[FOM] Re: Contrasting Methodologies

[Harvey Friedman]

Reply to Holden.

On 9/30/03 11:00 PM, "tom holden" <thomas.holden at balliol.ox.ac.uk> wrote:

I wrote:
 
>> "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."

Holden wrote:
> 
> This seems naïve to me.

This seems naïve to me.

>When Chow's teacher builds the machine scepticism is
> in no way dodged.

As I said before, the main point of my posting regards contrasting
methodologies. Here is an amplification of what I said.

A good deal of the crucial steps needed to do serious f.o.m. regarding the
foundations of simple-rule-following is already contained in the work done
by developers of computer technology in order to get computer's to carry out
simple-rule-following.

In particular, it is a quite nontrivial task to build a computer system from
scratch, starting from nothing, so that serious interaction between humans
and computers can take place - i.e., the computer can be given some even low
level human generated instructions, and it does what is intended. The stored
program idea is highly nontrivial to actually design and implement.

I thought I made it clear that I was not attempting to engage in a
discussion of skepticism directly, but only very interested in listening to
a discussion of skepticism, in order help generate subjects of permanent
value. 

>If the teacher is to succeed in educating Chow's "I" with
> her machine, she must [blah blah blah].

I can pick off one "subject" that I have already noticed (a very long time
ago) and done something about. I have long had theorems to the effect that
although induction principles cannot be derived purely deductively,
induction principles up to some very very very large numbers CAN be derived
purely deductively ***given suitable defining equations and axioms for
successor***. Of course, this is trivial - make a deduction of size the very
very large number. However, to actually carry that out, one needs some
induction on the same very very very large numbers, so that begs many
questions...

But the real RESULT is that this can be done with quite SMALL length, so
that one can easily physically create the proof and read the proof. This is
a nontrivial fact, taking some argument. So there is one stark contrast
between initial segments of the nonnegative integers such as
{0,1,...,2^2^2^2^2^2^1000} - to which this result applies - and the full
initial segment {0,1,2,3,4,...}. For example, it can used to give a proof
WITHOUT INDUCTION that for numbers in that large initial segment,

x+y = y+x
x dot y = y dot x
x dot (y+z) = x dot y + x dot z
x dot x = y dot y + y dot y has no nontrivial solutions.

And it obviously is of permanent intellectual value. Also, the phenomenon
demonstrably runs out with much longer initial segments of N. That also
takes some work. 

The ideas behind this work have definitely been around for some time, and
probably this is regarded as part of the folklore - I think some forms have
been published. 

If someone complains that this is only vaguely related to any issues of
skepticism being discussed, I could, presumably, work to find some other
results of permanent value that are closer to the issues at hand, if I
wanted to. And after someone complains again that it is still only vagely
related..., then I can repeat the process, coming closer. All the time
leaving a permanent value trail of f.o.m. work.

> Perhaps such a reply is precisely what Friedman was objecting to in his
> discussion of the back and forth of such discussions,

Of course.

>but ironically
> Friedman's argument lends support to the sceptical side.

I don't have any opinion about that. If it is true, then the nonskeptical
side would have a perfectly good defense, which you could attack, etc.

If you like the back and forth, then we all gain. But I will not directly
participate - only indirectly.

>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.

Anything can take hold for a while if you work enough at the presentation -
until the "refutation" comes, and then you will have to modify the
presentation accordingly, which can be done, etc.
> 
> 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.)

Concerned about? Ridiculous, from my methodology. Of course I am VERY
concerned about it if I want to do something of permanent value connected
with it.
> 
> 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 (?) [blah blah blah].

A (new?) presubject? Give a theory which tells us why certain "machine
instructions" are elemental, and others are not. This is close to what is
called Church's Thesis research, which I still think is in for some big
breakthroughs.

Or give a theory which tells us why no such interesting distinction can be
made. 
> 
> 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...
> 
I like it, except for the opening statement "This seems naïve to me", which
seems naive to me. 

Harvey Friedman