Vincent Edward Smith - St. Thomas on the Object of Geometry (Aquinas Lecture 18)
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St. Thomas on the Object of Geometry (Aquinas Lecture 18)
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St. Thomas On the Object of Geometry Aquinas Lecture ; 1953
author
:
Smith, Vincent Edward.
publisher
:
Marquette University Press
isbn10 | asin
:
0874621186
print isbn13
:
9780874621181
ebook isbn13
:
9780585197944
language
:
English
subject
Thomas,--Aquinas, Saint,--1225?-1274, Geometry.
publication date
:
1954
lcc
:
QA447.S5 1954eb
ddc
:
189.4
subject
:
Thomas,--Aquinas, Saint,--1225?-1274, Geometry.
Page iii
The Aquinas Lecture, 1953
St. Thomas on the Object of Geometry
Under the Auspices of the Aristotelian Society of Marquette University
By Vincent Edward Smith, Ph.D.
MARQUETTE UNIVERSITY PRESS MILWAUKEE 1954
Page iv
COPYRIGHT 1954 BY THE ARISTOTELIAN SOCIETY OF MARQUETTE UNIVERSITY
Page v
Prefatory
The Aristotelian Society of Marquette University each year invites a scholar to deliver a lecture in honor of St. Thomas Aquinas. Customarily delivered on the Sunday nearest March 7, the feast day of the Society's patron saint, these lectures are called the Aquinas lectures.
In 1953 the Society had the pleasure of recording the lecture of Vincent E. Smith, Ph.D., professor of philosophy at Notre Dame University. Dr. Smith was educated at Xavier University in Cincinnati; the University of Fribourg, Switzerland; Institutum Divi Thomae, Cincinnati; Harvard University, and the Massachusetts Institute of Technology. He received his Ph.D. degree from the Catholic University of America where he taught from 1946 to 1948. He has been on the philosophy faculty at Notre Dame since 1950.
Page vi
During World War II he was a radar engineer in the U.S. Navy and received a commendation from Fleet Admiral Nimitz for developmental work in radar counter measures. After the war he served with the U.S. Naval Technical Mission in Europe.
Dr. Smith has been editor of The New Scholasticism, the journal of the American Catholic Philosophical Association, since 1948.
He is the author of the following books: Philosophical Frontiers, Catholic University Press, 1947; Idea-Men of Today, Bruce, Milwaukee, 1950; Philosophical Physics, Harper and Bros., 1950; Footnotes for the Atom, Bruce, Milwaukee, 1951.
He is also the author of numerous articles which have appeared in scientific and philosophic journals of Europe and the United States.
To the list of his writings the Aristotelian Society has the honor of adding St. Thomas on the Object of Geometry.
Page 1
St. Thomas on the Object of Geometry
Mathematical knowledge has always been haunted by ambiguity and is the easiest, among human sciences, to be carried to extremes. Science and art, liberal and practical, inductive and deductive, the work of both the intellect and the imagination, mathematics is akin to logic, physics, and metaphysics and has at various times impersonated each of them. Hovering between physical and metaphysical knowledge, it affects the whole order of human sciences when its object1 is deformed, and among the lessons of modern thought is the power of mathematics to pass for other sciences and its impotence to reconstruct a single one of them.
The nature of geometry is one of the
Page 2
great classical problems in philosophy, worth re-stating in any age because the answer contains truth for every age. It is tempting to hold on the one hand that a material thing is only geometrical or, on the other, that geometry has nothing to do with reality at all. What is the abstraction peculiar to mathematics? What is the quantified being so abstracted and regarded not simply as quantified but as being? What is the continuum specifically envisioned by geometry? This combination of questions can turn the lock to reveal the object of geometrical science, but like all questions, they must first be properly raised before they can be effectively solved. Our quest for the object of geometry will have three stages: the nature of mathematical abstraction; an analysis of quantified being first in terms of being and then in terms of quantity; and finally the problem of the continuum.
In the spectrum of present-day philosophy, there are two extreme views about
Page 3
the nature of geometry. On the one hand, there is the over-empirical view, encouraged by Einstein,2 where geometry descends into a branch of physics and submits to experimental test. On the other hand, there is the over-formal view of Hilbert, Russell, and the logical empiricists, where mathematics is raised into a kind of logic.3 Within that wide sweep of present-day opinion, a third possible position, that of Pythagoras and Plato, is no longer visible. Yet from the case argued by Aristotle against this Pythagorean-Platonic view and continued, clarified, and completed by St. Thomas Aquinas, there is much to be learned about the object of geometry and the bond between geometry and the real.
Geometry is one of the two distinct species in the genus of mathematical sciences.4 It studies continuous quantity, like lines, planes, and solids, while arithmetic is the science of discrete quantity, like number. Between geometry and arith-
Page 4
metic, there are various mixed sciences, but there is no distinct science of quantity as such apart from its character as continuous or discrete.5 To study quantity as an accident related to substance belongs to metaphysics, and to metaphysics, as the ruling science of philosophy, belongs the hard subterranean work of charting out the object of geometry.
If geometry is a science, it must demonstrate, and demonstration, like all syllogisms, is a movement of the mind from principles to conclusions. Before demonstrating, however, whether in geometry or in any other science, the mind must acquire, by w ay of determination, the causes and principles that make demonstration possible.6 In the first two books of the
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