Academic journal article The Review of Metaphysics

Numerical Foundations: Arithmetic as Episteme

Academic journal article The Review of Metaphysics

Numerical Foundations: Arithmetic as Episteme

Article excerpt

I

MATHEMATICS HAS HAD ITS SHARE of historical shocks, beginning with the discovery by Hippasus the Pythagorean that the integers could not possibly be the elements of all things. (1) It was much the same with Kurt Godel's Incompleteness Theorems, which presented a serious (even fatal) obstacle to David Hilbert's formalism, and Bertrand Russell's own discovery of the paradox inherent in his intuitively simple set theory.

More recently, Paul Benacerraf presented a problem for the foundations of arithmetic in "What Numbers Could Not Be" and "Mathematical Truth." (2) Drawing out difficulties he found in mathematical realism, Benacerraf ended up proposing his own structuralist view of mathematics, according to which numbers could not be considered objects at all, but mere placeholders, wholly defined in terms of the structures within which they were found. (3) While we do not intend to work through the intricacies of structuralism or other types of mathematical nonrealism, or even of any of the many other competing theories in contemporary philosophy of mathematics, our question concerns problems one finds in an earlier, and perhaps more naive, view of mathematics which, although a realism of its own, begins with seemingly more problematic assumptions than these. (4)

Classically expressed, the question is: do numbers exist in the world, whether as substances or as attributes of things? These are among the things Benacerraf holds numbers could not be; still, we hope to dispel some of the confusion which might bring one prematurely to reject this account. Most precisely, we wish to address the problem whether there can be a science of arithmetic, as that term is understood by Aristotle and Thomas Aquinas. (5)

II

As Aristotle points out at the outset of Posterior Analytics, all teaching and learning must proceed from preexistent knowledge. (6) In the case of scientific knowledge ([TEXT NOT REPRODUCIBLE IN ASCII]), the preexistent knowledge is of different sorts. We must already know that certain propositions, the premises of the demonstration ([TEXT NOT REPRODUCIBLE IN ASCII]), are true. We must also know the meaning of certain terms, such as the subject and what we are predicating of it in the conclusion. Finally, we must know that the subject exists. (7)

   It is necessary to have preexistent knowledge of two sorts: for in
   some cases it is necessary to assume that something is, in others
   what the term is, and in still others both. For example, of
   "concerning everything, either the affirmation or the denial is
   true" [we assume] that it is, of "triangle" [we assume] what it
   means, and of "unit" [we assume] both what it means and that it is.
   (8)

Thomas elaborates upon these preconditions for demonstration:

   That about which we seek scientific knowledge through demonstration
   is a certain conclusion, in which a proper attribute is predicated
   of a subject, which conclusion is inferred from certain principles.
   And because our understanding of simple things precedes our
   understanding of complex things, we must know the subject and the
   attribute in some way before we can know the conclusion. So, too,
   we must first know the principle from which the conclusion is
   inferred, since a conclusion is made known through knowing the
   principle. (9)

While Aristotle says that we must already know, of both the subject and the premise, that they are, it is important to note that "being" has many senses, one of which is being in the sense of being true. (10) Of the principles of demonstration, we can know that they are, that is, to use Aristotle's own example, we know that the principle of excluded middle is true. This, then, is the first type of preexistent knowledge required in a demonstration.

Another type is our understanding of the meaning of a term, which has special reference to what Thomas calls the "proper attribute":

   We are able to know what the attribute is since, as is shown in
   this same book, accidents do have a kind of definition. … 
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