[FOM] Are Proofs in mathematics based on sufficient?
Irving
ianellis at iupui.edu
Thu Jul 15 22:51:27 EDT 2010
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.
It is easiest to dispense with the first point quickly. Russell's
principal theme here was that symbolic logic is the exemplary means for
rigorously and systematically developing -- as well as teaching --
geometry. His concern was not that Euclid made mathematical errors, but
that the Elements did not present theorems in strict and rigorous
accordance with logical reasoning. (This is not to say that the
propositions presented by Euclid were not properly ordered, and indeed
one could readily follow the thought processes that led from one to the
next; moreover, each one was in most cases close enough to its
predecessor and to its successor to allow an intuitive grasp of the
transition from one to the next. But the justifications and
explanations that attended the stepwise development were not explicitly
based upon logical inferences.
Since Russell's "On Teaching Euclid" (Mathematical Gazette 2 (May
1902), 165-167) is relatively unfamiliar, I hope I might be forgiven
for a comparatively extensive quotation to make my point.
Against the concept of Euclid's Elements as a masterpiece and exemplar
of logical reasoning, because Euclid's "logical excellence is
transcendent," Russell began in his essay "On Teaching Euclid" by
asserting that this claim "vanishes on a close inspection. His
definitions to not always define, his axioms are not always
indemonstrable, his demonstrations require many axioms of which he is
quite consconscious. A valid proof retains its demonstrative force when
no figure is drawn, but very many of Euclid's earlier proofs fail
before this text." Among the examples of problems are the first
proposition, which assumes, without warrant, the intersection of the
circles used in the construction; another example is the fourth
proposition, which Russell calls "a tissue of nonsense", given that
supperposition is "a logically worthless device," and a logical
contradiction arises when, taking the triangles as spatial rather than
material, one engages the idea of moving them, while, if taking them as
material, they cannot be supposed to be perfectly rigid and thus, when
superposed, they are certain to be slightly deformed from their
previous shape.
Through most of the history of mathematics, Euclid was accounted the
ideal exemplar of sound mathematical reasoning. This was the way it was
presented at Cambridge University when Russell studied mathematics.
Non-Euclidean geometries were not taught at Cambridge while Russell was
a student, although elsewhere, Hilbert and others like him were not
only examining other geometries and attempting to structure them
axiomatically in accordance with explicit inference rules (the American
postulate theorists, e.g. Huntington, with his "A Set of Postulates for
Abstract Geometry, Expressed in Terms of the Simple Relation of
Inclusion" (1913), as well as Italian and German mathematicians, such
as Pieri, with his "I principii della geometria di posizione, in
sistema logico deduttivo" (1898), "Della geometria elementare come
sisterma ipotetico-deduttivo" (1899) and "Sur la geometrie envisagee
come un systeme purement logique" (1901), and Pasch, with his
Vorlesungen ueber neuere Geometrie (1882), were or had been working on
sets of axioms for various geometries); and it is against this
background and within this milieu that Russell, for his graduate
fellowship, underook his philosophical study of metageometry, published
the following year as his Essay on the Foundations of Geometry (1897).
In the remainder of my reply to the first question, I will tentatively
establish an at least partial link between it and question (4).
The question of how to consider Euclid’s Elements -- as an axiomatic
system or as a formal deductive system, if we define a formal deductive
system as an axiomatic system with explicit inference rules, will
depend in part on whether one considers Aristotle's syllogistic logic
as providing explicit inference rules (e.g., whether the Barbara
syllogism is understood, taken in its most general form, as itself an
inference rule or as a valid argument structure -- or at least
functions as if it were an inference rule, but which does not provide
more than a psychological and metaphysical explanation of how valid
reasoning is to proceed; or if the Laws of Identity, of
Non-Contradiction, and Excluded Middle are considered as inference
rules or metalogical principles).
The second, purely historical consideration in how Euclid is to be
understood has been a matter of debate among specialists. There are
essentially two schools of thought on the matter, and so far as I am
presently aware, no consensus. One school argues that Aristotle
specifically wrote his Analytics as a justification for the methods of
demonstrations which Euclid utilizes; the other that Euclid
deliberately proceeded in the demonstrations in his Elements in
accordance with the syllogistic rules devised by Aristotle in the
Analytics. (The sub-question is whether Euclid proceeded in his proofs
on the categorical or the hypothetical syllogism). The one thing both
schools agree upon is that, in explaining the mechanics of the
syllogism, Aristotle frequently employs geometrical examples to
illustrate his points. The possibility of Aristotle undertaking his
work in constructing logic as a justification for the method of proof
employed by Euclid was dependent upon the older chronology, which had
Euclid as Aristotle's contemporary, with his dates thought to have been
ca. 356-ca. 300 B.C.) Consider Aristotle's example, in Bk. II, Chapt.
17 Prior Analytics: "it is not perhaps absurd that the same false
result should follow from several hypotheses, e.g. that parallels meet,
both on the assumption that the interior angle is greater than the
exterior and on the assumption that a triangle contains more than two
right angles." See, e.g. Henry Desmond Pritchard Lee, "Geometrical
Method and Aristotle's Account of the First Principles", Classical
Quarterly 29 (1935), 113-129; Benedict Einarson, "On Certain
Mathematical Terms in Aristotle's Logic", American Journal of Philology
57 (1936), 33-54, 150-172; John Corcoran, "A Mathematical Model of
Aristotle's Syllogistic", Archiv fuer Geschichte der Philosophie 55
(1973), 191-219; Alfonso Gomez-Lobo, "Aristotle's Hypotheses and the
Euclidean Postulates", Review of Metaphysics 30 (1977), 430-439; Rick
Smith, "The Mathematical Origins of Aristotle's Syllogistic", Archive
for the History of Exact Sciences 19 (1977-78), 201–209.
Having already gone on at considerable length, I hope I might be
forgiven if I stop here for now and take up (2) and (3) in a future
posting.
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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