[FOM] Re: Sharp mathematical distinction between potential and actual infinity? + Re: John Baez on David Corfield's book

[tom holden]

Friedman writes:

"In a way, the computer revolution solved this problem [of scepticism about
rule following], in that it "taught" computers how to carry out such rules
for any reasonable length input."
 
This seems naïve to me. When Chow's teacher builds the machine scepticism is
in no way dodged. If the teacher is to succeed in educating Chow's "I" with
her machine, she must first be sure that the machine itself is "properly
educated." Maybe she requests the source code, but still there is room for
scepticism, maybe the processor was designed by an errant student who
thought 100 + 1 was 102 (or 100*2^100+1=..), hence the processors "INC"
instruction (and every other) needs to be thoroughly tested. So she sits
down for a year or twenty and checks the INC instruction with every 32-bit
integer, and so on with the other instructions. But still this does not help
as there is no guarantee that the processor isn't created such that the
10'th/2^10'th/... INC instruction executed after turning on the computer
acts as the teacher would expect a DEC instruction to act. So the teacher
phones up Intel and requests the physical make up of the machine, and swats
up on the relevant laws of physics. This though, will never lead to a defeat
of rule scepticism. Half the point of Wittgensteinian rule scepticism is
that scepticism about traditionally "analytic a priori" matters is just as
easy to formulate as scepticism about synthetic matters, including of course
physics.

Perhaps such a reply is precisely what Friedman was objecting to in his
discussion of the back and forth of such discussions, but ironically
Friedman's argument lends support to the sceptical side. That the arguments
will go on as long as someone is espousing a realist/ absolutist/
foundationalist philosophical position, means precisely that such positions
will always leave a place for scepticism to take hold.

My suggestion is that issues of rule-following are not something for the
mathematicians (even those working in foundations) or computer scientists to
be too concerned about. (See Maddy's general version of the fit philosophy
to mathematical practice position.)

Koskensilta's reply to Chow is precisely the kind of mathematical attempt at
answering the philosophical question, which seems ill advised to me.
Complexity measures all (?) come down to length of program and computation
time on a suitable universal machine. And even if such machines do not have
instructions as obviously vulnerable to scepticism as INC, their
instructions have still been chosen to coincide with the ones which seem
simple to us. Take the Turing machines one "merged" instruction of write
value, move left or right and set new state. Why should the one instruction
program "if in start state and current cell is blank, write one and move
right, setting state back to the start state" write 1111111111111111111...
rather than 1010101010... or 1101001010101010101010101011001... Our choice
of the Turing machine's instruction rather than one of the rogue ones is
based on our "subjective" considerations of its simplicity: the same
considerations Koskensilta hoped to justify by appeal to Kolmogorov
complexity and Turing machines.

I hope this reply has not been embarrassingly uninformed (and that it gets
through this lists editorial process...) as it was my first venture out of
silent lurker-ship...

Tom Holden