> In the end it was shown that the PP was independent of the
> other axioms, and in fact you can get non-contradictory and interesting
> versions of geometry by assuming either that there are multiple parallel
> lines through a point or no parallel lines.
Yes. A statement expressible within a formal axiomatic system is
independent of the system if neither the statement nor its negation
can be proven within the system.
But a new tendency in (the philosophy of) mathematics, called
"experimental mathematics", is emerged. There is, also, the
"Journal of Experimental Mathematics".
See also J. Horgan, "The Death of Proof", Scientific American,
269 (1993), p. 74-82, and D. Zeilberger, "Theorems for a Price:
Tomorrow's Semi-Rigorous Mathematical Culture", Not. Amer.
Math. Soc., 40 (1993), p. 978-981.
There are, also, many mathematical propositions for which only
computer proofs are available, in example the proof of four-colour
And, of course calculations may yield considerably different
values, depending on machines and on softwares. Try, i.e.,
to iterate x[n+1] = 3,95 x[n] (1-x[n]).
There are, also, very subtle situations. Goedel and Cohen
have shown that neither the CH (Continuum Hypothesis)
nor its negation can be proven within the ZF (Zermelo &
Fraenkel) set theory plus the AC (Axiom of Choice).
Now let f(n) be defined on the set of natural numbers and
f(n) = 1 [if the CH is true]
f(n) = 0 [if the CH is false]
Now f(n) is computable because there is the double algorithm above,
but the trouble is that we can not know which is the correct one,
that is to say which of the two algorithms actually is the one
that computes f(n). Just because the CH is *independent* of
the standard framework for mathematics (the ZF plus the AC).
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