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|Geometry||Thiel I 287
Geometry/Protogeometry/Inhetveen/Thiel: Often, an "operational" model of geometry is mentioned, whereby it must be taken into account that the properties thus acquired can only be realized if they are idealized. (> Accuracy).
There is an attempt of a "protogeometry": "a circular-free method of size comparison" (Inhetveen)
In order to satisfy the circular freedom, we have to deal without the need for recourse to geometrical "devices" in the production of forms on bodies.
The simplest operation with two bodies K1 and K2 is to bring them into contact with each other.
The relation of the touch is symmetrical. Two bodies each have at least one possible contact point.
Further bodies K3 and K4 can then always be constructed, so that K3 contacts K1 at the point where K2 previously did this. "Imitation", "Replace". Inhetveen has called this the "weak transitivity". The subjungat requires three rather than two antecedents.
Definition "weaker"/Thiel: weaker means in mathematics less prerequisite.
We extend our regulations to the touching of two bodies, not only at individual points, but in all parts of a given surface part. The bodies (Definition) "fit" then in these pieces.
These formulas are statements about bodies, but they are not sentences about bodies that we have before us in our body world. In this way, we make statements about the production targets we are pursuing. Inhetveen describes it as "aphairetic" (from drawing, taking away) criteria for the quality of a technical realization. They lie protogeometrically before the theory of geometric forms.
Now there are the terms of the "fitting" as well as derived from that one of the original and imprint. Fit: "protogeometrically congruent".
For technical purposes, however, one would not only like to be able to shape bodies in such a way that they fit one another, but also fit a third one. Or that each of them fits the other.
Definition weak transitivity of the fitting: every body must match a copy of itself (since it cannot be brought to itself in a situation of fitting).
Definition "copy stable": the definition says nothing about how a body is made to fit with any copy, and in fact it can happen in different ways .... + ... I 291
Folding axes, rotational symmetry, mirror symmetry are derived protogeometrically. Terms: "flat", "technical line" (=edge), "complementary", "supplementary wedges", "tipping", "edge".
The methods are considered. The transition from protogeometry to geometry takes place in two abstraction steps. We do not look at the methods and consider the results in geometry.
No reference is made to tools. By the way, there are devices that are more effective than compasses and rulers: two "right-angle hooks" cannot only achieve all constructions that can be executed with compasses and rulers, but also those which lead analytically to equations of third and fourth degrees.
The angle bisector can be constructed by means of a copy.
((s)Fitting/((s): Equality in forms does not lead to fit: E.g. plugs fit on sockets, but not sockets on sockets and not plugs on plugs.)
Protogeometry defined, geometry proves. (> Proof).
If geometry is to be the theory of constructible forms, then we have to take into account this independence (which can be described as "quantity invariance" (> measure)) and do so with the, in constructive science theory, so-called
Form principle: If two additional points P', Q' are obtained by a construction extending from two points P, Q, then each figure obtained by means of a sequence K1... Kn of construction steps from P' zbd Q' is geometrically indistinguishable from the figure to which the same construction steps of P and Q lead.
A whole series of important statements of classical geometry can only be proved by using this principle. For example, the squareness of the fourth angle in the Thales' theorem can be assured in a purely protogeometric manner just as little as the uniqueness of the parallels to a given straight line through a point outside.
Only the Euclidean geometry knows forms in the explained sense in such a way that figures are equal in form if they cannot be distinguished and no application of the same consequences of further steps of construction makes them distinguishable.
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