ZusammenfassungDas Axiomensystem der ebenen projektiven Geometrie ist dual in dem Sinne, dass es bei Vertauschung der Begriffe « Punkt » und « Gerade » in sich übergeht. Daraus folgt, dass mit jedem Satz auch der duale Satz aus den Axiomen beweisbar ist. Dagegen kann aus der Dualität des Axiomensystems nicht geschlossen werden, dass in einem Modell mit jedem Satz auch der duale Satz gilt; noch weniger folgt, dass ein Modell eine eineindeutige Abbildung zulässt, welche Punkte and Geraden unter Erhaltung der (...) Inzidenz vertauscht. Im Falle der projektiven Geometrie sind solche Modelle bekannt; für die einfache Typentheorie wird gezeigt, dassa die Existenz solcher Modelle äquivalent ist mit der Widerspruchsfreiheit des Systems « New Foundations ».Zusatz. Der folgende Satz beantwortet die beiden in der Arbeit aufgeworfenen Fragen: Eine vollständige Theorie mit einem Automorphismus hat ein Modell, welches einen entsprechenden Automorphismus zulässt. Insbesondere ist somit NF widerspruchsfrei, wenn die einfache Typentheorie mit den zusätzlichen Axiomen S ≡ S* widerspruchsfrei ist.RésuméLe système d'axiomes de la géométrie projective plane est dualistique dans le sens qu'il se transforme en lui‐même si I'on y échange les notions « points » et « droite ». Cette dualité entraîne un parallélisme des théorémes et de leurs démonstrations. Par contre, la dualité d'un système d'axiomes ne nous permet pas de conclure au parallélisme des énoncés qui soient vrais dans un modéle donné du systéme; moins encore peut‐on insérer I'existence d'une transformation biunivoque échangeant points et droites et conservant la relation d'incidence. Pour la géométric projective, de tels modèles sont bien connus; pour la théorie simple des types, I'existence d'un tel modèle est équivalent a la cohérence du système « New Foundations ».Remarque. Le théoréme suivant résout les deux problèmes proposés dans le travail: Toute théorie complète admettant un automorphisme possède un modèle admettant un automorphisme correspondant. NF est par conséquent cohérent si la théorie simple des types avec les axiomes supplémentaires S ≡ S * est cohérente.The axiom system of plane projective geometry is dual in the sense that it is transformed into itself by exchange of the notions « point » and « line.» It follows that for every theorem the dual sentence is also a theorem. However, from the duality of the axiom system one cannot conclude that in a model the truth of a sentence implies that of the dual sentence; even less can one conclude that each model admits a 1‐1‐transformation interchanging points and lines and preserving the incidence relation. For projective geometry, models of this kind are well known. For the simple theory of types it is shown that the existence of such models is equivalent to the consistency of « New Foundations. »Additional remark. The following theorem answers both of the questions proposed in the paper: If it is complete, then a theory with an automorphism has a model with a corresponding automorphism. NF is therefore consistent if simple theory of type with the additional axioms S ≡ S* is consistent. (shrink)

SummaryIn order to use modern infinitary mathematics in the natural sciences, human beings have developed a corresponding picture of the world. The gnomes have made a similar step for their trade — books e.g. are sold and bought in infinite sets. In a gnomic‐human dialogue the problems arrising from such attitudes are discussed and a path from modern to postmodern mathematics is sketched.

Zwei ältere Herren, einer Biophysiker, der andere Psychologe, treffen sich, um über die Vergangenheit zu sprechen. Das könnte eine ganz alltägliche Geschichte sein. Allerdings handelt es sich bei den beiden Protagonisten um die Väter des Radikalen Konstruktivismus. Der Bitte ihre Verlegers, ein Tonband mitlaufen zu lassen, kamen sie gerne nach. Ein Glücksfall, wie sich im Nachhinein herausstellt. in einem ungemein spannenden, quicklebendigen Dialog werden komplexe wissenschaftliche Fragen auf allgemeinverständliche Weise thematisiert: Was haben Kurzwellen und Kühe gemein? Warum vergräbt ein Eichhörnchen (...) eine Nuss? Was verbindet diese Frage mit der folgenden: Was kommt zuerst - Sprache oder Bewusstsein? (shrink)

Chapter Growing up Constructivist: Languages and Thoughtful People What is radical constructivism? It is an unconventional approach to the problems of ...

Problem: How can constructivists speak of social interaction or communication with others, when, as they claim, their experiential world is their own construction? This question is frequently asked and is perfectly reasonable. The present paper is intended as an answer. Solution: After providing an outline of the constructivist approach to perception and the generation of recognizable objects in the experiential field, I argue that “others,” too, can be explained as an individual’s creation; a creation, however, that is just as constrained (...) by the condition of viability as are the physical objects with which we furnish our world. Consequently, “society” too can be considered an individual construct rather than an ontological given. Benefits: The exposition may help to clarify the constructivist position with regard to social interaction and communication. (shrink)

The existence of objective knowledge and the possibility of communicating it by means of language have traditionally been taken for granted by educators. Recent developments in the philosophy of science and the historical study of scientific accomplishments have deprived these presuppositions of their former plausibility. Sooner or later, this must have an effect on the teaching of science. In this paper I am presenting a brief outline of an alternative theory of knowing that takes into account the thinking organism's cognitive (...) isolation from 'reality'. This orientation was proposed by Vico at the beginning of the 18th century, disregarded for two hundred years, and then propounded independently by Piaget as a developmentally grounded constructivist epistemology. The paper focuses specifically on the adaptive function of cognition, Piaget's scheme theory, the process of communication, and the subjective perspective on social interaction. In the concluding section it then suggests some of the consequences the shift of epistemological presuppositions might have for the practice of teaching. (shrink)

Purpose: The paper contributes to the naturalization of epistemology. It suggests tentative itineraries for the progression from elementary experiential situations to the abstraction of the concepts of unit, plurality, number, point, line, and plane. It also provides a discussion of the question of certainty in logical deduction and arithmetic. Approach: Whitehead’s description of three processes involved in criticizing mathematical thinking (1956) is used to show discrepancies between a traditional epistemological stance and the constructivist approach to knowing and communication. Practical implications: (...) Reducing basic abstract terms to experiential situations should make them easier to conceive for students. (shrink)

Purpose: An attempt to interpret the fragments of Parmenides as a first suggestion of a constructivist epistemology. Today, two and a half millennia later, no one can be sure of what exactly Parmenides had in mind. Method: Reviewing the varying translations of acknowledged experts and paying attention to what Plato said in his Parmenides dialog. Findings: We cannot be certain of any interpretation, but an epistemologically unbiased review of the translations shows that Parmenides may well have believed that experience is (...) the domain accessible to human reason and the structure of the real world lies beyond it. Implications: Forerunners of constructivist ideas may have been overlooked because the interpreters of ancient texts were stuck in dogmatic views of “being.” When studying classical texts written in languages that are no longer spoken, it is fruitful to consult many different translations. (shrink)

Purpose: The notion of dichotomy is central to Josef Mitterer’s work and he uses the term as a portmanteau. My paper characterizes the specific dichotomies he describes, uses C. K. Ogden’s work on “Opposition” to classify them, and reviews attempts to overcome incompatible oppositions in other disciplines. Approach: Conceptual analysis in an attempt to show some of the conceptual differences in the various types of opposition. A “sampler” indicates possible divisions. Findings: From the constructivist point of view, the notion of (...) dichotomy is a complex one and must be divided into separate types, not all of which can be discarded in rational discourse. Implications: From this author’s perspective, Mitterer’s publications present a powerful stand against the tradition of realism and lead one to hope that his next will be a primer of non-dualistic discourse. (shrink)

Purpose: Radical constructivism holds that experiential reality is created by each individual. As a way of thinking, it unquestionably belongs to the theories of knowledge that are called “subjectivist” and “relativist”. This paper deals with the Italian philosopher Adriano Tilgher’s analysis of the relation between relativism and fascism and examines the possible impact of this connection on constructivism and its view of ethics. Approach: Conceptual analysis and the demonstration of a contradiction in Tilgher’s argumentation. Findings: A review of the ethics (...) inherent in Kant’s categorical imperative shows a tendency towards a subjectivism that may hint at anarchism but in no way implies fascism. Implications: This investigation should make it more difficult for critics of constructivism to relate it to fascism. (shrink)

Open peer commentary on the target article “Arguments Opposing the Radicalism of Radical Constructivism” by Gernot Saalmann. First paragraph: Gernot Saalmann freely admits that what he presents “as ‘radical constructivism’ is necessarily simplified and distorted, as there are considerable differences amongst the authors, who, in addition, have changed their opinions in the course of time.” (§3). This is an ominous opening. Conflating the critique of the different views of individual constructivists makes it difficult for individuals to respond. Given that Saalmann (...) refers thirty-eight times to Wendel, Maturana/ Varela, Rusch, and myself, and only sixteen times to four other delinquents, it would have made things easier, if he had dealt with the four principal ones individually. (shrink)

Open peer commentary on the target article “How and Why the Brain Lays the Foundations for a Conscious Self” by Martin V. Butz. Excerpt: I feel that the use of terms such as “code,” “information,” and “symbol” for neural constellations that are not further described defeats the intention of the analysis presented because unless these terms are explicitly given specific neurobiological definitions they inevitably suggest the presence of a consciously reflecting agent.

Excerpt: Love – as Ovid pointed out long ago – is an art. It has to be constantly created and requires persistent vigilance, care, and thoughtfulness. This is very clear from a constructivist point of view. The partner is always what we experience of him or her. We have abstracted him or her from our own experiences and therefore he or she is our construction and not, for example, a thing in itself which exists independently from us. And it is (...) this person who has been constructed by us, who has been credited with a past by us, who stirs up expectations in us. If these expectations are not met, we are disappointed and tend to blame the other person. We forget that actually we alone are responsible for how we think of our partner because how she or he acts and speaks can always be interpreted in a variety of different ways. (shrink)

From the constructivist perspective, science cannot transcend thedomain of experience. Scientific theories are seen as models that helpto order and manage that domain. As the experiential field expands,models are replaced by others based on novel conceptual constructs. Thepaper suggests the substitution of viability or functional fit forthe notions of Truth and objective representation of anexperiencer-independent reality. This by-passes the sceptics'incontrovertible arguments against certain real-world knowledge andproposes the Piagetian conception of cognition as the function thatgenerates ways and means for dealing with (...) the world of experience. (shrink)