[lit-ideas] Re: Euclid's Theorem
- From: Robert Paul <rpaul@xxxxxxxx>
- To: lit-ideas@xxxxxxxxxxxxx
- Date: Sun, 09 Jun 2013 15:54:11 -0700
JL wrote
Euclid said, somewhat out of the 'blue':
"There are infinitely many prime numbers".
Not really.
From
<http://primes.utm.edu/notes/proofs/infinite/euclids.html>
The ancient Greeks also did not have our modern (sic) notion of
infinity. School children now easily understand lines as infinite, but
the ancients were again more concrete (in this regard). For example,
they viewed lines as segments that could be extended indefinitely (not
something infinite that we view just part of). For this reason Euclid
could not have written "there are infinitely many primes," rather he
wrote "prime numbers are more than any assigned multitude of prime numbers."
This seems a plausible account.
An English translation of the Elements Book IX, proposition 20.
<http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html>
In, The Man Who Knew Infinity: A Life of the Genius Ramanujan, by Robert
Kanigel (1992), there's an elegant little proof that there can be no
greatest prime. Indeed, it's such a short proof that I'm now wondering
whether it's sufficient to prove what I remember its setting out to prove.
Robert Paul,
beside the wine-dark sea
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