Aristotle and Aquinas on the Freedom of the Mathematician

ARISTOTLE AND AQUINAS ON THE FREEDOM OF THE MATHEMATICIAN IT IS NOT unusual to find contemporary mathematicians who claim to have an unlimited degree of freedom in their discipline. Some even maintain that they can study (at least symbolically) anything and everything. The mathematician , they say, simply posits any definitions he pleases concerning any group of symbols and relations among them, defines the operations thereupon, and then proceeds logically. Needless to say, these mathematicians do not consider themselves bound in any way to treat entities which resemble real physical things. (Indeed, they not infrequently give the impression that they have little or no concern as to whether their mathematical considerations have any application to physical reality.) Nor do they consider mathematics to be a science of abstracted quantity in the traditional sense, fearing that to assert this would needlessly restrict the range of their science. The purpose of this essay is not to pass judgment on the claims of today's mathematicians regarding freedom in their science. I intend rather to investigate the philosophies of mathematics of two much earlier men, Aristotle and Thomas Aquinas, both of whom considered mathematics to be a science of quantity , in order to determine the degree of freedom each allowed the mathematician in his science. Specifically, I will show that the medieval theologian's doctrines contain significant advances in this area over those of his Greek predecessor. Moreover, it will be suggested that to designate mathematics as a science of quantity, as these two thinkers do, still allows for a tremendous degree of freedom on the part of the mathematicianthough it is not claimed that either man envisioned, or would agree with, the degree of freedom claimed by some mathematicians today. 231 THOMAS C. ANDERSON I. THE QUESTION Let us begin by returning to a point just mentioned, that for both Aristotle and Aquinas mathematics is considered to be a science of quantity. Let us hasten to add, however, that the quantity studied in mathematics is, according to both thinkers, a quantity not found as such in real things but a quantity abstracted from such things. As is well known, this abstraction involves mentally setting aside all the nonquantitative atributes of things and retaining only their quantitative ones. In his famous text of the M etaphyf!ics, a text which Thomas repeats with approval in his Commentary, the Stagirite speaks of the mathematician "stripping away" all features of things but their quantitative attributes, ... the mathematician investigates abstractions (for before beginning his investigation he strips off all the sensible qualities, e. g., weight and lightness, hardness and its contrary, and also heat and cold and other sensible contrarieties, and leaves only the quantitative and continuous, sometimes in one, sometimes in two, sometimes in three dimensions, and the attributes of these qua quantitative and continuous, and does not consider them in any other respect, ...1 Of course, it is precisely because of this mental abstraction, or subtraction, that the quantities studied in mathematics are said by both men to acquire their specific features as immobile, nonsensible, free from time and place and from sensible matter, and often possess less than three dimensions. And yet, though the features of abstract mathematical quantities and quantified things are radically different, this does not mean that these quantities are totally dissimilar; indeed, both philosophers stress that it is in fact the quantities of physical things that the mathematician studies. However, they add-it is not as quantities of phyl!ical things that they are studied. One text of Aristotle's which makes this clear is the following: 1 Metaphysics, XI, 3, 106la !'l9-36. Thomas's commentary is In XI Metaphysics, L. 3, !'l!'lO!'l. FREEDOM OF THE MATHEMATICIAN 233 Obviously physical bodies contain surfaces and volumes, lines and points, and these are the subject-matter of mathematics.... Now the mathematician, though he too treats of these things, nevertheless does not treat of them as the limits of a physical body, nor does he consider the attributes indicated as the attributes of such bodies. That is why he separates them; for in thought they are separable ....2 Thomas Aquinas makes exactly the same point in his commentary on this...