To symbolise is to produce signs that have meaning. A sign that has meaning is a signifier, whatever its nature. The universe of the symbolic is thus the set of all past and present signifiers and all possible signifieds. The symbolic function is what enables us to associate in a stable fashion one psychic state with another. That is how English speakers repeatedly associate the word ‘cow’ with the animal the word designates. The symbolic function is pre-inscribed – that is to say genetically inscribed – in our psyche and in our entire body. It is involved in all forms of thought and action humankind has invented until now and will be in all those we invent in the future.
All signifiers are signs that refer to something other than themselves, whereby this thing takes on meaning for everyone who produces or receives these signs. In 1910, Ferdinand de Saussure had envisaged a discipline, semiology, that would study and explain ‘the social life of signs’, and which would thus reach beyond linguistics; this was a limitless project that got no further than its ambitious beginnings. The difficulty lay in the fact that nearly everything can be or become the sign of something, which makes a general, global classification of signs nearly impossible. C. S. Peirce tackled the problem without ever resolving it.1 He analysed signs from three standpoints: the sign itself (sound, gesture, mental representation, spoken word, written form, etc.); the sign as it relates to its ‘object’; and the sign as it relates to its interpreter. We shall restrict ourselves to commenting on the three categories into which Peirce divided signs according to their relation with their object: signs as indications, or ‘indices’; signs as ‘icons’; and signs as ‘symbols’. But let us not forget that all these signs have meaning and that all are therefore ‘symbols’.
An index is a sign that has a physical connection with the object it indicates. It is the paw print of a bear in the snow, in which a tracker will read the animal’s size, its sex, the time elapsed since its passage and the direction in which it is moving. The mental image of the bear will immediately spring to mind, call up, through inner speech, the word for the animal and unconsciously reactivate all of the cultural representations and emotions connected with the bear in his society and in his personal experience. It is a trace of blood a killer leaves at the scene of his crime, which will allow the police to identify his DNA. It is the involuntary signs of weakness, good health or despair, produced by the body of those who are sick, desperate or joyful, but which the doctor, friend, et cetera observes and interprets.2 Smoke rising from a chimney means that someone has made a fire. A bear’s paw print, blood, and smoke are all signs in the present that refer to past events; they are all ‘indices’. But for a driver, the red light at an intersection means stop, do not go through; the green light means it is possible to go. A finger on the lips is a discreet signal to another to keep quiet. Here the sign concerns the future.
Peirce termed ‘icons’ all signs that possess a certain formal resemblance to the object they refer to, such as van Gogh’s self-portrait, or the giant portraits of Mao Zedong or Stalin. Another kind of icon is a road sign bearing the silhouette of a wild boar or a deer at the entrance to a forest, which tells drivers of the risk of suddenly coming upon one or several of these animals. Another type of icon is that which adorns Orthodox churches, representing the Virgin Mary and the Child Jesus, or the archangel Gabriel announcing to Mary that she would bear a son who would be the Messiah long awaited by the Jewish people. Yet another icon is the mental image of the bear I have in my head.
The relationship of similitude between a silhouette of a deer and the animal on a road sign is easy 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. 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.