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

On Tue, Apr 08, 2003 at 09:35:29PM -0400, John Cowan wrote:
> Nokta Kanto scripsit:
>
> > 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.
>
> The typical approach is to build up numbers other than natural numbers
> in pieces.
[...]
> Real numbers are sets of rationals; pi is {3/1, 31/10, 314/100,
> 3141/1000, 31415/10000, ...}.
While this statement is mathematically correct, it obscures[1] the
hideously huge leap of logic that is required to attain to the ranks of
the Real Numbers.[2] This innocent-looking definition of real numbers
actually involves chockfuls of dark arcana and black magic[3], and even
summons the Big Brother of Infinity himself, who is also known as
Infinity, except that he is infinitely more infinite than Infinity.[4]


[1] That is to say, abstracts away. :-P

[2] As opposed to the lowly underlings called the Natural Numbers. Being
natural and mundane, y'know...

[3] Including the Axiom of Infinity, which one must take with an
infinitely large grain of salt. :-P

[4] If this totally loses you, take comfort in the fact that this is all
built from empty sets.[5] Well actually, that could make it all the
scarier.

[5] That's why you should never trust a mathematician; he is liable to
make an infinite mountain out of an empty molehill.

[snip]
> It's all artificial, but it can afford to be:  as long as the various
> rules of arithmetic are defined so that the meanings of operators are
> preserved, all is well.  We have to distinguish between numeric equality
> and set identity, of course:  (1, 0) = 1, but they are not the same set.
[snip]

That's why 1+1=0, and 1+1=1, and 1+1=2, simultaneously.[6] Depending, of
course, on what + and = happen to mean at the time.[7] :-P


[6] Except on your math exam.

[7] Well, it also depends on what '0', '1', and '2' represent at the time.
But we won't go there, since only insane people[8] are capable of
understanding that.

[8] Mathematicians.


T

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