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Re: [PrimeNumbers] The history of the primality of one

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  • Paul Leyland
    ... To the Pythagoreans, primality and irrationality were closely related ... Although odd by modern standards, it arose from their fundamentally geometric
    Message 1 of 9 , Feb 3, 2012
      On Thu, 2012-02-02 at 19:42 +0000, Chris Caldwell wrote:
      > I have a couple undergraduate students researching the history of the
      > 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.

      To the Pythagoreans, primality and irrationality were closely related
      --- 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
      intellectual activities.


      Paul
    • djbroadhurst
      ... But then some Germans disagreed, circa 1900: http://www.peterhug.ch/lexikon/primzahl/13_0390 http://de.academic.ru/dic.nsf/meyers/111736 Des goûts et des
      Message 2 of 9 , Feb 3, 2012
        --- In primenumbers@yahoogroups.com,
        Paul Leyland <paul@...> wrote:

        > The Greeks' concept of number makes good linguistic sense
        > and tallies quite well with modern English language.

        But then some Germans disagreed, circa 1900:

        http://www.peterhug.ch/lexikon/primzahl/13_0390
        http://de.academic.ru/dic.nsf/meyers/111736

        Des goûts et des couleurs, on ne dispute pas?

        David
      • djbroadhurst
        ... V. A. Lebesgue designated 1 as prime on page 5 of his 1859 textbook: http://books.google.co.uk/books?id=ea8WAAAAQAAJ He is not to be confused with Henri
        Message 3 of 9 , Feb 3, 2012
          --- In primenumbers@yahoogroups.com,
          Chris Caldwell <caldwell@...> wrote:

          > For Goldbach, Lebesgue, and Lehmer, it was a prime.

          V. A. Lebesgue designated 1 as prime on page 5 of his
          1859 textbook:
          http://books.google.co.uk/books?id=ea8WAAAAQAAJ

          He is not to be confused with Henri Lebesque,
          who was not born until 1875.

          David
        • djbroadhurst
          ... Here is a thumbnail biography: http://www.les-mathematiques.net/phorum/read.php?17,323622 ... David
          Message 4 of 9 , Feb 3, 2012
            --- In primenumbers@yahoogroups.com,
            "djbroadhurst" <d.broadhurst@...> wrote:

            > V. A. Lebesgue designated 1 as prime on page 5 of his
            > 1859 textbook:
            > http://books.google.co.uk/books?id=ea8WAAAAQAAJ

            Here is a thumbnail biography:
            http://www.les-mathematiques.net/phorum/read.php?17,323622
            > Un mathématicien méconnu aujourd'hui mais qui a
            > joué un rôle important dans la première moitié du XIX ème.

            David
          • Phil Carmody
            ... Fundamental, pervasive, and perverted. It is, after all, a language in which the singular thou has been jetisoned for the plural you , and similarly the
            Message 5 of 9 , Feb 3, 2012
              --- On Fri, 2/3/12, Paul Leyland wrote:
              > 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.

              Fundamental, pervasive, and perverted.

              It is, after all, a language in which the singular 'thou' has
              been jetisoned for the plural 'you', and similarly the plural
              'they' adopted as a singular when trying to avoid mentioning
              gender. And we can't really decide whether we want companies
              or bands to be singular or plural - is Nokia going down the
              pan, or are Nokia going down the pan? (Which does seem to
              correlate strongly with pondianness.)

              Phil
              (who recently left a country where in "5 boys", "boys" is *not* plural?!?!)
            • djbroadhurst
              ... Is/are Manchester ******* United singular/plural? Meanwhile, back in the archives: on page 252 of http://www.gutenberg.org/ebooks/31246.html Rouse Ball
              Message 6 of 9 , Feb 3, 2012
                --- In primenumbers@yahoogroups.com,
                Phil Carmody <thefatphil@...> wrote:

                > And we can't really decide whether we want companies
                > or bands to be singular or plural

                Is/are Manchester ******* United singular/plural?

                Meanwhile, back in the archives: on page 252 of
                http://www.gutenberg.org/ebooks/31246.html
                Rouse Ball indicates that Mersenne may
                have condered 2^p - 1 to be prime for p = 1:

                "In the preface to the Cogitata a statement is made about
                perfect numbers, which implies that the only values of p not
                greater than 257 which make N prime, where N = 2^p - 1, are
                1, 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257.."

                However, Chris's students should check the original Latin for this.
                So far they are expected to be adept in Greek, Latin, Italian,
                German, French and English. Maybe Euler wrote something
                relevant in Russian:
                http://books.google.co.uk/books?id=cJTkQxTvGa4C

                David
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