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

[Harvey Friedman]

Reply to Chow.

On 9/29/03 4:51 PM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:

> People disagree on what exactly Kripkenstein's argument is.  I will
> give my version.  ...
> 
> 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

And I respond by saying that "I am baffled by stories such as this".

It does not seem easy for you to explain the point of the story. In fact,
"the standard model of N" and "simple-rule-following-such-as-this" seem much
less baffling for anyone than the point of the story.

For here is an obvious reaction to the story. I am sure that you say that
this "misses the point of the story."

First of all, any long string that it fails at is, in reality, so long, that
you will not come up with it (the string). It has to be of length at least,
roughly, the number of bytes of memory, which, nowadays, can easily be
arranged to be in the trillions, or even quadrillions. You may not live long
enough to write down any string this long.

Secondly, the machine does not produce S from S. Instead, it produces no
output at all. Or it produces "error message: string too long". Are you
still confused?

The first point is that assuming computers are big enough and people don't
live long enough, the situation never arises. The second point is that the
computer should not be giving any misleading answer that needs any awkward
explanation involving physics.

YOUR DEFENSE: We'll, I (the I of Chow's story) might think that because the
computer doesn't answer on this long string, or gives an error message, that
the rule is undefined at long strings.

MY REATTACK. Blah blah blah.

Perhaps I am not a good judge of how awe inspiring this story is, since I am
not confused about "the standard model of N" and I am not confused about the
concept of a "rule", or at least such clear rules. I am far more baffled
about the point of the story. E.g., what is the effect of the fact that
couldn't happen?

YOUR DEFENSE: Well, Friedman is saying that it couldn't happen because of
some beliefs about human life etc., and how does he know that for sure? And
isn't that besides the point? It *could* happen.

MY REATTACK. Blah blah blah.

OK, this can go back and forth for several generations of philosophers. But
is this intellectually productive or not?

I know one great admirer of Wittgenstein's work at a famous Philosophy Dept,
who thinks his philosophy of mind and language is inspiring, but thinks his
philosophy of mathematics is actually bad.

Of course there is another possible point to the story. This may be more
fruitful, intellectually.

It seems rather subtle to set up the foundations of simple-rule-following,
so that it satisfies certain criteria of clarity and self containment. This
might lead to interesting foundational work provided some reasonable
criteria for the required clarity and self containment can be laid out
properly - as Wittgenstein never did, to my knowledge. I have never seen
such criteria laid out properly by anyone.

In a way, the computer revolution solved this problem, in that it "taught"
computers how to carry out such rules for any reasonable length input. But
can this "solution" be reformulated in foundational form? Is it worth doing?

NOW - what is the point of ***MY*** story?

I am not sure. Here's a try.

I am questioning the standard methodology employed in philosophy. Does it
appear that for every good move there is a good countermove? For every good
countermove there is a good countercountermove? For every good attack there
is a good defense? For every good defense there is a good attack?

I LIKE to see it - the back and forth with no expectation of resolution -
but I like to not be in it, and instead (hope to) pick off systematic
subjects of permanent value.

Harvey Friedman