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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.