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

Purpose: To explore possible innovations that constructivism and its epistemological participatory position offer to philosophy, in particular to the age‐old problem of grounding epistemological assumptions. Design/methodology/approach – The paper follows von Foerster’s account of the participatory position as an epistemological stance. It tries to explain why it is called a “position” rather than “insight” or “theory.” Constructivist (participatory) concepts are explored and related to “classical” philosophical debates such as the “Münchhausen trilemma.” In the conclusion the paper sketches possible ways of how to apply the answers of the participatory position to the philosophical discourse. Findings: The paper points at the possibility to go beyond the insurmountable boundaries dividing different epistemological positions one continuously encounters when searching for the appropriate epistemological starting point. As a result, one cannot expect answers to be universally valid. The paper takes that into consideration. It argues that most philosophical attempts are first order changes (revolutions) as they seek truth and exclude alternative views at the same time. Following von Foerster, the paper suggests second order changes that lead from truth to trust. This transition allows a peaceful coexistence of all philosophical systems. Turning from truth (and belief in analytical clarity) to trust and, consequently, from objectivity to responsibility, it may become possible to transcend the epistemological barriers. Originality/value – The paper contributes to finding a possible direction for the future of discourses in philosophy and many humanities in order to overcome the incompetence of philosophy of finding final answers.

Key words: Cognition, epistemology, undecidability, participatory position, constructivism

Purpose: The purpose of this paper is to examine shared principles of “irreducibility” or “undecidability” in second-order cybernetics, architectural design processes and Leibniz’s geometric philosophy. It argues that each discipline constructs relationships, particularly spatio-temporal relationships, according to these terms. Design/methodology/approach – The paper is organized into two parts and uses architectural criticism and philosophical analysis. The first part examines how second-order cybernetics and post-structuralist architectural design processes share these principles. Drawing from von Foerster’s theory of the “observing observer” it analyses the self-reflexive and self-referential modes of production that construct a collaborative architectural design project. Part two examines the terms in relation to Leibniz’s account of the “Monad”. Briefly, developing the discussion through Kant’s theory of aesthetics, it shows that Leibniz provides a “prototype” of undecidable spatial relations that are also present in architectural design and second-order cybernetics. Findings: The paper demonstrates that second-order cybernetics, architectural design and metaphysical philosophy enable interdisciplinary understandings of “undecidability”. Practical implications: The paper seeks to improve understanding of the geometric processes that construct architectural design. Originality/value – The paper explores interdisciplinary connections between the disciplines, opening up potential routes for further examination Its analysis of the aesthetic and geometric value of the Monad (rather than its perspectival value) provides a particularly relevant link for discussing the aesthetic production and experience of spatial relations in second-order cybernetics and contemporary architectural design.

Key words: Cybernetics, design, architecture, geometry

Exact isomorphisms with formal systems which employ the standard linguistic notations are established for Spencer-Brown’s primary algebra and F. J. Varela’s calculus for self-reference. The primary algebra is “essentially isomorphic” with classical propositional calculus, and the calculus for self-reference translates isomorphically into an axiomatization of S. C. Kleene’s three-valued logic of partial recursion. This correlates Varela’s “autonomy” with “total recursive undecidabilily,” and it suggests the use of Kleene-Varela type systems for discussing “mechanically unknowable” or empirically untestable system properties.

Key words: Laws of form, self-reference, paradox, many-valued logics, recursive undecidability, mechanical unknowability

Exact isomorphisms with formal systems which employ the standard linguistic notations are established for Spencer- Brown’s primary algebra and F. J. Varela’s calculus for self-reference. The primary algebra is “essentially isomorphic” with classical propositional calculus, and the calculus for self-reference translates isomorphically into an axiornat- rzauon or S. C. Kleene’s three-valued logic of partial recursion. This correlates Varela’s “autonomy” with “total recursive undecidability,” and it suggests the use of Kleene-Varela type systems for discussing “mechanically unknowable” or empirically untestable system properties.

Key words: Laws of form, self-reference, paradox, many-valued logics, recursive undecidability, mechanical unknowability.

We present two original computational models – globular universe and autopoietic automata – capturing the basic aspects of an evolution: a construction of self–reproducing automata by self–assembly and a transfer of algorithmically modified genetic information over generations. Within this framework we show implementation of autopoietic automata in a globular universe. Further, we characterize the computational power of lineages of autopoietic automata via interactive Turing machines and show an unbounded complexity growth of a computational power of automata during the evolution. Finally, we define the problem of sustainable evolution and show its undecidability.