1 = 1 and 1 + 1 = 2s ```Name: rick kiper Status: N/A Age: N/A Location: N/A Country: N/A Date: Around 1995 ``` Question: Why does 1 = 1 and why does 1 + 1 = 2? Replies: Did Bertrand Russell not find a way to prove this using the postulates of symbolic logic, Principia Mathematica? Otherwise, I would say because we define counting numbers to have these properties. Send in the real mathematicians . . . rtopperIn Principia (Whitehead & Russell) "1" is defined as the class of all unit classes; probably a good book to avoid for most readers. Other possibili- ties include Frege's "Foundations of Arithmetic" or references to Peano's Postulates. Grappling with the axiomatic logic is not suitable for most of us and these readings are difficult. Let me attempt an inadequate simplifi- cation. Frege would say that a number "belongs to a concept" and is an extension of that concept and then statements about numbers correspond to "identities of concept". In the sense of counting, one apple and one orange represent identical concepts even if apples are not oranges, so we write 1 = 1. Now 1 + 1 or for that matter 1 + 1 + 1 + 1 + 1 + 1 (repeating) are represented in the sense of identity by other symbols which, if arabic numerals are used, we write as 2 or 5. The symbols are just compact notation for representation of an identical "concept". Frege stated that number is not anything physical nor is it subjective. This may not be very satisfying but deep philosophical questions about numbers may always remain unanswered in very satisfying ways. tee Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (help@newton.dep.anl.gov), or at Argonne's Educational Programs