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I have this naive intuition that graphs are a universal encoding scheme, all theorems of category theory can be expressed as graphs, with the implication there

I have this naive intuition that graphs are a universal encoding scheme, all theorems of category theory can be expressed as graphs, with the implication there is a massive unifying leap forward in maths that will result from expressing existing problems in terms that are consistent with such a representation. Maybe it's literary sci fi handwaving, but there must be a rules based level of abstraction that contains all we can conceive of and express. These diagrams appear to be an example of it.

posterboy 7y ago
> Maybe it's literary sci fi handwaving, but there must be rules based level of abstraction that contains all we can conceive of and express. > rules based level of abstraction that's not literature, that's bad grammar. Sorry.
- motohagiography 7y ago
  Please.If only egregious crimes against the language were the bar.
zozbot234 7y ago
What kinds of graphs? There are a number of graphical representations that turn out to be incredibly useful in category theory, but they're far from equivalent. IIRC, https://arxiv.org/abs/1803.05316 https://arxiv.org/abs/1803.05316 introduces some of them.
- motohagiography 7y ago
  Just saying objects and morphisms. The types of each are found in CT, which abstract-up to these, but are bounded/encompassed by the limits we know of regarding elementary ideas like cliques, cycles, paths, colouring, various morphisms. Compression, folding, encoding, and other information theory ideas have analogues represented as these, and the bottom up/depth first search of the problem space seems to come at it from the wrong direction. We're well into indulging Internet crank territory here but I'm naively speculating that for every known theorem, there is a consistent object/morphism representation of it, and the rules we're getting good at finding for these relationships (graphs) will yield insights we were missing for lack of an encompassing consistent abstraction.
carapace 7y ago
Are you channeling Grothendieck? :-) https://en.wikipedia.org/wiki/Alexander_Grothendieck#Mathematical_work https://en.wikipedia.org/wiki/Alexander_Grothendieck#Mathema...
nixpulvis 7y ago
What language expresses itself in the most intuitive way is a somewhat undefined problem as far as I understand it. Humans are fuzzy messy objects, our languages too. Lambda calculus is partially fun just because of the minimal extent of its description.
galaxyLogic 7y ago
Graphs are a natural way to express structures, of things being composed of parts which are somehow related to each other connected to each other. What are some alternatives to graphs? Well some kind of language, say a programming language. But underlying any program written in any programming language is a graph of the parts of the program and their connections to each other. For instance a variable may be written in one section of the code and read in another, and if it is the same variable then those locations in code are part of the graph showing the data-flow of the program. So it is my intuition too that graphs are the fundamental representation of reality as we see it, the parts of reality and the connections and relations between those parts. Knowledge is knowledge about containers and parts and how those related to each other. We can describe it in natural language or perhaps in algebra but really the picture we are expressing in any language is that of wholes and parts and their connections. This is my intuition about it I don't know if such a general intuition is much good in actual theory formation. Theories are graphs I think.
- galaxyLogic 7y ago
  I guess what I'm after here is Semantic Web, which was supposed to be able to represent all knowledge, and would be a graph.
fyp 7y ago
Since there's a correspondence between proofs and programs (Curry-Howard), maybe you're just looking for a graph-based visual programming language. But given the failure (of adoption) of dataflow programming languages I don't think there will really be that much of a jump in understanding just by visualizing stuff as graphs.
visarga 7y ago
> I have this naive intuition that graphs are a universal encoding scheme I have thought of the same thing in the domain of neural nets. You want to understand an image? Build a graph of the objects and their relations. You want to understand a text? Turn it into a graph. Need to simulate the environment? You can represent it as a graph. Want to mine a huge database of common facts? Represent it as a graph (database of triplets subject-relation-object). Even the execution of a program could be better conceptualised as a graph. Graphs seem like an universal format. Graph Neural Nets have been a blooming subfield lately, ever since Thomas Kipf's paper [1] which in three years has accumulated over 2300 citations. The main difference between classical neural nets and GNNs is that GNNs have permutation invariance. You can rename the nodes of the graph and still have the same graph. Previously neural nets only had translation invariance (CNN) and time invariance (RNNs). Graphs would solve the combinatorial explosion by virtue of their permutation invariance - a problem which limits previous models of neural nets. Instead of learning all possible combinations of input objects you learn the pairwise relations, then you can generalise those relations to new configurations. On a side note, the insanely popular transformer architecture (based on soft attention) is a kind of implicit graph, where the connectivity is evaluated based on the dot product similarity of the key and query objects. [1] https://scholar.google.ro/scholar?hl=en&as_sdt=0%2C5&q=kipf+Semi-supervised+classification+with+graph+convolutional+networks&btnG= https://scholar.google.ro/scholar?hl=en&as_sdt=0%2C5&q=kipf+...
- xorand 7y ago
  Thanks for the link to Kipf's paper, I was not aware of it. I'm a mathematician working on graph rewrite systems, I would like to explore in the NN direction, in case you're interested contact me.
- slifin 7y ago
  I'm definitely coming around to graph first systems like Fulcro and Datomic, with Datahike being a great Datomic introduction
ratmice 7y ago
Similarly all propositions in modal logic can be expressed as kripke models, which have quite the correlation to graphs As good a reference as I could find... https://rkirsling.github.io/modallogic/ https://rkirsling.github.io/modallogic/
xorand 7y ago
I think, similarly, that there is an underground graphical rewriting formalism for all of mathematics, category theory included. Category theory is somehow a step towards it, but we don't keep categories in our heads, only the graph reducer.

posterboy 7y ago

> Maybe it's literary sci fi handwaving, but there must be rules based level of abstraction that contains all we can conceive of and express. > rules based level of abstraction that's not literature, that's bad grammar. Sorry.

motohagiography 7y ago
Please.If only egregious crimes against the language were the bar.

zozbot234 7y ago

What kinds of graphs? There are a number of graphical representations that turn out to be incredibly useful in category theory, but they're far from equivalent. IIRC, https://arxiv.org/abs/1803.05316 https://arxiv.org/abs/1803.05316 introduces some of them.

motohagiography 7y ago
Just saying objects and morphisms. The types of each are found in CT, which abstract-up to these, but are bounded/encompassed by the limits we know of regarding elementary ideas like cliques, cycles, paths, colouring, various morphisms. Compression, folding, encoding, and other information theory ideas have analogues represented as these, and the bottom up/depth first search of the problem space seems to come at it from the wrong direction. We're well into indulging Internet crank territory here but I'm naively speculating that for every known theorem, there is a consistent object/morphism representation of it, and the rules we're getting good at finding for these relationships (graphs) will yield insights we were missing for lack of an encompassing consistent abstraction.

carapace 7y ago

Are you channeling Grothendieck? :-) https://en.wikipedia.org/wiki/Alexander_Grothendieck#Mathematical_work https://en.wikipedia.org/wiki/Alexander_Grothendieck#Mathema...

nixpulvis 7y ago

What language expresses itself in the most intuitive way is a somewhat undefined problem as far as I understand it. Humans are fuzzy messy objects, our languages too. Lambda calculus is partially fun just because of the minimal extent of its description.

galaxyLogic 7y ago

Graphs are a natural way to express structures, of things being composed of parts which are somehow related to each other connected to each other. What are some alternatives to graphs? Well some kind of language, say a programming language. But underlying any program written in any programming language is a graph of the parts of the program and their connections to each other. For instance a variable may be written in one section of the code and read in another, and if it is the same variable then those locations in code are part of the graph showing the data-flow of the program. So it is my intuition too that graphs are the fundamental representation of reality as we see it, the parts of reality and the connections and relations between those parts. Knowledge is knowledge about containers and parts and how those related to each other. We can describe it in natural language or perhaps in algebra but really the picture we are expressing in any language is that of wholes and parts and their connections. This is my intuition about it I don't know if such a general intuition is much good in actual theory formation. Theories are graphs I think.

galaxyLogic 7y ago
I guess what I'm after here is Semantic Web, which was supposed to be able to represent all knowledge, and would be a graph.

fyp 7y ago

Since there's a correspondence between proofs and programs (Curry-Howard), maybe you're just looking for a graph-based visual programming language. But given the failure (of adoption) of dataflow programming languages I don't think there will really be that much of a jump in understanding just by visualizing stuff as graphs.

visarga 7y ago

> I have this naive intuition that graphs are a universal encoding scheme I have thought of the same thing in the domain of neural nets. You want to understand an image? Build a graph of the objects and their relations. You want to understand a text? Turn it into a graph. Need to simulate the environment? You can represent it as a graph. Want to mine a huge database of common facts? Represent it as a graph (database of triplets subject-relation-object). Even the execution of a program could be better conceptualised as a graph. Graphs seem like an universal format. Graph Neural Nets have been a blooming subfield lately, ever since Thomas Kipf's paper [1] which in three years has accumulated over 2300 citations. The main difference between classical neural nets and GNNs is that GNNs have permutation invariance. You can rename the nodes of the graph and still have the same graph. Previously neural nets only had translation invariance (CNN) and time invariance (RNNs). Graphs would solve the combinatorial explosion by virtue of their permutation invariance - a problem which limits previous models of neural nets. Instead of learning all possible combinations of input objects you learn the pairwise relations, then you can generalise those relations to new configurations. On a side note, the insanely popular transformer architecture (based on soft attention) is a kind of implicit graph, where the connectivity is evaluated based on the dot product similarity of the key and query objects. [1] https://scholar.google.ro/scholar?hl=en&as_sdt=0%2C5&q=kipf+Semi-supervised+classification+with+graph+convolutional+networks&btnG= https://scholar.google.ro/scholar?hl=en&as_sdt=0%2C5&q=kipf+...

xorand 7y ago
Thanks for the link to Kipf's paper, I was not aware of it. I'm a mathematician working on graph rewrite systems, I would like to explore in the NN direction, in case you're interested contact me.
slifin 7y ago
I'm definitely coming around to graph first systems like Fulcro and Datomic, with Datahike being a great Datomic introduction

ratmice 7y ago

Similarly all propositions in modal logic can be expressed as kripke models, which have quite the correlation to graphs As good a reference as I could find... https://rkirsling.github.io/modallogic/ https://rkirsling.github.io/modallogic/

xorand 7y ago

I think, similarly, that there is an underground graphical rewriting formalism for all of mathematics, category theory included. Category theory is somehow a step towards it, but we don't keep categories in our heads, only the graph reducer.