Aquinas, Leibniz, and Spinoza all agree that there is a God (G) that created[1] a Creation (C) which we are a part of.
One major way Aquinas (and Leibniz) differ from Spinoza is in how determinate C may be. Spinoza says C is determined by G; Aquinas says there are many possible C's for any given G. (Leibniz splits the difference and says there are many possible C's, but in our particular case, our G has created the best[2] possible C.)
The reconciliation: let G and C be in an adjoint relationship, such that we have functions picking out the maximum C any given G may create, and the minimum G that can create any given C.
Now, if you are Aquinas, G is omnipotent, and hence has the possibility to create other C's, but our C is the maximum[3] one.
On the other hand, if you are Spinoza, G determines C[4], so it is trivially maximal. (The maximum of a singleton being the unique element)
Does that make sense?
Question: do there exist Gods that are incapable of creating any creation, or Creations that are impossible for any god to create? If so, need we replace "maximum" and "minimum" above by LUB and GLB? What other situations (eg. gods or creations being only domains rather than lattices) would also require further abstraction?
:: :: ::
[0] my apologies for any non-standard notation. I tried to find a survey paper on "Algebraic Theology" so I could follow the existing notation, but failed to find any concrete instances in this (currently only platonic?) field.
[1] Spinoza is accused of "pantheism": the heresy of identifying God and the Creation. Reading him according to a Galois-theoretic model, he would be innocent of this heresy, for when he says "God, or, the Universe", he is simply using metonymy, for in his model God and Creation are dual, so (being in a 1:1 relationship) one determines the other. Note that in general, not only are duals not identical, they're not even isomorphic.
[2] but cf Voltaire, Candide (1759)
[3] if you are Leibniz, the order in which it is maximal corresponds to the traditional "worst" "better" "best" order. I don't know Leibniz well enough to say if he had a total, or only partial, order in mind; presumably in his model if there are several maximal "best" creations they would all be isomorphic?
Exercise for the reader: work out the Leibnizian metaphysical adjunction.
[4] turning this arrow around, C determines G, which explains why Einstein would say he believed in the "God of Spinoza", and chose to base his research by thinking about C, unlike the medieval colleagues of Aquinas, who spent a lot of time and effort trying to work the arrow in the other direction, hoping to come to conclusions about C in starting by thinking about G.
> The Galois theory normally taught in graduate-level algebra courses ... involves a Galois connection between the intermediate fields of a Galois extension and the subgroups of the corresponding Galois group.
Instead of speaking informally, of abuse of Galois Theory, I should have spoken formally, of abstraction of Galois Theory.
Aquinas, Leibniz, and Spinoza all agree that there is a God (G) that created[1] a Creation (C) which we are a part of.
One major way Aquinas (and Leibniz) differ from Spinoza is in how determinate C may be. Spinoza says C is determined by G; Aquinas says there are many possible C's for any given G. (Leibniz splits the difference and says there are many possible C's, but in our particular case, our G has created the best[2] possible C.)
The reconciliation: let G and C be in an adjoint relationship, such that we have functions picking out the maximum C any given G may create, and the minimum G that can create any given C.
Now, if you are Aquinas, G is omnipotent, and hence has the possibility to create other C's, but our C is the maximum[3] one.
On the other hand, if you are Spinoza, G determines C[4], so it is trivially maximal. (The maximum of a singleton being the unique element)
Does that make sense?
Question: do there exist Gods that are incapable of creating any creation, or Creations that are impossible for any god to create? If so, need we replace "maximum" and "minimum" above by LUB and GLB? What other situations (eg. gods or creations being only domains rather than lattices) would also require further abstraction?
:: :: ::
[0] my apologies for any non-standard notation. I tried to find a survey paper on "Algebraic Theology" so I could follow the existing notation, but failed to find any concrete instances in this (currently only platonic?) field.
[1] Spinoza is accused of "pantheism": the heresy of identifying God and the Creation. Reading him according to a Galois-theoretic model, he would be innocent of this heresy, for when he says "God, or, the Universe", he is simply using metonymy, for in his model God and Creation are dual, so (being in a 1:1 relationship) one determines the other. Note that in general, not only are duals not identical, they're not even isomorphic.
[2] but cf Voltaire, Candide (1759)
[3] if you are Leibniz, the order in which it is maximal corresponds to the traditional "worst" "better" "best" order. I don't know Leibniz well enough to say if he had a total, or only partial, order in mind; presumably in his model if there are several maximal "best" creations they would all be isomorphic?
Exercise for the reader: work out the Leibnizian metaphysical adjunction.
[4] turning this arrow around, C determines G, which explains why Einstein would say he believed in the "God of Spinoza", and chose to base his research by thinking about C, unlike the medieval colleagues of Aquinas, who spent a lot of time and effort trying to work the arrow in the other direction, hoping to come to conclusions about C in starting by thinking about G.