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.
Hoffjann O. (2017) Die Wahrheitsspieler. Strategische Kommunikation als Spiel [The truth players: Strategic communication as a game]. In: C. K. & K. K. S. M. J. N. (eds.) Realism – relativism – constructivism. Proceedings of the 38th International Wittgenstein Symposium in Kirchberg. De Gruyter, Berlin: 59–72.
Strategic communication plays are in ‘between’ and their messages have a paradox character. The paper identifies these paradoxes using the example of truth. On the one hand, a basic doubt in the possibilities of recognizing and in absolute concepts such as truth is a condition for entering a strategic communication play. On the other hand, truth is implicitly or explicitly alleged in these situations. And, finally, strategic communication plays can – despite the ‘great doubt’ – have in medium term at least such an effect that descriptions are recognized as truth. The paper describes the epistemological questions from a non-dualistic perspective (Mitterer 1992, 2001). Thereto, the concept of the strategic communication play is developed with the help of the play term used by Bateson (1956, 1985) and of the frame analysis (Goffman 1980).
This is column number 10. In this column we shall discuss a recent relative [1], [2] of the Russell paradox, the Metagame Paradox. This paradox is related to a set theoretic paradox about well-founded sets, the Well-founded Set Paradox. These two paradoxes are both related to the basic nature of any observing system that would include itself in its own observations. I give you these paradoxes and a comment on the nature of the observer. Judge them for yourself. Near the end of the column we show how, by turning the paradox around, there are new proofs of uncountability of certain infinities. These proofs have the remarkable quality that they use a bit of imaginary reasoning, only to have it vanish just as it appears! It is this role of the imagination that is central to our theme. We reason by an imaginary detour. And we obtain a real answer.
Kauffman L. H. (2012) The Russell Operator. Constructivist Foundations 7(2): 112–115. https://cepa.info/253
Context: The question of how to understand the epistemology of set theory has been a longstanding problem in the foundations of mathematics since Cantor formulated the theory in the 19th century, and particularly since Bertrand Russell articulated his paradox in the early twentieth century. The theory of types pioneered by Russell and Whitehead was simplified by mathematicians to a single distinction between sets and classes. The question of the meaning of this distinction and its necessity still remains open. Problem: I am concerned with the meaning of the set/class distinction and I wish to show that it arises naturally due to the nature of the sort of distinctions that sets create. Method: The method of the paper is to discuss first the Russell paradox and the arguments of Cantor that preceded it. Then we point out that the Russell set of all sets that are not members of themselves can be replaced by the Russell operator R, which is applied to a set S to form R(S), the set of all sets in S that are not members of themselves. Results: The key point about R(S) is that it is well-defined in terms of S, and R(S) cannot be a member of S. Thus any set, even the simplest one, is incomplete. This provides the solution to the problem that I have posed. It shows that the distinction between sets and classes is natural and necessary. Implications: While we have shown that the distinction between sets and classes is natural and necessary, this can only be the beginning from the point of view of epistemology. It is we who will create further distinctions. And it is up to us to maintain these distinctions, or to allow them to coalesce. Constructivist content: I argue in favor of a constructivist perspective for set theory, mathematics, and the way these structures fit into our natural language and constructed speech and worlds. That is the point of this paper. It is only in the reach for absolutes, ignoring the fact that we are the authors of these structures, that the paradoxes arise.
Excerpt: The papers focus on two classes of learning problems: learning causal relations, from observing co- occurrences among events and active interventions; and learning how to organize the world into categories and map word labels onto categories, from observing examples of objects in those categories. Causal learning, category learning and word learning are all problems of induction, in which children form representations of the world’s abstract structure that extend qualitatively beyond the data they observe and that support generalization to new tasks and contexts. While philosophers have long seen inductive inference as a source of great puzzles and paradoxes, children solve these natural problems of induction routinely and effortlessly. Through a combination of new computational approaches and empirical studies motivated by those models, developmental scientists may now be on the verge of understanding how they do it.
Pörksen B., Loosen W. & Scholl A. (2008) Paradoxien des Journalismus. Theorie – Empirie – Praxis. Festschrift für Siegfried Weischenberg (=Paradoxes in journalism. Theory, empirical research, and practice. Festschrift for Siegfried Weischenberg. VS Verlag für Sozialwissenschaften, Wiesbaden.
Operating with paradoxes seems to infringe scientific rules, which try to avoid paradoxes as false argumentation. Both constructivism and system theory do not ignore logic paradoxes and practical dilemma situations. Rather, observing paradoxes theoretically and solving di-lemma situations practically is typical for constructivist research programmes (cf. Watzlawick, systemic therapy etc). The constructivist way of thinking in terms of paradoxes can be applied to journalism re-search (theory) and journalism (practice). Journalists have to cope with conflicting expecta-tions and demands in practice, and journalism researchers cannot ignore these dilemmas and the ways of overcoming them in theory-building. This volume collects almost fifty authors contributing relevant issues in journalism research which are more or less paradox in struc-ture. Although many of the authors are not committed to a constructivist or system-theoretical perspective, they manage to describe typical paradoxes and how these paradoxes can be “solved”. As this volume is also a festschrift for Siegfried Weischenberg, a prominent journal-ism researcher in Germany, it closes with an extensive interview the editors conducted with Weischenberg on major issues in journalism research and practice.
Schiltz M. (2006) Power and the Third Paradox. Cybernetics & Human Knowing 13(1): 49–70.
In Talcott Parsons’ general theory of social systems, power was discussed as a medium that is potentially functionally equivalent to the medium of money. In later versions of social systems theory, notably Niklas Luhmann’s theory of society, this line of thought is conspicuously absent. In this article, I argue that Luhmann’s theory, especially in its exploration of selfreference and paradox, is well-equipped to develop Parsons’ insights to a point of higher complexity. I show how power can be discussed from the viewpoint of its three paradoxes. The ‘third paradox’ must be set apart from the former two, as it concerns the non-ontological nature of power (and money). It is explained as the original paradox of all modernity.
Stenner P. (2005) An Outline of an Autopoietic Systems Approach to Emotion. Cybernetics & Human Knowing 12(4): 8–22. https://cepa.info/3290
This paper presents the outline of an autopoietic systems approach to emotion. Distinctions are drawn between organic, psychic and social system types on the basis of the work of Serres, Luhmann and Tomkins. Literature on emotion is reviewed and three forms of reductionism identified, each of which corresponds to one of these system types. A case is then made that emotions are in fact threshold phenomena at the interstices between these system types, and distinctions between affects, emotions and proto-communication are proposed accordingly. These constitute forms of structural coupling best grasped in terms of parasitism and paradox. Emotions, affects and proto-communication de-paradoxify the paradoxes that we must know what we cannot know and share what we cannot share.
Teubner G. (1988) Introduction to autopoietic law. In: Teubner G. (ed.) Autopoietic law: A new approach to law and society. Walter de Gruyter, Berlin: 1–11. https://cepa.info/6410
Excerpt: Is the practice of legal reasoning bound to end in “strange loops,” “tangled hierarchies,” and “reflexivity dilemmas” (Hofstadter, 1979: 692; 1985: 70)? Is the legal process nothing but a closed cycle of recurrent legal operations: “computation of computation of computation…” (von Foerster, 1981: 296)? And are the social dynamics of the legal system based upon the “paradoxes of self-reference” (Wormell, 1958; Quine, 1976)? Up to now, the intricate problems of self-referential relations have not been part of the discourse of lawyers; they have been discussed outside the law, in logic, linguistics, cybernetics and general systems theory. Now the theory of legal autopoiesis is importing the logic of self-referentiality into the legal world. Legal autopoiesis breaks a taboo in legal thinking – the taboo of circularity. Legal doctrine, legal theory and legal sociology have all regarded circularity as a subject not to be broached. Circular arguments have been viewed as petitio principii forbidden by the iron law of legal logic. Legal autopoiesis now presumes to invalidate this iron law by transferring circularity from the world of ideas to that of hard facts. The message is that circularity is not a flaw in legal thinking which ought to be avoided (Fletcher, 1985: 1263), but rather that the reality of law consists of a multitude of circular processes.
Thyssen O. (2004) Luhmann and Epistemology. Cybernetics & Human Knowing 11(1): 7–22. https://cepa.info/3319
In his development of an universal systems theory, Niklas Luhmann runs into a number of epistemological problems, above all the problem of the existence of the external world, the problem of other minds and the problem of paradoxes. These problems are not primarily treated as philosophical problems. Instead Luhmann, influenced by Quine and Maturana, develops a ‘natural’ epistemology, where philosophical problems are observed as operations in social systems, which do not allow themselves to be blocked in their flow of operations. Instead of philosophical solutions Luhmann finds empirical–or sociological–solutions. Side by side with sophisticated constructivist and theory-laden descriptions of systems operations, Luhmann also resorts to everyday language and common sense, so that it is possible to find two epistemological tracks in Luhmann.