# Is there a complete translation of "Arithmetices principia" by Peano?

If I understand correctly, Arithmetices principia: nova methodo exposita is one of the most important works covering the axiomatisation of arithmetic. I was surprised that no translation into English (or any other modern language for the matter) was available.

I just found on Github someone who translated part of it but that's it.

Did I miss something?

• According to the Wikipedia entry on his axioms it seems that a work by Jean van Heijenoort From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press. ISBN 9780674324497. which contains a translation of Peano, Giuseppe (1889). Arithmetices principia, nova methodo exposita [The principles of arithmetic, presented by a new method]. An excerpt of the treatise where Peano first presented his axioms, and recursively defined arithmetical operations. Fratres Bocca. pp. 83–97. I have no way of checking that which is why it is a comment not ananswer. Sep 11, 2022 at 12:55
• The van Heijenoort translation does not appear to be complete, finishing at 6 Division. The translation in Kennedy, Selected Works of Giuseppe Peano (University of Toronto Press), is supposed to be complete.
– abo
Sep 11, 2022 at 13:02

We see one complete English translation of Peano's Latin text Arithmetices principia: nova methodo – which is only 49 pages in its entirety, with just twenty pages numbered in:

### VII The principles of arithmetic, presented by a new method (1889)*

This remarkable little booklet (given here in its entirety) contains the first statement of Peano's best-known achievement, the postulates for the natural numbers. The essential ideas were first published by Dedekind, as Peano very generously points out in a later work, but there can be no doubt of the originality of Peano's work, and we have his own statement that he saw Dedekind's work only when Arithmetices principia was going to the press.

Following the preface, there is an introductory section on logical notation. In it we find, for the first time, the symbol E to indicate membership and J to indicate inclusion, and a note on the necessity to distinguish these two. The first section of the Arithmetices principia proper contains the famous five postulates, along with four others dealing with the sign =. These postulates were slightly modified in later writings.

The bulk of the work is written in mathematical and logical symbols. The preface and explanatory notes are in Latin, and the title is Latin, or almost - 'arithmetices' is transliterated Greek (the completely Latinized form would be 'arithmeticae').

PREFACE

Questions pertaining to the foundations of mathematics, although treated by many these days, still lack a satisfactory solution. The difficulty arises principally from the ambiguity of ordinary language. For this reason it is of the greatest concern to consider attentively the words we use. I resolved to do this, and am presenting in this paper the results of my study with applications to arithmetic. […]

ending with

46'. $$a \in \text K \text Q . \supset . \text E \text Ea = -(\text Ea \cup \text L \text Ea)$$.

— "Selected Works of Giuseppe Peano", Translated and edited, with a biographical sketch and bibliography, by Hubert C. Kennedy, George Allen & Unwin: London, 1973, pp 101–134.

Please forgive the substandard ASCII-like rendering. It matches compared to the last lines in his original booklet: