Nature and the Process of Mathematical Abstraction

THE THOMIST A SPECULATIVE QUARTERLY REVIEW OF THEOLOGY AND PHILOSOPHY EDITORS: THE DoMINICAN FATHERS oF THE PROVINCE oF ST. JosEPH Publishers: The Thomist Press, Washington 17, D. C. VoL. XXIX APRIL, 1965 No. fl NATURE AND THE PROCESS OF MATHEMATICAL ABSTRACTION 1 T HE PRESENT paper is conversant with the relation of scientific thought to reality, both in mathematics proper and in the mathematical interpretation of nature. Our main issue can be formulated as follows. Does the subject of physics retain, at least within the phase of the fundamental definitions, the character of a thing of nature? Or should it be said that from the beginning the physicist deals with subjects modified by mathematical treatment? From both a doctrinal and an historical point of view a reflection upon 1 This article by Professor Yves R. Simon, originally dictated to his wife during a stay at the University of Chicago Clinics and completed at home in South Bend with the help of his secretary, Mrs. Pauline Ryan, was composed during the two last years of severe illness. A few very minor changes have been made in the text by its editor, Dr. Edward D. Simmons, Marquette University, according to his own judgment or because of minute corrections brought to his attention by Mrs. Yves Simon and seconded by Mr. Powell Boyd, of Albuquerque, and Dr. John 0. Riedl, Marquette University. 117 118 YVES R. SIMON Cartesian idealism seems to be an appropriate approach to this ISSUe. Let it be recalled, first o£ all, that in the philosophy o£ Descartes, thought does not apprehend things directly; its operations are primarily relative to ideas, which are modes o£ the thinking self. It cannot be taken £or granted (that is, held uncritically) that ultimately thought transcends its own modes and attains reality. Every experience o£ illusion reminds us that there are modes o£ thought which do not resemble any part or aspect o£ the real world. The transition £rom idea to thing raises a problem. There would be no such problem i£ ideas were merely instrumental in the attainment o£ things. Again, there would be no such problem i£ the philosophy o£ Descartes were an absolute idealism and held that the objects o£ our ideas are not possessed o£ any being distinct £rom their being known. But in the system o£ Descartes there exists a world o£ reality faithfully represented by some o£ our ideas, and we shall know this real world with perfect certainty i£ only we make the difference between the ideas which actually represent things and those which lead nowhere. Clarity and distinctness are the needed criterion. The idea o£ extension is clear and distinct, the idea o£ red and that o£ sweet are not. Extension is real, whereas sense qualities are nothing else than modes o£ consciousness. Thanks to the criterion o£ clarity and distinctness, a causal inference enables the mind to go beyond its own modes and to attain things. The theory o£ ideas is only the first and best known aspect o£ Cartesian idealism. We still have to inquire into the kind o£ reality represented by the clear and distinct ideas, £or this reality may itself be affected by some development o£ the idealistic principle. On this subject, there is a most enlightening passage £rom the Treatise on the World where Descartes discusses the definition o£ motion.2 He quotes Aristotle's definition , " act o£ a being in potency as such," and declares it nonsensical. Then he goes on to indicate that when he himself 2 Le Monde ou le Traite de la Lumiere, c. 7 (Oeuvres de Descartes, Adam and Tannery, Paris, 1909, vol. 11, p. 89). NATURE AND THE PROCESS OF MATHEMATICAL ABSTRACTION 119 speaks of motion he has in mind what the geometricians mean when they say that a line is generated by the motion of a point or a surface is generated by the motion of a line. These expressions are unmistakable: the concept of motion, in the physics of Descartes, expresses a physical reality which has already been processed by mathematical abstraction. This conception of the physical world warrants Descartes' well-known paradox: " I do not accept in physics any principles that are...