Extract from Chapter 6 of Social Constructivism as a Philosophy of Mathematics, Albany, New York: SUNY Press, 1997. (Footnotes and references in original omitted here.)
One 'mystery' or central problem that any philosophy of mathematics must account for is that of the existence and nature of the objects of mathematics, and indeed their objectivity itself. This might be rephrased as the question: how to obtain the benefits of Platonism without adopting its questionable ontology?
The social constructivist approach to this problem is that the objects of mathematics are among the social constructs of mathematical discourse. As Rotman (1993: 140) puts it: "one can say that mathematical objects are not so much 'discovered out there' as 'created in here', where 'here' means the cultural circulation, exchange, and interpretation of signs within an historically created and socially constrained discourse."
According to the social constructivist view the discourse of mathematics creates a cultural domain within which the objects of mathematics are constituted by mathematical signs in use. Mathematical signifiers and signifieds are mutually interacting and constituting, so the discourse of mathematics which seems to name objects outside of itself is in fact the agent of their creation, maintenance and elaboration, through its use. Furthermore, because of the historically constituted nature of the objects of mathematics and the discourses in which they subsist, mathematicians join and learn to participate in a pre-existing and already populated realms of discourse. By doing this they are appropriating and recreating their own corner of this world, as well as ultimately contributing to its maintenance. Elsewhere I describe some of the ways that language and linguistic practices contribute to this phenomenon psychologically (Ernest 1991). Here I discuss the relationship between the objects of mathematics and texts.
In Chapter 2 I recounted Rotman's (1988) semiotic account of the success of Platonism as philosophy for the working mathematician. He argues that it views mathematics as a full sign system, comprising both signifiers and signifieds, and thus accommodates human signifying activity concerning shared, objective meanings, which is essential for mathematical practice. In this respect Platonism was judged superior to formalism and logicism which prioritize mathematical signifiers, or intuitionism, which prioritizes the signified. Thus Platonism, according to Rotman, treats mathematical knowledge as inseparably tied up with mathematical texts (and their objective meanings).
Popper (1979) links objectivity and abstract objects with texts in one of his defences of World 3. He argues that even if all machines and tools disappeared from the world (World 1) and all knowledge of science and abstract concepts from individuals' minds (World 2), then persons (or even intelligent creatures from space) would be able to reconstruct our scientific knowledge from texts. Thus the contents of texts is objective knowledge, and according to Popper, there is a unique objective meaning associated with scientific texts, and independent of the physical world or subjective knowledge. Popper makes it clear that he thinks discussions of meaning are futile - preferring to discuss truth instead - but he does describe World 3 as including the contents of journals, books and libraries. For him such texts have unique contents and it is these that populate World 3.
There is one particular strength of Popper's account (and Plato's) which needs to be acknowledged: its recognition of the strong autonomy of the objects of mathematics and of the population of World 3 in general. The social constructivist account has to explain how something so apparently flimsy as a shared fiction created by texts (in context) can become so solid, robust and autonomous as World 3 objects. There are also number of powerful objections to Popper's view to be put, but before embarking on a further critique I wish to elaborate the consequences of his view further.
The traditional Platonistic or Popperian view is that objective knowledge, information, propositions, meanings and the objects of mathematics are independent entities existing in a some superhuman realm (World 3 or the realm of Plato's Forms). Furthermore these abstract objects may be referred to or signified in multiple ways. For example, '2', '1+1', '99-97', '4/2', '200%', and '1.999...' all signify the number two. Similarly, the different sentences 'Two is the only even prime number.' and 'The unique even prime number is two.' are normally understood to express the same proposition, i.e., to have identical senses and references. These are almost trivial illustrations of a virtually ubiquitous phenomenon of ambiguity. The way that such ambiguities are accommodated is to distinguish signifier from signified, and then to attribute a many-one relation between the signifying expressions (terms or sentences) and the signified 'entity', which is a mathematical object or proposition, respectively.
Another traditional problem lies in distinguishing between '2', '2', '2' and '2', for example. Each of these is a numeral for two, but drawn differently, or as here, presented in a different typeface. This problem is dealt with by distinguishing signifier tokens - the physical inscription or utterance - and signifier types - the intended symbol (or the class of tokens standing for it). As before we have a many-one relation: many actual and possible inscribed tokens standing for or instantiating the intended signifier.
In the two cases described, the signifier and signified is each regarded as an object. Furthermore, it is an abstract object, except in the case where the signifying relation is one of direct reference, and the signified referred to is a physical object, action or event. In Popperian terms, although signifier tokens are part of World 1 (the physical world), signifier types are in World 3 (the domain of objective knowledge). Likewise, except when signifieds are physical objects, actions and events (World 1) or mental states (World 2), they are abstract objects (World 3). Thus, since in all but trivial mathematics signifier types and signifieds are abstractions, they are both inhabitants of World 3 and part of objective knowledge. An outcome of this view is that World 3 objects are prioritized - certainly in mathematics - and that Platonism or mathematical realism is virtually inescapable.
Discussing mathematical signifiers or texts, on the one hand, and the mathematical objects, contents or knowledge signified by them, on the other hand, raises the issue of the synonymy (and non-synonymy) of texts or signifiers, and the identity and diversity of the abstractions signified, respectively. A Platonistic account of such relations prioritizes the realm of the abstract. Consequently, mathematical signifiers are synonymous if they refer to the same mathematical object, i.e., have the same signifieds. Mathematical inscriptions can be regarded as identical if they represent the same signifier type. There is nothing wrong with these definitions except that the establishment of identity, synonymy or equivalence is based on recourse to abstract and intangible World 3 objects. An unsolved problem for metaphysics (insoluble in my view) is how the identity and diversity of abstract objects might be ascertained without recourse to signifiers or discourse. For those, such as myself, Bloor (1984), Rotman (1993) and the many others wishing to question or unwilling to accept the ontological assumptions of Platonism, and the epistemological problems it brings, this is deeply problematic.
Platonism and Popper's World 3 are perspectives that arise out of ontological considerations, but they have profound consequences for epistemology. Following Frege's introduction of predicate logic and its semantics, the reference of a mathematical term is its value, which for a numerical term is its numerical value, and the reference of a sentence is its truth value. But to know the truth value of a sentence is to have information about its epistemological status. For example, to know that the sentence 'x3-3x2+3x-1 = 0 ® x=1' is true, is to know that 1 is the solution to the cubic equation. Likewise, to know the numerical value of a complicated term or functional expression is to know the outcome of a calculation. Thus knowing what some mathematical expressions signify is to have substantial mathematical knowledge. Furthermore the implicit realism in prioritizing the abstract signifieds of mathematics suggests that answers and truth-states pre-exist their human discovery. According to this perspective, the values exist, and it is only human wit that lags behind in discerning what these values of terms and sentences are. Thus in this account the identity of terms arises from their sharing the same value, and the equivalence of sentences arises from their sharing the same truth values.
Social constructivism rejects the priority and prior existence of abstractions and thus overturns these notions of identity and equivalence. If embodied physical existence is taken as the bedrock of ontology, it is tokens which are the empirically real signifiers. Signifier types are cultural artifacts - the sign intended, and to a greater or lesser extent understood - when signifier tokens are perceived. But this notion of the intended signifier or sign underlying each inscription or utterance is something that arises from human agreement in language games and forms of life. That is, from a social constructivist perspective, it is a secondary and derived phenomenon (albeit more important). Likewise, signifiers have ontological priority over the signified - especially in mathematics, for the signifiers can be inscribed and produced, or at least instantiated, whereas the signified can only be indicated indirectly, always mediated through signifiers. The questions, 'When do two signifier tokens represent the same signifier type?', and 'When do two signifiers indicate the same signified?' must be answered differently from this perspective. The social constructivist answer is that the accepted practices of mathematical language games are what determine the answers. The identity and equivalence of linguistic forms and expressions that are admitted vary according to time, community and context, and are not given once for all. Indeed the history of mathematics can be seen partly in terms of the growth and elaboration of relations of equivalence of linguistic expressions, as a brief excursion into this history illustrates.
In 5000 years of written mathematics, a number of areas of mathematics have grown immensely. There has been, first of all, the great profusion of domains of mathematical knowledge, which can be described as mathematical theories, language games and contexts. According to Høyrup (1994), mathematics first emerged as a discipline through the unification of the three protomathematical practices of primitive accounting, practical geometry and measurement. This took place late in the 4th millennium BCE in Mesopotamia through the common application of numeration and arithmetic to these applications in scribal training. By the late twentieth century the number of distinct modern mathematical subspecialisms was estimated at 3400 (Davis and Hersh 1980), indicating the explosive growth of the discipline.
Second, within each of these specialisms, and shared across many of them, is the large and still-growing range of mathematical symbols, diagrams, inscriptions and notations. Beginning with simple symbols serving as numerals, mathematical notation has grown over history into a very elaborate set of special and dedicated symbols, icons and figures. Indeed, so important is this development in supporting mathematical calculation, reasoning and conceptualisation, that the history of mathematics might be viewed as the history of the development of mathematical symbolism.
Third, an intrinsic part of each mathematical language game is a set of rule-based symbolic transformations which are regarded as preserving some features of mathematical expressions or texts as invariant. These include identity transformations of terms and equivalence transformations of sentences and formulas. Many of these transformations are built into the uses of symbols irrespective of context, and are thus shared across many mathematical language games. Thus identity (used in expressing the identical transformations of terms) is an equivalence relation with the properties of reflexivity, symmetry, transitivity and substitutivity, which is probably used in all mathematical language games.
Platonism and mathematical realism suggest a reading of the history of mathematics in which increasingly refined sets of mathematical signifiers have evolved to describe the universe of mathematics. The social constructivist account reverses this prioritization and argues that the development of the increasingly elaborate systems of mathematical symbolism have helped bring into being and scaffold the imaginary universes of mathematics ideas.
Structural invariance is widely established as a central feature in both the content and the development of modern mathematics (Weyl 1947). Thus continuous functions, homeomorphisms, homomorphisms, isomorphisms, isotopies, and other structure preserving maps, functions and functors; their properties and the relations between their domains and codomains, are pivotal in the mathematics of the past 200 years. Likewise, in the elaboration, extension or abstraction of theories in the historical development of mathematics, certain designated central relations and structures are preserved faithfully, and the extent of this preservation is often the deciding factor in community acceptance (Corfield 1995).
Above I discussed the following problems of identity (and the traditional solutions). On what basis are different signifiers of mathematics regarded as identical? Under what conditions are distinct signifiers regarded as having the same signified. The social constructivist answer is that the conditions for identity depend on traditions and rules for the identification of signifiers located in language games and situated in mathematical forms of life. In particular, in mathematics there are usually explicitly permitted sequential transformations of symbols which convert a signifier into an equivalent one. Thus terms are converted into equal terms, equations are converted into equivalent equations, and more generally, sentences and formulas are transformed into equivalent sentences and formulas. This raises the question: What is the basis for the permitted transformations? The answer is normally that they are transformations that preserve the signifieds, or at least some central feature or property of signifieds such as truth value. Thus permitted transformations of numerical expressions and terms are those that keep the numerical values of expressions and terms invariant. Permitted transformations of sentences and formulas are those which preserve truth values. Typically, such transformations are introduced and defined constructively, with a certain number of basic transformations admitted (as demonstrable preservers of value signified), and compound transformations defined inductively as finite combinations of the basic ones. However, in the limit, classical mathematics is not inhibited from introducing infinite sequences of operations provided there is a guarantee that some underlying invariant is preserved. Thus equivalence transformations of mathematical expressions are admitted stepwise because it can be shown that they preserve a central feature of the fictional objects which are accepted as the underlying signifieds.
One area of especial importance for mathematics which exploits these issues is that of (explicit) definition. This is a stipulation of identity (of terms) or of equivalence (of sentences and formulas) in which the definiendum is substitutable for the definiens salve veritate (i.e., truth value preserving). In fact, because a definition is a stipulated synonymy, meaning as well as truth are preserved in such substitutions.
There is a series of developments in modern philosophy of direct bearing on the issues of mathematical texts and the objects they signify. The position of Quine (1970) on the meanings of texts is well known. He argues, first of all, against the idea that the meaning or sense of a sentence is a proposition. His argument is that propositions cannot be individuated satisfactorily, and therefore it makes no sense to claim they exist. His second and earlier claim is the more general argument that synonymy is indefinable (Quine 1951). In justifying this he first shows that the boundaries of the concept (relation) are ill-defined and cannot be rendered precise. He also shows that substituting synonyms for each other need not in all cases preserve truth values, which is a sine qua non of synonymity. Third, he argues for meaning-holism, i.e., the view that the meaning of a linguistic expression cannot be divorced from its linguistic context or the background theory.
Although Quine's position is quite distinct from that of Wittgenstein, his holism supports the latter's view that the meanings of textual objects cannot in general be individuated, as Platonism seems to require. Wittgenstein's holist conception of meaning as use in specific contexts is a central underpinning conception of social constructivism.
It is worth mentioning that there is a whole modern tradition stemming from modern continental philosophy including post-modernism (Derrida, Lyotard, Norris), post-structuralism (Foucault, Lacan, Dreyfus) and Hermeneutics (Gadamer, Ricoeur, Palmer) which has deeply influenced modern literary criticism (Eagleton, Fish) concerning the thesis that texts do not have unique signifieds. However much the meaning of an expression appears transparent, literal or unique, in adopting a favored or apparently correct interpretation "a thousand possibilities will always remain open" (Derrida 1977: 201).
In writing, the text is set free from the writer. It is released to the public who find meaning in it as they read it. These readings are the product of circumstance. The same holds true even for philosophy. There can be no way of fixing readings... (Derrida in Anderson et al. 1986: 124)
Derrida of course takes up an extreme position on the indeterminacy of textual meaning, namely that meaning is infinitely open. However a broad consensus exists within this tradition and beyond, including the rhetoric of the sciences and the social studies of science movements, that it is a mistake to assume the existence of a unique meaning or 'correct' interpretation of a text.
Support for this thesis can also be found within mathematics from the generalized Lowenheim-Skolem theorem (Bell and Machover 1977). For this theorem states that any set of sentences expressed in a first-order language which has an infinite model, also has models of every infinite cardinality. This means that first-order axiomatizations of Peano arithmetic admit uncountable models of the natural numbers, for every infinite cardinality. Another consequence is that first-order axiomatizations of the field of real numbers and of set theory admit denumerable models (although they would appear uncountable from within the theory itself, Schoenfeld 1967). These are well known and explicable but nevertheless counterintuitive results indicating that even the most precise theories of mathematics cannot uniquely determine their meanings or domains of interpretation (Tiles 1991).
The conclusion to be drawn is that the signifiers of mathematics do not correspond to unique signifieds. The relation is one-many, not a mapping, let alone a one-to-one correspondence. Mathematical language is not therefore a map, model or 'mirror' of a Platonic or World 3 reality (Rorty 1979). The case I want to argue is that signifieds are individuated instead by the admission of a class of textual transformations accepted as preserving the signified. The signifieds themselves of mathematics are unreachable, except through other signifiers. Ultimately, the accepted notion that there is something autonomous and real behind the signifiers is the result of the reification of abstract objects which are part of mathematical culture.
In Ernest (1991) I argued that the objects of mathematics are reifications in three senses: philosophical, sociological and psychological. I shall discuss the thesis that the objects of mathematics are psychological reifications in the next chapter, when I treat subjective knowledge in mathematics.
Sociologically, there is a tradition epitomized by Marx, in which concepts are understood as reifications which become cultural objects and things in themselves. "..the productions of the human brain appear as independent beings endowed with life, and entering into a relation both with one another and with the human race." (Marx 1867: 72) He argues that the form of products becomes reified and fetishized into an abstract thing: money, value or commodity (Lefebvre 1972). This argument has been extended to show how mathematical objects and knowledge are abstract reifications of more concrete conceptions and operations, and the transitions to the more abstract and autonomous objects of mathematics can be identified historically (Hadden 1994, Restivo 1985, 1992).
In philosophy there are a number of traditions that are relevant. First of all, there is a tradition going back to ancient philosophy concerning problem of universals. Aristotle's view is that the objects of mathematics do not exist apart from their instances. Similarly, the nominalism of the Schoolmen, especially William of Ockham, suggests that concrete individuals are what is first known, and the abstract concept corresponding to a class of individuals is a name whose meaning is identical with its extension, i.e., the individuals it represents. Universals or abstract concepts are derived from the individuals they represent and may thus be seen as reifications. It is impossible to do justice to the problem of universals in the history of philosophy in a paragraph. However, these indications suffice to show that there is a long tradition of rejecting realism in mathematics and which instead views abstract concepts as based on the particulars to which they apply.
Another tradition is that of intuitionism, which regards the objects and sentences of mathematics as representing constructions, i.e., the construction of an individual or the claim that a proof of a sentence can be constructed, coupled with guidance on how they should be implemented (Troelstra and van Dalen 1988). An alternative reading is that objects and sentences represent promises by the utterer that such constructions have been or will be made (Heyting 1956). Thus according to this perspective, mathematical objects are constructions. Going beyond this position Machover (1983) and Davis (1974) argue that the objects of mathematics are reifications: reified constructions, as the former puts it. This explanation is intended not only to account for the genesis of mathematical concepts, but also to account for the nature of mathematical objects, and the truth conditions associated with their properties, i.e., as an ontological and epistemological account.
Rotman's (1993) semiotic theory of mathematics also interprets mathematical inscriptions as recipes, instructions, or claims about the outcomes of procedures. This is based on Peirce's ideas and those of modern semiotics as well as arising from a linguistic analysis of mathematical texts.
Moving from a realist to a social constructivist account of the objects of mathematics brings with it certain problems. For all their weaknesses, Platonism and realism in mathematics offer an account of an autonomous realm of mathematical objects. Objects found there (projected there by the imagination, according to social constructivism) have apparent - but nonetheless convincing - stability, 'solidity', autonomy.
The claim that I am making in this section is that the objects of mathematics, as well as the theorems and other expressions of mathematical knowledge, are cultural constructions. In terms of their genesis these represent the reification of more primitive operations, but they are no less convincing for that. Like many other cultural artifacts such as moral rules and taboos, money and the value attached to objects and work, Shakespeare's plays, Beethoven's symphonies, the gods and saints of organized religions, Bugatti cars, they appear to have an essential 'inner nature' which is inevitable, necessary and transcends the contingent. Furthermore, each in this more or less arbitrary list of cultural objects is woven into multiple layers of networks of social usage, expectation and necessity which provides stability and 'solidity' buttressing their autonomy. Such ramifications and bonds are even stronger in the case of the objects of mathematics, being those of logical necessity, and where shifts in the material basis of the objects can be perceived as irrelevant.
Overall, the claim is that the ontology of mathematics is given by the discursive realm of mathematics, which is populated by cultural objects, which have real existence in that domain, just as money does in the domain of human economic affairs. In an influential remark (Wright 1980, Shanker 1987), Kreisel (1965) has claimed that the key issue is not so much the existence of mathematical objects so much as the objectivity of mathematical knowledge that is at stake. This objectivity is clear (under the reinterpretation of objectivity as cultural and social à la Bloor 1983 and Harding 1986). My claim is that mathematical discourse as a living cultural entity, creates the ontology of mathematics.
Within the philosophy of mathematics and philosophical logic there is also a widespread if technically formulated view that discourses entail, and perhaps even create, the universes of objects they refer to. In particular, a mathematical theory or discourse brings with it a commitment to the objective existence of a set of entities. Quine (1948) identifies bound individual variables within an informal or formal mathematical theory as giving the clearest indications of its ontological commitments.
Model theory is the branch of mathematics (or mathematical logic) concerning the interpretation of formal mathematical theories within the domain of mathematics itself. It is glossed in terms of mathematical realism and the correspondence theory of truth by Tarski (1936) and others. However, from a social constructivist perspective, the provision of formal interpretations of mathematics within itself serves to illustrate the mutual constitution of signifier and signified in mathematics, and how mathematical discourse, if not actually self-referential, constitutes a closed system. This account finds support from some technical developments within mathematics itself. For example, Henkin (1949, 1950) derived completeness proofs for the first-order predicate calculus and the theory of types in which he constructs models of the language in terms of the objects of the language themselves. Thus he makes the objects of the language serve a dual role, as both linguistic entities (signifiers), and as the objects (self-) signified. In this case, the signifiers not only create the signified, but they are one and the same, but viewed from different perspectives.