2012-03-15 An interdisciplinary peer-reviewed academic e-journal dedicated to constructivist issues raised by philosophy a well as the natural, human, and applied sciences. Special Issue “Constructivism In and About Mathematics” 1782-348X http://www.constructivistfoundations.info 2012-03-15 Van Kerkhove B. Van Bendegem J. P. Constructivist Foundations 7 2 97 103 Context: As one of the major approaches within the philosophy of mathematics, constructivism is to be contrasted with realist approaches such as Platonism in that it takes human mental activity as the basis of mathematical content. Problem: Mathematical constructivism is mostly identified as one of the so-called foundationalist accounts internal to mathematics. Other perspectives are possible, however. Results: The notion of “meaning finitism” is exploited to tie together internal and external directions within mathematical constructivism. The various contributions to this issue support our case in different ways. Constructivist content: Further contributions from a multitude of constructivist directions are needed for the puzzle of an integrative, overarching theory of mathematical practice to be solved. http://www.univie.ac.at/constructivism/journal/7/2/097.vankerkhove 2012-03-15 Quale A. Constructivist Foundations 7 2 104 111 Context: the position of pure and applied mathematics in the epistemic conflict between realism and relativism. Problem: To investigate the change in the status of mathematical knowledge over historical time: specifically, the shift from a realist epistemology to a relativist epistemology. Method: Two examples are discussed: geometry and number theory. It is demonstrated how the initially realist epistemic framework – with mathematics situated in a platonic ideal reality from where it governs our physical world – became untenable, with the advent of non-Euclidean geometry and the increasing abstraction of the number concept. Results: Radical constructivism offers an alternative relativist epistemology, where mathematical knowledge is constructed by the individual knower in a context of an axiomatic base and subject items chosen at her discretion, for the purpose of modelling some part of her personal experiential world. Thus it can be expedient to view the practice of mathematics as a game, played by mathematicians according to agreed-upon rules. Constructivist content: The role played by constructivism in the formulation of mathematics is discussed. This is illustrated by the historical transition from a classical (platonic) view of mathematics, as having an objective existence of its own in the “realm of ideal forms,” to the now widely accepted modern view where one has a wide freedom to construct mathematical theories to model various parts of one’s experiential world. http://www.univie.ac.at/constructivism/journal/7/2/104.quale 2012-03-15 Kauffman L. H. Constructivist Foundations 7 2 112 115 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. http://www.univie.ac.at/constructivism/journal/7/2/112.kauffman 2012-03-15 Cariani P. Constructivist Foundations 7 2 116 125 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. http://www.univie.ac.at/constructivism/journal/7/2/116.cariani 2012-03-15 Gwiazda J. Constructivist Foundations 7 2 126 130 Context: The infinite has long been an area of philosophical and mathematical investigation. There are many puzzles and paradoxes that involve the infinite. Problem: The goal of this paper is to answer the question: Which objects are the infinite numbers (when order is taken into account)? Though not currently considered a problem, I believe that it is of primary importance to identify properly the infinite numbers. Method: The main method that I employ is conceptual analysis. In particular, I argue that the infinite numbers should be as much like the finite numbers as possible. Results: Using finite numbers as our guide to the infinite numbers, it follows that infinite numbers are of the structure w + (w* + w) a + w*. This same structure also arises when a large finite number is under investigation. Implications: A first implication of the paper is that infinite numbers may be large finite numbers that have not been investigated fully. A second implication is that there is no number of finite numbers. Third, a number of paradoxes of the infinite are resolved. One change that should occur as a result of these findings is that “infinitely many” should refer to structures of the form w + (w* + w) a + w*; in contrast, there are “indefinitely many” natural numbers. Constructivist content: The constructivist perspective of the paper is a form of strict finitism. http://www.univie.ac.at/constructivism/journal/7/2/126.gwiazda 2012-03-15 Loeb I. Constructivist Foundations 7 2 131 140 Context: It is often suggested that the methodology of the programme of Constructive Reverse Mathematics (CRM) can be sufficiently clarified by a thorough understanding of Brouwer’s intuitionism, Bishop’s constructive mathematics, and classical Reverse Mathematics. In this paper, the correctness of this suggestion is questioned. Method: We consider the notion of a mathematical programme in order to compare these schools of mathematics in respect of their methodologies. Results: Brouwer’s intuitionism, Bishop’s constructive mathematics, and classical Reverse Mathematics are historical influences upon the origin and development of CRM, but do not give a full “methodological explanation” for it. Implications: Discussion on the methodological issues concerning CRM is needed. Constructivist content: It is shown that the characterisation and comparison of varieties of constructive mathematics should include methodological aspects (as understood from their practices). http://www.univie.ac.at/constructivism/journal/7/2/131.loeb 2012-03-15 Van Bendegem J. P. Constructivist Foundations 7 2 141 149 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. http://www.univie.ac.at/constructivism/journal/7/2/141.vanbendegem 2012-03-15 Constructivist Foundations 7 2 150 153 Upshot: This section lists publications related to constructivist approaches – constructivism, second-order cybernetics, enaction, non-dualism, biology of cognition, etc. – that recently have been published elsewhere, and which the reader of the journal might find interesting. The entries are ordered alphabetically and clustered according to their respective primary disciplinary backgrounds or application. The increasingly extending bibliography can be consulted at http://www.constructivistfoundations.info/bib/ http://www.univie.ac.at/constructivism/journal/7/2/150.review