The pragmatist view is that thought has no other end than producing belief, and belief no other end than producing habit. Thus, belief is a specific rule of action, and habit is the practical consequence of belief. One is tempted to say that this applies even to the most abstract ideas. However, such a statement would be inaccurate. In truth, the pragmatist view allows us to see that abstract ideas are meaningful only by virtue of their practical consequences. Take the proposition:
- Harmonic infinite series are divergent
This statement may seem abstract. However, its practical meaning lies hidden in technical language. In fact, an equivalent statement is:
- The sum of the inverse of all natural numbers does not approach a definite quantity.
An even more sober equivalent—one that does not appeal to the strange idea of adding all elements of an infinite set—is the following:
- There is no definite quantity whose difference with the sum of a certain amount of inverse natural numbers can be arbitrarily reduced by increasing that amount.
The third statement is equivalent to the first, and yet holds its meaning in greater disclosure. The reason why statement 3 is comprehensible is that it places the proposition’s weight upon a practical consequence. If I may rephrase it one last time, the statement is simply that
- The sum of a certain amount of inverse natural numbers increases indefinitely when that amount increases indefinitely.
In other words, the statement “harmonic series are divergent” is a practical statement about the expected effect or outcome of a particular action; namely, adding a great number of a specific kind of quantity. My belief in the divergence of harmonic series is my belief in the nonexistence of a quantity satisfying the specific property stated in proposition 3. The habit my belief in the statement informs is nothing but the set of actions I would undertake under all possible situations involving such a sum. This is the practical bearing of the statement “harmonic series are divergent,” and that practical bearing is the whole realm of its meaning. Of course, a mathematician does not delve, like we did, in reformulating statements such as proposition 1. The meaning of each term in the proposition being well understood, he is always directed to the empirical realm disclosed in propositions 3 and 4. But these empirical notions are the meaning of statement 1, and the four statements above differ from one another in a strictly linguistic manner.
The same logic brings the meaning of notions such as limit or infinity to a clearer bearing than that brought about by the prejudicial notion according to which such expressions are simply operative symbols that happen to work. It is true that if a symbol “works”—i.e., if it permits us to draw valid conclusions in a systematic manner—then it is either the expression of something that is true or of a certain relationship that is true. Infinity, for example, when dissected as we did when dealing with harmonic series, is not a quantitative concept in itself—for speaking of an infinite quantity is as valid as speaking of an indeterminate determination—. It rather expresses a certain relationship among the terms and the outcome of a sum, in the case of series, or the behavior of a sequence in the case of limits—where sums and sequences are themselves a useful representation of concrete realities. Furthermore, the concept of infinity arises from a negative necessity, for the fundamental notion of limit deals with the relationship between a certain sequence and some definite quantity that—to be intentionally redundant—limits the sequence. When such a certain quantity does not exist, infinity becomes conceptually useful to express that nonexistence—since, wherever the idea of an unlimited sequence arises, so does the notion of a tendency to unlimited quantities. When such a certain quantity does exist, we can by definition approximate it by means of the sequence to any arbitrarily small degree —and so, when the notion of a limited sequence comes to mind, the idea of an unlimited approximation is also consequential.
Thus, we find that seemingly obscure mathematical concepts are made perfectly clear not only without the need of metaphysics, but precisely because of the absence of metaphysical confusions. It is the general rule—though I can not argue in favor of this position here—that the more denuded a doctrine is of metaphysics the greater its chances are of finding truth, and mathematics is no exception.
Take for example a classical objection made to the notion of infinity, according to which infinity corresponds only to an “ontological totality”, the unlimited universe not conceived as made of parts but as a single absolute reality. The reason, it is said, is that no two infinities could exist, for if that were the case one would limit the other, which is clearly a contradiction. But firstly, infinity is not an ontological concept—this is hopefully clear due to our previous elaborations. Thus, not only mathematics deals not with the “existence” of several infinities, but the claim that a single infinity exists—the universe, or any form of universal metaphysical unity—is meaningless. The weight of reality falls only upon individual, concrete objects, in the same way the meaning of a proposition depends entirely on the meaning of its terms. And secondly, many infinities can (and actually do) exist when the concept is properly understood; that is, when it is treated in terms of countability—not ontology. This is so clear that it hardly requires further elaboration. If you consider the set of natural numbers and the set of even numbers, you will find both to be countless. However, there are infinite mappings associating each element in the second with two elements in the first.
An equally absurd objection is that according to which the only “real” numbers are integers —or, in some cases, natural numbers. This is proposed because non-integer numbers are either divisions among non-divisible numbers or arbitrary expressions that can never be achieved by means of adding or subtracting the unity. This objection touches a grain of truth; that is, that unity (in the arithmetical sense) is the only real quantity. This is true in the sense that individual objects are the only objective reality, whereas collections of objects constitute already some form of cognitive elaboration. However, the arithmetical unity is an arbitrary value, and can be defined only by means of an exterior phenomena —that is, with a total disregard of logic.