[FOM] vagueness in mathematics?

tchow tchow at alum.mit.edu
Mon Feb 13 18:24:42 EST 2017

Stewart Shapiro wrote:

> Mathematics goes to great lengths to avoid any kind of vagueness
> or indeterminacy. In what sense has it succeeded or not succeeded?
> Doesn't vagueness enter in to almost every other subject?

In my view, the question of whether mathematics successfully avoids 
vagueness boils down to the question of whether Kripkenstein is silly or 
not. The tacit consensus among mathematicians is that Kripkenstein is 
silly, and so we have access to a bedrock non-vague system of syntactic 
rules on which everything else can be solidly built.

Below is my personal explanation of what Kripkenstein is, copied from a 
MathOverflow answer I wrote some time ago.



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? If you have previously already agreed on some basic rules then 
you can build on those to define new rules, but how do you get started? 
It's hard to get more basic than "append a 1."

Perhaps you could try building a physical device that optically scans 
its input and writes a 1 next to it. But any physical device will 
eventually fail to implement your intended "append a 1" rule when it 
reaches a certain physical limit, so the device doesn't unambiguously 
communicate your intended rule to the Martian either.

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.) Even more radically, we could wonder whether the notion of 
a "rule" is incoherent. Perhaps there really is no such thing as a 
"rule" in the sense of some unambiguous finite description of something 
that applies to an infinite number of cases.

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