[FOM] Are Proofs in mathematics based on sufficient?

[Irving]

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