La freccia e il cerchio
anno 6, numero 6, 2015
pp. 26-28
Carlo Sbordone, Aldo Trione
The destiny in the number
I. An infinity to be defined
SBORDONE
Let’s begin with one sharp question: is the universe mathematical or philosophical? I would answer that it is a bit of both, as it is impossible to make sharp distinctions. Yet, mathematics and philosophy have now become two disciplines with very few contact points. Only seldom is a mathematician –myself included– a philosopher. And it is difficult to find a philosopher who is knowledgeable when it comes to mathematics: usually, in fact, he ignores the most basic concepts of this discipline, concepts which should be part of the cultural luggage of any scholar, such as the one of limit, of the sum with infinite addends, of the derivative…
TRIONE
Agreed, fine. If I intend to search convergences and mutual curiosities, I need to go back to a remote past. I then think of the comparison between science and poetical wisdom, traceable in ancient legends, marked by an indissoluble and necessary relationship between number and imagination. Our culture has moved within the horizon of John’s statement, who says: “In the beginning was the Word.” But this Word was, somehow, the number. And this number was wisdom, knowledge, imagination. myth, project, illusion. The number outlined the destiny of the entire cosmos, of poetry, of art…
SBORDONE
No one would find that surprising, of course. Certainly not Archimedes and Pythagoras. Or, in more recent times, Leibniz and Newton, who also held astrological and cosmological knowledge. Nor would Einstein, a scientist who gave indisputable contributions be surprised: after him, the perception of time and space has irremediably changed. However, Einstein represented already an exception in a more and more hyperspecialised context.
TRIONE
Shall we try to draw some boundaries? Dedekind maintains: the number exists in order to count. Dedekind, a great mathematician, a scientist, and, I would add, a philosopher. It sounds like a paradox. Without soaring into the strategies of art, he confirms: “No, the number is there for us to count.” Yet, the number is a wide concept. What if I were to say with similar vivid simplicity, “the word is there for us to express ourselves”? Perhaps it is true that we need to search for an original word, the one Heidegger talks about. Or perhaps we, our words and rites, are that word. I wonder: is there a number that is set conceptually at the beginning, as origin? Has it ever been formulated?
SBORDONE
1, 2, 3 are the numbers set at the origin. We got to the zero much later. It is true, the goal of numbers is being able to count: so, the first is number one. But then there are number 2 and number 3, numbers which are familiar but not obvious. When we think of contemporary science, the numbers that are used are 10 to the 50th, which means billions of billions of billions…The list of numbers gets wider and wider.
TRIONE
You mean that numbers are not measurable, that they are in fieri?
SBORDONE
They are infinite. The set of numbers used to count is infinite, but it is necessary to define this concept of infinity, as it differs depending on the context, whether we are talking mathematics, poetry or physics. As regards mathematics, a further distinction seems crucial, that between infinite and finite sets (the five fingers of my hand, for example). What do we mean by finite set? We mean that this set cannot be placed in one-to-one correspondence with a part of it: I will never be able to find a one-to-one correspondence between the fingers of my right hand and those of my right hand de- void of the thumb. However, there is the possibility of a one- to-one correspondence between the fingers of my right hand and those of my left hand. Now, the set that cannot be place in a one-to-one correspondence with a part of it is called fi- nite. The set of all natural numbers (one, two, three, four…), on the other hand, can be put into a one-to-one correspond- ence with a part of it, for example with even numbers, therefore it is infinite.
TRIONE
In mathematics, do we need to define infinity case by case?
SBORDONE
Not at all: infinity is definite.
TRIONE
There is a profound difference. In mathematics “infinites” own a conceptual finiteness, but in philosophy and aesthetics infinity is one, no-one and one hundred thousand..
SBORDONE
A Pirandellian kind of infinity….
TRIONE
A never-ending infinity. An infinity that does exist, but within which problems, structures, wide meanings are reshaped.
SBORDONE
A continuously indefinable infinity?
TRIONE
Yes, exactly. In mathematics, instead, we can, as you were rightly saying, move inside an accomplished organisation, which takes into account the analysis of the number, of his finiteness, of his specificity. As for the forms of myth and imagination, they are not storable. No one is able to stop this infinity, because it holds its own law inside, which is similarly infinite: the law of the cosmos.
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