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added the following to the Baruch Spinoza and Spinoza’s System pages:
Spinoza seeks, in the vein of deep unification programs in mathematics and natural science, to find structural uniformities behind the segregative diversities analytic philosophy so prizes. Spinoza is “musical” in this search for unity and unification of all being(s), prizing a motif (as [a tragic soul mate of Spinoza’s], Grothendieck, so wisely dubbed it three hundred years later) - a single structural archetype idea, recurring throughout the theory of cognition, the theory human (and not “intentional’) action, the study of the mathematics of space-time, the nature of God and the interlocked natures of power and right - in short a motif unifying all beings and all actions in Nature.
Everything in Its Right Place: Spinoza and Life by the Light of Nature, Joseph Almog, pg xi
Spinoza’s System does not exist. Spinoza’s system does.
;-) I did a redirect!
single structural archetype idea
I had asked about this elsewhere: what’s the historical road really from Plato’s Ideas in plural to a singular The Idea which is explicit in WdL and, maybe, as your quote here claims, in Spinoza.
I note however, for what it’s worth, that by WdL the quoted statement would be held as false: substance as understood there is not the idea and I think the lecture comment on Spinoza is quite explicit in pointing out a lack of such.
Also, I must say that I find the suggested connection with Grothendieck’s motives similarly not fine-grained enough to be satisfactory. It is true that in both cases we have an abstract general and its realizations as concrete particulars, but on the one hand it is “substance” in Spinoza’s (and Hegel’s) sense, and on the other it is…. something else.
But since Grothendieck’s motives are something that do connect directly to topos theory, I have a suggestion: while we are adding stuff on philosophy to the nLab, we should try – for this to be productive – to consistently follow Lawvere’s lead. Lawvere’s lead is what brings an POV to philosophy (and be it just a 1-POV) and hence makes it fit well on the Lab. So following that lead, we are to substantiate claims that this philosophical concept is the same as that philosophical concept by first arguing for a formalization of either in categorical logic, then providing some evidence that this formalization is sensible, and then finally proceeding by actual mathematical analysis.
I have a picture of how this works such that both Spinoza’s “substance” and Grothendieck’s “motif” sit in the same picture. And when I look at that, both come out in distinctively different places.
Namely for Spinoza’s substance: following Lawvere we read Hegel’s determinations of being as a given system of adjoint modalities in some category. Then we find with Hegel that this passing to the “essence” means that it reflects on itself, and the obvious sensible way to formalize this internal self-reflection is by what it is verbatim in (higher) category theory: the type universe. Then finally, by Hegel, Spinoza’s substance is the “unity of being and essence”, hence the unity of the adjoint modalities and the type universe, hence the substance is the cohesive -topos. It’s as easy as that :-)
Now in there we find Grothendieck’s motives by looking at its tangent -topos and considering the induced dependent linear type theory. Motives are then the linearizations of non-linear types by push-forward of the linear types parameterized over them
So Grothendieck’s motives are “built out of” the substance, as “everything” is, but they are in no way somehow the same.
That’s my picture anyway, when I try to continue along the lines that Lawvere has suggested.
Now, of course, as with every formalization of every informal concept, there will always be room for debate of the sensibility of the formalization steps. But in any case, if you disagree with this analysis, then the productive way to proceed would be, I’d suggest, not to just disagree, but to in turn propose an alternative formalization and then see what happens.
In other words, in order for us not to wander off too far out into arbitrariness of vague ideas, I am suggesting that the course of the action is to consider the philosophy, yes, but only and always as a backdrop on which to explore various structures in categorical logic.
I agree with Urs. That’s just a vague analogy that Almog’s making. Why not use on the mathematical side the case of a theory and its models or a group and its representations? Is it just because there has been a certain difficulty in understanding what motifs are?
Now in there we find Grothendieck’s motives by looking at its tangent ∞-topos and considering the induced dependent linear type theory. Motives are then the linearizations of non-linear types by push-forward of the linear types parameterized over them
So Grothendieck’s motives are “built out of” the substance, as “everything” is, but they are in no way somehow the same.
I agree that Almog is just making a vague analogy. Seeing as Urs demonstrated that, when made explicit along Lawverian lines, this analogy does not make sense, I removed it from the nLab entries.
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