Ernest, Paul (1998). Social Constructivism as a Philosophy of Mathematics. Reviewed by James A. Telese

Ernest, Paul (1998). Social Constructivism as a Philosophy of Mathematics. Albany, NY: State University of New York Press. Pp. xiv + 315 ISBN 0-7914-3588-1     $59.50 Reviewed by James A. Telese University of Texas, Brownsville June 15, 1998 Paul Ernest makes an important contribution to the philosophy of mathematics by arguing that traditional epistemology has not ensured the infallibility of mathematical knowledge. His thesis is that the genesis of mathematical knowledge is a human endeavor, a process in which historical origins and social context play important roles. Throughout the book where appropriate, Ernest provides a point-counterpoint approach when presenting his ideas, as he anticipates and responds to possible objections. His argument is centered around two "maverick" philosophies established by Lakatos and Wittgenstein. Lakatos's view regarding mathematical proofs and Wittgenstein's "language games," are woven together into a coherent view of mathematics called social constructivism. Social constructivism, which includes intuitionism, is an extension of constructivism and other philosophies, such as fallibilism. Readers who are unfamiliar with certain aspects of the philosophy of mathematics, such as absolutism, formalism, and constructivism have no need to avoid the book because the author presents these various views clearly enough as he lays the foundation for his own brand of social constructivism. This review will focus on Ernest’s social constructivism and its sources in Lakatos and Wittgenstein. Ernest believes that mathematical knowledge is influenced by human activity and contends that mathematical knowledge is situated within and grows out of a community composed of individual mathematicians. The key word is "community." An emblematic quote would be the following: "all knowledge is rooted in basic human knowledge and is thus connected by a shared foundation ... human agreement is the ultimate arbiter of what counts as justified knowledge" (p.48). The implication is that since mathematical knowledge is a product of the social nature of the mathematical community, then the development of mathematical knowledge cannot occur without human activity. It is the role of the philosophy of mathematics "to reflect on, and give an account of, the nature of mathematics" (p. 51). He offers six criteria for judging the adequacy of a philosophy of mathematics: mathematical knowledge and the role of proof critical evaluation of theories objects of mathematics applications activities of mathematicians both past and present how mathematical knowledge is passed on from one generation to the next Ernest develops his view through a thorough explication of these criteria, which are used to support the idea that the historical origins and social context of mathematics should be admitted as proper concerns of the philosophy of mathematics. Criteria one and three represent the focus of traditional epistemology and ontology. Criterion two is included because it is concerned with the usual form of mathematical knowledge. And through criteria four and five, Ernest extends the traditional boundaries of the philosophy of mathematics and helps legitimize the idea that mathematics is related to other areas of human knowledge. Criteria six relates to mathematics education and is concerned with how mathematical knowledge passes from one generation to the next, how it is learned, and the dialectical relationship between individuals and existing knowledge. Ernest’s account thus includes external issues such as the history, genesis and practice of mathematics, along with internal issues such as the justification of mathematical knowledge itself. Ernest prepares the reader by offering a thorough critique of a kind of absolutism that has been the dominant view of mathematical knowledge and the traditional philosophy of mathematics. Absolutism makes two basic assumptions: 1) mathematics qua knowledge is in principle separable from other human activities, and 2) mathematical knowledge and logic are absolutely valid and infallible. These assumptions mean that, for the absolutist, the truths of mathematics and logic are absolute and infallible. However, Ernest’s compelling critique of absolutism proceeds by establishing that mathematics and logic are indeed fallible by their very social nature. In a deductive system such as mathematics, certain established rules and statements are assumed to be true while "logical rules of inference preserve truth" (p. 14). In the absolutist view, axioms are assumed and the rules of inference follow a formal syntax. For Ernest, though, in the mathematical community proofs result from a social discourse in which participants agree to accept or deny particular theories. The validity of a deductive proof is therefore based on the "transmission of truth." Ernest establishes that proofs are outcomes of socially agreed upon sets of rules and mathematical objects, such as definitions, axioms, and theorems. Proofs are made up of "deductive logic and definitions which are used with an assumed set of mathematical axioms or postulates . . . to infer mathematical knowledge" (p.3). Hence, proofs--the heart and soul of the creation of mathematical knowledge--are themselves socially derived. This claim premises Ernest’s social constructivist philosophy. Further support for this proposition is found in Lakatos and Wittgenstein. The fallibilism of mathematical knowledge was first addressed by Lakatos in 1978. The underlying notion is that knowledge, based on human opinion or judgment, has the possibility of losing its truth or necessity. Through the door of fallibilism, the author offers a naturalistic account of mathematics. Ernest's argument is further strengthened by incorporating constructivism with intuitionism. He realizes that although intuitionism is a well established constructivist philosophy of mathematics, its weakness lies in the fact that axioms and associated proofs derived from intuition cannot stand alone in establishing absolute truth based on subjective beliefs. Consequently, Lakatos incorporates general aspects of constructivism and fallibilism into a theory where the standards for rigorous proof in mathematics change and evolve over time. As Ernest suggests, a careful analysis of Wittgenstein is necessary for an understanding of social constructivism. For Wittgenstein, mathematics is one among many language games which possess certain features that must be followed in order to participate, such as rules, behavioral patterns and linguistic usage. Meaning is thereby established from social patterns. In order to participate in a language game, one must accept certain rules in order to apply terms in social discourse. The rules and form of the game may change but are learned through participation in the game. In the mathematical community, agreements regarding the rules of mathematical proof and establishing theories arise from a shared language. This suggests that mathematicians have established guidelines for the verification of proofs. Depending upon social conditions, the guidelines may change as a result of a new approach to mathematical proofs. For example, in the past, proof by computer would have been unacceptable, but the community has recognized the legitimacy of computer generated proofs; thus, what counts as a rigorous proof can change due to particular advances such as technology. The language game of mathematics includes informal mathematical practices as well as formal practices. As in any language, the syntactic rules are important for understanding to occur among the participants. The form of the rules and their acceptance evolve within context-related linguistic and social practices. Mathematical knowledge is created in the minds of individual mathematicians, participating in language games, each of whom idiosyncratically construct meaning. In this case, "mathematics is constructed by the mathematician and is not a preexisting realm that is discovered" (p.75). Aspects of mathematics that are considered to be inventions include mathematical propositions, theorems, mathematical concepts, and the forms mathematical expression may take. The social nature of mathematics is established by meanings derived from the context-related linguistic and social practices. Consequently, mathematical knowledge is founded on human persuasion and acceptance. In support of Wittgenstein, Ernest employs Lakatos's theory of Quasi-Empiricism, which claims that mathematical knowledge is fallible and corrigible. Mathematical knowledge, represented by proofs, is considered to be the outcome of a language game. Ernest elaborates Lakatos's view of the evolution of mathematical knowledge in terms of the Logic of Mathematical Discovery (LMD): a cyclic view of the development of proofs. There are three stages. The first stage is the called the "beginning" stage where primitive proofs develop. The second stage is called the "mid-cycle" where proofs and refutations occur. The third stage is the labeled "new beginning." During the third stage, improved conjectures are developed through criticism, analysis and strengthening of the proof and of the primitive conjecture, leading to the new conjecture. From this perspective, proofs grow from a process of conjecture- refutation-new conjecture. Proofs develop within the social community of mathematicians and are considered mathematical knowledge when they are fully established through a testing process. The social nature of mathematics is visible when a proof is presented to a body of mathematicians. The proof is placed under scrutiny and faced with either acceptance or rejection if flaws are present in the proof, a new and improved version is then presented. The cycle continues in a similar fashion until there is agreement. Mathematical knowledge is, thus, tentative, continually being tested, and not necessarily established by rigorous proof alone because the guiding assumptions are based upon human agreements that are capable of changing; conjectures, proofs and theories arise from a communal enterprise that includes both informal mathematics and the history of mathematics. Ernest believes that the LMD offers a unified theory of mathematical development with four functions. He sees the first function as a philosophical description accounting for the genesis and justification of mathematical knowledge and the detailed epistemological pattern of mathematics. The second function is historical and describes the actual development of mathematics. A third, a heuristic function provides guidelines for practicing mathematicians to follow. The fourth function is related to pedagogy, and describes a method for teaching mathematics that parallels its historical development. The role of informal mathematics is pivotal for both Lakatos and Ernest. Within the mathematical community, informal mathematical theories are first established and then explained through formal means. This process refines and redefines the mathematical concepts, conjectures, and proofs that function as a filter for mathematical knowledge- -a filter that may itself be reformulated in response to the criticism of the mathematical community. Since it remains possible that resultant proofs may or may not be successful, it seems that the potential falsification of formal mathematics ultimately depends upon the informal mathematical theories. The formation of mathematical knowledge thus relies heavily upon discourse and rhetoric. Ernest contends that the LMD reflects a dialectical logic of human conversation and interaction and is therefore a language game with alternating voices in dialogue. Language is crucial to social constructivism, as knowledge grows through language and the shared meanings it makes possible. According to Ernest, "... the social institution of language . . . justifies and necessitates the admission of the social into philosophy at some point or other" (p. 131). Mathematics has many conventions which are basically social agreements on definitions, assumptions and rules. Objective knowledge rests within the social framework due to linguistic rules and practices. For social constructivism, mathematical knowledge and logic are epistemologically social phenomena that includes language, negotiation, conversation and group acceptance. As a result, "social constructivism accounts for both the 'objective' and 'subjective' knowledge in mathematics and describes the mechanisms underlying the genesis . . . of knowledge socially" (p. 136). So, objective knowledge in mathematics is taken to be what is accepted and warranted by the mathematical community. So the real purpose of proofs is to persuade the mathematical community to accept certain claims. Ernest argues further that the transmission of mathematical knowledge is a key aspect of social constructivism. There must exist a social context for the learning of mathematics, and items of mathematical knowledge differ in significance and meaning depending on that context. Similarly, learners' knowledge of mathematics is related to shared social activities. The function of mathematical symbol systems as semiotic tools must be socially acquired and mastered if students are to gain mathematical knowledge and competence. Semiotic tools have their own rhetorical features and become part of the overall social constitution and context of the classroom. The acquisition and use of mathematical knowledge are not readily detachable from social activities and are often elicited as a result of an engagement with a specific social situation. One such situation is the mathematics classroom, which also has its language games and communally-shared social forms of life. A problem, however, lies in the fact that there are significant variations in the rhetorical demands of teachers in different contexts. For example, through participation in the high school algebra language game, beginning algebra students must learn how to use variables in mathematical sentences with their own syntax and semantics and represent common language in mathematical form, use various tools, such as graphs, tables, or calculators. The language is acquired through shared meanings. Ernest views the mathematics classroom as a micro- community composed of individuals using their personal knowledge of mathematics and mathematics education for directing learning conversations, presenting knowledge to learners, and to participating in a dialectical process of criticism and warranting others' knowledge claims. The school context is one part of the creative/reproductive cycle of mathematics. The other part is the academic context. In each context, there is action associated with human activity such as conversation/negotiation and a textual movement of mathematics, the publication of mathematical knowledge open to public criticism. Overall, Ernest’s scholarship is deep and insightful. It is also wide: he uses the ideas of many other thinkers such as Dewey, Foucault, and Vygotsky to lend greater support for his social constructivism. Ernest helpfully reminds us that mathematics is not independent of social concerns. External circumstances such as funding for research, developments in adjacent fields such as theoretical physics, or technological developments such as those in numeration and digital computing, have an impact on problems, solutions, concepts, standards for proof, and the overall shape of mathematics as a discipline. Mathematics as a cultural activity is not devoid of human failings. It cannot develop or evolve without human thoughts, actions, reactions, acceptances and revisions. Mathematics and society shape one another, and by recognizing this social constructivism represents an advance in the philosophy of mathematics that is, for this reviewer, a welcome departure from the cold and unyielding nature of the traditional view. About the Reviewer Jim Telese University of Texas, Brownsville School of Education 80 Fort Brown Brownsville, TX 78520