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.

Cyzman M. (2017) On the non-dualizing rhetoric: Some preliminary remarks. In: Kanzian C., Kletzl S., Mitterer J. & Neges K. (eds.) Realism – relativism – constructivism. De Gruyter, Berlin: 17–29. https://cepa.info/4198

In the reception of Josef Mitterer’s writings up to now, there are two predominant types of motifs: the radical constructivist background of his philosophy and the ontological and epistemological foundations and consequences of non-dualism. The critics are focused rather on some problematic consequences of non-dualism, ranging from the problem of infinite regress up to the thesis assuming that Mitterer’s philosophy presupposes a world reduced to descriptions. However, these two types of readings are founded on dualizing assumptions which are not coherent with non-dualism. \\Thus, in the present paper I interpret non-dualism in the frame of non-dual-ism, based on non-dualizing assumptions. I argue that non-dualism is a rhetorical project resulting in far-reaching consequences in the field of academic and scientific debates, poetics and practice of negotiations and deliberations, as well as in ordinary discourse. Non-dualism fulfills Richard Rorty’s dream of culture as a never-ending conversation in which the argument of power is successfully replaced by the power of argument. Mitterer makes transparent the rhetorical techniques performed in the dualizing discourse (not only in situations of conflict) in order to present an alternative – the non-dualizing mode of discourse. Mitterer’s philosophy – reread in the context of Rorty’s pragmatism, Foucault’s conception of discourses, Perelman’s new rhetoric – offers the new vocabulary (in Rorty’s meaning) which may change the practice of speaking

Based upon the shape-space formalism, a model of an idiotypic network including both bound and free immunoglobulins is simulated. Our point of interest is the network development in the context of self antigens. The investigations are organized around simulations initiated by various spatial configurations of antigens; the behavior of the system with respect to antigens is analyzed in terms of morphogenetic processes occurring in the shape space. For certain values of the parameters, the network expands by traveling waves. The resulting spatial pattern is a partition of the shape space into zones where introduction of an antigen entails an infinite growth of the clones binding to it, and into zones where, on the contrary, the anti-antigen idiotypes decrease. Among the parameter combinations tested, some produce a partition that remains static whereas others produce a partition that changes in time. For other values of the parameters, the patterns generated do not partition shape space into zones; in these cases, it is observed that the system systematically explodes when an antigen is present.

Freeman W. H. (2000) Brains create macroscopic order from microscopic disorder by neurodynamics in perception. In: Arhem P., Blomberg C. & Liljenstrom H. (eds.) Disorder versus order in brain function. World Scientific, Singapore: 205–220. https://cepa.info/2702

The essential task of brain function is to construct orderly patterns of neural activity from disorderly sensory inputs, so that effective actions can be mounted by the brain, a finite state system, to deal with the world’s infinite complexity. Two schools of thought are described, that characterize distinctive sources of the order within brains, one passive, the other active. These schools have profoundly influenced ways two groups of contemporary neuroscientists design their experiments and process their data, so that they have very different perspectives on the roles of noise and chaos in brain function.

Friston K. & Frith C. (2015) A duet for one. Consciousness and Cognition 36: 390–405. https://cepa.info/5877

This paper considers communication in terms of inference about the behaviour of others (and our own behaviour). It is based on the premise that our sensations are largely generated by other agents like ourselves. This means, we are trying to infer how our sensations are caused by others, while they are trying to infer our behaviour: for example, in the dialogue between two speakers. We suggest that the infinite regress induced by modelling another agent – who is modelling you – can be finessed if you both possess the same model. In other words, the sensations caused by others and oneself are generated by the same process. This leads to a view of communication based upon a narrative that is shared by agents who are exchanging sensory signals. Crucially, this narrative transcends agency – and simply involves intermittently attending to and attenuating sensory input. Attending to sensations enables the shared narrative to predict the sensations generated by another (i.e. to listen), while attenuating sensory input enables one to articulate the narrative (i.e. to speak). This produces a reciprocal exchange of sensory signals that, formally, induces a generalised synchrony between internal (neuronal) brain states generating predictions in both agents. We develop the arguments behind this perspective, using an active (Bayesian) inference framework and offer some simulations (of birdsong) as proof of principle.

Context: The infinite has long been an area of philosophical and mathematical investigation. There are many puzzles and paradoxes that involve the infinite. Problem: The goal of this paper is to answer the question: Which objects are the infinite numbers (when order is taken into account)? Though not currently considered a problem, I believe that it is of primary importance to identify properly the infinite numbers. Method: The main method that I employ is conceptual analysis. In particular, I argue that the infinite numbers should be as much like the finite numbers as possible. Results: Using finite numbers as our guide to the infinite numbers, it follows that infinite numbers are of the structure w + (w* + w) a + w*. This same structure also arises when a large finite number is under investigation. Implications: A first implication of the paper is that infinite numbers may be large finite numbers that have not been investigated fully. A second implication is that there is no number of finite numbers. Third, a number of paradoxes of the infinite are resolved. One change that should occur as a result of these findings is that “infinitely many” should refer to structures of the form w + (w* + w) a + w*; in contrast, there are “indefinitely many” natural numbers. Constructivist content: The constructivist perspective of the paper is a form of strict finitism.

Janew C. (2014) Dialogue on alternating consciousness: From perception to infinities and back to free will. Journal of Consciousness Exploration & Research 5(4): 351–391. https://cepa.info/1059

Can we trace back consciousness, reality, awareness, and free will to a single basic structure without giving up any of them? Can the universe exist in both real and individual ways without being composed of both? This metaphysical dialogue founds consciousness and freedom of choice on the basis of a new reality concept that also includes the infinite as far as we understand it. Just the simplest distinction contains consciousness. It is not static, but a constant alternation of perspectives. From its entirety and movement, however, there arises a freedom of choice being more than reinterpreted necessity and unpredictability. Although decisions ultimately involve the whole universe, they are free in varying degrees also here and now. The unity and openness of the infinite enables the individual to be creative while this creativity directly and indirectly enters into all other individuals without impeding them. A contrary impression originates only in a narrowed awareness. But even the most conscious and free awareness can neither anticipate all decisions nor extinguish individuality. Their creativity is secured. Relevance: This article includes major constructivist concepts like operational closure and openness, individual and alternating perception, creativity, and a non-dualistic theory of everything.

Kauffman L. H. (1987) Imaginary values in mathematical logic. In: Proceedings of the Seventeenth International Conference on Multiple Valued Logic. IEEE: 282–289. https://cepa.info/1842

We discuss the relationship of G. Spencer-Brown’s Laws of Form with multiple-valued logic. The calculus of indications is presented as a diagrammatic formal system. This leads to new domains and values by allowing infinite and self-referential expressions that extend the system. We reformulate the Varela/Kauffman calculi for self-reference, and give a new completeness proof for the corresponding three-valued algebra (CSR).

Le Moigne J.-L. (1997) Les épistémologies constructivistes: Un nouveau commencement I: Le constructivisme en procès. Sciences de la société 40: 215–238. https://cepa.info/7747

Epistemology is defined as “the study of the constitution of valuable knowledge” (J. Piaget). That is to say, as the study of the “foundations” of scientific knowledge; a definition which leads to an infinite recursion, those foundations being themselves knowledge. To interrupt this infinite recursion, one adopt the collective affirmation of a “convention”. During the least hundred years, the epistemic convention of positivism and of realism were the basis on which were established the value and validity of scientific knowledge. When, thirty years ago, the epistemic convention of constructivism reappeared, it provoked an understandable perplexity amongst the supporters of the positivist and realist convention who believed that its monopolistic position was legitimate. Also they proposed a “proceeding against constructivism”, discussion of which will reveal the content of the thesis. This epistemological proceeding constitutes for us an introduction to the discussion of the genesis and of the organization of the constructivist (epistemological) convention, in its formulation at the end of the XXth century.

This article analyses the work of Robert Rosen on an interpretation of metabolic networks that he called (M, R) systems. His main contribution was an attempt to prove that metabolic closure (or metabolic circularity) could be explained in purely formal terms, but his work remains very obscure and we try to clarify his line of thought. In particular, we clarify the algebraic formulation of (M, R) systems in terms of mappings and sets of mappings, which is grounded in the metaphor of metabolism as a mathematical mapping. We define Rosen’s central result as the mathematical expression in which metabolism appears as a mapping f that is the solution to a fixed-point functional equation. Crucially, our analysis reveals the nature of the mapping, and shows that to have a solution the set of admissible functions representing a metabolism must be drastically smaller than Rosen’s own analysis suggested that it needed to be. For the first time, we provide a mathematical example of an (M, R) system with organizational invariance, and we analyse a minimal (three-step) autocatalytic set in the context of (M, R) systems. In addition, by extending Rosen’s construction, we show how one might generate self-referential objects f with the remarkable property f(f)=f, where f acts in turn as function, argument and result. We conclude that Rosen’s insight, although not yet in an easily workable form, represents a valuable tool for understanding metabolic networks.