automorphisms

The basis elements of the automorphisms are e_a = ?

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In his hands, it turned from the study of substitutions in formulas and of functions invariant under such substitutions into the study of field extensions and corresponding automorphisms (Stein 1988, earlier Mehrtens 1979b).

Dedekind's Contributions to the Foundations of Mathematics Reck, Erich 2008

“NFU can be understood as the theory of certain nonstandard models of Zermelo-style set theory with automorphisms”

Quine's New Foundations Forster, Thomas 2006

Now, the mathematical notion of symmetry demands that there shall be given some group of automorphisms.

SYMMETRY AND ASYMMETRY SALOMON BOCHNER 1968

The symmetry of a figure is “interesting,” or “mean - ingful,” or “relevant,” mathematically or aesthetically, if the automorphisms of the underlying group are

SYMMETRY AND ASYMMETRY SALOMON BOCHNER 1968

It is again a relative concept and, again, a group of automorphisms of the space is involved.

SYMMETRY AND ASYMMETRY SALOMON BOCHNER 1968

From this approach, an “obvious” asymmetry is frequently a regular symmetry with respect to a group of automorphisms that are not strict orthogonal trans - formations but obvious distortions of such.

SYMMETRY AND ASYMMETRY SALOMON BOCHNER 1968

A figure is not just symmetric tout court, but it is symmetric relative to a given group of automorphisms, which, in a logical sense, has to be given first.

SYMMETRY AND ASYMMETRY SALOMON BOCHNER 1968

By this is meant that any point P of the space can be carried into any other point Q by at least one of the automorphisms.

SYMMETRY AND ASYMMETRY SALOMON BOCHNER 1968

In order to arrive at a general mathe - matically oriented notion of symmetry, it is also neces - sary to consider not only single automorphisms but certain assemblages of automorphisms called groups

SYMMETRY AND ASYMMETRY SALOMON BOCHNER 1968