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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. These form, for mathematicians, a field of idealities and norms that envelop them and which they expand by their discoveries, but which always transcends them and remains open to other systems of relations. Mathematicians’ acts make them, as producers of universal knowledge, universal subjects.

      Furthermore, the development of mathematics since the end of the nineteenth century has increasingly freed it from all reference to the empirical intuition that was originally connected with Euclidian geometry and with the physics that had been associated with it. But even in Euclidian geometry, there was, as Descartes pointed out, the distinction between the figure of a triangle and its concept. And non-Euclidian geometries have completely eliminated the intuition of the three-dimensional space in which we move.⁸ Geometry has thus moved towards increasing abstraction and formalism, without it ever being possible to call into doubt either the reality of its new objects or the truth of their demonstrations. Paradoxically – given its ideal character – mathematics is still the area of knowledge that attests to the capacity of the human mind to produce objective knowledge that transcends time and cultural worlds. Descartes knew this – even if he postulated that mathematical truth was, in the final analysis, grounded in God (the Christian God) when he wrote: ‘For if it happened that an individual, even when asleep, had some very distinct idea, as, for example, if a geometer should discover some new demonstration, the circumstance of his being asleep would not militate against its truth.’⁹

      To conclude this all-too-rapid analysis of the symbolic function, allow me to underscore five important features.

      1. Any sign always either stands for something else, or is a function of something else (Hjelmslev).¹⁰

      2. A sign always obeys a code and is a coded access to a referent. A sign therefore transmits information to those who know the code.

      3. A sign as signifier can refer to one or several signifieds at the same time.

      4. Signs and the necessary access codes can never contain more information or other types of information than that invested in them by their original inventers and by all those who in turn put them to new uses.

      5. The symbolic function exceeds the mind and the body. It pervades everything people do, everything people invest with meaning: churches, temples, statues, mountains, the sun, the moon, et cetera.

      All signs, whatever they may be – including mathematical symbols¹¹ – were imagined before being used.

CHAPTER

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