[lit-ideas] Re: Euclid's Theorem
- From: Donal McEvoy <donalmcevoyuk@xxxxxxxxxxx>
- To: "lit-ideas@xxxxxxxxxxxxx" <lit-ideas@xxxxxxxxxxxxx>
- Date: Mon, 10 Jun 2013 11:35:12 +0100 (BST)
________________________________
From: Robert Paul <rpaul@xxxxxxxx>
>JL wrote
> Euclid said, somewhat out of the 'blue':
>
> "There are infinitely many prime numbers".
Not really.>
Accepting that, the question is whether what he wrote amounts to the same thing.
>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.">
Is a line that can be extended indefinitely [even in concrete segments] not an
infinite line or at least potentially infinite? Likewise if "prime numbers are
more than any assigned multitude of prime numbers" does this not amount to
saying there are infinitely many primes (as more than "any assigned multitude"
means more than any finite multitude, and the only multitude that is more than
any finite multitude is an infinite multitude)?
In mathematical terms, isn't any distinction between the Greek and the modern
idea of "infinity" a distinction without a difference in terms of the above
being interchangeably true?
Donal
Btw beware any book that asserts "School children now easily understand lines
as infinite..."
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