Open peer commentary on the article “Cybernetics, Reflexivity and Second-Order Science” by Louis H. Kauffman. Upshot: This commentary reflects broadly on the concept of eigenform and reflexive domains, focusing on the idea that second-order science is neither the same as nor completely distinct from ordinary living.

I discuss what it means to comprehend in Gasparyan’s theory that defines language as an eigenform. For that, I raise questions concerning the conditions of mutual understanding of signs, ….

Open peer commentary on the article “Cybernetics, Reflexivity and Second-Order Science” by Louis H. Kauffman. Upshot: One of cybernetics most confusing and least understood concepts is that of the eigenform. With this article Kauffman has enabled a clear understanding of the concept as “the coherence of a situation that allows a distinction to be made.”.

The hypothesis of this paper is that language is one more Eigenform, the “external” description of which is impossible. It follows that the application of second-order cybernetics to Eigenform might be adequate. In this article, I would like to concentrate on one relatively small aspect of the idea of Eigenform suggested by Foerster, Kauffman and Spenser-Brawn. I will use Foerster`s recursive approach namely that neither observer nor the thing observed can precede each other, but instead mutually assume (...) each other. In my research, language would stand in the place of the observer, and in the place of the thing – the world itself. To specify further, I focus on the problem of noting, and in particular, on how signs correspond to things. Therefore, I will try to show that the sign and the object do not precede one another and do not exist isolated from each other, but on the contrary, condition each other. Thus, language creates the world of objects, but in turn, it is created by the world. Thus it is a model of self-referentiality, which arranges the form of language as Eigenform as the idea of “mutual referentiality,” as we see it in the relation between language and the world. (shrink)

Upshot: Eigenform is a precondition for distinction and distinction is a precondition for eigenform. While my target article discussed eigenform and reflexivity, it could just as well have discussed distinctions and the emergence of distinctions. This theme was implicit in many of the comments. We make this circularity explicit for the sake of a deeper understanding.

Context: Recent decades have seen the development of new branches of semiotics, including biosemiotics, cognitive semiotics, and cybersemiotics. An important feature of these concepts is the ….

Context: By proposing to regard objects as “tokens for eigenbehavior,” von Foerster’s seminal paper opposes the intuitive subject-object dualism of traditional philosophy, which considers objects to be instances of an external world Problem: We argue that this proposal has two implications, one for epistemology and one for the demarcation between the natural sciences and the humanities. Method: Our arguments are based on insights gained in computational models and from reviewing the contributions to this special issue. Results: Epistemologically, von Foerster’s proposal (...) suggests that what is called “reality” could be seen as an ensemble of eigenforms generated by the eigenbehavior that arises in the interaction of multiple dynamics. Regarding science, the contributions to this special issue demonstrate that the concept of eigenbehavior can be applied to a variety of disciplines from the formal and natural sciences to the humanities. Its universal applicability provides a strong argument for transdisciplinarity, and its emphasis on the observer points in the direction of an observer-inclusive science. Implications: Thinking in eigenbehavior may not only have implications for tearing down the barriers between sciences and humanities, a better understanding of eigenbehaviors may also have profound effects on our understanding of ourselves. This also opens the way to innovative behavior design/modification technologies. (shrink)

Context: Second-order cybernetics and its implications have been understood within the cybernetics community for some time. These implications are important for understanding the structure of scientific endeavor, and for researchers in other fields to see the reflexive nature of scientific research. This article is about the role of context in the creation and exploration of our experience. Problem: The purpose of this article is to point out the fundamental nature of the circularity in cybernetics and in scientific work in general. (...) I give a point of view on the nature of objective knowledge by placing it in the context of reflexivity and eigenform. Method: The approach to the topic is based on logical analysis of the nature of circularity. Mathematics and cybernetics are both fundamentally concerned with the structure of distinctions, but there can be no definition of a distinction without circularity, since such a definition would itself be a distinction. The article proceeds by explicating the structure of reflexive domains D where the transformations of the domain are in one-to-one correspondence with the domain itself. Results: I show that every element of a reflexive domain has a fixed point. This means that eigenforms arise naturally in reflexive domains. Furthermore, a reflexive domain is itself an eigenform (at a higher level. This supports the context of a second-order science that would study domains of science as part of a larger cybernetic landscape. Implications: The value of the article is in its concise reformulation of the scientific endeavor as a search for eigenforms in reflexive domains. This new view of science is promising in that it includes the former worlds of apparent objectivity and it embraces those newer worlds of science where the theories and theorists become active participants in the ongoing process of creating knowledge. Constructivist content: I argue that the perspective of reflexive domains constitutes a new way to think about the practice of science, with observers deeply imbedded, and objectivity understood as the mutual search for eigenforms. (shrink)

The phrase ‘mathematical foundation’ has shifted in meaning since the end of the nineteenth century. It used to mean a consistent general theory in mathematics, based on basic principles and ideas to which the rest of mathematics could be reduced. There was supposed to be only one foundational theory and it was to carry the philosophical weight of giving the ultimate ontology and truth of mathematics. Under this conception of ‘foundation’ pluralism in foundations of mathematics is a contradiction.More recently, the (...) phrase has come to mean a perspective from which we can see, or in which we can interpret, much of mathematics; it has lost the realist-type metaphysical, essentialist importance. The latter has been replaced with an emphasis on epistemology. The more recent use of the phrase shows a lack of concern for absolute ontology, truth, uniqueness and sometimes even consistency. It is only under the more modern conception of ‘foundation’ that pluralism in mathematical foundations is conceptually possible.Several problems beset the pluralist in mathematical foundations. The problems include, at least: paradox, rampant relativism, loss of meaning, insurmountable complexity and a rising suspicion that we can say anything meaningful about truth and objectivity in mathematics. Many of these are related to each other, and many can be overcome, explained, accounted for and dissolved by concentrating on crosschecking, fixtures and rigour of proof. Moreover, apart from being a defensible position, there are a lot of advantages to pluralism in foundations. These include: a sensitivity to the practice of mathematics, a more faithful account of the objectivity of mathematics and a deeper understanding of mathematics.The claim I defend in the paper is that we stand to learn more, not less, by adopting a pluralist attitude. I defend the claim by looking at the examples of set theory and homotopy type theory, as alternative viewpoints from which we can learn about mathematics. As the claim is defended, it will become apparent that ‘pluralism in mathematical foundations’ is neither an oxymoron, nor a contradiction, at least not in any threatening sense. On the contrary, it is the tension between different foundations that spurs new developments in mathematics. The tension might be called ‘a fruitful meta-contradiction’.I take my prompts from Kauffman’s idea of eigenform, Hersh’s idea of thinking of mathematical theories as models and from my own philosophical position: pluralism in mathematics. I also take some hints from the literature on philosophy of chemistry, especially the pluralism of Chang and Schummer. (shrink)

Gasparyan shows the relationship of eigenform with semiosis. In agreement with her, I discuss these ideas from my own viewpoint.