|Axiom: principle or rule for linking elements of a theory that is not proven within the theory. It is assumed that axioms are true and evident. Adding or eliminating axioms turns a system into another system. Accordingly, more or less statements can be constructed or derived in the new system. > System.|
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|Axioms||A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967
Axiomatics/d'Abro: This new science was developed mainly by the formalists Hilbert and Peano.
Hilbert/d'Abro: Examples of Hilbert's typical claims:
1. Two different points, A and B, always form a straight line.
2. Three different points, A, B, and C, which do not lie on a straight line, always form a plane.
3. Of three points lying on a straight line, there is one and only one between the other two.
4. If the segment AB is equal to the segments A'B 'and A''B'', then A'B' is equal to A''B''.
The N.B. of Hilbert's postulates: points, lines, and planes are not the only quantities which satisfy these relations: with some imagination others can be found.
E.g It originally refers to plane geometry and can be given a different meaning: circles as new lines, with angles as distances.
All relations are fulfilled, so the new model and the old (Euclidean) model can be regarded as different models or so-called "concrete representations", both corresponding to the postulates.
It may seem absurd, but Hilbert warns against assigning a priori certain characteristics to the points and lines which he mentions in his postulates.
We can replace the words point, straight, plane, in all postulates by letters a, b, c. If we then employ points, lines, and planes, we obtain the Euclidean geometry, if we employ others, whose relations, however, must be the same, we have a new model between point, lines and planes. They are isomorphic.
For example, the new elements are expressed by a group of three of numbers and by algebraic terms which relate these numbers to one another.
He had this idea when he chose cartesian coordinates instead of points, lines and planes.
The fact that the new elements, here numerical, satisfied Hilbert's postulates, proves only that the simple geometrical ways of concluding and the Cartesian method are equivalent to analytical geometry.
This proves the logical equivalence of the geometric and arithmetic continuum.
Long before Hilbert, mathematicians had realized that mathematics has to do with relationships, and not with content.
With Hilbert's postulates, we can create the Euclidean geometry, even without knowing what is meant by point, line and plane.
The achievements of axiomatics:
1. They are of invaluable value, from the analytical as well as from the constructive point of view.
2. It has shown that mathematics is about relationships and not about content.
3. It has shown that logic itself cannot confirm the consistency.
4. It has also shown that we have to go beyond axiomatics and have to show their origin.