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Science Forum Index » Logic Forum » Rucker on Infinitesimals (was '0.999... = 1')
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| Paul Holbach |
 Posted: Sun Jan 04, 2004 5:19 pm |
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Guest
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"In our ordinary real number system, we say that the number K with
decimal expansion .99999... is the samas 1. An informal argument for
this is sketched below:
10K = 9.999...
- K = .9999...
____________
9K = 9
K = 1
But maybe this argument is misleading. What if there is some number,
call it 1 - 1/omega, that is greater than any finite string .9...9 of
nines, yet less than 1? If K were actually equal to 1 - 1/omega, the
informal argument used in the last paragraph would not work, for this
argument overlooks the fact that the difference between 10K and 10 is
ten times as great as the difference between K and 1. There is a
residual infinitesimal quantity below that does not get canceled out:
10K = 10 - 10/omega
- K = 1 - 1/omega
_________________
9K = 9 - 9/omega
K = 1 - 1/omega
Intuitively, nothing could be more natural than to go ahead and talk
about 1/omega, 1/Aleph-1, and so on. Just as we move from the natural
numbers to the fractions and then on to the reals, should we not be
able to move from the whole ordinal numbers to some richer number
field?
Curiously, Cantor himself was very much opposed to this step. When a
fellow mathematician attempted to use Cantorīs transfinite numbers to
develop a theory of infinitely small quantities, Cantor accused him of
trying to 'infect mathematics with the Cholera-Bacillus of
infinitesimals'. Cantor even constructed a proof that no number can be
infinitesimal. This proof, however, is just as circular and worthless
as finitist attempts to prove that no number can be infinite. In both
cases, the desired conclusion is smuggled in as part of the definition
of 'number'.
Why was Cantor so vehemently opposed to infinitesimals? In his
valuable essay, 'The Metaphysics of the Calculus', Abraham Robinson
suggests that Cantor already had enough problems trying to defend
transfinite numbers. It seems likely that, consciously or otherwise,
Cantor deemed it politically wise to go along with othodox
mathematicians on the question of infinitesimals. Cantorīs stance
might be compared to that of a pro-marijuana Congressional candidate
who advocates harsh penalties for the sale or use of heroin. Yet, as
we shall see, there is almost as much justification for infinitesimals
as there is for Cantorīs transfinite ordinals.
Formally speaking, it is as consistent to say that there is a number
between all of .9, .99, .999, ... and 1 as it is to say that there is
a number greater than all of 1, 2, 3, ... . And just as we go on to
find more and more ordinals piled atop one another, we can go on to
find more and more infinitesimals squeezed beneath each other.
[...]
But so great is the average personīs fear of infinity that to this day
calculus all over the world is being taught as a study of *limit
processes* instead of what it really is: *infinitesimal analysis*.
As someone who has spent a good portion of his adult life teaching
calculus courses for a living, I can tell you how weary one gets of
trying to explain the complex and fiddling theory of limits to wave
after wave of uncomprehending freshman.
I often think of C. H. Hintonīs words from a similar context:
'How pleasant it would be to let pass away some of the verbiage I
learnt at school--learnt because teachers must live, I suppose. The
apeing and prolonged caw called grammar, the cackling of the human hen
over the egg of language--I should like to unlearn grammar.'
But there is hope for a brighter future. Robinsonīs investigations of
the hyperreal numbers have put infinitesimals on a logically
unimpeachable basis, and here and there calculus texts based on
infinitesimals have appeared [*]."
[*: - Keisler, H. J. (1976). /Elementary Calculus/. Boston: Prindle,
Weber & Schmidt.
- Henle & Kleinberg (1978). /Infinitesimal Calculus/. Cambridge,
Mass.: MIT Press.]
[Rucker, Rudy (1995). /Infinity and the Mind/. Princeton, NJ:
Princeton University Press. (pp. 79/80 + 87)]
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