Re: OT: mathematicians (Was: Re: Results of Poll by Email No. 27)

On Tue, Apr 08, 2003 at 09:07:27PM -0400, Nokta Kanto wrote:
> On Tue, 8 Apr 2003 11:32:44 -0400, H. S. Teoh <hsteoh@...>
[snip]
> >And even the idea of sets itself would be overly complex (mathematically
> >speaking) if we allowed sets of arbitrary objects; therefore in formal
> set
> >theory, sets don't contain anything except other sets. By the same
> >argument, the only relational operation on sets is the "contains"
> >relation.  The most basic assumption of set theory (axiom #1) is that at
> >least one set exists. This is usually understood to mean the empty set.
> >
> >Putting these together, you can think of 0 as the empty set, 1 as the set
> >that contains the empty set, 2 as the set containing the 0 and 1 (i.e.,
> >the empty set and the set that contains the empty set), ad infinitum. In
> >other words, the entire field of mathematics is made of empty sets; all
> >theorems are ultimately just pronouncements about empty sets. Now we know
> >where mathematicians dispose their cups after consuming all that
> >coffee.[2]
> >
> >:-P
> >
>
> It hardly seems complete. What is -1? The largest set contained by the
> empty set? I didn't think mathematicians would like a system that couldn't
> be extrapolated to a more general number space.
[snip]

Well, these are just the natural numbers, you see. The point is that you
can define integers and fractions in terms of natural numbers and
operations on natural numbers. Think about how you learn to add and
subtract negative numbers--the techniques you learned are in fact just
manipulations of natural numbers, plus a little bookkeeping of the
occasional negative sign. Mathematicians don't care about the mechanics of
performing these operations; all they care is that they're able to define
integer operations in terms of natural number operations, and derive
properties of these operations based on that. You just do that once, then
you abstract away the messy details and think of integer operations as
atomic.

The entire set of integers contain exactly the same number of elements as
the natural numbers, so all you need is a scheme to map integers to
natural numbers and vice versa. One straightforward mapping is simply to
map all even natural numbers N to N/2, and all odd natural numbers to
-(N+1)/2. How addition and subtraction are done on these "natural"
integers is left as an exercise for the reader. ;-)

Similarly, since there are exactly as many rational numbers as natural
numbers, all you need to do is to find a 1-to-1 mapping between natural
numbers and fractional numbers. This is the same as putting all rational
numbers in a sequence, 'cos that way you just map the first element of the
sequence to 0, the second to 1, the third to 3, etc.. One way to do this
is to lay out fractions in an infinite table of numerator vs. denominator:

1   2   3   4   5   6   ...
1/2 2/2 3/2 4/2 5/2 6/2 ...
1/3 2/3 3/3 4/3 5/3 6/3 ...
1/4 2/4 3/4 4/4 5/4 6/4 ...
1/5 2/5 3/5 4/5 5/5 6/5 ...

Then you traverse the table in a diagonal, zigzag pattern:
        1, 1/2, 2, 1/3, 2/2, 3, 1/4, 2/3, 3/2, 4, 1/5, ...

Of course, there are some duplicates, like 2/2 = 1, 2/4 = 1/2, etc., but
we'll just assume you're smart enough to drop duplicates from the real
sequence. The point is, the table contains all possible rational numbers
(plus a few duplicates), and there is a consistent algorithm that covers
the entire table if you repeat this process to infinity. So this shows
that there are at least as many natural numbers as there are fractional
numbers.

But we also know that there are at least as many fractional numbers as
natural numbers. Therefore, the only possible conclusion is that there are
just as many natural numbers as there are fractional numbers,
counter-intuitive as that may seem.

Of course, the point behind all this is that firstly mathematicians want
to establish the *existence* of sets that contain all the rational
numbers, or all the integers. This is what has been shown thus far. Of
course, with the proposed representations, it would be quite cumbersome to
define numerical operations on members of these sets. But that's OK; we
don't care about the internal structure of set members; we can label them
however we want, and define operations[1] in terms of the labels.


[1] How to define these operations is, of course, another matter
altogether. They would have to be defined in terms of operations on the
natural numbers, of course. That's just a matter of appropriately mapping
the operands to the natural numbers. I.e., in a similar way you learn to
add/multiply fractions in school. If you think about it, you aren't really
dealing directly with fractions; you're dealing with pairs of natural
numbers using manipulations (on natural numbers) that are isomorphic to
how fractions behave. Once you've defined fractional operations, you
abstract away the messy details and think of them as atomic operations.

Mathematicians don't have to physically build anything, you see, so once
they prove something, they can reuse it thereafter without worrying about
the details ever again. Of course, this is unpalatable to the Engineer,
since the engineer is concerned about the efficiency of the design[2],
even if engineers also abstract things the same way. But mathematicians
live in an empty set[3], so they're not concerned about such mundane
things as time and space efficiency.

[2] Being what determines if something can actually be built physically in
finite time and space, of course. The mathematician couldn't care less if
something can't be actually built; all he cares is its *theoretical*
possibility. That's why you should trust a mathematician when he says
something is impossible; but when he says something is possible, take it
with a grain of salt[4].

[3] That is to say, a vacuum. :-P

[4] The size of which depends on which axioms[5] he has invoked while
making the statement.

[5] There is an axiom known as the Axiom of Infinity. The size of salt to
use in this case is left as an exercise for the reader. :-P


T

--
If it's green, it's biology, If it stinks, it's chemistry, If it has numbers
it's math, If it doesn't work, it's technology.