[FOM] Simpson on Tymoczkoism
Sandy Hodges
SandyHodges at alamedanet.net
Thu Oct 2 17:31:24 EDT 2003
Ron Rood writes:
----
However, I do not think that one can unconditionally say that
mathematical propositions are true. For example, if the axiom of
parallels is
true, then (let us assume) its negation is false. Therefore, if I know
the axioms of Euclidean geometry, then I do not know (indeed, cannot
know) those of non-Euclidean geometry. But that seems absurd.
----
I take it that Rood declines to assert:
Given a line and a point outside it, there is exactly
one line through the given point which lies in the
plane of the given line and point,
so that the two lines do not meet.
I wonder if he would be willing to assert:
IF, given a line and a point outside it there is exactly
one line through the given point which lies in the
plane of the given line and point,
so that the two lines do not meet,
THEN the angles of any triangle add up to two right angles.
(I'm sorry to be so prolix, but there is a distinction between asserting
the axiom of parallels and asserting that the axiom of parallels is
true. It would have been quicker for me to write "If Euclid's axioms
are true, then the angles of a triangle add up to two right angles."
but if one can't, as Rood asserts, say unconditionally that mathematical
propositions are true, then "If Euclid's axioms are true" is a phrase
that makes no sense. (Or means something unexpected - if Euclid's
parallel axiom is not the sort of thing that can be true or false, then
the claim that it is true is simply false).)
I don't know if Rood will be willing to assert the above claim (the
indented one starting "IF, given ...") but if he is, then he is prepared
to assert logical facts, although unwilling to assert Euclidean ones.
I think that for the debate about foundationalism, as long as one is
prepared to assert some logical or mathematical fact unconditionally, it
does not matter that one declines to assert others. If Rood is
willing to assert the above indented "IF, ..., THEN ..." claim, and
thinks it is true, and that it is something he knows, then there is
propositional knowledge.
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Sandy Hodges / Alameda, California, USA
Note: This is a new address as of 20 June 2003
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