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Re: Archaeopteryx flight
<No, a mathematical proof is a logically constructed sequence of statements
that extends from a set of statements known or assumed to be true to a
proposition whose truth is established by the existence of the sequence.
There is no circularity. Here the operative phrase, of course, is "logically
The assumptions in this statement include the premises (the 'statements
known or assumed to be true') and also the logic which connects these
premises to a conclusion 'whose truth is established'. After the argument
is complete, a proposition has become a conclusion, no?
Anyway, the logic itself is subject to question, and may be proven either
outright incorrect or ambiguous.
Talking about Kant's antinomies, a web essay notes:
More recently Quine has defined an antinomy as a paradox which 'produces a
self-contradiction by accepted ways of reasoning. It establishes that some
tacit and trusted pattern of reasoning must be made explicit and
henceforward be avoided or revised.' Such revision, Quine says, involves
'nothing less than a repudiation of part of our conceptual heritage'.
If a proposition and its opposite can both be proven, or both be proven
false, there's something wrong with the logic. In many cases, the problem
proves to be confusing interpretation with observation...
So, I suggest that 'truth is established by the existence of the sequence'
is overstating it a bit.