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.

Key words: Foundations of mathematics, verificationism, finitism, Platonism, pragmatism, Gödel’s Proof, Halting Problem, undecidability, consistency, computability, actualism

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.

Key words: mathematical constructivism, conceptualism, internalism vs externalism, meaning finitism, mathematical practice

Context: The main context of this study is the shift of programming education from professional development to general education. Problem: The article deals with methods, environments and approaches to teaching programming to everyone. Method: Conceiving programming education as concept building by creating pupils’ mental models in selected didactical environments that are constructed to allow pupils to focus on the given problem and, at the same time, to have the structure of a set of similar short tasks of increasing difficulty with the same underlying concept. Design-based research on the evaluation of curricular materials created according to this method. Results: Specified principles of creation of appropriate materials for teaching programming, intervention conducted with these materials and experience from a pilot research study of teaching that contains signals of how difficult it would be to change teachers’ minds to make them willing to accept and implement this approach in their teaching. Implications: The article focuses on applying the theory from mathematics education to a different field. The results could be beneficial for programming curricula education creators; a qualitatively new generation of textbooks on programming education for pupils from an early age could be created using this approach. Future research could focus on teachers’ beliefs and the changes to these beliefs when teaching programming in this way. Constructivist content: The theory used has its origin in mathematical constructivism and is based on the work of Papert and Hejn. It could bring experience in applying a proven theory originally used in another discipline. Key words: Computer science, programming education, junior high school, concept building, Scratch.