4 ms·

In particular, abuse of Galois Theory makes it possible to reconcile Spinoza with Aquinas.

by 082349872349872 2y ago

esafak 2y ago
Go on.
- koolala 2y ago
  Reconcile Liebniz too. I agree this sounds awesome.
  - 082349872349872 2y ago
    Leibniz does get a mention, but it'll already take a while to explain these two, so I'll leave his full reconciliation as an exercise for the reader ;)
    - koolala 2y ago
      ;) A mention is enough. Monads are LIFE!
      - 082349872349872 2y ago
        Every mathematical monad comes from an adjunction, which unfortunately implies they're not as windowless as Leibniz would like? (or does it merely imply that in the Leibnizian setting the structure is always external and never internal?) Can we make a mathematical monad out of the adjunction presented above? Let g(C) be the minimum G and c(G) be the maximum C in the model above, then we certainly have g(C) = g(c(g(C))) and c(G) = c(g(c(G))), which formally suggest we may be able to do something. Exercise: does the triple (T, μ, η) exist? what about (G, δ, ϵ) in the other direction? If they do exist, what are they, for Aquinas, Leibniz, and Spinoza?
novosel 2y ago
Why would they need to be reconciled? Or, indeed, Galoas abused?
- 082349872349872 2y ago
  They don't need to be reconciled, but they do differ, which suggests we might be able to find a framework which reconciles them. Galois Spinoza Aquinas in Google yields (for me at least) the following HN thread: https://news.ycombinator.com/item?id=39885475 https://news.ycombinator.com/item?id=39885475 , in which an attempt at "Algebraic Theology" not only provides such a subsuming (thanks to Galois Theory) framework, answering that particular question in the affirmative, but also raises many other questions which might be amusing to pursue. (I could summarise those Q's in this thread, if any of you all are more interested than the downvoters were) [I will summarise that thread further up in this one, but as it was months ago it may take me an hour or two to page everything back in]
082349872349872 2y ago
To expand[0] upon that claim: 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.
- 2y ago

In particular, abuse of Galois Theory makes it possible to reconcile Spinoza with Aquinas.

esafak 2y ago

Go on.

koolala 2y ago
Reconcile Liebniz too. I agree this sounds awesome.
- 082349872349872 2y ago
  Leibniz does get a mention, but it'll already take a while to explain these two, so I'll leave his full reconciliation as an exercise for the reader ;)
  - koolala 2y ago
    ;) A mention is enough. Monads are LIFE!
    - 082349872349872 2y ago
      Every mathematical monad comes from an adjunction, which unfortunately implies they're not as windowless as Leibniz would like? (or does it merely imply that in the Leibnizian setting the structure is always external and never internal?) Can we make a mathematical monad out of the adjunction presented above? Let g(C) be the minimum G and c(G) be the maximum C in the model above, then we certainly have g(C) = g(c(g(C))) and c(G) = c(g(c(G))), which formally suggest we may be able to do something. Exercise: does the triple (T, μ, η) exist? what about (G, δ, ϵ) in the other direction? If they do exist, what are they, for Aquinas, Leibniz, and Spinoza?

novosel 2y ago

Why would they need to be reconciled? Or, indeed, Galoas abused?

082349872349872 2y ago
They don't need to be reconciled, but they do differ, which suggests we might be able to find a framework which reconciles them. Galois Spinoza Aquinas in Google yields (for me at least) the following HN thread: https://news.ycombinator.com/item?id=39885475 https://news.ycombinator.com/item?id=39885475 , in which an attempt at "Algebraic Theology" not only provides such a subsuming (thanks to Galois Theory) framework, answering that particular question in the affirmative, but also raises many other questions which might be amusing to pursue. (I could summarise those Q's in this thread, if any of you all are more interested than the downvoters were) [I will summarise that thread further up in this one, but as it was months ago it may take me an hour or two to page everything back in]

082349872349872 2y ago

To expand[0] upon that claim: 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.

2y ago