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Aristotle’s actuality and potentiality
Late Ivan Supek (Croatian theoretical physicist, writer and a bit of philosopher and political activist), who was a student of Heisenberg, often emphasized that because of this aspect of Aristotle’s picture of the world, Aristotle’s point of view would better suit quantum world than Platon’s which is otherwise more dominant in traditional scientific Weltanschauung. I listened to one lecture of him as a student dedicated to this topic, at a Croatian Academy of Sciences workshop related to Ruđer Bošković (whose theory of forces influenced Bohr according to Bohr himself).
If there is any reference, I’d be interested in having a look.
He definitely wrote something about that idea but I believe in Croatian only. Maybe with some effort some summary can be found in English somewhere.
With the expected arrival of directed HoTT next week, what can we say of how the analogue of our possible worlds story should go?
So we have in the ordinary case the adjoint triple generating the necessity and possibility (co)monads:
$(\exists_W \dashv W^\ast \dashv \forall_W) \;\colon\; \mathbf{H}_{/W} \stackrel{\stackrel{\forall_{w \colon W}}{\longrightarrow}}{\stackrel{\stackrel{W^\ast}{\longleftarrow}}{\underset{\exists_{w\colon W}}{\longrightarrow}}} \mathbf{H} \,.$Presumably in the directed case we have similar maps between a slice of some $(\infty, 2)$-topos and itself.
Let’s keep things simple. So a classic example of the undirected case sees a set of worlds, $W$, and then $W$-dependent propositions. We also might consider $W$-dependent sets and look at sections or the total space.
In the directed case, we might take $W$ to be a poset of worlds. A $W$-dependent proposition is presumably an upper set. Then adjoints to base change, $W^{\ast}$, are the limit and colimit over $W$, I take it. And we might consider presheaves of $(\infty, 1)$-categories over $W$.
This set-up then generates 2-(co)monads which resemble necessity and possibility, and so on right up to dependence on an $(\infty, 1)$-category or a morphism between two such. I take it there wouldn’t be anything new here, but considering things modally might be interesting.
added pointer to
for the observation that S5 Kripke semantics is given by base change adjoint triples.
have touched wording, typesetting, and hyperlinking in the section Via S4 modal logic, in the hope of streamlining a little more