Thread by @prathyvsh

This is one of the most important talks I have come across recently. It critiques the discrete alphabet oriented encoding which Longo locates as the LaPlacian computational paradigm for its determinism as being the reason behind many artificial bounds imposed on human inquiry.

https://twitter.com/patternatlas/status/1286555755978452992

Yet to arrive at a tractable way of expression for this in my programming work. But computation can be expressed as a traversal on a topological space. Here’s Longo’s paper that talks about topologies for computation: di.ens.fr/users/longo/fi…

Finally a step closer towards this direction. The paper “Topological Interpretation of Interactive Computation” by Emanuela Merelli and Anita Wasilewska shows how a loop we describe with symbols when programming becomes an actual loop in space! arxiv.org/abs/1908.04264

Add together the idea that currying composes cross sections of a manifold together and one might be able to start to piece together a picture of what programming can be interpreted to be doing with space:

https://twitter.com/prathyvsh/status/1190219676363771905

This has been something brewing in the back of my mind, but great that it has got explicit expression here. Programs have a counterfactual-supporting causal structure and can provide a causal model of the external world: onlinelibrary.wiley.com/doi/epdf/10.11…

These two threads by Jules are well worth a read in this direction of machines ∩ topology:

1/

https://twitter.com/_julesh_/status/1070020821241004033

2/

https://twitter.com/_julesh_/status/1313492035433791495

This conversation made me think of the idea of describing the possible states of a system using types:

https://twitter.com/mimblewabe/status/1156654690886737920

It rings close to Yaron Minsky’s quote: “Make illegal states irrepresentable”.

Also cf. types as spaces:

https://twitter.com/EliSennesh/status/1182670329954492417

Going by the looks of it, this is converging into a thesis where I need to start talking about the “Dynamics of Computation”. That is, the idea that computation is visual merges directly into chaos and dynamics:

https://twitter.com/prathyvsh/status/1187669411936686080

Where all this lead up to is, executions of computer programs can be thought of as taking distinct trajectories on a higher dimensional object in a particular geometric space. Similar to what we see here.

Source: dsweb.siam.org/Media-Gallery/…

Having a structural/spatial interpretation of computation means that we can understand interactive computations as topological structures.

Here is a note on comparing recursion with recurrence in Poincaré map:

Fixed points in dynamical systems and computation can now be given a unified treatment by taking the spatial perspective!

This is a pretty interesting paper for it shows how a regular language automata can interpret a space of data described topologically. A point in the space of data can be interpreted as a path of the automata: semanticscholar.org/paper/Non-loca…

Found Emanuela’s talk that goes over the concept of topological interpretation in a bit more detail up here. A Topological Approach to Compositionality in Complex Systems: simons.berkeley.edu/talks/emanuela…

This is a really accessible talk by Mario Rasetti on what their motivation and vision is for the topdrim.eu project is. The idea is to create a sort of Maxwellian equations for studying complex systems:

Wittgenstein style “picture theory of computing” is indeed a direction which I think is fertile and few advancements has been made towards. Helps ground the polysemy aspect of information without a context.

Source: iopscience.iop.org/article/10.108…

This rings close to the model of computation I was seeking. Compute by describing the symmetries of the space you operate on! Here’s the relevant paper that details the approach — “Computational Mechanics: Pattern and Prediction, Structure and Simplicity”: csc.ucdavis.edu/~cmg/papers/cm…

I need more time to ground exactly how, but there’s a sense in which axiomatic systems like Boolean Algebra are just like this: you describe a set of invariants, describe the possible (a)symmetries, and out comes a system that you can explore and build constructs with.

Can’t help but relate this with the description of computation as wave function collapse on a Hilbert space as Peter Rodgers put it:

I genuinely thought it was hyperbole at the time, but can start to see how constructs of physics describe computation.

Another find in this direction: “Data Structures as Topological Spaces” — lacl.fr/~michel/PUBLIS…

This idea of treating graph network points as lying on a manifold I feel could be helpful when representing the space of computation. Transport distance seems to be used as the metric that generates this geodesic for the network.

Source: arxiv.org/pdf/1708.07904…

TIL {multiply, application, complex, complicated, imply} all share the word pleḱ (Proto-Indo-European word meaning to fold/plait) as a root word. Multiply seen this way becomes a verb like synonym of the noun manifold.

A critical insight I gained recently: Category Theory studies “functoriality” of objects in higher dimensional spaces and formalizes the relations immanent in projections/lifts of them. Here’s an awesome illustration from Kostecki that illuminates this: fuw.edu.pl/~kostecki/ittt…

The notion of classifying theories by their toposes feels like a really powerful idea. What I am most excited about is, what geometrical intuition can such classifying toposes provide: glass-bead.org/article/the-th…

Strongly feel this structural invariant description style is missing in software engineering today. Best we get is bi-interpretation where we go from one language to another. Describing constraints and then letting a language unfold around it could be a really fruitful approach.

TIL about the work of Andrée Ehresmann: en.wikipedia.org/wiki/Andr%C3%A…

With Vanbremeersch, she has developed a model for living systems using an approach that pioneers ‘dynamic’ category theory of evolving memory systems which can analyze complexity, emergence and self-organization.

This is a notion I am slowly warming upto: Category theory as process theory. Ehresmann’s work patently expresses how CT can be used to model complex dynamic systems. She aces the Glass Bead Game and how! glass-bead.org/article/the-gl…

It is very interesting to note that Turing devised a topological proof for establishing that every written alphabet is finite: divisbyzero.com/2010/05/27/tur…

This looks like an interesting two part series introducing the link between modal logic and topology:

https://twitter.com/serokell/status/1217811595025420288?s=21

Thanks to @MaineFrameworks for the hint! https://t.co/saRfnS8rP0

Dictionary relating concepts from topology and computation: cs.bham.ac.uk/~mhe/.talks/po…

Neat introductory article from @johncarlosbaez on Topos Theory. It has got a beginner to advanced set of book recommendations towards the end which is worth treasuring: math.ucr.edu/home/baez/topo…

@johncarlosbaez Unearthed yet another crazy good paper during my spelunking: This one elucidates the duality between syntax and semantics as a manifestation of duality between algebra and geometry.

Source: arxiv.org/pdf/2001.04952…

Steve Huntsman and Michael Robinson of the algebra/geometry adjoint paper along with Jimmy Palladino has written a survey on the uses of topology in computer science here: arxiv.org/pdf/2008.03299…

There are a few different ways to define a topological space. But the characterization in this book: amzn.to/2Xg6XRB as transformations that preserve the adjacency relation made me realize that it can also be understood from an information theoretical viewpoint.

Topological transformations preserve the adjacency relation between elements in that no new locations (topos) are generated/destroyed in the structure. From an information theoretical perspective this can be seen as preserving the amount of information in the original structure.