24029Re: [PrimeNumbers] The history of the primality of one
- Feb 3, 2012On Thu, 2012-02-02 at 19:42 +0000, Chris Caldwell wrote:
> I have a couple undergraduate students researching the history of theTo the Pythagoreans, primality and irrationality were closely related
> primality of one. For example, most of the early Greeks did not
> consider one to be a number, so one could not be a prime number for
> them. (A few considered primeness a subcategory of oddness, so two
> wasn't prime either!) As we move forward to the middle ages and
> later it is quite a mixture. For Cataldi, Euler, Gauss, and Landau,
> one appears not to be a prime. For Goldbach, Lebesgue, and Lehmer, it
> was a prime.
--- so closely related that they were almost identical concepts.
Although odd by modern standards, it arose from their fundamentally
geometric viewpoint and, in particular, from the concept of
measurability. To the Greeks, a number was necessarily greater than
one. Most on this later.
Think of a unit as being an unmarked ruler. A prime number is something
which can be measured only by a unit but is immeasurable by any other
number. A composite can be measured not only by a unit but also by
other numbers. This view is actually rather close to the modern
definition of a prime.
A rational is a length which may be measured by a unit if it is first
multiplied (i.e. multiple copies of the rational are placed end to end)
by a number.
The Greeks' concept of number makes good linguistic sense and tallies
quite well with modern English language. When we speak of "a number of
objects" or "a number of occurrences", we almost invariably refer to
more than two of them. One is not a number in this linguistic sense and
English, in common with most other languages, distinguishes between
singular and plural in a way which is both fundamental and pervasive.
That last statement also indicates why two is not really a number
either. English doesn't have much of the dual case left, but it still
distinguishes between one, two and many in constructs such as the
comparative and superlative, and the use of words and phrases such as
"either this or that but not both" and "among the options are".
Fascinating stuff if you like the history of the development of
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