FOM: Proper Names and the Diagonal Proof
Dean.Buckner at btopenworld.com
Mon Jun 24 16:39:16 EDT 2002
Let's think of numerals as a system of generating new proper names,
plus the assumption that each name designates a different thing. I don't
really care about what kind of things these are.
Let's think of the mathematical operators as ways of constructing further
proper names. For example "two times four". The skill of mathematics is
working out when an expression formed using the operators, designates the
same thing as an expression that does not use the operators. Thus the
expression just mentioned designates the same thing as "eight".
Mathematicians will get excited by this, pointing out perhaps that
(a) every expression using "--- times ---" can be correlated with some
non-operator expression ("one hundred times two" with "two hundred" &c),
such that they signify the same thing.
(b) this is not always possible with other operators. E.g. the operator
"---divided by ---" is such that "four divided by two" co-signifies with
"two", but "three divided by two" with nothing. Similarly for "the x such
that x times x equals ---"
This takes us a little too far into mathematics however. Let's assume just
that we have a method of generating an indefinite number of names "one",
"two", "three" ..., and that we have an indefinite number of operations we
can perform on any name in this series, to produce a new name -
"the square root of one", "the square root of two", "the square root of
three" and so on.
Crucial to this assumption is that we have a way of explaining exactly what
the operation is, i.e. we have a way of putting together a finite sequence
of English (or French) words so that mathematicians are able to understand
the meaning of the new name.
Then, some interesting questions. (i) Is the number of operations that we
can form strictly limited? (ii) Even if unlimited, is there a way of
matching up each operation with the sequence 1,2,3, ...? (iii) something
1. the square root of ---
2. the length of the line whose distance from the same point = 1
3. the definition of the number "e" which I can't remember
4. Any other definitions of strange operations on numbers
If we can list all the (1-place) operators out like this, then it is simple
to prove that we can match the names formed by any
operator+any numeral, with the numerals. Equally simple to extend
the proof to operators that have two or three or more places. Hence, even
if different names signify different things, there are enough ordinary
names (i.e. numerals) to go round, to match the more complex names.
Cantor's proof has it otherwise. It says that we cannot "match" a set of
ordinary names (numerals) to a set of names formed by infinite decimal
expansion. Each row is "complete": it is a proper name formed out of an
"infinite" collection of digits. It assumes not just an infinite collection
of names, but an infinite collection of names with infinite number of parts.
But it's not clear which object is named by any name in the the sequence,
since it is not clear what the name actually is. Even if we suppose that
"the square root of two" names something, what does "1.23245467 ..."
name,when the dots signify not a specific method of carrying on the
sequence, but no specific method at all?
I'm happy to assume that, given that a meaningful expression of English can
formed using a finite sequence of expressions, using words drawn from a
finite vocabulary, we can signify a different thing with each different
expression. I'm not happy to assume any more than that, unless I
know what I'm meant to assume.
I'm not arguing against Cantor's proof, I'm saying I don't understand what
it's meant to prove.
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