be expressed in program code, and program code can be sent in a message (which makes it linguistic code.)
- They may link each other, form groups, with outward and internal protocols.
- Processes may chose to run/interpret process, emulate each other, receive "update" messages.
What would the ability to encode a process interpret correspond to, in CT? enriched categories/n-category theory?
Does this exist in semiotics? Some form of "reflection"
From programming languages, I would reach for "staged computation" which can be shown to correspond to modal operators in logic.
So this kind of made me think of operads, from the point of view of "networks" and sending information between nodes, as well as being able to substitute structure into nodes. has anyone in this stream ever thought about this type of thing?
(of course a central focus of my work is on operads, so it's perhaps not surprising that I see them everywhere)
(the basic idea of an operad is that it's some collection of "operations" (which you could think of as programs or functions if you wanted) that have some kind of typed inputs and typed outputs, with rules for how to compose these operations)
The operads I constructed in Network Models are specifically about constructing "communication networks" where you're supposed to think about information traveling on edges between the nodes.
@Joe Moeller oh nice!
There has been work done on operads of graphs where the operad composition is sticking a graph inside the node of another graph, and then flattening this out to a single graph somehow.
Joe is referring to his paper here: https://arxiv.org/abs/1804.07402
There's a book called Set Operads in Combinatorics and Computer Science by Miguel Méndez which talks about this stuff a lot. I haven't had a chance to dig too deep into it though.
@Joe Moeller that seems a lot like properads (the stuff with graphs)
I've never gotten around to finding out exactly what those are either!
Oh properads aren't hard if you know what operads are.
They're just operads but where you have multiple outputs as well as multiple inputs
Hey – @Jonathan Beardsley gave a ‘justification’ definition per @Joe Moeller
lol
That is a seriously useful idea!
What did I do? What's a 'justification' definition?
Oh is a 'justification' definition one in which you say 'just?'
Okay so @Joe Moeller can you tell me intuitively what one of these functors, these "network models" is supposed to be? like i understand the definition. how are they used? is one of these lax monoidal functors F:S-->Mon supposed to be something like "F(n) is all possible networks of type F on n nodes?"
Yes, exactly.
That's the set. The monoid structure specifies what happens when you overlay two such networks with the same set of nodes.
The lax structure of the functor tells you how to build a new one from two by setting them side by side.
okay nice. so is "overlaying" networks a thing that happens a lot, in practice? I guess it's one way you could model something like "adding an edge"
Right, think of it as adding edges. The original motivation is like communication networks, possibly different forms of communications, on the same map. Nodes are like people or boats or operation bases.
So you can suddenly establish a line of communication with a node that was previously out of range or something.
interesting. and so in the operad, the networks themselves become the "operations?"
i guess of a one colored operad
Networks are the operations, but the colors are a little odd to me. The colors are the number of nodes...
Right.
Hmm, if I were to choose to think of it as a properad rather than an operad, I think I'd be able to express something like partial evaluation of combining operations. Not sure why I would though.
Well, the properad idea is somehow a little different.
I mean I think you can still do something, but I'm not sure it's going to really be related. Maybe we should talk about it in a different topic though, b/c this isn't really cybernetics anymore (or is it???)
Not sure. It's vaguely CS-ish.
I think the idea I have is closer to what you mentioned before, something about inserting graphs into other graphs.
Oh yeah, I was thinking of some other properad structure.
I mean, so there's a sort of natural "cooperation" here, which is something like "dismembering" networks
Right! A very common observation in the algebraic combinatorics of networks, I think.
How would you make this like a properad?
Well you'd need to get really precise about what the cooperations are I suppose. But basically, instead of maps (as in an operad) from you have maps going the other direction (i.e. this thing is a coalgebra for the composition product)
I guess I hadn't even thought about properads as consisting of cooperations. I was just thinking multi-in multi-out. But that does make sense. So then maybe one of these like takes a bunch of graphs, sticks them together, and then rips them apart (probably in all possible ways)?
Yeah that's right. I mean I was just trying to think about what the cooperations were, before even thinking about what the "corollas" were
Also, yeah, since you don't really have any canonical way to do it in general, you just kind of "produce all possible subthings"
Maybe because it's a bit late at night, and I haven't thought about this stuff in a long time, I'm sort of blanking on some of the details.
One issue with this in your current construction of these network model operads is that you've only got one index, n, on everything, which is the number of nodes in the network. So I'm not sure you can, like, pull a properad structure out of that. Unless you just kind of, freely generate it or something
Yeah, I mean I'm not particularly attached to network models as a mathematical object. If the ideas call for a different initial data, then I'm fine with following that.
I'll try again (one day will read up on operads): you could say, the people on this stream are nodes, lines between are some common language, or protocol we share. That gives a network. But we want to understand how meaning is made. If somebody coined a new term, or made a joke, and that enables new common language or protocol, it seems there needs to be more than one level. I was thinking n-category, too, but not well-versed in the topic.
I mean the network should enter the picture, and an operad, too, probably (there is some process algebra, and from what I can tell operads are the symmetric monoidal way of doing lawvere's version of algebraic theories). We have this federation of processes who communicate and make new symbols.
The question how new symbols become into being is subject to George Spencer Browns "Laws of Form". The basic unit of meaning-production is the distinction, the difference that makes a difference. I think it has some relations to category theory and would be happy to explore them. The nothingness which comes before the first distinction is very similar to the "primordial ooze" which often comes up in CT.
I guess it’s not true Moeller justification until it is phrased as something as just an X in the category of Y – that’s the gold standard
@Johannes Drever I just read the wikipedia summary of "Laws of Form", and thought a bit about this. The part where I will agree is the logics: that one should be able to build entire systems of logical reasoning on the notion of apartness. One pointer is https://ncatlab.org/nlab/show/apartness+relation and from programming languages (PL), there is "separation logic" which applies this idea to reason about memory safety in the presence of pointer aliasing. I think if anybody has doubts about this, it becomes a lot more compelling when you think about the road more often travelled, which is to take identity and equality as primitive. I have this draft blog post about the formulation of Church's simple theory of types (simply typed -calculus which only has equality as a primitive notion (Andrews ). There is a rich literature on how identity and equality are fundamental to Martin-Loef type theory and HoTT.
The part where I'd want to differentiate (can't help the meta-irony) is to equate logic and meaning. I would say, creation of new signs and meanings is probably accompanied by a common (or common enough) understanding of rules, attributes, and yes, logical axioms and systems of reasoning, but there is a lot more to it. For example, I would guess that the inadequacy of existing systems motivates creation of new ones.
Let's take "justification" as a symbol whose complex meaning is currently "in production". I followed some lovely threads on twitter, where the frequent use of "just" is discussed as being sometimes lucid, often well-meaning, but for newcomers can also be alienating since it presupposes understanding and can be (mis)understood as patronizing. The ironic use of "justification" is catchy, but I want to call out that the attitude - that while attractive to specialists, one should take care of not alienating newcomers - is such a powerful, constructive thing which carries inherent meaning; and still we'd have a hard time capturing all that using logic. Not all use of "just" will be bad and alienating, but calling out "justification" will always be helpful. I think since this appears in the context of ACT community discussing category theory, I think this is something where we flip between the object logic/language "a properad is ..." and the meta-level (or maybe the meta meta level). There is a pragmatic aspect here.
I do think taking Cybernetics as an angle, identify/apartness as a primitive to build formal systems / reasoning is a rich angle, and the need to adapt/create new systems is a challenge to any system. The closest I have seen is Piaget's developmental theory of knowledge (I somehow dislike the term 'genetic epistemology' but it is how he called his theory), where children have a model of reality which they adapt, and they do this for different kinds of knowledge.
Thanks for the valuable input, @Burak Emir ! Discussing LoF as separation logic makes sense, I'll look into the nLab article. David Ellerman wrote a memo explaining LoF as partition logic (http://www.ellerman.org/a-note-on-spencer-browns-algebra/). I think this is a common theme that people say: "What you are doing sounds interesting, but actually it's just the same thing what I'm doing and what XYZ was doing for a long time". I have a grudge with this kind of argumentation, because I'm trying to convince OOP-people that FP is great and call it the "Turing tarpit". So is LoF something alien which has additional merits to other formalisms or is it just an obscure blip in the history of mathematics? I don't know and I also don't care. I just appreciate the beautiful book GSB wrote.
And yay for Piaget, this is exactly the stuff I'm looking for. Piaget did look into category theory his later years, as outlined in "Piaget’s Category-Theoretic Interpretation of Cognitive Development: A Neglected Contribution".
I didn't follow the discussion on "just" and "justification", but it sounds promising. Equating meaning with logic may indeed be messy and not helpful. But LoF goes to a place before any distinction between content-level and meta-level. Actually chapter 8 of LoF is called "re-uniting the two orders", where the equation of what GSB calls "content" and "image" is laid out. I try to relate this to the "primordial ooze" which occurs in CT.
These here about "just" and "justification". Meaning-production as it happens! Please don't take too seriously :-)
https://twitter.com/_julesh_/status/1227621935124811781
https://twitter.com/_julesh_/status/1237057980115255307
I looked up Fourier transform on nLab as a joke, looking for "it's just X in the category of Y", but it's actually a totally normal page about Fourier transforms that hardly mentions categories at all. I feel let down https://ncatlab.org/nlab/show/Fourier+transform
- julesh (@_julesh_)I remember this thread and I was asking a question back then: https://twitter.com/comandingo/status/1227925464632692743?s=20 Actually I wanted to know something there, but now it just sounds like a bad pun :grinning_face_with_smiling_eyes:
@_julesh_ @danghica @KyleCranmer Is it always “X is just a Y in the category of Z” or are there other patterns where “just” is just?
- Johannes Drever (@comandingo)@Johannes Drever wrote:
I think this is a common theme that people say: "What you are doing sounds interesting, but actually it's just the same thing what I'm doing and what XYZ was doing for a long time".
The whole topic of formal logics is like this, and it is interesting because it is a "Turing tarpit" that existed before Turing published his celebrated 1936 Entscheidungsproblem paper. I think on the one hand, academic on research in formal logic is concerned with expressiveness and limits of reasoning. So "anything that can be expressed in language/system X can be expressed in Y" is an insight worth stating.
Now category theory has been called the "structure mathematics" and also something like an ultimate achievement of the axiomatic method (since everything that is expressed follows from a few axioms). Also, a hundred years ago, every PhD student came up with their own system of formal reasoning (some parallels to PL research today where everybody builds their own language and/or type system sound).
Jokes aside, in programming languages "you can write X as just Y" is basically everywhere. I think a more generous way to look at "just" is that showing alternative ways of expressing the same things is something close to analogy (with the difference that analogy is not always regarded as proof, while X is just Y is often something that can be proved).
Saunders Mac Lane did actually try to work on the principia mathematica in his PhD. He was disappointed because he didn't get very far. The reason was that he just made some kind of "linear transformation" and just reduced the complexity by a factor of n. (I'm not sure where I got this story from, maybe McLarty ). But there are actually transformations which are not only linear, I'm thinking in terms of compression.
McLarty gives a more reasonable account of the story in The Last Mathematician from Hilbert's Gottingen: Saunders Mac Lane as Philosopher of Mathematics :
In practice, though, Mac Lane found that the way to radically shorter proofs—and to previously infeasible proofs—is not through abbreviation or apt details. It is through new concepts. His dissertation had introduced the concept of the ‘leading idea’ of a proof, which was itself meant to be a leading idea for further work in logic. He soon found leading ideas that still guide work in algebra and topology today. They grew from where he did not expect them.
I actually have Laws of Form on a bookshelf around here somewhere. Maybe I should crack it open again. I think I brought it up on the nforum a while back and kinda got shot down.
Jonathan Beardsley said:
I actually have Laws of Form on a bookshelf around here somewhere. Maybe I should crack it open again. I think I brought it up on the nforum a while back and kinda got shot down.
Maybe the notes on chapter 12 are a good place to start. He describes how distinctions are written on a sphere instead of a plane, and then proceeds to explain how different positions of the observer lead to different expressions, which are distinct and not equivalent.
Oh here it is, just found it... everyone is probably being very reasonable, but as a second year grad student I was a bit cowed by the discussion.
https://nforum.ncatlab.org/discussion/4047/cybernetics-and-homotopy-theorycategory-theory/#Item_18
I don't have access to the library these days due to lockdown, and shelf space is a scarce resource in my home; but I'm quite curious now that I see how several folks connect this book to the 'cybernetics' topic.
The origin event, May 30, 2018
@Joe Moeller
Idea: the expression of a mathematical concept in categorical terms should be called "justification". A Lie group is JUST a group object in Diff. The term "justification" pairs nicely with "categorification".
@John Baez
That is cute. This also explains why people hate category theorists.
Moeller
This is a joke I feel could be taken too far. A concept which hasn't been expressed in CT is "unjustified".
https://mobile.twitter.com/CreeepyJoe/status/1001902385172627458
Jonathan Beardsley said:
Oh here it is, just found it... everyone is probably being very reasonable, but as a second year grad student I was a bit cowed by the discussion.
https://nforum.ncatlab.org/discussion/4047/cybernetics-and-homotopy-theorycategory-theory/#Item_18
Thanks for sharing this! The second comment by Urs Schreiber is pretty awesome and insightful.
I imagine that this was what Spencer Brown was after. A kind of creation myth of the world from algebra of symbols. I am not sure if this is a reasonable thing to do. But I think I do understand the inner conditions that could make one try to do this.
is it wrong to think that "justification" is just the localization of a mathematical object of "Math" into "Category Theory"?
http://www.rutherfordjournal.org/article030107.html The Structures of Computation and the Mathematical Structure of Nature
by Michael S. Mahoney ... has a few extracts from a paper by Schützenberger on cybernetics, and generally the whole piece touches on cybernetics topics (e.g. "complex adaptive systems").
Re: justification, was hoping that "X is just Y in category of Z" could be an example of signifier (the name X) and signified (the concept), which is coded with a different sign (Y) and embedded in a context of meaning Z.
Yes! “Justification” shows a connection between between CT and semiotics that seems alternately profound and trivial …
Let me share this nice quote from Girard "On the meaning of logical rules I : syntax vs.
semantics"
Syntax is about language and semantics (from ση˜µα, sign) is about the interpretation of
signs, an activity that includes
Heinz von Foerster "Objects: Tokens for (Eigen-)Behaviors"
I shall talk about notions that emerge when the organization of sensorimotor interactions (and also that of central processes (cortical-cerebellarspinal, cortico-thalamic-spinal, etc.)) is seen as being essentially of circular
(or more precisely of recursive) nature. Recursion enters these considerations whenever the changes in a creature’s sensations are accounted for by
its movements (si = S(mk)), and its movements by its sensations (mk = M(sj)). When these two accounts are taken together, then they form “recursive
expressions,” that is, expressions that determine the states (movements, sensations) of the system (the creature) in terms of these very states (si =
S(M(sj)) = SM(sj); mk = M(S(mi) = MS(mi)).
[...]
... cognition computes its own cognitions through those of the other: here is the origin of ethics.
I write in this thread though it has been asleep for about a year now, since I see @Johannes Drever and @Jonathan Beardsley have been speaking openly about Laws of Form and the other contributors seem to have appreciated. Moreover I see everybody knows Varela. So I think it is the right place to ask something without being cowed.
Besides all the criticism one can make to Spencer Brown style and work, I think the truly interesting thing about Laws of Form is that it tries to formally model the act of drawing a mark on a sheet of paper, i.e. making a distinction, and this is an act which coincide with its own result.
This aspect is shared by other concepts in natural languages (at least those coming from Latin, which are those I know, I would be curious about other roots...). Take for instance "organization", which points both to an "it" and to the "process leading to it". Similar are "abstraction", "(re)cognition", "assertion","judgement","sight" and many other. All these share the fact of being "perfect in themselves", and may be couterposed with other, which have a "progressive" nature, such as "walk", for instance. Somehow this recalls the distinction between "energheia" and "kinesis" in Aristotle's terms.
In a sense, the distinction is the archetype of all these concepts and in my opinion Spencer Brown's attempt to formalize it is valuable, despite the mixed feelings it arouses in the math community.
Now, probably all I written above is very well known by all of you here. What I would like to ask is:
Which mathematical concepts offer, in your opinion, a semantical domain to interpret the act of distinction?
Which mathematical structure embodies this identification of act and result?
Can we find answers in categories or generalizations?
I think these would be interesting question to answer, and that our difficulty to relate with Spencer Brown ideas comes from thinking in terms of sets, functions, relations. A function can stand for a process, but it will always detach act and result: it will always be some black box which converts input in output, and we can tell input (domain) output (codomain) and process (the function as a subset of Domain X Codomain).
Instead, I think we would need something different, in a sense something which groups ore lambda algebras have, where each element can act on the structure itself (this is I think what Kauffman does with magmas). Or a kind of graph, where somehow vertexes can be edges.
All the above is quite imprecise, but I would like if somebody could share their thoughts about it
I find myself returning to Ashby's Introduction every now and again, more and more often since the 90s when I returned to grad school in a systems engineering program to continue my work on a class of “intelligent systems as models of scientific inquiry” called Inquiry Driven Systems. Traces of what I've been doing along those lines can be found on the following Survey pages from my blog.
• Survey of Cybernetics
• Survey of Inquiry Driven Systems
Regards,
Jon