Mathematics is the science of relationships

Seduction, games and democracy, between entities and numbers: a conversation with Chiara Valerio.

Andrea Zanni has a degree in mathematics, and is a digital librarian at MLOL. He has been very active in wiki projects for some time, he was president of Wikimedia Italy. He lives in Modena.

C.hiara Valerio was born in Scauri in 1978, she is a writer and editor of Italian fiction for Marsilio. He has studied mathematics for many years, and has a Ph.D. in probability calculus.

His latest book is a small pamphlet, published by Einaudi: Mathematics is politics . Blending essay, civil controversy and autobiography – not unlike his previous book Human History of Mathematics –, Valerio proposes a parallelism between mathematics and democracy, explores the analogies between two apparently very different spaces but in reality based on the collective construction of a meaning starting from axioms and shared rules: two “systems of rules, which create communities and work on relationships “. Both in mathematics and in democracy, every concept of truth can only be separated from a context, from commonly accepted definitions and from present conditions that may change in the future. They are human, cultural constructions, which must be continually re-discussed. Despite some sometimes risky analogies, the emulsion between apparently incompatible substances holds out above all thanks to the personal and civil objective of the book: to re-establish, in times of pandemic and quarantines, the individual and collective plan,

Andrea Zanni: In Mathematics is politics you propose an unprecedented parallelism between mathematics and democracy: you write that both are based on “systems of rules, they create communities and work on relationships”. Because mathematics is a place of plurality, of rules decided in a transparent way and consequences that must be clear to everyone, in a democratic way. It is an inversion of the cliché of mathematics as the custodian, on the other hand, of every definitive, monolithic, despotic exactness and truth. In fact, you say that math is not what people think.

Chiara Valerio: It’s true, I wanted to say – also by nature, because of a note of character – that authority has nothing to do with rules. I have studied for many years a discipline in which the rules are strict, depend on the system in which you are moving, and they are shared, because without that sharing you cannot even talk to others, and I finally understood that in reality mathematics first of all, it is the discipline farthest from the principle of authority that I could come across, and that above all authority and rules – even if they seem the same – have a very different nature.

Authority has to do with a vertical view of the management of things, rules have to do with a horizontal management of things. So this book tries to say that both democracy and mathematics have some interest in the horizontal axis . This is the parallelism. Among other things, time is historically represented on the abscissa axis, therefore on the horizontal axis.

Then there are the differences, between mathematics and democracy. Mathematics does not deal with human beings: the entities of mathematics are not people. The entities of democracy yes: this obviously on a scale we say “three-dimensional” does not allow parallelism, but from the point of view of language, the language that democracy represents and that mathematics represents, I still find this analogy convincing. Even now that I have become the one who reads the book, and I am no longer the one who wrote it.

AZ: One of the words that came to mind initially, while reading, is the word game, precisely in the mathematical sense of “game theory”: the one in which you develop a system, give it rules and start the spark of life , to see what happens.

And if you want to go a step further, a great thing about the games is that you realize that there is enormous freedom, even with very few rules. That is, the generation of worlds, the generation of spaces is gigantic, even in a seemingly simple game such as John Conway’s Game of life . Or, if I think of more philosophical things, the idea that all the complexity of the universe can be contained in a few computational rules that we still have to discover, as Stephen Wolfram has been proposing for twenty years.

CV: One of the first things I gave my grandson is a small tower in Hanoi , a wooden puzzle with some discs that have to be moved from one stake to another according to a rule, which is in fact a recursive algorithm. Already this simple game immediately teaches us that there is something to do with the pieces and something else that has to do with the combination of those pieces. And the fun is in the combination.

When we were children – I was born in 1978 – there were fantastic games like the Galton table , in which many small spheres are let slide on a plane, go through a kind of funnel and are finally hindered by many pegs that make them accumulate on bottom according to a bell curve.

What are the principles governing the Galton table? Essentially gravity, and some pegs. From gravity and the rungs, from a children’s game, a curve emerges which we rely on to describe the trends of very different phenomena. You quote Game of life and Wolfram: you remind me that we are talking about somehow recursive systems and all recursive systems start from four very simple rules – certainly simpler than gravity and the pegs – start from that single ribbon imagined by Alan Turing for his machine , and in which we can say that we are completely immersed, without reaching the excesses of the Matrix.

The possible moves in a Turing machine are step forward, step back, write in the cell, clear the cell. And the Turing machine is the basis of computers and the modern world. It doesn’t take many rules to give birth to a varied and complex life.

After writing a code, it takes a program that compiles that code, processes it, executes it. If the commands are few, the compiler programs are complex, if the commands are many, the compiler programs are light. I mean that the rules, from the tower of Hanoi, to the coding commands, are just a part of the game, the funniest part and how you combine the rules, what you decide to do with them, which world draw the rules and which ravines of that world they seem not to exist, and then they do exist.

The way you combine the rules makes life somehow. And it also makes a democratic life. This I meant in the book. Not having a rule is also a rule. We do not leave a system of conventions: we are not beings who can live outside a system of arbitrary and conventional signs.

AZ: A strange thing is that one might read Euclid’s five postulates or read the Italian Constitution and in some way they resemble each other, as systems of rules, as “generators of spaces”. The interaction – in this case the interaction between rules – is what creates the complexity.

One might see what jurists call the “conjunction”, or the result of the joint interpretation of two or more norms, as a vector space generated by the laws. On the other hand, it is the most difficult part: in this sense, jurisprudence is the study of the interaction of some laws in a legal space, mathematics is the study of the interaction of some rules in a mathematical space.

CV: You say that rules are generators of spaces, and yes, of course they are, we can almost mathematize it. Over the centuries they have realized, for example, that the space generated by Euclid’s postulates could change when, in fact, the postulate of parallels changed. The fifth postulate, but it took centuries and centuries to dare to say it, is logically independent of the others, it could be changed and generate another geometry. The postulates are not exactly what in geometry is called “linearly independent vector system”, but they resemble us.

So the rules, in this case the postulates, properly and even metaphorically generate a space. Here, considering the constitution a “system of linearly independent vectors” is more complicated, because precisely the entities of the Constitution are human beings. Perhaps it is not even necessary to arrive at the Constitution to understand that in the spaces generated by the constitutional articles there is no linear independence … Even with regard to the tables of the law, the Ten Commandments of Moses, it cannot be said that they are linearly independent. A person can give false testimony but not cheat on his wife, he can take God’s name in vain but not kill, stuff like that. They could be interdependent. But I’d like to be able to see in 4 or 5 dimensions to understand it better… it would be enough to be able to see time maybe.

AZ: This idea of ​​finding a core of rules is very fascinating, like saying if one could  dry the constitution down to its prime numbers, to stay in mathematical metaphors. Axioms that are totally independent but that are sufficient, are necessary and sufficient to generate this “space of the Constitution”. But it doesn’t seem like a thing of this world to me.

CV: Also because we know from the third century BC that prime numbers are infinite, while we certainly know that that system of rules could not be infinite.

AZ: Going back to the speech above, you say that when there is recursion there is something interesting. Or even better: when there is something interesting there is recursion. This is the thesis of the famous book Gödel, Escher, Bach , in which Douglas Hofstadter shows how at the center of very different things such as life, consciousness, paradoxes, Bach’s music or Escher’s graphics, there is always the self-referentiality, a process that looks at itself. Even Gödel’s incompleteness theorem says natural numbers, among the simplest things we know, are however complex enough to generate a mathematics in which we know that there will be true but not provable theorems. Something that until Gödel we didn’t even think was possible.

In a certain sense, even with the fifth postulate of parallels it was realized that what  appeared to be a pillar, one of the five that had to support the scaffolding of geometry, was actually a door. And there was a universe out there, a non-Euclidean universe.

CV: It also took us nearly two thousand years of obsessive thinking to understand the issues surrounding the parallel postulate. Over the years I have become convinced that it has to do with parallel transport. From the geometric point of view, the expression parallel transport means the translation of a vector while keeping the angle that the vector forms with the curved surface constant. If the angle with the surface is constant, the objects I move – as in the technical drawing tables at school – do not deform. Here, I really think, as I wrote in the book, that the question of parallels has to do with parallel transport, therefore with the idea that our truths keep the form that human beings have given them. If parallel transport did not exist geometrically, projecting the human being from here to eternity does not mean that it maintains the anthropomorphic form. So somehow in my opinion the question of the parallel postulate is a good example to say that we are vain beings but that this vanity also sometimes leads us to great discoveries.

Obviously recursion – and I believe that after all it has always come from there, from something that has nothing to do with it – however it has an attitude of representation -, Proust’s Recherche : where there is repetition there is a sense of reality. Why this? Partly because we are ruminative beings: despite all our digital devices and all the speed that is imposed on us, it does not make us go back to things – and that coming back to things seems almost a waste of time – in reality, of course, we return to things. Neurologically melancholy because the memory area of ​​the past is the same as the imagination of the future.

So recursion somehow interprets an attitude of human thought, as we know it today, in 2020; the aptitude to rethink. We rethink – sometimes with nostalgia, we have remorse, we have second thoughts about truths that we thought accepted, even sentimental truths. Recursion somehow does this. It is true that as Hofstadter says when there is something interesting there there is recursion. But I say that when there is something  recursive at the bottom there is something extremely human.

AZ: A beautiful thing that Paolo Zellini , another mathematician and essayist says , is that the algorithm belongs to men: it lives in space and time, in a space of memory and in a time of execution, as opposed to the “mathematics of the gods ”, That of numbers – abstract entities that live in Plato’s hyperuranium. So the algorithm is a chimera, divine and human, it is no coincidence that it is the basis of our current age.

CV: And they also show you how infinity is made! Mathematically, we only really understood this with the tool of algorithms. Before, infinity was a much more abstract entity, with algorithms it becomes more concrete again.

AZ: Returning to recursion, it occurs to me that Norbert Wiener, the founder of cybernetics, who placed the concept of feedback (and therefore of recursion) at the basis of all complexity, said so too. Instead of having a black box with inputs and outputs , with feedback the output is transformed back into input , and changes how the machine itself works. This mechanism is the basis, and we have been discovering it for decades, of all that is life, consciousness, the universe, everything.

CV: The thing that has always amused me is that you have conjectured the feedback by working on the accuracy of American missiles, to hit targets at sea. So in addition to the genesis of the modern world, we also owe the game of naval battle to Wiener. He formalized a gesture that each of us naturally knew.

For me he is special, one of the greatest mathematicians of the century. My PhD professor had studied with one of his students, so I found myself with all his books when I was very young, and reading them was a literally lysergic experience: when I was reading God and Golem I felt like I had taken LSD .

AZ: You speak of mathematics as a study of relationships, therefore also of networks, which are one of the fundamental metaphors of this century, especially in these pandemic times.

CV: After so many years of study, I think that mathematics is essentially a science that deals with the relationships between entities and numbers. Numbers and numbers, entities and entities. And that if it were presented in this way, it would be natural to think of why it is a discipline that is learned step by step and which therefore, learned step by step, inoculates you, without explaining it, but building it, a principle of causality. The principle of causality is a great compass when you live in a recursive and networked world. The principle of causality is practically innate, comes with the naturals. I saw him with my nephew Francesco, after saying mom and dad, my sister taught him to count. Count to five, on the fingers of one hand. So we learn to count to 5 even before learning Italian grammar,

The question of teaching is a crucial question has to do with something that  I read many years ago in a book I loved very much, Possession by Antonia Byatt (translated into Italian by Anna Nadotti for Einaudi). Possession is a novel in which, in my opinion, Byatt wanted to insert her otherwise unpublished Victorian poems (despite being Antonia Byatt). Around the poems he builds a kind of spy story where there are two researchers, who depend on two professors, one English and another American.

When the two meet to explain to the BBC what is happening, what earthquake is going through English literature, that Christabel LaMotte, who lived with a woman, was instead engaged to Randolph Ash, and even had a  daughter – and begins to talk. Blackadder, the Englishman, at one point Leonora Stern, the American, stops him and tells him: “Stop, James, do it sexy”, and begins to say the same thing, but differently.

So I think that telling math as a science of relationship is doing  what Leonora Stern did with James Blackadder about Victorian literature: trying to make it captivating. This is because we learn to speak essentially to seduce and be seduced. Because formal language is one of the greatest seductions!

So much so that it leads to mysticism: if it were not one of the greatest seductions we have available, why do all mathematicians at the end of their life tend to the “theorem of God”? Also Gödel, or Ennio De Giorgi; Why does Paolo Zellini, our great mathematician and theorist, consider transfinite numbers to be real? Here: how would these researches be reached if formal language were not a form of maximum seduction? Going down to the lower floors, to the levels not of study, but of training – culture is always confused, which is a single gesture, an intention, with education which is a collective gesture, a constitutional right – here, level of training, why can’t it be pointed out that mathematics has this form of seduction based on the fact that it is the science of relationships? I would like to say it,

AZ: You tell me one thing that struck me at the beginning with Caccioppoli’s anecdote: ” a student, during an exam with stunted answers, confesses to the professor that he is in love with mathematics and the professor replies, in Neapolitan: – Guaglio ‘ , but no one ‘reciprocated’ “. It made me laugh because I myself often said: “I like math, it’s me that she doesn’t like”.

But you say something that I personally am not yet sure I agree with, which is that mathematics is democratic, just study. We tend to study it poorly, but everyone can access it. This is obviously true, and the common idea that mathematics is just the stuff of geniuses and predestined is wrong.

I say so, but I’m not so convinced. I remember very well the effort I made on math books. Perhaps because I approached mathematics in a “wrong”, very philosophical and romantic way: which is one of the most wrong approaches, because then you clash with a whole university system that is anything but romantic or literary: you have five exams, four of algebra, four of geometry, etc.

The fact is this: we are not used to thinking in rarefied spaces, with few elements. Otherwise it would be easier to do chess endings, when you have a king, a rook and two pawns. Certainly more difficult than making an opening because the elements are few, but the space and the possible interactions are more. As Feynman said, there is a lot of space : not physical, but mathematical, combinatorial. Here, I find this thing often “not very human”, because we are not at all good at putting order in abstract spaces, when there is very little margin for error. And in my opinion this makes math easier for people who, for a thousand reasons, are more capable and are not afraid of these spaces.

CV: I really like the example of the chessboard because it allows me to say one thing. I  liked playing chess; I was not particularly good, but my ability lay in the fact that when I started playing, the pieces, for me, were not only the chess pieces, but also the squares. As I lost the pieces, I gained squares, so I had a kind of invariant number of pieces.

You said earlier “it reminds me of the effort I had to study mathematics”: I too did a lot of effort, I’m not one of those who came to understand Hilbert’s spaces by flying. I certainly got there swimming, with extreme effort. But I also remember the satisfaction of when I landed and what it allowed me after.

I was a child – maybe my kind helps me – full of fairy tales, where to get to conquer the princess, you have to climb the tower. So the idea of ​​overcoming very high obstacles to get where you want to go has been told to me since I was a child. So maybe there is a gender predisposition on the contrary – compared to what you think – for mathematics: because the obstacle does not scare you, there is no other way.


AZ: Actually I have had many colleagues in university, mathematics is often studied by girls, so much so that I have never really understood the prejudice that it was a “male” thing. Probably there remains a big obstacle – the classic glass ceiling – to become professors or researchers … so often girls do not do research but go to teach at school. Perhaps it is therefore always the same speech. As with writing, those who can keep their asses on the chair, those who can study can do it.

On the other hand, provoking a little, the concept of male gaze could also be applied to Euclid’s fifth postulate. It took us centuries to interpret it differently: at a certain point there had to be someone thinking differently, letting themselves go to the different.

CV: From tomorrow I will say that I thought this thing up, I have already appropriated it. Why is it so …  I thought about it just a few days ago, when I was doing an interview on the radio. Pietro Greco, presenter of Radio3 Scienza , asked me: “But do you think there is a difference between the male brain and the female brain?”

And for the first time I understood and replied: “In my opinion, no. But we wait two thousand years to see if that part of the relational brain that has been denied to us in the last 1500 will be able to recover it in the next centuries. ” This too is a mathematical thing: to think that we are beings – male or female – who do not live alone, and that what is around us is a form of intelligence. Others are the part of our brain outside of us.

I like this about mathematics, you can very little deny that you know what you know. Thinking about it gives me a fresh sense of responsibility. If you know one thing, you cannot long deny that you know it. It changed you, like the feedback , it changed the system.


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