Berkowitz G. C., Greenberg D. R. & White C. A. (1988) An approach to a mathematics of phenomena: Canonical aspects of reentrant form eigenbehavior in the extended calculus of indications. Cybernetics and Systems: An International Journal 19(2): 123–167.

Self-reference and recursion characterize a vast range of dynamic phenomena, particularly biological automata. In this paper we investigate the dynamics of self-referent phenomena using the Extended Calculus of Indications (ECI) of Kauffman and Varela, who have applied the ECI to mathematics, physics, linguistics, perception, and cognition. Previous studies have focused on the algebraic structure of the ECI, and on form dynamics using only the arithmetic of Spencer-Brown. We here examine the temporal behavior of self-referent or reentrant forms using the full power of the ECI to represent tangled hierarchies and multiple enfolded dimensions of space-time. Further, we explore the temporal convolution of static and recursive states in coherent fluctuation, providing a foundation for going beyond the Turing model of computation in finite automata. Novel results are presented on the structure of reentrant forms and the canonical elements of form eigenbehavior, the characteristic self-determined dynamic inherent in reentrant forms.

Bonawitz E., Gopnik A., Denison S. & Griffiths T. L. (2012) Rational randomness: The role of sampling in an algorithmic account of preschooler’s causal learning. In: Xu F. & Kushnir T. (eds.) Advances in child development and behavior. Volume 43. Academic Press, Waltham MA: 161–191.

Probabilistic models of cognitive development indicate the ideal solutions to computational problems that children face as they try to make sense of their environment. Under this approach, children’s beliefs change as the result of a single process: observing new data and drawing the appropriate conclusions from those data via Bayesian inference. However, such models typically leave open the question of what cognitive mechanisms might allow the finite minds of human children to perform the complex computations required by Bayesian inference. In this chapter, we highlight one potential mechanism: sampling from probability distributions. We introduce the idea of approximating Bayesian inference via Monte Carlo methods, outline the key ideas behind such methods, and review the evidence that human children have the cognitive prerequisites for using these methods. As a result, we identify a second factor that should be taken into account in explaining human cognitive development the nature of the mechanisms that are used in belief revision.

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.

By assuming the bunching process of electrons in the drift region of a klystron as being space periodic, rather than time periodic, it is possible to describe the debunching mechanism in a finite diameter beam on a large signal basis. It is shown that such beams have a discrete spectrum of oscillations with the plasma frequency as series limits. This makes it impossible to assign a unique bunching parameter to all electrons. These results check with a velocity‐phase analysis of high‐density electron beams carried out with a beam analyzer at a frequency of 3000 Mc.

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.

Gwiazda J. (2012) On Infinite Number and Distance. Constructivist Foundations 7(2): 126–130. https://constructivist.info/7/2/126

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.

McCulloch W. S. (1945) A heterarchy of values determined by the topology of nervous nets. Bulletin of Mathematical Biology 7: 89–93. https://cepa.info/2830

Because of the dromic character of purposive activities, the closed circuits sustaining them and their interaction can be treated topologically. It is found that to the value anomaly, when A is preferred to B, B to C, but C to A, there corresponds a diadrome, or circularity in the net which is not the path of any drome and which cannot be mapped without a diallel on a surface sufficient to map the dromes. Thus the apparent inconsistency of preference is shown to indicate consistency of an order too high to permit construction of a scale of Values, but submitting to finite topological analysis based on the finite number of nervous cells and their possible connections.

Reprinted in: McCulloch R. (ed.) (1989) Collected works of Warren S. McCulloch. Intersystems, Salinas CA: 467–472

Müller K. H. (2011) The Two Epistemologies of Ernst von Glasersfeld. Constructivist Foundations 6(2): 220–226. https://constructivist.info/6/2/220

Purpose: The article pursues three aims. First, it intends to differentiate between two different approaches for knowledge studies, namely an empirical and a normative mode. In a second move, two different epistemologies in the work of Ernst von Glasersfeld will be introduced under the labels of “Epistemology I” and “Epistemology II.” Epistemology I relates to empirical research, Epistemology II is normative in nature. Third, the article makes the point that while Ernst von Glasersfeld’s Epistemology II has already been presented in a finite and mature form, his empirical analysis of cognitive processes still provides a rich pool of tools and designs that should be further developed and advanced in the years and decades ahead. Method: The article is analytical in nature, identifying the different building blocks and relational networks of von Glasersfeld’s two epistemologies. By this, the article intends to contribute to a further advancement of von Glasersfeld’s Epistemology I. Results: The main finding lies in recognizing the radical and innovative elements of von Glasersfeld’s Epistemology I and on the still-challenging research designs of Epistemology I.

Peschl M. F. (1997) The representational relation between environmental structures and neural systems: Autonomy and environmental dependency in neural knowledge representation. Nonlinear Dynamics, Psychology. and Life Sciences 1(2): 99–121.

In this paper it will be shown that in neural systems with a recurrent architecture, the traditional concepts of knowledge representation cannot be applied any more; no stable representational relationship of reference can be found. That is why a redefinition of the relationship between the states of the environment and the internal representational states is proposed. Studying the dynamics of recurrent neural systems reveals that the goal of representation is no longer to map the environment as accurately as possible to the representation system (e.g., to symbols) It is suggested that it is more appropriate to look at neural systems as physical dynamical devices embodying the (transformation) knowledge for sensorimotor integration and for generating adequate behavior enabling the organism’s survival. As an implication the representation is determined not only by the environment, but highly depends on the organization, structure, and constraints of the representation system as well as the sensory/motor systems which are embedded in a particular body structure. This leads to a system relative concept of representation. By transforming recurrent neural networks into the domain of finite automata, the dynamics as well as the epistemological implications become more clear. In recurrent neural systems a type of balance between the autonomy of the representation and the environmental dependence/influence emerges. This not only affects the traditional concept of knowledge representation, but has also implications for the understanding of semantics, language, communication, and even science.

Context: Strict finitism is usually not taken seriously as a possible view on what mathematics is and how it functions. This is due mainly to unfamiliarity with the topic. Problem: First, it is necessary to present a “decent” history of strict finitism (which is now lacking) and, secondly, to show that common counterarguments against strict finitism can be properly addressed and refuted. Method: For the historical part, the historical material is situated in a broader context, and for the argumentative part, an evaluation of arguments and counterarguments is presented. Results: The main result is that strict finitism is indeed a viable option, next to other constructive approaches, in (the foundations of) mathematics. Implications: Arguing for strict finitism is more complex than is usually thought. For future research, strict finitist mathematics itself needs to be written out in more detail to increase its credibility. In as far as strict finitism is a viable option, it will change our views on such “classics” as the platonist-constructivist discussion, the discovery-construction debate and the mysterious applicability problem (why is mathematics so successful in its applications?). Constructivist content: Strict finitism starts from the idea that counting is an act of labeling, hence the mathematician is an active subject right from the start. It differs from other constructivist views in that the finite limitations of the human subject are taken into account.