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From: paulv@boring.UUCP
Newsgroups: net.math
Subject: Re: Beyond Exponentiation
Message-ID: <6303@boring.UUCP>
Date: Thu, 31-Jan-85 10:27:02 EST
Article-I.D.: boring.6303
Posted: Thu Jan 31 10:27:02 1985
Date-Received: Thu, 7-Feb-85 03:39:46 EST
References: <186@ihnet.UUCP>
Reply-To: paulv@boring.UUCP (Paul Vitanyi)
Organization: CWI, Amsterdam
Lines: 35
Summary: 
Apparently-To: rnews@mcvax.LOCAL

In article <186@ihnet.UUCP> eklhad@ihnet.UUCP writes:
><  where no function has gone before  >
>
>what is the next function in the following sequence?
>Allow me to use '^' for exponentiation.
>	y = 2
>	y = 2*x
>	y = 2^x
>	???
>Each operation evolve from an iteration of the previous (originally).
>
>Karl Dahlke    ihnp4!ihnet!eklhad

It is perhaps interesting that such matters were the main subject
of Archimedes' "The Sand Reckoner". In that treatise (<100BC),
Archimedes addresses the question of very big numbers, basically
by continueing the process above. He convinces king Golon that
the number of grains of sand needed to fill up the universe tightly
is not infinite but can easily be counted. A quick way to obtain
an upper bound on the needed number is by using the above process.
	On the way he enunciates the so-called "Axiom of Archimedes",
definition 4 Book 5 of the "Elements" of Euclid, which serves
to derive the whole theory of proportion and is at the very
foundations of the calculus. The row between Newton and Leibnitz
about whether or not to accept this axiom (resulting in a
final victory for Newton & Archimedes by the success of Weierstrass's
epsilon-delta method for defining limits and so ground the
calculus accepting the Axiom) has its present day revival with the emergence of
Nonstandart Analysis which rejects the Axiom.
	Archimedes also mentiones in passing the theory of
Aristarchos of Samos (<300BC), who claimed that the sun stood still like the
fixed stars and the earth revolved around her.