In Seminar III: The Psychoses (in [L2], I82), Lacan positions the speech of the analyst "somewhere in A", the lower right corner of the schema L, (cf. "Le séminaire de la lettre volée", [L1], p.66).

Following [L3] and [L4], in the present work we apply Lacan's four discourses to the communication of mathematics, locating them into a progressive construction of the schema L.

The first section below corresponds to the segment connecting the speech of the master, as S, to the speech of the university, as other. The second section applies the imaginary relation of the schema L, (a'-a) to the communication between the hysteric discourse, as ego, and the university.

The latter two sections express the main idea of the present paper: the enjoyment (or, more commonly, the lack of enjoyment) of mathematics is determined by the interaction of the discourse of the master with the hysteric's speech and the university (what we call "the authoritarian drift"), while the jouissance, as in imprecise applications of aesthetics to mathematics, is a consequence of attempts to identify mathematics with the speech of the analyst ("le mort qui parle"). With respect to the schema L, the sections "the authoritarian drift" and "le mort qui parle", both in reltion to the university and the hysteric discourse, correspond, respectively, to the upper-left triangle S-a'-a and the lower-right triangle a'-a-A.

To emphasize the position of the subject as the agent we often denote the speech of the hysteric as the creative speech.

THE EDUCATION OF GOVERNMENT

The signifying chain is the agent of the speech of the university. In mathematics, it is usually a set of numbers, polynomials, functions, operators: analytic, geometric or algebraic structures.

The other, referent of the university speech, animates the work of the speaker/listener. The effort of students and teachers, mathematicians or laypersons, their research, relies on a desire of a yet-to-be-determined knowledge.

In opposition to social sciences, in mathematics a signifier is not recognized as a subject for another signifier, and the signifying chain is based on a formal understanding of the fundamental signifier. The fundamental signifier, for example, is viewed as an overarching concept of the signifying chain as a whole: the concept of a limit of infinitely close points on a line is the fundamental signifier of the signifying chain of real numbers and mathematical analysis; the algorithm of division is the fundamental signifier of rational functions; the concept of a number as a solution of a polynomial equation (with integral coefficients) in the case of' algebraic numbers; the concept of an element of a space as an operator on the space for a vector space of operators; etc.

The goal of the university, education, is shaped as the production of a subject who speaks reads and writes about numbers, polynomials, operators, etc. The logical bar of mathematics determines how the symbolic universe of this speaking subject is quantified, analyzed and synthesized.

Turning the clock back to the speech of the master, traditionalists view the communication of mathematics ideas in terms of failure and success. Tests, homework, grades and awards organize the unfolding of the signifying chain. The fundamental signifier is understood as the order of the teacher, the hard work of a student or the super-ego of a mathematician.

Enjoyment of mathematics is thus suspended between a powerless subject of research and education, and an equally powerless subject determined by the tasks of government and self-government. This is not the jouissance of mathematics that we are looking for.

IMAGINARY PARADOXES

The university speech faces its creative counterpart, the discourse of hysteric, in an imaginary juxtaposition. In a didactical context, the university confuses the demand of existence in the hysteric's speech with the need for further education. In response to such demand, mathematicians, on the one hand, stretch the signifying chain into an increasing sequence of steps, clarifications, further details and applications, etc. and, on the other hand, dismiss large portions of the hysteric's speech as insignificant gibberish.

Conversely, the hysteric confuses the university with the cause of his/her discontent. For example, what the university introduces in the mathematics discourse (in textbooks, lectures, software, etc.) with the explicit purpose of overcoming a symbolic barrier, may be viewed by the hysteric (implicitly) as an imaginary attempt to thwart the development of a creative discourse. Paradoxically, the proliferation of images in the university discourse presupposes the insignificance of mathematics symbolic discourse. For the hysteric, at least on a symbolic level, mathematics is to be enjoyed as little as possible.

A similar paradox occurs in scholarly descriptions of creativity in mathematics. For example, with very few exceptions (see, for example, [W]), the use of imaginary words such as "beauty" and "elegance" in mathematics is not supported by a detailed presentation of the relationship between aesthetics and mathematics. As in the hysteric's speech, imprecise imaginary creativity is opposed to the routine work of reason. Again, the explicit goal to lighten up the communication of mathematics with pictures of "beauty" and "elegance" is thwarted by the implicit assumption of the obscurity of the symbolic discourse.

In Lacan's formalization of the four discourses, this is the transposition of impossibility and impotence that turns the speech of the university directly into the speech of the hysteric. What the university views as impossible, the relationship of the signifying chain with the object a, the hysteric sees as powerless, and, conversely, what the hysteric views as impossible, the relationship of the barred subject with the fundamental signifier, the university sees as powerless.

THE AUTHORITARIAN DRIFT

The prototypical position makes the master's speech the obvious intermediary between the hysteric and the university. Through the master, the discourse of the university relapses into the traditionalists' understanding of the demand of the subject as a need of strong government. And consequently, through the master, the hysteric's speech understands the signifying chain as a set of orders, drills and rote learning.

The speech of the master provides enjoyment through the organization of creativity and education. For a student, this enjoyment can be passing a test, graduating, receiving awards, etc. In each case the speech of the master is the measure of the (lack of) empowerment of the subject.

For an educator, a similar enjoyment is difficult: teaching requires much more flexibility than learning. Moreover, in mathematics, the higher the level of education, the more esoteric the subject, the harder it is to test, rank and reward teachers.

For a mathematician, the enjoyment provided by the speech of the master is impossible. The multiplicity of structures turns the communication of mathematics ideas into a sort of inverse of the master's speech. The understanding of the fundamental signifier as an order cannot be much more than a joke as in the anecdote of proving theorems by saying "it is so because I say so". Furthermore, the Subject beneath the order is also little more than an anecdotal joke as in the case of a (famous) mathematician considering the visit of a (Roman) soldier an opportunity to work on lines and circles.

On the imaginary level of the previous section, the speech of the master increases the mistrust of mathematics in the speech of the hysteric and, vice versa, in mathematics, deepens the lack of understanding of the creative discourse.

LE MORT QUI PARLE

On the symbolic level, the discourse of the analyst sets a hole in the position of the agent: a vacuum, a space underneath the symbols, a gap before two measurements, a pause between spoken sounds In this discourse the other is the subject.

In mathematics education, the speech of the analyst appears in controversial teaching methods that require a teacher "not to assist students in creating [mathematical] proofs, but to respond to whatever the student says" (the Moore Method in [K], pp. 32-33). Faced by a mathematical proof yet-to-be-discovered, for both the student's speech and the teacher's response, the subject is in the position of the other.

On a scholarly level, mathematics provides symbolic representations of the hole supporting the object a: the empty set, complement of the whole, the number zero, generator of the entire symbolic universe, the vacuum space. What the speech of the hysteric views as the (imaginary) background of any signification, mathematicians, through the speech of the analyst, view as the foundation of any scientific discourse.

Conversely, the opposition to the discourse of mathematics takes the form of a horror vacui, fear of a blackboard with nothing written on it, fear of a blank sheet of paper (and of all the mathematics that can be written on it). Mathematics is correctly identified with a tautological speech of the vacuum, the Other (underneath? underlying?) in the scientific discourse.