Cariani P. (2012) Infinity and the Observer: Radical Constructivism and the Foundations of Mathematics. Constructivist Foundations 7(2): 116–125. https://cepa.info/254

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.

Diettrich O. (1997) Sprache als Theorie: Von der Rolle der Sprache im Lichte einer konstruktivistischen Erkenntnistheorie. Papiere zur Linguistik 56(1): 77–106. https://cepa.info/5340

Theories and languages have in common that they aim at describing the world and the experiences made in the world. The specificity of theories is based on the fact that they code certain laws of nature. The specificity of languages is based on the fact that they code our worldview by means of their syntax. Also mathematics can be considered as theory in so far as it codes the constituting axioms. Language can achieve the objectivity postulated by analytical philosophy only if it can refer to a mathematics and logic being objective in the sense of platonism and based on a definitive set of axioms, or if the world-view concerned is definitive and based upon an objective (and therefore definitive) set of laws of nature. The first way is blocked by Goedel’s incompleteness theorem. The objectivity of the laws of nature being necessary for going the second way is questioned as well by what is called the constructivist evolutionary epistemology (CEE): the perceived patterns and regularities from which we derive the laws of nature is considered by the CEE to be invariants of inborn cognitive (sensory) operators. Then, the so called laws of nature are the result of cognitive evolution and therefore are human specific. Whether, e.g., we would identify the law of energy conservation which in physics results from the homogeneity of time, depends on the mental time-metric generator defining what is homogeneous in time. If cognitive operators are extended by means of experimental operators the result can be expressed in classical terms if both commute in the sense of operator algebra (quantitative extensions). Otherwise results would be inconsistent with the classical worldview and would require non-classical approaches such as quantum mechanics (qualitative extensions). As qualitative extensions can never be excluded from future experimental reasearch, it follows that the development of theories cannot converge towards a definitive set of laws of nature or a definitive ‘theory of everything’ describing the structure of reality. Also the structures of mathematics and logic we use have to be considered als invariants of mental operators. It turns out that the incompleteness theorem of Goedel has to be seen as analogy of the incompleteness of physical theories due to possible qualitative experimental extensions. Language, therefore, cannot be considered as an objective depiction of independently existing facts and matters but only as a theory generating propositions being consistent with our world-view. The competence of language is based on the fact that the mental mechanisms generating the ontology we use in our syntax are related to those generating our perceptions. Similar applies to the relationship between the operators generating perceived and mathematical structures enabling us to compress empirical data algorithmically (i.e. to transform them into mathematically articulated theories) and then to extrapolate them by means of the theory concerned (inductive inference). An analogue mechanism establishes our ability to compress verbal texts semantically (i.e. to reduce them to their meaning) and then to extrapolate them (i.e. to draw correct conclusions within the framework of the meaning concerned). This suggests a modified notion of meaning seing it as a linguistic analogy to theories. Similar to physical and mathematical theories also languages can be extended qualitatively particularly by means of metaphorical combinations of semantically noncompatible elements. The development of languages towards it actual richness can be seen as a process of ongoing metaphorosation. this leads to some parallels between verbal, cultural and genetic communication.

Van Kerkhove B. & Van Bendegem J. P. (2012) The Many Faces of Mathematical Constructivism. Constructivist Foundations 7(2): 97–103. 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.