By Ladislav Kvasz
The e-book bargains a reconstruction of linguistic suggestions within the background of arithmetic. It argues that there are at the very least 3 ways within which the language of arithmetic should be replaced.
As representation of adjustments of the 1st type, known as re-codings, is the advance alongside the road: artificial geometry, analytic geometry, fractal geometry, and set idea. during this improvement the mathematicians replaced the very manner of creating geometric figures.
As representation of alterations of the second one style, referred to as relativization, is the advance of artificial geometry alongside the road: Euclid’s geometry, projective geometry, non-Euclidean geometry, Erlanger software as much as Hilbert’s Grundlagen der Geometrie.
Changes of the 3rd sort, referred to as re-formulations are for example the adjustments that may be obvious at the assorted variants of Euclid’s parts. probably the simplest recognized between them is Playfair’s swap of the formula of the 5th postulate.
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Extra resources for Patterns of Change: Linguistic Innovations in the Development of Classical Mathematics
Integrative power – what sort of unity and order the language enables us to conceive there, where we perceived just unrelated particular cases in the previous stages, 5. Logical boundaries – marked by occurrences of unexpected paradoxical expressions, 6. Expressive boundaries – marked by failures of the language to describe some complex situations. The evolution of the language of mathematics consists in the growth of its logical and expressive power – the later stages of development of the language make it possible to prove more theorems and to describe a wider range of phenomena.
There- 30 Re-coding as the First Pattern of Change in Mathematics fore the close connection between knowledge and sight which formed the core of the Greek epist´eme, was lost. The Greek word theoria is derived from theoros, which was a delegate of the polis who had to oversee a religious ceremony without taking part in it. A theoria was what the theoros saw. This means that for the Greeks a theory is what we see when we observe the course of events without participating in them. Thus the metaphor behind the Greek notion of theoretical knowledge was view from a distance.
Here we see one of the fundamental strategies of dealing with paradoxes. It consists in the extension of the language and in the reinterpretation of the paradox so that it becomes a positive fact. Nevertheless, it took many centuries until the incommensurability was resolved inside of the symbolic language itself. 24 Re-coding as the First Pattern of Change in Mathematics Geometrical language, in contrast to the language of arithmetic, is an iconic and not a symbolic language. Its expressions are pictures formed of points, line segments and circles, rather than formulas formed of linear sequences of arithmetic symbols.