[FOM] Re: Sharp mathematical distinction between potential and actual infinity?

[Aatu Koskensilta]

Timothy Y. Chow wrote:

> On Mon, 29 Sep 2003, Vladimir Sazonov wrote:
> 
>>I do not know what is "Kripkensteinian" skepticism.
> 
> 
> People disagree on what exactly Kripkenstein's argument is.  I will
> give my version.  (If you are an expert on Kripkenstein and think I've
> got him all wrong, please just think of what follows as "Chow's argument"
> rather than taking me to task for faulty exegesis.)
> 
> If someone tells me that he's baffled by "the standard model of N," then
> I respond by saying that I am baffled by the concept of a "rule."  For
> example, someone gives me the rule, "Given any string, append 1 to the
> end to get a new string."  Someone tries to explain this rule to me by
> giving me lots of examples.
> 
>    139085   ->  1390851
>    1        ->  11
>    101010   ->  1010101
> 
> I look at all these examples, am puzzled for a while, and then my eyes
> light up.  "Ah!  I see.  Let me try an example of my own."
> 
>    12401    ->  124015
> 
> Strangely, my attempt meets with disapproval.  For some reason, I have
> failed to grasp the rule that I am supposed to learn.  So my teacher
> builds a physical machine that implements the rule.  I study the machine
> diligently.  After my study, I am able to do the "right thing" for all
> the examples that the machine can do.  Of course, the machine is finite.
> For a certain very long string that I abbreviate by "S" it gives the
> following result.
> 
>    S        ->  S
> 
> I dutifully imitate this behavior, but am baffled when my teacher tells me
> that I've got it wrong---that this behavior is an "exception" due to the
> "limited memory" of the machine and that when the memory limit is reached
> then I have to extrapolate the rule to
> 
>    S        ->  S1
> 
> Seeing my blank stare, the teacher tries to teach me some physics
> --concepts of isotropic and uniform space, and laws, and so on.  The
> apparent hope is that by understanding the principle behind the
> construction of the machine I will understand how to "extrapolate"
> correctly.  But I can't seem to learn the laws of physics the way I'm
> supposed to.  Everything is fine as long as I can compare to an example
> that is handled by some given finite machine or finite list of examples,
> but as soon as I have to "extrapolate" I screw up.  The teacher keeps
> making excuses like, "Oh, the machine broke" or "Oh, the machine ran
> out of power."  Secretly, I believe that they're just mystics trying
> to make my life difficult.  What "rule"?  They're just making things
> up as they go along.

In popular IQ tests one often encounters sequences of small numbers and 
is asked to produce the next (few) number(s), and most "smart" people 
are able to give the "correct" answer to most of these questions. There 
will always be those who object that the infinite sequence of numbers 
generated by the rule "behind" these numbers is not at all determined by 
any initial sequence. Similarly, the operation plus is not determined by 
any finite sample of correct additios, and so someone might come and 
claim that we've been all quadding and adding all the time, without 
knowing it.

When debating the IQ test issues, someone is almost inevitably bound to 
come forth and claim that although no infinite sequence is determined by 
the segment, there is only a handful of sequences with sufficiently low 
Kolmogorov complexity, and possibly even an unique sequence with minimal 
Kolmogorov complexity containing the initial segment.

Setting aside the question of whether Kolmogorov complexity in general 
has anything to do with rule following or grasping an infinite sequence, 
I'd like to point out that there's a very real sense in which if someone 
instead of adding 1 after 10000 starts adding 2 is doing something which 
  is more "complex" than the process of just adding 1. Of course, what 
is more complex depends on what you know and what you're familiar with, 
and thus Kolmogorov complexity is not really an appropriate metric.

The idea that one can extrapolate from a finite sample into a rule 
correctly requires some sort of a complexity metric, so that there is 
something that decides in favour of one way of continuing 
(extrapolating) instead of some other. I don't know what this metric 
might be, but in your example, your teacher could point out to you that 
you are extrapolating from the samples a sequence which would require a 
more complex rule than the one they prefer. Possibly you would even 
understand this, given that you seem to be able to communicate with your 
teacher.

-- 
Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus