Philosophy Dictionary of ArgumentsHome
| |||
|
| |||
| Geometry: Geometry is the branch of mathematics that deals with the shapes, sizes, and positions of figures. See also Mathematics, Arithmetics, Forms. _____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. | |||
| Author | Concept | Summary/Quotes | Sources |
|---|---|---|---|
|
Christian Thiel on Geometry - Dictionary of Arguments
I 287 Geometry/Protogeometry/Inhetveen/Thiel: There is often talk of an "operative" model of geometry, whereby it must be borne in mind that the properties captured in this way can only be realized if they are idealized. >Accuracy. I 288 There is the attempt of a "protogeometry" "circle-free method of size comparison" (Inhetveen) In order to satisfy the requirement of freedom from circles, we have to do without any geometric "devices" when producing shapes on bodies. I 289 The simplest operation with two bodies K1 and K2 is to bring them into contact with each other. The relation of touching is symmetrical. Two bodies each have at least one possible point of contact. Then further bodies K3 and K4 can always be constructed, so that K3 touches K1 at the point where K2 used to do so. "Imitation" "Replace". Inhetveen has called this "weak transitivity" because the subject requires three antecedents instead of two. I 289 Def "Weaker"/Thiel: means less demanding in mathematics. I 289/290 We extend our determinations to touching two bodies not only at individual points, but at all points of a given surface piece. The body definitions then "fit" together in these pieces. These formulas are statements about bodies, but they are not sentences about bodies that we have in front of us in our body world. We thus make statements about the manufacturing goals we pursue. Inhetveen describes them as "aphaetic" criteria for the quality of a technical realization. They lie protogeometrically before the theory of geometric forms. I 290/291 Now there are the terms of "fitting" as well as the original and impression derived from them. Fitting: "protogeometrically congruent". For technical purposes, however, one would not only like to be able to shape bodies in such a way that they fit, but also to fit a third person. Or that each of them also fits on the other. Def Weak transitivity of fitting: each body must fit to a copy of itself (since it cannot be brought to itself in a situation of fitting). Def "impression stable": the definition says nothing about how a body is brought to fit with any copy, and in fact this can happen in different ways...+...I 291 I 293 Folding axes, rotational symmetry, mirror symmetry are derived protogeometrically. Terms: "flat", "technical straight line" (= edge), "complementary", "supplementary wedges", "tilting", "edge". (...) The procedures are considered, the transition from protogeometry to geometry takes place in two abstraction steps. We ignore the methods and consider the results in the geometry. I 299 No reference is made to tools at any point. By the way, there are devices that are more effective than compasses and rulers: two "right-angle hooks" can achieve not only all constructions that can be done with compasses and rulers, but also those that lead analytically described to third-degree and fourth-degree equations. The bisector can be constructed using a copy. I 300 Protogeometry defined, geometry proven. >Proof. If geometry is to be the theory of constructible forms, then we have to consider this independence (describable as "size invariance" (>measurements)) and do this with what is known as the Form principle: if two further places P', Q' are obtained by a construction starting from two further places P,Q, each figure obtained by a sequence K1...Kn from construction steps of P' zbd Q' is geometrically indistinguishable from the figure to which the same construction steps starting from P and Q lead. I 301 A whole series of important statements of classical geometry can only be proved by using this principle. For example, the perpendicularity of the fourth angle in the theorem of thalas cannot be determined purely protogeometrically, nor can the uniqueness of the parallels to a given straight line be determined by a point outside. >Measurement. Only Euclidean geometry knows forms in the explained sense, in such a way that figures are identical in form if they cannot be distinguished and no application of the same consequences of further construction steps makes them distinguishable._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
||
Ed. Martin Schulz, access date 2024-04-20