A characteristic feature of mathematical texts is the extensive use of symbols, which differ both visually and syntactically from the 'verbal' part of the text; unless there is a very specific relationship between the mathematical knowledge of the author and the reader of the text, the text (in particular the symbols) will remain 'closed' to the reader. In semiotic language, the symbols will remain uninterpreted signifiers, and the whole text will, at best, convey the impression of containing an impenetrable discourse. This situation will occur for almost any combination of reader and text, as even experts of mathematics are able to decode only a very small fraction of the mathematical text available (say, in their departmental library).

In (Winsløw, 1997) I have tried to sketch a general framework for the analysis of mathematical discourse, based on a transformational theory interrelating the surface and deep structures of the text under investigation. This is in a sense a 'top-down' approach in that it presupposes a certain level of consensus regarding the interpretation of signs, at least between speakers and analyst. To the non-mathematician, such an assumption may sound like a somewhat risky or even naive basis for analysis; for a mathematician, I believe this is not so. Indeed, most of mathematical discourse would be impossible without this basic assumption of common reference, while, of course, many difficulties arise in such discourse from its occasional failure.

The purpose of this note is to describe a concrete example of a semiotic structure within advanced mathematics, and to extract from this description some conjectures about the particularity of semiosis in a mathematical context.

My concrete example is the paper, written in collaboration with Uffe Haagerup (Odense University, Denmark), entitled The Effros-Maréchal topology in the space of von Neumann algebras (Haagerup; Winsløw, 1997). I will try to unpack the basic semiotic landscape in which the discourse of this paper takes place; to describe, in 'backwards' direction, the genesis of its signs though a finite number of semiotic transformations.

The title of the paper is, from a syntactic point of view, of the form NounPhrase1 Preposition NounPhrase2, where by NounPhrase2 I understand the entire string 'the space of von Neumann algebras'. The determinate form of both noun phrases may convey the impression that both refer to mathematical entities which are supposed to be known to the reader; but this is not the case. Instead, the paper is about defining and studying a referent of NounPhrase1, in the mathematical context referred to by NounPhrase2 (which, indeed, is assumed to be familiar to the reader). Since I do not intend to provide a detailed discussion of the actual content of the paper, I shall concentrate on the 'de-semiosis' of NounPhrase2.

One way to approach the problem, which indeed seems necessary although not sufficient for a 'de-semiosis', is the chain of formal reductions (by definition in terms of 'simpler' concepts) explaining the setting in terms of more general mathematical knowledge. Here we reduce NounPhrase2 to structure of functions between natural and complex numbers, leaving undefined a few technical terms (marked *) which do not matter for our further analysis here. 'The space of von Neumann algebras' should be understood as 'the set of all von Neumann algebras'. But what is a von Neumann algebra?

A von Neumann algebra is a set of operators on Hilbert space, which satisfies *certain conditions. By Hilbert space I mean here the set of square summable functions from the set of natural numbers to the set of complex numbers. A function from natural to complex numbers is 'square summable' if the sum of the square of the moduli of the values of the functions is finite; the square root of this sum is called the norm of the function. A bounded operator on Hilbert space is a *map from Hilbert space into itself which preserves the *linear structure of Hilbert space.

In short, we may now re-express NounPhrase2 as: 'the set of all collections (with a certain property) of functions (with a certain property) from the collection of all functions from natural to complex numbers (with a certain property) into itself.' Besides the three instances of '(with a certain property)', only some basic set-theoretical notions, and complex and natural numbers, are left undefined. Notice here that we use the words 'set' and 'collections' as synonyms, but both are used to create variation!

The point of the above is that we see some pattern in the abstract reduction on the reference of NounPhrase2. If we use the following short hand notation:

COL(...) : All collections of ...

COL*(...) : All collections of ... with a certain property

F*(A,B) : functions from A to B, with a certain property

N: the set of natural numbers

C: the set of complex numbers

then it could be written

COL* ( F*( F*(N,C) ), F*(N,C)) ) ) ).

The whole title then says that we are studying a topology on this set. To define a topology on a set amounts to specifying, in some way, a certain set of subsets of the set (satisfying the axioms of a topology). So, the paper is really about a set of subsets of the set indicated by the symbol string above. So, the paper is about an element of

COL*(COL ( COL* ( F*( F*(N,C) ), F*(N,C)) ) ) ) )).

At the deepest level of this expression, we find the symbols N and C, being themselves signifiers for 'collection' type entities of numbers.

None of this symbolism appears in the paper, of course; while the expression above indicated (parts of) the semiosis underlying it, it is clearly inconvenient to use repeatedly (as in communication). Thus, instead of F*(N,C) the symbol H is used, while the whole thing is referred to verbally (by NounPhrase1). In general, signifiers are chosen as follows:

• lower case Greek letters: elements of H (Hilbert space)
• lower case Roman letters: operators on H
• upper case Roman letters: von Neumann algebras
• script letters: collections of von Neumann algebras (i.e., subsets of vn(H): set of all von Neumann algebras)

These roles of notational inventory are, as far as the first three are concerned, standard in the area. The list reflects the ascending 'magnitude' of the mathematical signifieds by the increasing symbolic 'power' (for instance, upper case letters dominate lower case ones). Furthermore, each of the symbols have an 'upward' and a 'downward' connector of signification. For instance, a lower case Greek letter signifies on the one hand a, possibly complicated, complex function (in this case on the set of natural numbers) in which arguments (in this case, numbers) could be fed to produce a result, and which may interact with other functions of the same kind ; on the other hand, it represents a potential 'in-put' for operators on the Hilbert space, something for them to 'act' on. In this way, the mathematical sign interlocks in the two vertical directions as well as in a horizontal direction (with signs at the same level).

There are two points regarding mathematical semiosis which I would like to make from the preceding example

1. Mathematical signs have, as signifieds, transformations of signifiers of other (groups of) mathematical signs.
2. The typology of these transformations may quite limited, and may be roughly divided in 'fusion' type (such as 'taking collections of', 'taking functions between...') and separation types (such as 'take subset according to some restriction').

Ultimately, a mathematical sign is therefore not to be understood merely as a simple pair of signifier and signified. It is to be understood as an ascending sequence of 'elementary signs' as illustrated below:

The question, how long is that sequence --- could it be infinite --- may challenge the philosophical mind. However, all that is visible in mathematical discourse are 'tails' of such sequences --- as illustrated in the example above.

Notes.

1. An alternative title, which could have suggested this more clearly, would be A topology in the space of von Neumann algebras, but as the new concept described in the paper is deeply inspired by works of the two mathematicians Odile Maréchal and Edward Effros, we wanted to indicate this in the title, and it would be hard to combine an indeterminate form of NounPhrase1 with the mentioned proper names without making the title even more complicated and hard to read. In other words, the title is formed by the desire to express a combination of several meanings (represented by Effros, Maréchal, topology, the space of von Neumann algebras) in a reasonably short title.
2. And of course: what has it to do with John von Neumann? The interested reader could consult the preface of (Takesaki, 1979) for an appetizer.
3. For instance, via the 'inner product', taking a pair of functions to the sum of the twisted product of their values, where a twisted product of two complex numbers is the product of the first number and the complex conjugate of the second. The inner product structure is essential for the geometry of Hilbert space, as it defines what it means for two elements of it to be orthogonal.

References.

Haagerup, U.; Winsløw, C. (1997) The Effros-Maréchal topology in the space of von Neumann algebras. To appear in: American Journal of Mathematics.
Hodge, R. & Kress, G. (1988) Social semiotics. Cornell Univ. Press, New York.
Rotman, B. (1988) Toward a semiotics of mathematics. Semiotica 72, 1-35.
Takesaki, M. (1979) The theory of operator algebras I. Springer Verlag.
Winsløw, C. (1997) On the role of transformations in mathematical discourse. In preparation.

Carl Winsløw