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While concentrating on Category Theory as applied to the Web, I have also been following French Philosopher of technology Bernard Stiegler, as he was supportive of the creation of the Philosophy of the Web conference 10 years ago.
Recently he has been looking at the concept of locality which he argues has been repressed mostly after Aristotle, being overtaken since Plato with the notion of the Universal (The world of ideas), transferred into the Philosophers notion of God. Those philosophers who have tried to bring Locality back, such as Heiddegger with his concept of Dasein (being here), or Japanese philosopher Kitaro Nishida concept of place as a departure point, have had problematic relations with the Axis powers during the second world war. Yet Stiegler believes that since Schroedinger’s development in What is Life? of the concept of negative entropy as what fights locally entropy, forces us, if we are to take this seriously, to give a central position to locality. In any case the Web and the Internet create tensions between local cultures due to its creating ” a topological space without any distances” (see interview of late Michelle Serres). I’ll see if I can find a good English paper of Stiegler that makes these connections clearly.
Now the Topos meant place in Ancient Greek (or see also Topoi on Topos: The Development of Aristotle’s Concept of Place). So I was wondering if people here who had deeper intutions about Category Theory than me, can see some insights that Category Theory can bring on these topics. Of course I bring this up here as Topos is also an essential concept in Category Theory which I believe one can summarise as the allowing one to connect logic and topology.
Perhaps modal logic captures this relation to location better. David Corefield in his published Chapter 4 on Modal HoTT writes of adjoint modalities:
Choosing q equal to p, we see that a proposition sits between the images of the two operators (◻︎, p, ◇):
- necessarily true, true, possibly true following the pattern of
- everywhere, here, somewhere.
I have to be elsewhere, but, very briefly, note at modal type theory,
A Grothendieck ‘topology’ appears most naturally as a modal operator of the nature ‘it is locally the case that’.
Thanks. Btw. the link to Bill Lawvere’s paper you sent is broken. It is now available here Quantifiers and sheaves
Steve Awodey et al. on Topological Completeness of First-Order Modal Logic and Topos Semantics for Higher-Order Modal Logic is also relevant.
I’ve fixed the Lawvere link on a couple of pages, thanks.
Looking around to find where in English Bernard Stiegler concisely covers the issues of locality, possibility, and negentropy and anti-entropy, I found that he covers the entropy side in the 2019 lectures of the book Nanjing Lectures 2016-2019 in chapter 5 “Entropy, Negentropy and Anti-Entropy in Thermodynamics, Biology and Information Theory”. I am not knowledgeable on the literature of entropy or anti-entropy, so I myself am missing a lot of pieces to be able to explain clearly at this point. So the following are some impressions of mine.
So, I would say that life forms are evolving (anti-entropic) structures that needs to maximise access to possibilities, so as to avoid death by anticipating. These structures evolve to be more powerful, ie can surmount more situations dangerous possibilities they encouter. Those that do, can evolve into future states of the world, which from the point of view of the structure are not calculable (the unknown unknown). This would thus be a way of explaining Nietzsche’s Life as will to Power. But the notion of a structure has to be kept very open, as that itself is evolving. For example as a society we form structures where we can act together as one (technologically enhanced) organism, the way a cell is part of a body. In the case of humans the individual and collective individuation work much more strongly together: an individual makes a new contribution (in art, mathematics, law, cooking, …) and thereby transforms the society in which they live, and these the other societies in which they interact. Anyway, this way of putting things brings possibilities in, as fundamental, and probabilites as potentially epistemologically important but metaphysically secondary. This should not be surprising here due to the relation between coalgebras, processes, and modal logic.
What is important from this perspective is that systems that overly synchronise behaviour, desires or thought patterns, that don’t cultivate difference, and different systematic (argued) points of views (real differences), reduce this differentiation that is needed to be able to overcome change. All forms of knowledge (practical, theoretical, emotional,…) for humans that have externalised memory (exosomatic) are here important.
He has criticisms of reductions of this concept to information theory, which I don’t follow well.
Anyway I have seen discussions here thinking about entropy, but not much about negentropy or anti entropy. It would be interesting to see if the locality of anti-entropy (life or life supplemented by technology) that he mentions (referring to Schroedinger) can be tied to the locality we find in modal logic.
There is more on entropy from a philosophical and historical point of view on their internation.world website page Response to Guterres. This one is a lot more readable that the previous article I posted. A short extract below.
From the perspective of thermodynamics, biological situations are not at a maximum entropy and do not tend towards maximum entropy. The low and sometimes decreasing entropy of biological objects seems to “contradict” the second principle of thermodynamics, which states that entropy cannot decrease in an isolated system. However, biological situations, including the biosphere as a whole, are not isolated systems. Biological situations are open; they use flows of energy, matter, and entropy. At the level of the biosphere, the sun is the primary provider of free energy that is used by photosynthetic organisms. Therefore, biological situations do not contradict the second principle. A consequence is that biological organizations and, by extension, social organizations, are necessarily local and depend on their coupling with their surroundings. In organisms, the relationship between the inside and the outside is materialized and organized by semi-permeable membranes.
I bring this to the attention here, because it would be very interesting to see if there are Category Theoretic interpretations of what Stiegler is saying.
From the point of view of modal logic, which I have studied a lot, I can see very well where the notion of “it’s locally the case” attributed to Grothendieck above, comes from, as well as the “here, there, everywhere” from David’s Modal HoTT.
David Lewis used to call that the indexicality of modality. He argued in many places (see papers online) that “actually” functions as an indexical, the way “I”, “you”, “here” and “now” do: that is their referent is determined relative to the speaker. For example he argued in his book Counterfactuals, that to determine the truth of “If Kangaroos had no tails they would topple over” one needs to know from which world the speaker is speaking from, as that determines the closest possible worlds in which Kangaroos have no tails, which one can then check to see whether or not they topple over. But the same point can be made in terms of tense logic, since what will happen depends on what time one is speaking from. Similarly, what is far away, depends on where one is. Interestingly in Counterfactuals he also argued that definite descriptions could be looked at this way, arguing that what one is referring to depends on the salience of things in the phenomenological field of those in the discussion. Similarly what ought to happen may depend on one’s idea (and importantly perhaps also those around one) of what the ideal world is (which in this case won’t usually be the actual world, pace Leibniz.) All of these ideas do bring with them notions of locality.
Having read half of Kohei Kishida’s thesis Generalized Topological Semantics forFirst-Order Modal Logic1, it looks like David Lewis and Kripke are still pretty solid ground.
So one could argue that modal logic makes this dependence on the locality of the speaker important. Given that communication is about subjectivities in communication, one can see how one could come to the conclusion that this is the fundamental point of view.
Of course this is also what coalgebras do as they allow one to traverse from one state which is the index, to another one. As the saying goes Modal Logics are Coalgebraic.
If one then adds along the lines that Stiegler is (online courses in French) that living things are always locally based (and one could extend the notion of the Living along the lines of Ruth Millikan’s “Language, Thought and Other Biological Categories” to anything that can reproduce, so thoughts, tools, etc…
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