dmm | Lightning talks at Boston Tech Poetics meetup

The write-up for my talk:

1) Shaders are awesome. Shadertoy site, "The Book of Shaders" online book, etc.

See dmm.dreamwidth.org/20076.html

2) I am playing with neuromorphic computations with linear streams.

See anhinga.github.io

There are many ways to view this topic. One of the possible viewpoints: we want to synthesize animations, just like we synthesize digital music and audio: via composition of unit generators (invented by Max Mathews (Bell Labs, 1957)).

Some examples of that idea can be found in our Project Fluid: github.com/anhinga/fluid

I showed a Processing 2 run roughly corresponding to this recording: https://youtu.be/fEWcg_A5UZc

3) If there are questions afterwards, or if people wants to collaborate on this, one of the ways to contact me is the first author's e-mail here: arxiv.org/abs/1512.04639

(The meetup was on October 16 near Davis Square.

Boston Tech Poetics exists is Boston for many years, it used to be called Creative Coding at first.)

Flat | Top-Level Comments Only

From:

dmm

That's because Dana Scott can do it both ways: via Scott topology and via logic valued in Heyting algebras (so-called Omega-sets) :-)

But finding a nice way to integrate it all might be non-trivial, despite the fact that deep connections between them exist...

I'd love to try to bring them closer together :-)

From:

juan_gandhi

Hmm, interesting; I thought it was all closely related.

From:

dmm

These are two different ways to approach partially defined elements. That Dana Scott initiated both is rather funny :-)

On some level (e.g. Heyting-valued equalities vs partial ultra metrics valued in dual Heyting) they are very compatible, but on a deeper level differences start to show...

Bringing it all together would be interesting, but might be quite non-trivial...

From:

juan_gandhi

OMG, studying quantales. A big new world opening.

From:

dmm

Yes, I love those things...

From:

dmm

Interestingly enough, possible non-commutativity here evokes "non-commutative geometry" by Alain Connes, and non-idempotency makes one think of linear logic.

I am not sure what is known of the possible generalizations of the notion of topos, so that the internal logic would be a quantale rather than a Heyting algebra. I'd like to find out...

***

I should probably write here some pointers I know about quantale-valued logic (and Grothendieck topology-valued logic), and also some pointers I know about the categorification of Heyting-valued and quantale-valued logic in the sense of Lawvere.

Edited Date: 2019-10-23 01:29 pm (UTC)

From:

juan_gandhi

Yes; thanks a lot! It's all new to me. Reading.

From:

dmm

I'll do this as a series of comments (too much material; and I need to do some extra searches; the only thing I so far have no idea about at the moment is how to generalize a topos so that the internal logic becomes a quantale; but I'll try to search more on this as well; but other topics first).

Some connections with linear logic are noted here:

https://ncatlab.org/nlab/show/quantale#RelationToLinearLogic

( https://www.jstor.org/stable/2274953 )

I know that Ulrich Hoehle is focusing on relationships between quantales and non-commutative geometry in recent years:

https://www.researchgate.net/scientific-contributions/2024098557_Ulrich_Hoehle

(e.g. https://www.researchgate.net/publication/319284012_Non-commutativity_and_many-valuedness_The_topological_representation_of_the_spectrum_of_C-algebras )

To be continued...

From:

juan_gandhi

Thanks!

From:

dmm

Next, I'll write a bit about the start of the theory of "Heyting-valued sets" (sets with equality predicates valued in a Heyting algebra).

The prime example is, of course, a set of partially defined functions from a topological space to some fixed set. We say that the degree of equality of functions F and G is the interior of the set of points X, where F(X)=G(X). So the "=" predicate is valued in the algebra of open sets of the topological space in question.

Generally speaking, the domain of definition of F is some open set U, so the extent to which F is equal to itself is not "True" (the whole space), but only its domain of definition. So the function is equal to itself to the extent to which it is defined.

So, that's the example. The theory was first created in an unpublished but widely cited manuscript by Denis Higgs, "A category approach to Boolean-valued set theory", 1973. Google Scholar thinks there is 114 citations of this text. A lot of things are strange about the history of this manuscript, starting from its title (looking at the title one would think, and many people do make this mistake, that Higgs had only developed a theory of predicates valued in Boolean algebras; but that's not the case; this might have been his initial intention, or, perhaps, a name for a course he decided to create in U. of Waterloo; but the manuscript contains a theory of Heyting-valued predicates, and not a Boolean-valued special case). I'll write a bit more later about his manuscript.

(Wikipedia page about Higgs thinks that "Boolean" here stands for this: "In 1973, he generalised the Rasiowa-Sikorski Boolean models to the case of category theory."; so, basically, that he took the predicates valued in Boolean algebras by Rasiowa and Sikorski, and when he approached that categorically, the degree of generality he arrived at was Heyting-valued predicates (and more, see below).)

The first canonical text is a totally wonderful long paper (almost a mini-book), "Sheaves and logic" by Michael Fourman and Dana Scott in “Applications of Sheaf Theory to Algebra, Analysis, and Topology,” Lecture Notes in Mathematics, 753, Springer, 1979, pp.302–401. That's a text which is a delight to read.

Higgs, apparently, was a very interesting and remarkably brave person with a remarkable biography: https://en.wikipedia.org/wiki/Denis_Higgs

Speaking of Higgs' text, most of it, but not all, was published in his "Injectivity in the topos of complete Heyting algebra valued sets" (1984). I know that people were thinking about making his manuscript publicly available after his death in 2011, but have not done so yet, as far as I can tell.

One especially interesting part of that text by Higgs which remained unpublished (was not included into his 1984 paper) was the generalization from Heyting-valued predicates to predicates valued in Grothendieck topologies. I had no idea whether that part was "correct", but I took the liberty of making it publicly available in one of my slide decks, since I thought it was potentially very valuable, and quite interesting, see slides 44-47 of my https://www.cs.brandeis.edu/~bukatin/sumtopo2011.pdf

Edited Date: 2019-10-27 07:48 am (UTC)

From:

juan_gandhi

My fav topic, topos logic. And then, a topology, as a closure op in a logic.

From:

dmm

Ah, and what I say after slide 47 in that slide deck is that one should really try to generalize this to

https://en.wikipedia.org/wiki/Lawvere-Tierney_topology

I don't think this was done (or, at least, I am not aware of that).

From:

juan_gandhi

Oh, I did not get there yet.

For some reason I was always calling this topology, defined via closure, as Grothendieck topology.
A while ago I had classified them on finite posets; actually, it's extendable to well-founded posets in Boolean toposes.

From:

dmm

Да, это, конечно, гораздо более удобный и интуитивно понятный способ, чем конструкция Гротендика. Приятно, что он ещё и более общий...

Вот на что я никогда не смотрел: в обычной топологии есть довольно тривиальная двойственность, можно брать алгебры открытых множеств (то есть алгебры Гейтинга, которые народ ещё называет псевдо-Булевыми алгебрами), а можно брать алгебры замкнутых множеств (то есть алгебры Брауэра, которые народ ещё называет dual Heyting algebras или co-Heyting algebras).

Известно ли, как в контексте Lawvere-Tierney topology делается такого рода двойственность?

(Вопрос мой, конечно, показывает отсутствие близкого знамомства с этой конструкцией.)

From:

juan_gandhi

Я видел это дело, но никогда не заглядывал глубоко, что там вообще. Так-то конечно.

From:

dmm

Вот, мне осталось рассказать, что я понимаю про двойственность. Это, видимо, тоже займёт несколько комментариев, но восходит эта тема к другой работе Ловера, написанной в 1973-м году, "Metric spaces, generalized logic and closed categories":

http://tac.mta.ca/tac/reprints/articles/1/tr1abs.html

In this paper, we see a rather simple duality between quasi-metrics and fuzzy partial orders (or, between quasi-pseudo-metrics and fuzzy pre-orders). And, on top of that, one can interpret those structures as enriched categories (triangle inequality corresponds to transitivity and corresponds to categorical composition).

This is one of the most remarkable papers by Lawvere...