[FOM] vagueness in mathematics?

Timothy Y. Chow tchow at alum.mit.edu
Thu Feb 16 21:22:29 EST 2017

```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
```