Ask A Scientist Mathematics Archive

Author:      rick kiper
Why does  1 = 1  and why does  1 + 1 = 2?

Response #:  1 of 2
Author:      rtopper
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 . . .

Response #:  2 of 2
Author:      tee
In 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.