We are used to thinking of numbers as infinite. And, in fact, they are. Even as a child, I was drawn to mathematics precisely because of infinity. It doesn't fit in our heads, and yet we talk about it without much embarrassment. Strange. It was only in the twilight of my life, however, that I discovered that Letters are also Infinite. And I dare say: they belong to a type of Infinity greater than that of numbers.
In mathematics, infinity produces profoundly counterintuitive results. For example, some infinities are larger than others. "Infinity greater than infinity" sounds like a paradox, but it isn't. To understand this, however, requires some effort and patience. Some truths demand a price.
The first step is to understand how to compare infinities. Don't be alarmed, dear reader, the procedure is quite simple: just establish a correspondence between the elements of two sets. Let's take a concrete example. The set of natural numbers is infinite. The set of even numbers, although it is only a subset of the natural numbers, is also infinite. And both have the same cardinality. How can the part have the same size as the whole? It's counterintuitive, but mathematically true. Just associate each natural number with its double: (1,2), (2,4), (3,6)… to infinity. For every natural number, there is exactly one corresponding even number, and vice versa. QED [1]
But how can we demonstrate that one infinity is larger than another? If two infinite sets have the same size when their elements can be paired one by one, then one will be larger when such pairing proves impossible.
The most celebrated and elegant example is the demonstration that the set of real numbers has a greater cardinality than the set of natural numbers. Let us consider, without loss of generality, a simplified version of the reasoning. Imagine that a friend claims to have listed all possible infinite sequences of decimal numbers between 0 and 1, using only the digits 1 and 2, such as 0,12112… To show that this list is necessarily incomplete, it is enough to find a missing number in the list. This is done by constructing a new number from the “diagonal of the digits” of the list, as follows: take the first digit of the first number, the second of the second, the third of the third, and so on, replacing each 1 with a 2 and each 2 with a 1. The number obtained will differ from the first in the first digit, from the second in the second, from the third in the third, and so on. Therefore, it does not coincide with any element of the list. There will always be a missing number. It is thus demonstrated that there are infinitely many numbers greater than others. This is the famous Cantor Diagonal Argument (Georg Cantor, 1845–1918). The Cantor diagonal element is rebellious: it refuses to be confined to any supposedly complete list. Pause your reading for a moment and contemplate Figure 1.
For the Pythagoreans (6th to 4th centuries BC), “everything is number” was practically a dogma. At first glance, this statement may seem untenable to contemporary ears. However, some caution is advised. Modern science seems to confirm it in a surprising way, as the universe manifests a singular property [2] : physical phenomena insist on obeying mathematical structures. In other words, mathematical models manage to describe them with extraordinary precision. The success of this method in the so-called exact sciences ended up inspiring the attempt to “mathematize” all other forms of human knowledge. In my view, this is a gross error, but that is a matter for another reflection.
Let's return to Literature.
I am not referring, of course, to the alphabet or to words. Both are finite. I am referring to Literature in its highest sense. This expresses what numbers are incapable of communicating. The most skeptical reader might retort: What, exactly?
The first verse of Genesis says: “ In the beginning God created the heavens and the earth ” [3] . When I first read this passage, at the age of ten, I looked up and gazed out the window at the sky covered with clouds [4] . Only many years later, while studying the Patristic writings, did I discover that the word “heaven” in that passage designates first and foremost the order of invisible, spiritual realities. The verse, therefore, distinguishes not only two places, but two orders of creation: the spiritual world and the physical world. It took me many years to understand this Truth.
If numbers admirably translate the physical world, only [5] Letters can fathom the Spiritual world. It is there that all numerical infinities find their limit and fall silent [6] ; while Letters free themselves from the shackles of matter and sing the Transcendental. By resorting to the Freedom of Letters, by Poetic analogy with Cantor's Diagonal Argument, I conclude that they belong to an Infinity superior to that of numbers [7] , and that great Works are rebellious. No interpretation can definitively enclose them. Like the element of Cantor's diagonal, they always escape the list of senses that we considered complete. Numbers may be infinite; Letters, in turn, are inexhaustible.
I later recognized this other Truth. If the maturity I have today had been given to me in my youth, perhaps I would have pursued Literature. The infinity of numbers led me to Mathematics. The inexhaustible Infinity of Literature, that Infinity greater than infinity, would have swept me away with even greater force. [8]
(I dedicate this text to Laura, my daughter )
This text does not necessarily reflect the opinion of Unicamp.
[1] Quod Erat Demonstrandum.
[2] There is no epistemological need for the universe to be mathematically intelligible. The success of mathematics remains, to a large extent, a philosophical mystery.
[3] And in the New Testament we have: “ In the beginning was the Word, and the Word was with God, and the Word was God ” (John 1:1).
[4] In Manaus, due to the humidity produced by the Amazon rainforest and the large volume of water from the Amazon River, the sky is always covered with clouds, either rain-laden or as white as cotton. Upon arriving in Campinas, I was awestruck by the deep blue of a completely clear sky. I had never seen a sky without clouds.
[5] Obviously in contrast to numbers. Not in the sense of “exclusivity”, as we cannot forget the other Arts: music, painting, sculpture and architecture.
[6] Naturally, numbers have always been used to symbolize spiritual realities, from the Pythagoreans to the biblical tradition. I am referring here not to the symbol, but to the descriptive capacity proper to each language.
[7] Poetic Analogy dispenses with mathematical rigor to unveil Truths that transcend numbers.
[8] One of the greatest works of Brazilian Literature begins with Nothing and ends with Infinity. I suspect the author knew that Letters are Infinite.
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