What happens if you accept
1) uncorrelated asymmetries exist- i.e. agents in a game theoretic situation, or objects (constants) in a first order language 'know' themselves and can distinguish themselves from others. By 'know' I mean an epistemic process can be attributed to them such that this is the case- e.g. I impute a series of feline brain states to the neighbor's cat such that it would 'know' that it sat in the same place last Thursday afternoon and watched me then as it watches me now. Another way of saying this is that there is a 'constructible' epistemic sequence such that uncorrelated asymmetries can be identified within a game theoretic context or first order model theoretic language. Of course, we can only comment on these asymmetries in a second order language or within an intuitionistic theory.
2) rigid designation is restricted to what an object can know about itself- i.e. is first order 'constructible' or, if that is problematic, it is expressible as a 'lawlike' choice sequence. In other words, naive set theory is constrained from multiplying paradoxes because not just any property can be used to construct a set.
Can you also accept an 'End of Time axiom'? (ET)- id est the assumption ' that when mathematical activity has ended and all values of an arbitrary choice sequence α have been specified, it will turn out that α coincides with some classical “lawlike” (completely determined) sequence'? Moschavakis observes 'every classical lawlike sequence is extensionally equal to a choice sequence, and “at the end of time” intuitionistic and classical Baire (i.e. branching tree based) space will be indistinguishable.'
Before making up our minds, let us ask how we could benefit or what we might lose by taking this step.
One consequence would be that there must be a 'best' choice sequence for any given rational purpose. This corresponds to our intuition that if we had infinite time and infinite computational power etc. then there would always be a 'best' course of action whether for an individual or a set of individuals who are connected in some way. Moreover, because of our assumption of uncorrelated asymmetry there would be 'bourgeois strategies' corresponding to well defined (because of rigid designation) Hohfeldian incidents.
Surely, that's a good outcome? It appears to affirm our faith in the 'creating subject'- i.e. is humanistic, optimistic and open to new discoveries- as well as the fundamental intuition of Liberal Social Democracy.
In a lucid recent paper, Mark van Atten states- One may think of the Creating Subject as a generalisation avant la lettre of Turing’s computor: the generalisation consisting in the fact that the Creating Subject is not limited to making mechanical calculations, but can also engage in constructions that are potentially infinite, that depend on free choices, or on reflection on its own acts. Brouwer noticed that the Creating Subject can register not only what objects it has created so far but also how and when, and that this reflection can be exploited to demonstrate mathematical theorems
Why should 'created objects' not be seen as changing 'creating subjects'? Van Atten points out that Brouwer's theory could be seen as like Chomsky's notion of competence as opposed to performance. An ideal 'speaker-listener' could still be changed by who he is listening to. If beaten sufficiently, he may agree that an ungrammatical sentence is perfectly paradigmatic. Actually, this happens anyway, without coercion, if there is some collective benefit from doing so.
If there is a 'Hegelian' dialectical relationship between 'created object' and 'creating subject' then what is 'law-like' may look 'revolutionary' or 'anarchic', not lawlike at all.
At best, you can have a Platonic type theory where what is intensional and what is extensional changes without uprooting 'univalent foundations'. However, this can't be mindlessly turned into naive set theoretic propositions. It is intuitionistic and based on things which Socrates called 'synoida' (intuitionistic self-knowledge) and 'eudoxa' (conventions everybody accepts) which can't be well-defined.
Of course, one could stipulate for a one period economy. But in that case neither uncorrelated asymmetries nor rigid designation could arise. All you have is an imputation that what you have set up to observe is 'Rational', or 'Just', or a deep insight into whatever nonsense it is you profess.
No comments:
Post a Comment