[FOM] vagueness in mathematics?
Charlie
silver_1 at mindspring.com
Tue Feb 14 17:47:27 EST 2017
“Indeterminacy” is, indeed, reflected by Kripke in his book on Wittgenstein. Kripke’s own example seems clearer to me. He supposes that he may not have ever in the past performed the computation 68 + 57. Though he has no doubt that the result is 125, via the “plus” function ‘+', he encounters a “bizarre sceptic" who uses the “quus” function, which Kripke symbolizes as ‘⊕'. Kripke defines “quus” this way:
x ⊕ y = x + y, if x, y < 57
= 5 otherwise.
Kripke asks “Who is to say that this is not the function I previously meant by ‘+’ "?
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,…
A favorite example of mine is to ask what number “best” follows 1, 2, 4, 8, 16. Most of us would answer 32, thinking the pattern is to use the function 2^n, where n >0. However, one can argue that the next number should be 31, and the next after that is 57. In fact, if you follow the process sometimes called “successive subtractions” to reach the next “best” number, it does in fact yield 31 and then 57. This is reflected by the function:
(n^4 - 6n^3 + 23n^2 -18n + 24) / 24
which represents the number of regions in a circle cut off by straight lines connecting n points on the circumference of a circle, where each of the points is connected to every other by a line between them, creating some number of chords in the circle. For example, 1 point on the circle leaves the entire circle as a region. Connecting 2 points yield 2 regions. Connecting 3 points yield 4; 4 points yield 8, and so on, … until 6 points yield 31 regions, and seven yield 57, ….
Charlie Silver
> On Feb 13, 2017, at 3:24 PM, tchow <tchow at alum.mit.edu> wrote:
>
> 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.
>
> Tim
>
> ---
>
> 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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