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

From: | H. S. Teoh <hsteoh@...> |

Date: | Wednesday, April 9, 2003, 2:36 |

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
--
Hardware is like a candle, and software like the flame on the candle. Making
many candles actually requires materials and expertise; lighting many flames
needs only one initial flame and is easily done by anyone thereafter.
Hardware manufacturers will always be around; but there aren't many flame
manufacturers around these days.