Wednesday, 3 May 2017

Godel's God, Cantor's Paradise & Arrow's Impossibility.

By 'Deliberative Reason' we commonly denote abstract dispassionate ratiocination, militating to consensus on judgements of a universal kind. Following Kant, there has been a tendency to see Deliberative Reason as categorical and univocal rather than context and protocol bound. Thus the 'artificial reason' of the Courts has been considered to be akin, or convergent to, the 'natural reason' of the Scientists.

If the a priori truths of Pure Reason can be arrived at by Deliberative Reason, then it is likely that purely metaphysical judgements are related to fundamental theorems in mathematical politics or social choice- like Arrow's Impossibility result which in the opinion of some Law Professors, like Max Stearns, has fundamental implications for Constitutional Law & Jurisprudence.

Recently, some media controversy has been generated by the fact that Judges are increasingly relying on Artificial Intelligences (A.Is) and that the possibility exists that A.Is could replace Judges altogether.

Meanwhile, A.I has scored its first big success in Metaphysics-  proving the inconsistency of Godel's ontological proof.

Godel relied on an axiom such that the set of 'positive properties' is an ultrafilter. However, as is well known, this causes problems of self-difference, i.e. something both being and not being itself,  or else endangers 'accessibility' and entails 'modal collapse' (i.e. turns every true statement into a necessarily true statement as if this were the only possible world.)

Arrow's Theorem- which, for some reason that escapes me, is not regarded as nonsense- as extended to the infinite case by Kirman & Sondermann, such that the Arrowian Social Welfare Function (ASWF) is shown to be a non-principal ultrafilter,  can come to the rescue of Godel's proof because it supports self-difference- an invisible dictator is both different and the same as a visible dictator- and, allowing 'constructibility' to be endogenous, permits a laissez passer into Cantor's Paradise without, however, having to chose sides re. the Continuum Hypothesis. Inter alia, this means the underlying proof sequence grows faster than any possible algorithmic verification of modal collapse.

This gives rise to my claims- If we accept Arrow's Impossibility theorem is true (as opposed to wrong or meaningless) then we must accept God exists.

Argument- Let the choice of axioms for this proof be the result of an Arrowvian Social Choice Function (ASWF) over all possible rational beings as conceived by a given agent or set of agents.
Let us define God as a possible rational being who could always be dictatorial but is not necessarily so.
If Arrow's Theorem is true, then (because possibly rational beings are infinite (for e.g. by cloning)) then (by Kirman/Sondermann) God exists, as an 'invisible' Arrowvian Dictator over some subset of Deliberative Reason's domain for all possible rational beings. If Deliberative Reason can have a self-consistent form, a proof that God exists must be accepted by all Rational beings.

(Depending on one's attitude to the Continuum Hypothesis (CH) more could be predicated of this 'God'. Godel disliked CH, and-currently- we don't really think of it as an open problem so much as opening doorways.)

If Deliberative Reason can have a consistent expression and features Arrow's Theorem impredicatively then it must affirm that God exists . To assert otherwise, for any bien pensant votary of  Arrow's theorem, is either a dictatorial claim, elevating oneself above 'Deliberative Reason', or else a claim to be a not possibly rational being (in which case you are not admissible for the ASWF) and thus beneath the scope of Deliberative Reason.

What do you think?


Anonymous said...

Garbage in Garbage out.
Meanwhile, in the real world, a photonic Maxwell's Demon has been found which can charge a battery. What can Godel's God do for us?

windwheel said...

The quest for Godel's God motivates, for example, paradigm shifts in mathematical theories of quantum thermodynamics which can result in the invention of the sort of battery you mention.
Obviously, I don't mean it motivates me in that way coz I iz as stupid as shit. But, nothing's stopping you.