Monday, February 20, 2012

From Reuben Hersh's book "What is Mathematics Really"

From Reuben Hersh's book "What is Mathematics Really".

Any proof has a starting point. So a mathematician must start with some
undefined terms, and some unproved statements. These are "assumptions" or
"axioms." In geometry we have undefined terms "point" and "line" and the
axiom "Through any two distinct points passes exactly one straight line." The
formalist points out that the logical import of this statement doesn't depend on
the mental picture we associate with it. Nothing keeps us from using other
words—"Any two distinct bleeps ook exactly one bloop." If we give interpretations
to the terms bleep, ook, and bloop, or the terms point, pass, and line, the
axioms may become true or false. To pure mathematics, any such interpretation
is irrelevant. It's concerned only with logical deductions from them.
Results deduced in this way are called theorems. You can't say a theorem is
true, any more than you can say an axiom is true. As a statement in pure mathematics,
it's neither true nor false, since it talks about undefined terms. All mathematics
can say is whether the theorem follows logically from the axioms.
Mathematical theorems have no content; they're not about anything. On the
other hand, they're absolutely free of doubt or error, because a rigorous proof
has no gaps or loopholes.

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