[FOM] vagueness in mathematics?

[Arnold Neumaier]

On 02/14/2017 12:24 AM, tchow wrote:
> 
> In Wittgenstein's Philosophical Investigations, he makes an argument
> about "private languages" that Saul Kripke later interpreted in a
> certain way. The basic point is that it is very difficult, if not
> impossible, to pin down what a "rule" is. Imagine that you are trying to
> teach a Martian the syntactic rule, "append a 1 to the end of a string."
> The Martian looks puzzled so you give some examples:
> 
> 0 -> 01
> 101 -> 1011
> 0010 -> 00101
> 
> and so forth. The Martian seems to get the idea, and does a few examples
> to confirm with you. The first few examples look good, but then all of a
> sudden the Martian comes up with
> 
> 1111111 -> 111111110
> 
> Say what? Somehow it seems that the Martian hasn't gotten the rule after
> all. Or maybe the Martian has extrapolated from your examples to a
> *different* rule? How do you make sure you communicate the rule you
> intend? 

> No matter how you slice it, it seems that you can't guarantee that you
> have communicated your rule to the Martian, since any finite amount of
> interaction is consistent with infinitely many rules. Once we see this,
> we could take a more radical step and wonder, maybe *I'm* the Martian.
> Maybe all these years I've been assuming that I know what people mean
> when they specify syntactic rules, but actually I've just been lucky and
> haven't discovered the discrepancy between my understanding of what
> "append a 1" means and what everyone else means by it. (Here you can get
> a glimpse of where Wittgenstein's term "private language" comes into the
> discussion.) 

I discussed in Sections 1 and 3 of
   http://www.mat.univie.ac.at/~neum/ms/fmathl.pdf
the matter of communicating mathematics based on the assumption that
each user of mathematics has its own private language, and why it can be
done successfully - to the extend visible in our present mathematical
culture.


Arnold Neumaier