# Approach «Mathematical Constructivism»

Frans J. (2012) The Game of Fictional Mathematics. Review of “Mathematics and Reality” by Mary Leng. Constructivist Foundations 8(1): 126-128. Fulltext at https://cepa.info/821

Frans J.
(

2012)

The Game of Fictional Mathematics. Review of “Mathematics and Reality” by Mary Leng.
Constructivist Foundations 8(1): 126-128.
Fulltext at https://cepa.info/821
Upshot: Leng attacks the indispensability argument for the existence of mathematical objects. She offers an account that treats the role of mathematics in science as an indispensable and useful part of theories, but retains nonetheless a fictionalist position towards mathematics. The result is an account of mathematics that is interesting for constructivists. Her view towards the nominalistic part of science is, however, more in conflict with radical constructivism.

Gwiazda J. (2012) On Infinite Number and Distance. Constructivist Foundations 7(2): 126–130. Fulltext at https://cepa.info/255

Gwiazda J.
(

2012)

On Infinite Number and Distance.
Constructivist Foundations 7(2): 126–130.
Fulltext at https://cepa.info/255
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.

Loeb I. (2012) Questioning Constructive Reverse Mathematics. Constructivist Foundations 7(2): 131–140. Fulltext at https://cepa.info/256

Loeb I.
(

2012)

Questioning Constructive Reverse Mathematics.
Constructivist Foundations 7(2): 131–140.
Fulltext at https://cepa.info/256
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).

Van Bendegem J. P. (2012) A Defense of Strict Finitism. Constructivist Foundations 7(2): 141–149. Fulltext at https://cepa.info/257

Van Bendegem J. P.
(

2012)

A Defense of Strict Finitism.
Constructivist Foundations 7(2): 141–149.
Fulltext at https://cepa.info/257
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.

Van Kerkhove B. & Van Bendegem J. P. (2012) The Many Faces of Mathematical Constructivism. Constructivist Foundations 7(2): 97–103. Fulltext at https://cepa.info/251

Van Kerkhove B. & Van Bendegem J. P.
(

2012)

The Many Faces of Mathematical Constructivism.
Constructivist Foundations 7(2): 97–103.
Fulltext at https://cepa.info/251
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.

Waaldijk F. A. (2005) On the foundations of constructive mathematics: Especially in relation to the theory of continuous functions. Foundations of Science 10(3): 249–324.

Waaldijk F. A.
(

2005)

On the foundations of constructive mathematics: Especially in relation to the theory of continuous functions.
Foundations of Science 10(3): 249–324.
This article describes many foundational issues concerning what is known as constructivism in mathematics. First of all a flaw in the foundations of Bishop-style constructive mathematics, BISH, is discussed. A main theorem shows that the two current BISH definitions of “continuous function” are not equivalent within BISH, and that – together with the natural properties of “continuous function” – they imply the FT (fan theorem) axiom. The theorem sparks an investigation into the realm of topology and the axioms underpinning intuitionism (INT), classical mathematics (CLASS), recursive mathematics (RUSS) and BISH. Some new elegant axioms are introduced to prove theorems showing that CLASS and INT are closer than usually believed (“reuniting the antipodes”). The distance to RUSS is greater, due perhaps to a philosophical difference regarding “real world” phenomena. There is a connection with the old philosophical debate on determinism and perhaps with the debate in modern physics as well. The real-world experiment described in section 7 could cast an alternative mathematical light on this matter. Relevance: The article is entirely concerned with the foundations of constructive mathematics.

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