In the present essay we attempt to reconstruct Newtonian mechanics under the guidance of logical principles and of a constructive approach related to the genetic epistemology of Piaget and García (Psychogenesis and the history of science, Columbia University Press, New York, 1989). Instead of addressing Newton’s equations as a set of axioms, ultimately given by the revelation of a prodigious mind, we search for the fundamental knowledge, beliefs and provisional assumptions that can produce classical mechanics. We start by developing our main tool: the no arbitrariness principle, that we present in a form that is apt for a mathematical theory as classical mechanics. Subsequently, we introduce the presence of the observer, analysing then the relation objective–subjective and seeking objectivity going across subjectivity. We take special care of establishing the precedence among all contributions to mechanics, something that can be better appreciated by considering the consequences of removing them: (a) the consequence of renouncing logic and the laws of understanding is not being able to understand the world, (b) renouncing the early elaborations of primary concepts such as time and space leads to a dissociation between everyday life and physics, the latter becoming entirely pragmatic and justifed a-posteriori (because it is convenient), (c) changing our temporary beliefs has no real cost other than efort. Finally, we exemplify the present approach by reconsidering the constancy of the velocity of light. It is shown that it is a result of Newtonian mechanics, rather than being in contradiction with it. We also indicate the hidden assumption that leads to the (apparent) contradiction.

Stewart J. (2001) Radical constructivism in biology and cognitive science. Special Issue “The Impact of Radical Constructivism on Science” edited by Alexander Riegler. Foundations of Science 6(1–3): 99–124. https://cepa.info/3634

This article addresses the issue of “objectivism vs constructivism” in two areas, biology and cognitive science, which are intermediate between the natural sciences such as physics (where objectivism is dominant) and the human and social sciences (where constructivism is widespread). The issues in biology and in cognitive science are intimately related; in each of these twin areas, the “objectivism vs constructivism” issue is interestingly and rather evenly balanced; as a result, this issue engenders two contrasting paradigms, each of which has substantial specific scientific content. The neo-Darwinian paradigm in biology is closely resonant with the classical cognitivist paradigm in cognitive science, and both of them are intrinsically objectivist. The organismic paradigm in biology, based on the concept of autopoiesis, is consonant with the paradigm of “enaction” in cognitive science; the latter paradigms are both profoundly constructivist. In cognitive science, the objectivism vs constructivism issue is internal to the scientific field itself and reflexivity is inescapable. At this level, strong ontological objectivism is self-contradictory and therefore untenable. Radical constructivism is self-coherent; but it also rehabilitates a weak form of objectivism as a pragmatically viable alternative. In conclusion, there is an even-handed reciprocity between “objectivist” and “constructivist” perspectives Finally, the article examines the consequences of this conclusion for fields other than cognitive science: biology; physics and the natural sciences; and the human and social sciences.

Except in very poor mathematical contexts, mathematical arguments do not stand in isolation of other mathematical arguments. Rather, they form trains of formal and informal arguments, adding up to interconnected theorems, theories and eventually entire fields. This paper critically comments on some common views on the relation between formal and informal mathematical arguments, most particularly applications of Toulmin’s argumentation model, and launches a number of alternative ideas of presentation inviting the contextualization of pieces of mathematical reasoning within encompassing bodies of explicit and implicit, formal and informal background knowledge.

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.

Ziemke T. (2001) The construction of “reality” in the robot: Constructivist perspectives on situated AI and adaptive robotics. Foundations of Science 6(1): 163–233. https://cepa.info/4522

This paper discusses different approaches in cognitive science and artificial intelligence research from the perspective of radical constructivism, addressing especially their relation to the biologically based theories of von Uexküll, Piaget as well as Maturana and Varela. In particular recent work in ‘New AI’ and adaptive robotics on situated and embodied intelligence is examined, and we discuss in detail the role of constructive processes as the basis of situatedness in both robots and living organisms.