Understanding Divergent Series and Qualitative Mathematics

The quantitative sense of mathematics is so deeply rooted in our usual conception of numbers that when one dares to hint at their qualitative sense, one is generally misunderstood, risking being ridiculed or catalogued as eccentric. Yet, perfectly correct and accepted mathematical relations such as $$1 + 2^2 + 3^2 + 4^2 + 5^2 + \cdots = 0 \quad (10)$$ leave no room for any quantitative interpretation. Who would ever think that by indefinitely increasing a quantity of objects one would find oneself with none? From a quantitative point of view equation (10) is absurd, yet mathematically correct. In this series of articles we want to definitively challenge the unilateral interpretation of numbers as quantities. Since the topic would be proper to post-university mathematics, we want to present it here in a simple way without complications, allowing everyone to access this topic generally reserved for specialists. It would be easy for us to treat the topic in an absolutely rigorous way by introducing complex numbers, holomorphic functions, their identity with analytic functions, from here Liouville's theorem, the uniqueness of analytic functions and, finally, the idea of analytic continuation. From here we should then begin the real exposition by talking about divergent series, but all this would do nothing but alienate the reader unaccustomed to formalism and mathematical terminology, while boring the accustomed one. Anyone able to notice formal gaps will also be expert enough to be able to rigorously fill them alone and confirm the correctness of the exposition. For those instead who, intrigued by the topic, have the mathematical tools and the will to deepen, we recommend reading the now classic Divergent Series by G. H. Hardy. Let us begin by defining what a divergent series is. First of all, a series is a sum with an infinite number of terms. A series, according to elementary mathematics, can be convergent if it tends to a well-defined value and therefore possesses a finite sum; or it can be non-convergent, that is if it does not tend to a single well-defined value. If the series does not tend to a single well-defined value, it can have two notable behaviors: it can grow in its absolute value indefinitely and therefore be divergent, or have no limit and therefore be irregular. We immediately present three exemplary cases: $$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$$ We are in the presence of a convergent series. In this case, the sum of the series is a finite and known number, namely the number 2. By performing the sum of a large number of consecutive terms we notice, in fact, that this sum approaches asymptotically the number 2. $$1 - 1 + 1 - 1 + \cdots$$ In this case, if we sum the terms one by one, we obtain two distinct values, that is 0 or 1 depending on whether we sum an even or odd number of terms. Not having a unique limit this series is called irregular, specifically it is called oscillating because its value oscillates between the two values 0 and 1. $$1 + 2 + 4 + 8 + 16 + \cdots$$ By summing the terms of the series together we obtain a number that becomes ever larger. We are therefore in the presence of a divergent series. To indicate the indefinite growth of the series, the symbol of infinity $\infty$ is used. In what way a sum can be infinite. Before proceeding further we must give meaning to the divergence of a series, that is in what way we can say that $$1 + 2 + 4 + 8 + 16 + \cdots = \infty \quad (11)$$ It is clear that this is a conventional sign, but, since it can generate many misunderstandings, it is necessary to dwell for a moment on the analysis of this notation. Conceptually we must distinguish two fundamental types of infinity: the first is actual infinity, that is the total absolute affirmation of all qualities and therefore decidedly transcendent with respect to Creation; secondly, infinity as expression of a continuous becoming that expresses the sense of "indefinite" or "indeterminable". In this last case the symbol $\infty$ expresses a continuous succession of thesis and antithesis without the goal of synthesis. When we write $1+2+4+8+16+\cdots=\infty$ we are certainly considering infinity in its second meaning: in fact, from a mathematical point of view we are simply saying that, given any formulable number, the resulting sum is greater than such number and therefore is indefinite or indeterminable. But what is indeterminable? Clearly the quantity indicated by the series, not its quality which is essentially expressed by the number 2, since the series in question is nothing other than a series of powers of 2.