> >I have also provided a precise verbal mapping of the > instructions that reveal the logic of why the > products of both -ve x - and - x -ve are positive. > > Perhaps you can rationalize it - you can't prove it > in the mathematical sense without making prior > assumptions.
Just letting you know I use the same verbal logic as John Wallis, credited as the inventor of the number line.
Even the rule of signs was but a consequence of "the true notion of (arithmetic] Multiplication [which] is ... to put the Multiplicand, or thing Multiplied (whatever it be) so often, as are the Units in the Multiplier."
>From the latter definition Wallis argued consecutively that
a) multiplication of a negative multiplicand and positive multiplier involved no more than taking the multiplicand the specified number of times, and thus getting a negative sum;
b) multiplication of a positive multiplicand and a negative multiplier involved nothing more than taking the multiplicand away the specified number of times; and, finally,
c) multiplication of a negative multiplicand and a negative multiplier involved "taking away a Defect or Negative," which "is the same as to supply it" ? and thus getting a positive."
NOTE: My lettering for clarity.
SOURCE: Symbols, Impossible Numbers, and Geometric Entanglements. by Helena M. Pycior