Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

understanding proofs 34912.9 Understanding numeric substructuresAs a fourth and final example, I would like to consider a class of inferencesthat involve reasoning about the domains of natural numbers, integers, rationalnumbers, real numbers, and complex numbers, with respect to one another.From a foundational perspective, it is common to take the natural numbers asbasic, or to construct them from even more basic objects, like sets. Integerscan then be viewed as equivalence classes of pairs of natural numbers; rationalnumbers can be viewed as equivalence classes of pairs of integers; real numberscan be viewed, say, as sequences of rational numbers; and complex numberscan be viewed as pairs of reals. As McLarty points out in the first of his essaysin this collection, however, this puts us in a funny situation. After all, wethink of the natural numbers as a subset of the integers, the integers as a subsetof the rationals, and so on. On the foundational story, this is not quite true;really, each domain is embedded in the larger ones. Of course, once we havethe complex numbers, we may choose to baptize the complex image of ouroriginal copy of the natural numbers as our new working version. Even if wedo this, however, we still have to keep track of where individual elements‘live’. For example, we can apply the principle of induction to the complexcopy of the natural numbers, but not to the complex numbers as a whole;and we can divide one natural number by another, but the result may notbe a natural number. If we think of the natural numbers as embedded inthe complex numbers, we have to use an embedding function to make ourstatements literally true; if we think of them as being a subset of the complexnumbers, all of our statements have to be carefully qualified so that we havespecified what types of objects we are dealing with.The funny thing is that in ordinary mathematical texts, all this happens underthe surface. Textbooks almost never tell us which of these two foundationaloptions are being followed, because the choice has no effect on the subsequentarguments. And when we read a theorem that combines the different domains,we somehow manage to interpret the statements in such a way that everythingmakes sense. For example, you probably did not think twice about the factthat Lemma 1 above involved three sorts of number domain. The fact that theseries is indexed by n means that we have to think of n as a natural number(or a positive integer). Similarly, there is an implicit indexing of terms bynatural numbers in the use of ‘ ... ’ in the expression for the logarithm. Theproof, in fact, made use of properties of such sums that are typically provedby induction. In the statement of the theorem, the variable z is explicitlydesignated to denote a complex number, so when we divide z by n in the