Online research in philosophy

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. (shrink)

Context: A number of objections that are frequently raised in the literature against radical constructivism, including: the charge of solipsism, allegations of self-refutation, social and moral reservations, and the accusation that RC cannot explain the success of science. Problem: These four objections are sought to be refuted. Results: 1. Solipsism is only troublesome against the background of a realist ontological perspective. 2. The truth-value of any proposition is only defined relative to some ontological context, thus self-refutation, as constituting a logical (...) problem, does not arise. 3. Any ethical argumentation derives from one’s own personal views on ethical matters: their construction being a personal responsibility such that no one else can tell a person how to construct the “right ethics.” 4. In the relativist ontology of radical constructivism, a scientific theory is regarded as a model imposed on natural phenomena; its success is due to the capabilities of its constructor/scientists. Implications: It is found that the objections are based on an (overt or tacit) adoption of the antithetical viewpoint of scientific realism. In other words, radical constructivism is being criticised for not promoting a realist ontology. (shrink)

Context: Despite many obvious advantages (radical) constructivism has over other philosophies it has failed to become a mainstream philosophy that is widely taught and discussed. Problem: What are the reasons for this failure? Can we identify issues that make it difficult for scholars to accept and even embrace radical constructivist ideas? What is the best way to characterize, explain, and eventually refute objections? Method: By collecting articles from both proponents and opponents of radical constructivism the editors of the special issue (...) tried to present a range of answers to these questions. Results: Some problems are due to known objections to radical constructivism, in particular the idea that being responsible for one’s own constructions opens doors to a “whatever” attitude. Another important insight is that constructivism seems to resemble a river delta with ever branching new sub-disciplines that become increasingly incompatible with each other. Implications: The insights gained from the contributions may lead to a re-orientation of (radical) constructivism that will include less misunderstandings among its critics and to a higher acceptance in the academic community. (shrink)