Wednesday, December 15, 2010

Mathematics: Its Importance and Limitations

Mathematics is most co-natural to man and it is also the most exact science according to St. Thomas Aquinas, says Bernard I. Mullahy in his excellent thesis Thomism and Mathematical Physics. Armand Maurer on pg. xxxv of his translation of St. Thomas Aquinas's Division and methods of the sciences, a commentary on Boethius's De Trinitate questions V and VI, says that
St. Thomas also attributes a major role to reasoning in mathematics. In this respect it is like natural philosophy. The difference in their methods lies in the causes employed in reasoning. Mathematical demonstrations begin with definitions and principles, from which conclusions are deduced by way of formal causality. For example, a certain property of the triangle is shown to follow from its very definition. Unlike natural philosophy, mathematics does not demonstrate through final or efficient causes.
Fr. McCool, S.J., wrote in his book on the history of 20th century Thomism that (pg. 154-155):
A new edition of St. Thomas' In Librum Boethii de Trinitate, Questiones quinta et sexta, published in 1948, led to a major revision of the accepted understanding of St. Thomas' theory of abstraction [which that of Boethius inspired]. Apart from the abstractio totius, abstraction of a sensible whole from its particulars required for any form of conceptual thought, the only other type of abstraction proposed by St. Thomas confines itself to the level of mathematics. This is an abstractio formae, the mind's separation of the form of quantity from the rest of the sensible whole which the mind disregards in mathematics.
Thus we can see that medieval philosophers recognized the great importance and certainty of mathematics. Yet, according to Armand Maurer,
St. Thomas saw that there is a particular temptation to single out the mathematical method for this role [of being a common method for all the sciences, as Descartes wished], since it is the most exact and certain. But he warns against this, insisting on the specificity of method in each of the sciences [natural philosophy (physics in the broad sense), mathematics, and metaphysics; modern physics is a natural philosophy / mathematics hybrid, a scientia media or "middle science"].
St. Thomas warns about this in In II Meta. lect. 5, n. 335-337, which says, commenting on Aristotle's answer regarding "The Method to Be Followed in the Search for Truth:"

ARISTOTLE’S TEXT Chapter 3: 994b 32-995a 20

171. The way in which people are affected by what they hear depends upon the things to which they are accustomed; for it is in terms of such things that we judge statements to be true, and anything over and above these does not seem similar but less intelligible and more remote. For it is the things to which we are accustomed that are better known.

172. The great force which custom has is shown by the laws, in which legendary and childish elements prevail over our knowledge of them, because of custom.

173. Now some men will not accept what a speaker says unless he speaks in mathematical terms; and others, unless he gives examples; while others expect him to quote a poet as an authority. Again, some want everything stated with certitude, while others find certitude annoying, either because they are incapable of comprehending anything, or because they consider exact inquiry to be quibbling; for there is some similarity. Hence it seems to some men that, just as liberality is lacking in the matter of a fee for a banquet, so also is it lacking in arguments.

174. For this reason one must be trained how to meet every kind of argument; and it is absurd to search simultaneously for knowledge and for the method of acquiring it; for neither of these is easily attained.

175. But the exactness of mathematics is not to be expected in all cases, but only in those which have no matter. This is why its method is not that of natural philosophy; for perhaps the whole of nature contains matter. Hence we must first investigate what nature is; for in this way it will become evident what the things are with which natural philosophy deals, and whether it belongs to one science or to several to consider the causes and principles of things.


[St. Thomas's] COMMENTARY

335. For this reason one must be trained (174).

He exposes the proper method of investigating the truth. Concerning this he does two things. First (335), he shows how a man can discover the proper method of investigating the truth. Second (336), he explains that the method which is absolutely the best should not be demanded in all matters (“But the exactness of mathematics”) .

He says, first, that since different men use different methods in the search for truth, one must be trained in the method which the particular sciences must use to investigate their subject. And since it is not easy for a man to undertake two things at once (indeed, so long as he tries to do both he can succeed in neither), it is absurd for a man to try to acquire a science and at the same time to acquire the method proper to that science. This is why a man should learn logic before any of the other sciences, because logic considers the general method of procedure in all the other sciences. Moreover, the method appropriate to the particular sciences should be considered at the beginning of these sciences.

336. But the exactness of mathematics (175).

He shows that the method which is absolutely the best should not be demanded in all the sciences. He says that the “exactness,” i.e., the careful and certain demonstrations, found in mathematics should not be demanded in the case of all things of which we have science, but only in the case of those things which have no matter; for things that have matter are subject to motion and change, and therefore in their case complete certitude cannot be had. For in the case of these things we do not look for what exists always and of necessity, but only for what exists in the majority of cases.

Now immaterial things are most certain by their very nature because they are unchangeable, although they are not certain to us because our intellectual power is weak, as was stated above (279). The separate substances are things of this kind. But while the things with which mathematics deals are abstracted from matter, they do not surpass our understanding; and therefore in their case most certain reasoning is demanded.

Again, because the whole of nature involves matter, this method of most certain reasoning does not belong to natural philosophy. However, he says “perhaps” because of the celestial bodies, since they do not have matter in the same sense that lower bodies do.

337. Now since this method of most certain reasoning is not the method proper to natural science, therefore in order to know which method is proper to that science we must investigate first what nature is; for in this way we will discover the things which natural philosophy studies. Further, we must investigate “whether it belongs to one science,” i.e., to natural philosophy, or to several sciences, to consider all causes and principles; for in this way we will be able to learn which method of demonstration is proper to natural philosophy. He deals with this method in Book II of the Physics, as is obvious to anyone who examines it carefully.

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