David Pierce | Matematik | M.S.G.S.Ü.

The Foundations of Arithmetic in Euclid

Call it an essay, a monograph, or a book, it is 142 pages, size A5, 11-point type, text occupying (9/12)2 of each page, dated January 12, 2016:

The Introduction includes a summary of each of section of the book. Briefly, the four chapters concern:

  1. The philosophy of history, as developed by R. G. Collingwood in several books, as it pertains to the reading of Euclid.
  2. Some general findings about Euclid, mostly obtained in the process of leading a course in which students read Book I of the Elements.
  3. More specific findings about Euclid's number theory, as for example that he does prove rigorously the commutativity of multiplication in any ordered ring whose positive elements are well ordered.
  4. These findings, expressed in more modern terms, along with (in the first section) an argument that the last proposition of Book VII of the Elements is a later addition, precisely because it is too “modern.”

This version is the same as the above, but with less text on each page, because the box of text includes a header: 153 pages, size A5, 11-point type, text-with-header occupying 9/16 of each page, dated July 13, 2015:

Originally the first section of the fourth chapter was in the third chapter, and there was no fourth chapter: this draft was “On the Foundations of Arithmetic in Euclid” (April 17, 2015; 98 pages of size A5):

After preparing that draft, I wrote out the details of the proof of the commutativity of multiplication in another (draft) paper, “Commutativity of Multiplication in Euclid's Arithmetic” (May 5, 2015; 19 pages of size A5):

This shorter paper works out an argument that Euclid proves commutativity of ordinal multiplication in any well-ordered set that is closed under ordinal addition and ordinal multiplication, provided this addition is commutative.

Here is an earlier draft, of January 12, 2015; 97 pages of size A5:

Son değişiklik: Tuesday, 12 January 2016, 10:45:31 EET