Binghamton University The Open Repository @ Binghamton (The ORB) The Society for Ancient Greek Philosophy Newsletter 1983 Aristotle's Philosophical Principles of Mathematics Hippocrates George Apostle Grinnell College Follow this and additional works at: https://orb.binghamton.edu/sagp Part of the Ancient History, Greek and Roman through Late Antiquity Commons, Ancient Philosophy Commons, and the History of Philosophy Commons Recommended Citation Apostle, Hippocrates George, "Aristotle's Philosophical Principles of Mathematics" (1983). The Society for Ancient Greek Philosophy Newsletter. 5. https://orb.binghamton.edu/sagp/5 This Article is brought to you for free and open access by The Open Repository @ Binghamton (The ORB). It has been accepted for inclusion in The Society for Ancient Greek Philosophy Newsletter by an authorized administrator of The Open Repository @ Binghamton (The ORB). For more information, please contact [email protected]. I. ARISTOTLE'S PHILOSOFHICAL PRINCIPLES OF MATHEMATICS 1-J i ppooret fes G<'lZ>rae Arpos-fle. ''(;Jzsi-fl'"fl/ 11 IC/ g3 The definition of mathematics as the science which investigates the properties of quantities was first conceived by the ancient Greeks and was formulated later both philosophically and scientifically by Aristotle in his various works. Later mathematicians accepted this definition tacitly or by habit, for philosophical problems about mathematics as a whole were not much of a concern t0 them nor was the field of mathematics far. advanced in research to suggest alternative definitions. Early last century Gauss reaffirmed this definition in his treatise, � Foundations £!. Mathematics. From the latter part of last century until today, however, there has been a tendency away from this definition and in the direction of what was thought to be a better and more general definition; and in the opinion of most modern mathematicians and philosophers the old definition is too limited to cover modern mathematical research. The introduction of the so-called "non-Euclidean geometries" and of transfinite numbers, too, contributed somewhat to this tendency; for, it was t11ought, if the parallel postulate did not possess the absolute truth which was once attributed to it, it �ould have only hypothetical truth, if any truth at all, and today we find many if not most mathematicians and even philosophers taking the position that mathematics is not interested in the truth or falsity of its principles but on�y in consistent sets of postulates and the deduction of theorems from those postulates. As a consequence of such thinking there arose a number of new definitions which give the appearance of being general enough to include all actual and perhaps possible mathematical research. Peirce regarded mathematics �s the science which draws necessary conclusions, Russell identified mathematics with logic, Hilbert emphasized the symbolic nature of mathematics, and others posited such wide concepts for mathematical objects as order, intuition, relations, and the like. Now a fair criticism and evaluation of the old definition presupposes an 2 understanding of the terms in that definition and the principles according to which the formulation was made. Unfortunately, however, the critics failed on both counts; for they knew neither the meaning of those terms nor the principles according to which the formulation was made. First, I shall present Aristotle's definition of mathematics and the principles which he uses in formulating it; second, I shall discuss some a./501 definitions which have been given lately and their difficulties; and t:hil<i-, A I shall show that Aristotle's definition best fits modern mathematical research. DEFINITION: Mathematics is the science which investigates generically, specifically, and analogically the properties of quantities and whatever belongs to quantities. Aristotle did not give expressly this definition, but it can be gathered from what he says in his various works. Whether he wrote a work on mathematics or not is not known. Now the key terms in this definition are "science," "quantity," "generically," "specifically," "analogically," "belonging," and "property," 1''irst, let us turn to the term "science". It has two senses for Aristotle. Its main and narrow sense is: necessary knowledge of what exists through its cause. The other sense inc�udes the principles and the logical proofs of theorems from the principles. So the definition of science in this sense would be: universal knowledge of principles and demonstrations of properties from those principles under one genus of existing things or under one aim; and the definition of a theoremiould be: a demonstrated statement which signifies an attribute as belonging to a subject through the cause. In the case of mathematics, quantity is the genus. Now the principles in a science are four in kind. They are (1) the indefinable concepts, (2) the definitions, (3) the hypotheses, and (4) the axioms. The premises come from the definitions and the hypotheses; and as for the axioms, they are not premises but what some moderns call "directive" or "regulatory" principles which are used to de:nonstrate c •nclusions from premises. 3 Second, the meaning of the term "quantity" is clear to those who have read carefully Aristotle's Categories and Book Delta of the Metaphysics. 'Quantity' is a category, and its two immediate species are 'number' and 'magnitude'• ,and by "number" Aristotle means what nowadays call "a natural number which is greater than 211, Bertrand Russell, whose ignorance of the history of mathematics "' surp�sses even his ignorance of Aristotle's logic, chooses to reject the ancient definition in his Introduction to Mathematical Philosophy. I quote: "It used to be said that mathematics is the science of 'quantity'. 'Quantity' is a vague word, but for the sake of argument we may replace it by the word •number'. The statement that mathematics is the science of number would be untrue in two different ways. On the one hand, there are recognized branches of mathematics which have nothing to do with number - all geometry that does not use coordinates or measurement, for example; projective geometry and descriptive geometry, down to the point at which coordinates are introduced, does not have to do with number, or II even with quantity in the sense of greater and � ••• In this passage, Mr. Russell considers the word "quantity" as vague; and this shows that he did not do his homework. Again, he replaces the word "quantity" by the word "number" and chen tries to refute the ancient definition; and this is like setting up a straw man and then knocking him down. Again, the expression "mathematics is the science of quantity" was given as a definition and not as a fact, but truth and falsity do not apply to definitions; yet he regards the exfression as being untrue, that is, as false. Again, he is unaware of the fact that the relations of greater and less apply even to magnitudes in geometry which does not use coordinates, as in the case of plane geometry. There are other errors, but these should be enough for a logician. Third, let us turn to the term "property". It means an attribute which is not the essence of a subject but belongs to that subject and to no other subject. 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