[FOM] Are proofs in mathematics based on sufficient?
Irving
ianellis at iupui.edu
Mon Jul 19 22:43:21 EDT 2010
Monroe Eskew wrote that if Euclid had given correct informal proofs from
Hilbert's axioms, but still failed to specify any formal inference
rules, Russell would not have found any fault with the Elements. His
criticisms of I.1 and I.4 seem to be mathematical rather than
meta-mathematical points: Euclid did not have any continuity axioms,
and the superposition technique simply does not follow from anything
prior.
If I understand this aright, my response is that Russell
very explicitly denied that Euclid's "logical
excellence is transcendent". The claim on behalf of
Euclid's Logical excellence", Russell very plainly
assert, "vanishes on a close inspection," and the
reasons Russell lists for asserting this is that
Euclid's "definitions do not always define, his axioms
are not always indemonstrable, and his demonstrations
require many axioms of which he is quite unconscious,"
on the basis of which, I would suggest, his complaints
about Euclid have to do with the logical structure and
logical felicity of his demonstrations.
Turning from the specific examples which Russell gave, I
will readily admit that, at the time Russell wrote
to his conception of what is at stake philosophically, his
piece ("The Teaching of Euclid"), which was published in
1902, Russell had not yet attained the definitive position
of logicism with which we are familiar from his 1903
Principles of Mathematics. Russell began working towards
logicism in late 1900, but did not reach his definitive
position until late 1903, which led him to do last-minute
rewrites of parts of PoM in the galleys but after the
manuscript had already been delivered to the press. I
would contend, on the other hand, that there was enough
of logicism in Russell's foundational thinking at the
time he composed "The Teaching of Euclid" to suggest
that the distinction between "logical" and "mathematical"
were for him such that any difference between a logical
error and a mathematical error was essentially nil, in
particular as regards the questions of definitions that
define or do not define, the Independence vs. dependence of
axioms, and the completeness or incompleteness of axioms
for deriving from the axioms the mathematics that the set
of axioms, together with inference rules, are intended to
establish.
I had earlier attempt to send my response to the issues
which Prof. Eskew raised regarding computation/axiomatic system/
formal deduction and the question of the nature of proof, but
there may have been a transmission glitch of some sort, so I
will make another attempt shortly.
Irving H. Anellis
Visiting Research Associate
Peirce Edition, Institute for American Thought
902 W. New York St.
Indiana University-Purdue University at Indianapolis
Indianapolis, IN 46202-5159
USA
URL: http://www.irvinganellis.info
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