Problem: There is currently a great deal of mysticism, uncritical hype, and blind adulation of imaginary mathematical and physical entities in popular culture. We seek to explore what a radical constructivist perspective on mathematical entities might entail, and to draw out the implications of this perspective for how we think about the nature of mathematical entities. Method: Conceptual analysis. Results: If we want to avoid the introduction of entities that are ill-defined and inaccessible to verification, then formal systems need to avoid introduction of potential and actual infinities. If decidability and consistency are desired, keep formal systems finite. Infinity is a useful heuristic concept, but has no place in proof theory. Implications: We attempt to debunk many of the mysticisms and uncritical adulations of Gödelian arguments and to ground mathematical foundations in intersubjectively verifiable operations of limited observers. We hope that these insights will be useful to anyone trying to make sense of claims about the nature of formal systems. If we return to the notion of formal systems as concrete, finite systems, then we can be clear about the nature of computations that can be physically realized. In practical terms, the answer is not to proscribe notions of the infinite, but to recognize that these concepts have a different status with respect to their verifiability. We need to demarcate clearly the realm of free creation and imagination, where platonic entities are useful heuristic devices, and the realm of verification, testing, and proof, where infinities introduce ill-defined entities that create ambiguities and undecidable, ill-posed sets of propositions. Constructivist content: The paper attempts to extend the scope of radical constructivist perspective to mathematical systems, and to discuss the relationships between radical constructivism and other allied, yet distinct perspectives in the debate over the foundations of mathematics, such as psychological constructivism and mathematical constructivism.

Key words: Foundations of mathematics, verificationism, finitism, Platonism, pragmatism, Gödel’s Proof, Halting Problem, undecidability, consistency, computability, actualism

Problem: What is it that Ernst von Glasersfeld brought to mathematics education with radical constructivism? Method: Key ideas in the author’s early thinking are related to ideas that are central in constructivism, with the aim of showing their importance in math education. Results: The author’s initial thinking about constructivism began with Toulmin’s view of thinking as evolving. Ernst showed how Piaget’s genetic epistemology implied an epistemology that was not about ontology. Continuing with an analysis of the way radical and trivial constructivism were received by the mathematics education community, implications of Ernst’s ideas are considered. Implications: These include the need to consider major changes in ways content is introduced to children, to consider carefully the language used to describe children’s emerging mathematical ideas, and to consider new conjectures and also how we think about the foundations of mathematics. Ultimately the value of RC is the way it reinspires belief in the possibility and importance of human growth.

Key words: mathematics education, epistemology, language, Jean Piaget

Purpose: The paper contributes to the naturalization of epistemology. It suggests tentative itineraries for the progression from elementary experiential situations to the abstraction of the concepts of unit, plurality, number, point, line, and plane. It also provides a discussion of the question of certainty in logical deduction and arithmetic. Approach: Whitehead’s description of three processes involved in criticizing mathematical thinking (1956) is used to show discrepancies between a traditional epistemological stance and the constructivist approach to knowing and communication. Practical implications: Reducing basic abstract terms to experiential situations should make them easier to conceive for students.

Key words: foundations of mathematics, concept formation, conceptual semantics

In this paper our main objective is to interpret the major concepts in Wittgenstein’s philosophy of mathematics, in particular, language games and forms of life, from a social constructivist point of view in an attempt to show that this philosophy is still very relevant in the way mathematics is being taught and practiced today. We start out with a brief discussion of radical constructivism followed by a rudimentary analysis of the basic tenets of social constructivism, the final blow against absolutism – the soulless landmark of mathematics as often construed by the uninitiated. We observe that, the social constructivist epistemology of mathematics reinstates mathematics, and rightfully so, as “…a branch of knowledge which is indissolubly connected with other knowledge, through the web of language” (Ernest 1999), and portrays mathematical knowledge as a process that should be considered in conjunction with its historical origins and within a social context. Consequently, like any other form of knowledge based on human opinion or judgment, mathematical knowledge has the possibility of losing its truth or necessity, as well. In the third section we discuss the main points expounded in Wittgenstein’s two books, Tractatus Logico-Philosophicus and Philosophical Investigations, as well as in his “middle period” that is characterized by such works as Philosophical Remarks, Philosophical Grammar, and Remarks on the Foundations of Mathematics. We then briefly introduce the two main concepts in Wittgenstein’s philosophy that will be used in this paper: forms of life and language games. In the fifth and final section, we emphasize the connections between social constructivism and Wittgenstein’s philosophy of mathematics. Indeed, we argue that the apparent certainty and objectivity of mathematical knowledge, to paraphrase Ernest (Ernest 1998), rest on natural language. Moreover, mathematical symbolism is a refinement and extension of written language: the rules of logic which permeate the use of natural language afford the foundation upon which the objectivity of mathematics rests. Mathematical truths arise from the definitional truths of natural language, and are acquired by social interaction. Mathematical certainty rests on socially accepted rules of discourse embedded in our forms of life, a concept introduced by Wittgenstein (Wittgenstein, 1956). We argue that the social constructivist epistemology draws on Wittgenstein’s (1956) account of mathematical certainty as based on linguistic rules of use and forms of life, and Lakatos’ (1976) account of the social negotiation of mathematical concepts, results, and theories.

Key words: radical constructivism, social constructivism, forms of life, language games

Context: The question of how to understand the epistemology of set theory has been a longstanding problem in the foundations of mathematics since Cantor formulated the theory in the 19th century, and particularly since Bertrand Russell articulated his paradox in the early twentieth century. The theory of types pioneered by Russell and Whitehead was simplified by mathematicians to a single distinction between sets and classes. The question of the meaning of this distinction and its necessity still remains open. Problem: I am concerned with the meaning of the set/class distinction and I wish to show that it arises naturally due to the nature of the sort of distinctions that sets create. Method: The method of the paper is to discuss first the Russell paradox and the arguments of Cantor that preceded it. Then we point out that the Russell set of all sets that are not members of themselves can be replaced by the Russell operator R, which is applied to a set S to form R(S), the set of all sets in S that are not members of themselves. Results: The key point about R(S) is that it is well-defined in terms of S, and R(S) cannot be a member of S. Thus any set, even the simplest one, is incomplete. This provides the solution to the problem that I have posed. It shows that the distinction between sets and classes is natural and necessary. Implications: While we have shown that the distinction between sets and classes is natural and necessary, this can only be the beginning from the point of view of epistemology. It is we who will create further distinctions. And it is up to us to maintain these distinctions, or to allow them to coalesce. Constructivist content: I argue in favor of a constructivist perspective for set theory, mathematics, and the way these structures fit into our natural language and constructed speech and worlds. That is the point of this paper. It is only in the reach for absolutes, ignoring the fact that we are the authors of these structures, that the paradoxes arise.

Key words: distinction, Russell paradox, Barber paradox, sets, Cantor’s Theorem, classes, types

Context: Strict finitism is usually not taken seriously as a possible view on what mathematics is and how it functions. This is due mainly to unfamiliarity with the topic. Problem: First, it is necessary to present a “decent” history of strict finitism (which is now lacking) and, secondly, to show that common counterarguments against strict finitism can be properly addressed and refuted. Method: For the historical part, the historical material is situated in a broader context, and for the argumentative part, an evaluation of arguments and counterarguments is presented. Results: The main result is that strict finitism is indeed a viable option, next to other constructive approaches, in (the foundations of) mathematics. Implications: Arguing for strict finitism is more complex than is usually thought. For future research, strict finitist mathematics itself needs to be written out in more detail to increase its credibility. In as far as strict finitism is a viable option, it will change our views on such “classics” as the platonist-constructivist discussion, the discovery-construction debate and the mysterious applicability problem (why is mathematics so successful in its applications?). Constructivist content: Strict finitism starts from the idea that counting is an act of labeling, hence the mathematician is an active subject right from the start. It differs from other constructivist views in that the finite limitations of the human subject are taken into account.

Key words: mathematics, finite, largest number, infinite, limits, budget constraints