# Key word "numbers"

Context: The constructivist approach to the definition (or analysis) of the fundamental meanings of language in Ernst von Glasersfeld’s work. Problem: Has this approach achieved better results than other approaches? Method: Review of a book chapter by von Glasersfeld that is devoted to the analysis of the concepts of “unity,” “plurality” and “number.” Results: The constructivist approach to the semantics of the fundamental elements of language (some of which are fundamental for sciences too) seems to have produced positive results; moreover these are in a field where other approaches have produced results that do not objectively seem satisfactory.

Educational technology is firmly grounded in the rational tradition. However, there are growing numbers of educational technologists who consider themselves constructivist in orientation. In this paper I look at design in the field of educational technology through the lens of an enactive constructivist framework in order to locate trends that suggest a convergence with the enactive position as explicated in the works of Humberto Maturana and Francisco Varela. The enactive position provides a coherent framewok within which to guide constructivist practice.

Chilean biology covers a wide disciplinary spectrum, where evolutionary biology has been able to gain an outstanding presence, despite its notably small number of practitioners. In this regard, however, numbers may appear deceptive. For instance, a review of the papers published from 1983 to 1995 in the Revista Chilena de Historia Natural, which covers all naturalist disciplines, showed that only 4. 7 %dealt with evolutionary aspects sensu lato, a very low percent in comparison to dominant disciplines such as botany, zoology and ecology (Camus 1995) Of course, Chilean evolutionists publish in other Chilean and foreign journals, and thus the above figure is just a vague reference on the relative importance of evolution. Nevertheless, quantitative estimations could not capture the real importance or the impact of evolutionary knowledge on the formation of Chilean naturalists, regardless any explicit or implicit consideration in their own studies. Very likely, Chilean naturalists do see in evolution the ultimate foundation for their work, as an echo of that legendary statement by T. Dobzhansky, and partly as a result of a long darwinian tradition in Chilean universities. In fact, Manrfquez & Rothhammer ( 1997) documented that Darwin’s theory was already incorporated in some school texts as early as 1866, and in 1917 it was approved as part of the official educational program for public schools. This certainly lead to intense public debates between lay and catholic sectors, which lasted for about 60 years. However, Manrfquez & Rothhammer (1997) also mentioned that such a debate not only was virtually absent in Chilean universities, but darwinian theory, and even the basic tenets of the rising synthetic theory of evolution, were formally included in university curricula during the first decades of the 20�h century.

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.

The purpose of this essay is to sketch a picture of the connections between the concept self-reference and important aspects of mathematical and cybernetic thinking. In order to accomplish this task, we begin with a very simple discussion of the meaning of self-reference and then let this unfold into many ideas. Not surprisingly, we encounter wholes and parts, distinctions, pointers and indications. local-global, circulation, feedback. recursion, invariance. self-similarity, re-entry of forms, paradox, and strange loops. But we also find topology, knots and weaves. fractal and recursive forms, infinity, curvature and imaginary numbers! A panoply of fundamental mathematical and physical ideas relating directly to the central turn of self-reference.

This is the seventh column in this series on “Virtual Logic.” In this column I will discuss an imaginary machine devised by the logician Raymond Smullyan. Smullyan managed to compress the essence of Gödel’s theorem on the incompleteness of formal systems into the properties of a devilish machine. This column consists in two parts. In the first part we find a story/satire about such a machine, with the Smullyan structure at its core. In this story, the protagonist is bent on detecting a flaw in the machine and he operates with strict two-valued logic. In such logic a statement is either true or false. Thus we call the statement “If unicorns can fly then all numbers are less than pi.” true because it is not definitely false. In general “A implies B” is taken to be false only if A is true and B is false. This is the one significant case where “A implies B” must be false. All other cases, such as A false and B true are taken to be true. This is the classical logical convention. It works quite well in its own domain, but it has its limits. One of these limits occurs when there is a gradation of qualities. For example in statements about tall and short the truth is relative to your idea of this discrimination. Another limit is in the realm of self-referential statements. Certainly the Liar Paradox – “This statement is false.” is neither true nor false in any timeless sense.

The purpose of this column is to go underneath the scene of numbers as we know them and to look at how these operations of addition and multiplication can be built in terms of a little technology of distinctions and the void. This is a story as old as creation herself, and we shall take some time to point out a mythological connection or two as we go along.

This is column number 11 of our series on Virtual Logic. In this column we will discuss the mathematical subject of Newton’s calculus using the notion of infinitesimals as an imaginary form of number. A form that allows us to reason to real answers and compute limits that would otherwise be ineluctable. I shall also discuss “zero numbers,” a concept grounded in ordinary numbers that opens up the powers of zero so that 0, 0x0, 0x0x0,…. are all distinct forms of the void. We shall see that these subjects are related to one another and that they are both related to the perennial topic of this column – paradox resolution and the relation of imagination/imaginary values to properties of a system as a whole.

This paper is an exposition and extension of ideas begun in the work of G. SpencerBrown (Laws of Form). We discuss the relations between form and process, distinction and indication by the use of simple mathematical models. These models distill the essence of the ideas. They embody and articulate many concepts that could not otherwise be brought into view. The key to the approach is the use of imaginary Boolean values. These are the formal analogs of complex numbers – processes seen as timeless forms, then indicated (self-referentially) and re-entered into the discourse that engendered them. While the discussion in this paper is quite abstract, the ideas and models apply to a wide range of phenomena in mathematics, physics, linguistics, perception and thought.

This essay concerns fractal geometry as a bridge between the imaginary and the real, mind and matter, conscious and unconscious. The logic rests upon Jungs theory of number as the most primitive archetype of order for linking observers with the observed. Whereas Jung focused upon natural numbers as the foundation for order that is already conscious, I offer fractal geometry, with its endlessly recursive iteration on the complex number plane, as the underpinning for a dynamic unconscious destined never to become fully conscious. Everywhere in nature, fractal separatrices articulate a paradoxical zone of bounded infinity that both separates and connects natures edges. By occupying the ‘space between’ dimensions and levels of existence, fractal boundaries exemplify reentry dynamics of Varela’s autonomous systems, as well as Hofstadters ever-elusive ‘tangled hierarchy’ where brain and mind are most entwined. At this second-order, cybernetic frontier, the horizon of observers observing the observation process remains infinitely complex and ever receding from view. I suggest that the property of self-similarity, by which the pattern of the whole permeates fractal parts at different scales, represents the semiotic sign of identity in nature.

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