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Re:apples and oranges





For example, we might begin by defining the meanings and algebra of
trigonometric functions simply from the geometry of a right triangle. We
soon find it useful to extend all of this to angles larger than 90 deg.
We do this by DEFINING the meaning of sin(T), etc, for T>90 deg so that
we can carry over the same (generalized)identities, algebra, etc which
arose from right triangle geometry for only acute angles. The
generalization of the desired algebra motivates the definition, in the
interest of usefulness.

The same happens when we extend the algebra of the natural numbers to
include operations involving zero, negative numbers, irrationals,
complex, etc. Again, the heart of the issue is that this is a matter of
a freely chosen human creation, with usefulness as the determining
motivation (of course, within the bounds of logical consistency).

The realization that science is a humanly invented description of
reality, motivated by empirical and conceptual usefulness as opposed to
"truth", frees the learner from the nagging "mystery" lurking in such
matters.

-Bob

Bob Sciamanda


Perhaps you could give an example of what other choices we might have made
in inventing the negative or irrational numbers.

Richard Grandy
Philosophy
Rice University