8 ERRETT A. BISHOP I was careful not to call the number n defined above an integer. 0 Whether we do call it an integer is of no real importance, as long as we dis- tinguish it in some way from numbers such as n For instance we might 1 ' call no an integer and call n a constructive integer. The constructivists 1 have not accepted this terminology, in part because of Brouwerts influence, but also because it does not accord with their estimate of the relative impor- tance of the two concepts. I shall reserve the term "integer" for what a classicist might call a constructive integer, and put aside, at least for now, the problem of what would be an appropriate term for what is classically called an integer (assuming that the classical notion of an integer is indeed viable). Thus we come to the crucial question, "What is an integer? " As we have already seen, the question is badly phrased. We are really looking for a definition of an integer that will be an efficient tool for developing the full content of mathematics. Since it is clear that we always work with repre- sentations of integers, rather than integers themselves (whatever those may be), we are really trying to define what we mean by a representation of an integer. Again, an integer is represented only when some intelligent agent constructs the representation, or establishes the convention that some artifact constitutes a representation. Thus in its final version the question is, "How does one represent an integer? In practice we shall not be so meticulous as all this in our use of language. We shall simply speak of integers, with the understanding that we are really speaking of their representations. This causes no harm, because the original concept of an integer, as something invariant standing behind all of its representations, has just been seen to be superfluous. Moreover we shall not constantly trouble to point out that (representations of) integers exist only by virtue of conventions established by groups of intelligent beings. After this preliminary chatter, which may seem to have been unnecessary, we present our definition of an integer, dignified by the title of the Fundamental Constructivist Thesis Every integer can be converted in principle to decimal form by a finite, purely routine, process. Note the phrase "in principle. " It means that although we should be able to program a computer to produce the decimal form of any given integer,
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