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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.