[FOM] Re: Sharp mathematical distinction between potential and actual infinity?
Timothy Y. Chow
tchow at alum.mit.edu
Mon Sep 29 16:51:17 EDT 2003
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
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.
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