[FOM] Vagueness vis a vis clarity

[Ignacio Añon]

Chow wrote:

"The argument is not just that any finite sequence can be continued in
infinitely many ways.  The argument is that even after you say, with
as much precision and explicitness as the entire mathematical
community can muster, *exactly what the rule* for the sequence is, the
sequence is *still* not determined.  Any finite amount of natural
language conversation, gesticulation, drawing of pictures, training in
logic, building of computers and demonstration of their operation,
holding of international conferences, browbeating, and foaming at the
mouth will still fail to nail down even the simplest possible
so-called "rule." Everyone could agree on every instance of the
alleged rule that has ever come up in the history of mankind but it
could all just be a huge lucky coincidence."

Certain advances in complexity theory go even further than this...

Simon and Peter Schor showed in the nineties, that certain oracle
problems require exponential time in classical Turing machines, but
only polynomial time in quantum computers. This crucial result is
often ignored: notions like "Rule", "Set", "Function", might be
distorted when reduced to the Turing-Church paradigm of computation...

Obsessive clarity can be as intellectually dangerous as vagueness is,
if uncontrolled critically.

By giving the notion "Rule" a precise definition, you can reach a
level of clarity, that renders it trivial: mathematical culture seems
to be based on the human capacity to determine, critically, the
intelectual value of certain ideas, by observing their varying
degrees, not just of vagueness, but of trivial clarity...

Best


2017-02-17 0:22 GMT-02:00 Timothy Y. Chow <tchow at alum.mit.edu>:
>
> Charlie Silver wrote:
>
>>         To me, this is expressed simpler by asking what it means to
>>         "continue in the same way," which seems the point of Tim Chow's
>>         example.  For instance, if we begin with 1, 2, 3, and try to
>>         "continue in the same way," what stops us from this: 1, 2, 3, 1,
>>         2, 3, ?  Any finite sequence can be followed by any other. Another
>>         example would be 1, 2, 3, 101, 102, 103, 201,?
>
>
> There's more to Kripkenstein than that.  There is a glib response available to the challenge to "continue in the same way," which is that you have failed to say what "continue in the same way" means.  The glib response continues by saying that if you were to be more specific or precise, e.g., by saying that the sequence is the sequence of natural numbers, or the sequence is the sequence "1,2,3" repeated indefinitely, then the vagueness would disappear.
>
> The Kripkensteinian skeptic goes further, by insisting that *there is no way to specify a rule* that says what the rest of the sequence is.  For example, if you try to say that the rule is "1,2,3,1,2,3 repeated indefinitely" then I will skeptically ask, what do you mean by 1,2,3,1,2,3 repeated indefinitely?
>
> Tim: Is "1,2,3,1,2,3,4" an initial segment of "1,2,3,1,2,3 repeated indefinitely"?
>
> Charlie: Say what?  Of course not; didn't you hear me?  I said 1,2,3,1,2,3 repeated indefinitely, which means that the next number is 1, and the next one after that is 2, and the next one after that is 3.  That's the rule.
>
> Tim: A rule?  I thought that I knew what you meant by a rule, but now I'm not so sure.  It seemed quite clear to me that the rule "1,2,3,1,2,3 repeated indefinitely" meant that the next number was obviously 4.  But you're telling me that that "rule" dictates that the next number is 1? Weird.  What do you mean by a "rule" then?
>
> The argument is not just that any finite sequence can be continued in infinitely many ways.  The argument is that even after you say, with as much precision and explicitness as the entire mathematical community can muster, *exactly what the rule* for the sequence is, the sequence is *still* not determined.  Any finite amount of natural language conversation, gesticulation, drawing of pictures, training in logic, building of computers and demonstration of their operation, holding of international conferences, browbeating, and foaming at the mouth will still fail to nail down even the simplest possible so-called "rule." Everyone could agree on every instance of the alleged rule that has ever come up in the history of mankind but it could all just be a huge lucky coincidence.
>
> Tim
>
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