By Ladislav Kvasz
The booklet deals a reconstruction of linguistic ideas within the background of arithmetic. It argues that there are a minimum of 3 ways within which the language of arithmetic should be replaced. As representation of alterations of the 1st type, referred to as re-codings, is the advance alongside the road: artificial geometry, analytic geometry, fractal geometry, and set concept. during this improvement the mathematicians replaced the very means of creating geometric figures. As representation of alterations of the second one style, referred to as relativization, is the improvement of artificial geometry alongside the road: Euclid’s geometry, projective geometry, non-Euclidean geometry, Erlanger software as much as Hilbert’s Grundlagen der Geometrie. adjustments of the 3rd type, referred to as re-formulations are for example the adjustments that may be visible at the varied variations of Euclid’s components. maybe the simplest identified between them is Playfair’s switch of the formula of the 5th postulate.
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Extra info for Patterns of Change: Linguistic Innovations in the Development of Classical Mathematics (Science Networks. Historical Studies)
Logical Boundaries – Existence of Insoluble Problems There are three problems, formulated during the early development of Greek geometry, which turned out to be insoluble using just the el- 28 Re-coding as the First Pattern of Change in Mathematics ementary methods of ruler-and-compasses construction. These problems are: to trisect an angle, to duplicate a cube, and to construct a square with the same area as a circle. The insolubility of these problems was proved with modern algebra and complex analysis, that is, in languages of higher expressive and explanatory power than that of the language of synthetic geometry.
For this reason it is impossible in this language to express any general statement or write a general formula. The rules for division or for multiplication, as general statements, are inexpressible in this language. They cannot be expressed in the language, but only shown. For instance the rule that multiplication by 10 consists in writing a 0 at the end of the multiplied number cannot be expressed in the language. It can only be shown on speciﬁc examples such as 17 10 D 170, or 327 10 D 3270. From such examples one understands that the particular numbers are unimportant and one grasps the universal rule.
If we prove some statement for such a segment, in fact we have proved the statement for a segment of any length, which means that we have proved a general proposition. The segment of indeﬁnite length is not a variable, because it is an expression of the iconic and not of the symbolic language. ). Of course, any concrete segment drawn in the picture has a precise length, but this length is not used in the proof, which means that the particular length is irrelevant. This substantiates the interpretation of geometrical pictures as a language.