Autor: Flanagan, Josef

Buch: Quest for Self-Knowledge

Titel: Quest for Self-Knowledge

Stichwort: Zahlen, Rechenoperation

Kurzinhalt: Zahlenreihe 1, 2, 3 ..:Grundzahl 1; Zahlen als Wesen und Ergebnis von Rechenoperationen

Textausschnitt: 35/1 Students start out thinking that numbers have a single, fixed meaning, such as the number 4 which gets its assigned meaning from being counted or numbered after 3 and before 5. But, if you think of 4 as a sum (i.e., 3 + 1 = 4), then you are paying attention to the way the number is generated, which means that you have shifted attention away from the number and focused instead on the process of generating numbers through different operations. In this context the number 4 may be transformed from a sum (3 + 1 = 4) into a remainder (5 - 1 = 4), or into a product (2 X 2 = 4), or into a quotient (8 : 2 = 4). Once you understand that numbers are the way they are because of the operations that generated them, you can make the dramatic shift from defining terms or concepts to defining the operations that generate the terms or concepts. Instead of asking about the properties of a circle, you can ask about the properties of the operations that make circles to be the way they are. Thus, the definitions x2 + y2 = r2 may be identified as an equation in the second degree where the term 'second degree' refers to the type of operation that generates such geometrical forms as circles or squares or, more generally, numerical terms such as powers and roots. (25; Fs) (notabene)

36/1 This is what happened to mathematics in the nineteenth century. Mathematicians shifted their attention away from (i.e., abstracted from) the different types of numbers and started paying attention to the operations that generated those numbers. Thus, they made the discovery that the operations could be correlated to one another into a group, and that different groups generated different classes of numbers and had different ranges or powers of systematizing these objects. Finally, they grasped that there was a series of systems - like arithmetic, algebra, analytic geometry, and calculus - where successive systems were built upon prior systems by extending the range of objects that could be generated and combined with one another.1 (25f; Fs)

37/1 Such a series of systems that transformed and transcended prior systems provided scientists with the possibility of defining the notion of development in a radically new way. It was this possibility that Piaget and contemporary evolutionary theories have explored. More important, it is this distinction between an operation and the contents formed by that operation that has made possible the method that we are pursuing of inviting you to study your own operation of understanding, where attention shifts from 'the understood' to your own activity of understanding, which generates and forms the contents that you have understood. It is my purpose in this work not to study mathematics or science, but to study this act of understanding that has mediated and made possible the remarkable advances in these and other fields. (26; Fs)

38/1 Besides the kind of direct insights that we have been examining here, there is the rare, but even more unique, type of insight called 'inverse insight.' (26; Fs)