Sunday, 12 March 2023

Brouwer's Fan theorem vs Callard's fan dance

What passes for philosophy is, speaking generally, mere ignorance of developments in mathematical logic.

As a case in point, Agnes Callard- drawing on an argument given by Galen Strawson- writes in 'the problem of self-creation' -  

The new values, acquisition of which constitutes my act of self-creation, must be either continuous or discontinuous with the ones I already have.

This assumes the law of the excluded middle. Yet there is no reason why acts of self-creation may be neither continuous nor discontinuous or even some combination of the two. One reason for this is that self-creation- like intuition which depends only on verification- may be non algorithmic or non deterministic.

In any case, in our quantized reality, we expect to see things like stochastic 'cadlag' functions with 'jumps' generating a Skorokhod space- i.e one where time wiggles a bit. But this begs the question as to whether for any specific domain a discontinuous function could be have a representation as 'cadlag' and, through re-parametrization of the time axis, if only for some highly restricted purpose, there is a Polish space. However, at least in the social sciences, the thing is arbitrary.

There are very few meaningful or informative propositions which are only either true or false. Thankfully, Intuitionism does not have a law of the excluded middle. Brouwer's Fan & Bar theorems can be understood as relating to unbounded choice sequences. Self-creation, presumably, is based on choice. This also means that there can be a violation of the fixed point theorem even if there is a constructively compact convex set because a function may look continuous but not in fact be so. Thus, Choice sequences may converge to a fixed point without ever crossing it- though they may jump or tunnel across.

I think Errett Bishop's "Schizophrenia of Contemporary Mathematics' had well and truly thrown the cat of constructivism among the pigeons of classical American mathematics by the end of the Sixties. Thus, Callard is fifty years behind the times.

If they are continuous, I am not changing but rather working out the implications of the person I already was.

This does not follow. A choice may feature hysteresis of the above sort, but it may be purely ergodic or aleatory and 'discovery' based. On the other hand, maybe there are no 'lawless' choice sequences at 'the end of time'. 

If they are discontinuous and the new values contradict or come at a tangent to my old values, the change is not a product of my agency.

This is an arbitrary assertion. It can be countered by the allegation that the one making it is actually an emanation of the Nicaraguan horcrux of my neighbor's cat.  

I change, but I do not change myself.

Coz it was the feline horcrux which did that foul deed- right?  

This paradox, adapted from the work of Galen Strawson,

who doesn't know Math  

can be solved if we allow that the direction of value-dependence may be teleological: the aspirant’s values depend on, and are entailed by, those of the person she is trying to be.

That's backward induction, which is perfectly fine unless Knightian Uncertainty obtains- which it does, worse luck. In this case, though there is verification, there is non-computability or non determinism. Anyway, plenty of peeps set out to acquire God like powers. They failed. Sad.  

The aspirant does not fashion, control, or make the self she creates. Instead, she looks up to that self, tries to understand her, endeavors to find a way to her.

and writes boring, stupid, shite. But this would happen even if she is writing shite just to get tenure or to sell some shitty books. Callard's fan dance conceals not nudity but intellectual nullity. Still, she's on the spectrum and thus deserves affirmative action. 

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