to understand. On the other hand, for a non-Christian, the relationship is much less easy to perceive in the case of religious icons, whether Greek, Russian or Bulgarian. The portraits of Mary and Jesus are purely imaginary. They are idealised representations of figures who no doubt existed, but whose features no one has passed on to us. And the icon painter escalated the imaginary when he painted the archangel Gabriel (who we may believe to have never existed) in human likeness with wings on his back (a sign that he is ‘truly’ an angel). It is therefore hard for the non-Christian to understand the theological meaning of this picture, even if, independently, he can appreciate its beauty and admire it.
Let us move on to the signs Peirce classified as ‘symbols’, those for which there is no relationship of similitude between the sign and the thing signified, but an arbitrary relation determined by convention. Such are the words of a spoken language, and the same words if they are written, whatever the system used: alphabetic, ideographic, pictographic, et cetera. To these must be added the gestural systems that make up the different languages invented to communicate with the deaf and so that the deaf may communicate with each other. And then there is the drummed language of the Yangere people of Zaire, whose drumbeats send messages deep into the forest, for instance to tell a hunter to hurry back because his wife has just given birth. Nor should we forget Morse code and the various languages invented by sailors or soldiers to communicate their messages.
I could go on and on with examples of symbols. The blue, white and red French flag, the red flag of the communist parties bearing the hammer and sickle, symbols of the (hoped for) alliance between peasants and workers, or the black flag of the anarchists. These symbols refer to the existence of social groups that have chosen them as emblems of their identity, their values; though everyone knows that communists and anarchists hold opposite views on the role of the state and the nature of the society they would like to establish.
To conclude, I have chosen a strong symbol – ‘Je suis Charlie’ – which emerged in France following the assassination on 7 January 2015 by two terrorists, claiming to be from the Yemenite branch of al-Qaida, of twelve artists and editors from the satirical magazine Charlie Hebdo, together with two policemen. The symbol appeared in the streets, brandished by hundreds of thousands of people gathered to show their indignation, their anger and their solidarity with the victims, and above all their will and desire to defend a ‘sacred’ principle of the French Republic: freedom of speech.
They came, even if some demonstrators did not share the magazine’s irreverent tone, qualified by religious believers as blasphemous. In brandishing ‘Je suis Charlie’, they were identifying with the victims and defending a right that entailed the possibility for a journal like Charlie Hebdo to exist but which went far beyond the issue of its existence. We have the birth certificate of this symbol. It was invented on the spot by Joachim Roncin, the artistic director of a small free magazine, Stylist, who, shocked by the massacre, had written on his computer screen: ‘Je suis Charlie’.3 For a moment, Roncin worried that these words might offend the friends of Charlie. But colleagues reassured him, and a journalist tweeted the three words, which were immediately taken up and reproduced by tens of thousands of people. The success was tremendous. The symbol leaped borders and found itself carried in the streets or posted on the walls of Berlin, London, New York, Madrid and many other capitals by demonstrators who chose this means to attest that they shared the same values as the French who had taken to the streets in Paris. But following the assassination – after the Charlie Hebdo massacre – of four Jewish persons by Amedy Coulibaly, who claimed affiliation with Daesh, France also saw a counter-symbol, ‘Je suis Coulibaly’, diffused in social media networks by anonymous followers who adopted it to show their approval of these crimes and their agreement with the justifications advanced by their authors.
Among the varieties of symbols, mathematical symbols call for a separate treatment;4 these are numbers, geometrical figures, algebraic formulae and so on. They are of a completely different nature from those of the symbols discussed above: flags, emblems, slogans, writing systems and so on. Among the mathematical symbols, we must distinguish those that present a likeness, such as the isosceles triangle drawn on the board and analysed for the students by their math teacher, and the ‘linguistic’ symbols, such as a and b in the formula (a + b)2 = a2 + b2 + 2ab, or the Greek letter Π (pi), symbol for the number that represents the constant relation between the circumference of a circle and its diameter. If the drawing on the board may represent something for those who have no mathematical knowledge, the signs √, Π or the formula y = f(x) mean nothing to them. To understand these symbols, the person would have to become a mathematician and perform the conceptual operations that give them meaning. Failing that, all these symbols will remain a mystery, dead signs.
Let us come back to the example of the drawing of an isosceles triangle. The drawing is a physical representation of a mathematical ‘object’ that belongs to the field of Euclidian geometry and is defined by its axioms. Yet the word ‘object’, as Maurice Caveing showed, is inappropriate, for it carries various ontological representations and reifications.5 Mathematical objects are ideal ‘beings’, idealities that exist only in mathematical theories. Isosceles triangles are not found in nature, much less ‘transfinite’ numbers; the mathematical ideality known as ‘triangle’, therefore, cannot come from an act of perception but is the result of an operation of construction governed by rules. The triangle drawn is therefore one of the forms – they are infinite in number – that corresponds to the essential properties of the triangle as a ‘theoretical ideality’. The triangle cannot be drawn. The triangle, as a mathematical ideality, is unique. Its graphic representations, from the standpoint of shape (degrees of the angles) as well as size (length of the sides), are infinite.
What does the theoretical activity of a mathematician consist of, then? It means solving problems and proving theorems.6 Its objects are relations and systems of relations. By applying various types of conceptual operations to these relations, the mathematician builds mathematical objects, in other words, idealities, which are clusters of relations that themselves open onto other relations. Mathematical thinking thus works by ‘successive mediations that form a chain by connecting relations with each other’.7 The resulting relations, operations and idealities are expressed in a technical language belonging to the field of mathematics, a formal language made up of symbols that have no meaning for nonmathematicians. The plus (+) and minus (-) signs are symbols for the operations of addition and subtraction; the radical (√) is the symbol for the extraction of a root, which can be square (2√) or cube (3√), et cetera. Mathematical idealities, therefore, exist purely in and through this operational activity that supposes the mediation of its own language, which is universal. Certain symbols in the language of mathematics are borrowed from spoken languages, such as the words ‘group’, ‘ring’, ‘body’, ‘root’, ‘matrix’, ‘lattice’, or verbs like ‘extract’ and ‘extend’. But the nonlinguistic words and symbols are basically unequivocal, and their meaning depends strictly on the operations they express. Nothing in these symbols leaves room for the mathematician’s subjectivity; there is nothing equivocal about them that might invite the possibility of wordplay or a hermeneutic.
Each time a mathematician performs a sequence of operations and reactivates their meanings, they are no longer, as the standardised subject of these operations, the empirical self of everyday life. For the only way they can operate is by placing themselves within a domain of idealities constituted as a domain of preestablished truths, and the only way they can carry out this task is by submitting to the content of proven theorems. This is true for all mathematicians, French, Russian or Chinese. Once obtained, the results of mathematicians’ work, which is to produce and demonstrate truths that each can in turn repeat and verify, become both transcultural and transtemporal. They are now detached from the time, the society and the mathematician who first produced the demonstration, whether it was Euclid, Descartes, Hilbert or Cantor. The world of mathematics is thus one where each subject finds him- or herself in a relation of transparent reciprocal exchange with all other mathematicians, repeating the same operations and obtaining the same results.
