[FOM] Are proofs in mathematics based on sufficient evidence?
Irving
ianellis at iupui.edu
Fri Jul 16 13:00:44 EDT 2010
In my previous post, I began by saying that Monroe Eskew, Michael
Barany and Vaughan Pratt raise some important, and I think related,
questions, principally historical, and in particular concerning the
question of: (1) whether my interpretation of Russell's criticisms of
Euclid reflect what Russell may have in fact had in mind; (2); whether
the same criticisms that I claim Russell raised of Euclid could not
just as well be directed at other mathematicians; (3) where and how to
draw a line between a computation, an axiomatic system, a formal
deductive system; and (4) whether Euclid's Elements and Aristotle's
Analytics present deductive systems.
I might have added in my previous post that, not so long after Russell
in "On Teaching Euclid" questioned the stringency and correctness of
some of Euclid's proofs, William Betz in “Intuition and Logic in
Geometry" (The Mathematics Teacher II (1909-10), 3–31) argued that
Euclid was no longer the model of mathematical rigor that he had been
historically and that the prime exemplar was now Hilbert.
In that previous post dealt with (1) and (4), the latter in particular
in connection with (1). As I turn to a consideration of (2) and (3), I
would observe that these are similarly connected.
In the course of the discussions about Euclid, reference was had to
Reviel Netz's The Shaping of Deduction in Greek Mathematics: A Study in
Cognitive History and to Netz's understanding in that work of the
nature of Euclid's methods of demostration (and of Greek mathematics
generally). Examining reviews of Netz’s book reveals a mixed reaction
(I admit to relying on the reviews, having not yet myself gotten hold
of Netz's book). Historian and philosopher of logic Paolo Mancosu
noted, for example, that Netz uses "deduction" as virtually synonymous
with "argumentation". Historians of mathematics Len Bergren and Jens
Hoyrup, both specialists in ancient and medieval mathematics, find Netz
employing "deduction" to mean diagrammatic reasoning, and point out
that, in carrying out his demonstrations, Euclid often refers back to
previous results, but does not stipulate anything that we would today
recognize as an explicit logical chain of rules for proceeding from one
proposition to the next. If, as I suggested in the previous post,
formal deductive system = axiomatic system + inference rule(s)
with the inference rule(s) explicitly stated at the outset, and cited
in a proof for letting from one line of the proof to the next, then, in
our terminology, Euclid would be seen by each of the reviewers
(Mancosu, Bergren, Hoyrup) as providing us with an axiomatic system,
but not a formal deductive system. The other aspect of Netz's thesis,
as defined in the reviews that I have examined, argues that Euclid and
ancient Greek mathematics is "formal" in the sense only that Euclid and
his colleagues treated mathematical objects linguistically, rather than
taking a metaphysical position with regard to their ontological status.
Thus, for example, labeled diagrams were being dealt with, rather than
physical lines, circles, squares, etc., or, if they were Platonists, as
ideal lines, circles, squares, etc. having some kind of
extra-linguistic existence. (The reviews of I scanned are: Nathan
Sidoli, Educational Studies in Mathematics, Vol. 58, No. 2 (2005), pp.
277-282; Paolo Mancosu, Early Science and Medicine, Vol. 6, No. 2
(2001), pp. 132-134 ; Markus Asper, Gnomon, 75. Bd., H. 1 (2003), pp.
7-12 ; J. L. Berggren, Isis, Vol. 94, No. 1, 50th Anniversary of the
Discovery of the Double Helix (Mar., 2003), pp. 134-136; Jens Hoyrup,
Studia Logica: An International Journal for Symbolic Logic, Vol. 80,
No. 1 (Jun., 2005), pp. 143-147. (I should also perhaps mention that I
am otherwise unfamiliar with the work Sidoli and Asper.) The one point
upon which the reviewers agree is that Netz's book is an important
contribution to the literature on history of classical Greek
mathematics.)
In responding to the issue of whether Russell's criticisms of Euclid,
to the extent that they it is an axiomatic system rather than a formal
deductive system, or, as Russell asserted, not rigorously logical,
might also be applied to the work of other great mathematicians,
including those listed by Monroe, namely Gauss, Weierstrass, Cayley,
Cauchy, etc., I would begin by noting that, unlike Peano in the
Arithmetices principia to be a formal deductive system, or the claims
which were made on behalf of Euclid's Elements for its being THE
exemplary model of rigorous logical proof, mathematicians such as
Gauss, et alia, were not claiming to devise either formal deductive
systems or even axiomatic systems, but were, in the case for example,
of an Euler or a Gauss, working on solving specific mathematical
problems, and it has been recognized that much of their work was
"computational" (or, in the 18th century, taken as a synonym,
"algorithmic"). Even in the case of Weierstrass, the intent was to
build analysis upon the basis of a strict definition of the limit
concept, presented in terms of functions and built from the elements of
the real continuum (as an historical point, it should be noted (a) that
much of what we have in print of Weierstrass's rigorization of analysis
comes from the edition of his lectures by Dedekind; and (b) that it is
probably Otto Stolz who adopted and provided expositions of
Weierstrass’s approach, and starting in his textbook Vorlesungen ueber
allgemeine Arithmetik: nach den neueren Ansichten (1885-86) to develop
the foundations of Weierstrass's real analysis in the style of a formal
deductive system. This is NOT in the least to suggest that Frege's 1879
Begriffsschrift was not the first effort to undertake the foundations
of arithmetic and analysis within a formal deductive system, of course.
The differences are that (i) Frege's was the first explicit declared
effort to do so, and specifically in terms of logic, and (ii) his work
was for the most part ignored until Russell began to call attention to
it in his 1903 Principles of Mathematics.
I will, hopefully, in one more installment, have an opportunity to
explicitly take an historical look at the issue of (3), of whether, and
if so where and how, to draw a distinction between computational,
axiomatic, and formal deductive approaches or styles, or whether, and
if so how, they belong to a continuum.
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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