[FOM] Counterfactuals in relative computability theory

[W.Taylor at math.canterbury.ac.nz]

Quoting Matthias Jenny <mjenny at mit.edu>:

>  The only thing I would disagree with
> is that 'is an algorithm' is imprecise. I think it's informal, but not
> imprecise, unless 'imprecise' means the same thing as 'informal' here.

I would agree that "algorithm" is not imprecise in the (philosophical) sense
of "vague", which is used to apply to redness and other sorities properties.
So I doubt there are any "borderline" cases that Jenny requests.
(Unless OC finitist/feasibility considerations are considered.)

Still, it seems that any definition given (necessarily) in words, i.e. that
cannot be pared down to a strictly mathematical one, must retain *some*
element of some sort of imprecision, if only because natural language does so.

This is a very intuitive comment only, that I doubt I would be prepared to
defend strongly.

However; this and other posts have brought something to mind that might
be worth considering.  We have all been wondering whether, in some sense,
"algorithm-ness" is somehow equivalent to "effectiveness".  But there is
already a considerable literature on computability with respect to access
to various sets of naturals, i.e. to "oracles".   This idea surely easily
extends to algorithms with oracle-access.  This might help define the extent
to which some sort of borderline might exist.  And maybe even help to show
that the CT thesis is perhaps not so much about computability per se,
but more about the nature of "oracles"?

Bill Taylor


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