Weyl and Automorphism of Nature. Drunken Risibility.

In classical geometry and physics, physical automorphisms could be based on the material operations used for defining the elementary equivalence concept of congruence (“equality and similitude”). But Weyl started even more generally, with Leibniz’ explanation of the similarity of two objects, two things are similar if they are indiscernible when each is considered by itself. Here, like at other places, Weyl endorsed this Leibnzian argument from the point of view of “modern physics”, while adding that for Leibniz this spoke in favour of the unsubstantiality and phenomenality of space and time. On the other hand, for “real substances” the Leibnizian monads, indiscernability implied identity. In this way Weyl indicated, prior to any more technical consideration, that similarity in the Leibnizian sense was the same as objective equality. He did not enter deeper into the metaphysical discussion but insisted that the issue “is of philosophical significance far beyond its purely geometric aspect”.

Weyl did not claim that this idea solves the epistemological problem of objectivity once and for all, but at least it offers an adequate mathematical instrument for the formulation of it. He illustrated the idea in a first step by explaining the automorphisms of Euclidean geometry as the structure preserving bijective mappings of the point set underlying a structure satisfying the axioms of “Hilbert’s classical book on the Foundations of Geometry”. He concluded that for Euclidean geometry these are the similarities, not the congruences as one might expect at a first glance. In the mathematical sense, we then “come to interpret objectivity as the invariance under the group of automorphisms”. But Weyl warned to identify mathematical objectivity with that of natural science, because once we deal with real space “neither the axioms nor the basic relations are given”. As the latter are extremely difficult to discern, Weyl proposed to turn the tables and to take the group Γ of automorphisms, rather than the ‘basic relations’ and the corresponding relata, as the epistemic starting point.

Hence we come much nearer to the actual state of affairs if we start with the group Γ of automorphisms and refrain from making the artificial logical distinction between basic and derived relations. Once the group is known, we know what it means to say of a relation that it is objective, namely invariant with respect to Γ.

By such a well chosen constitutive stipulation it becomes clear what objective statements are, although this can be achieved only at the price that “…we start, as Dante starts in his Divina Comedia, in mezzo del camin”. A phrase characteristic for Weyl’s later view follows:

It is the common fate of man and his science that we do not begin at the beginning; we find ourselves somewhere on a road the origin and end of which are shrouded in fog.

Weyl’s juxtaposition of the mathematical and the physical concept of objectivity is worthwhile to reflect upon. The mathematical objectivity considered by him is relatively easy to obtain by combining the axiomatic characterization of a mathematical theory with the epistemic postulate of invariance under a group of automorphisms. Both are constituted in a series of acts characterized by Weyl as symbolic construction, which is free in several regards. For example, the group of automorphisms of Euclidean geometry may be expanded by “the mathematician” in rather wide ways (affine, projective, or even “any group of transformations”). In each case a specific realm of mathematical objectivity is constituted. With the example of the automorphism group Γ of (plane) Euclidean geometry in mind Weyl explained how, through the use of Cartesian coordinates, the automorphisms of Euclidean geometry can be represented by linear transformations “in terms of reproducible numerical symbols”.

For natural science the situation is quite different; here the freedom of the constitutive act is severely restricted. Weyl described the constraint for the choice of Γ at the outset in very general terms: The physicist will question Nature to reveal him her true group of automorphisms. Different to what a philosopher might expect, Weyl did not mention, the subtle influences induced by theoretical evaluations of empirical insights on the constitutive choice of the group of automorphisms for a physical theory. He even did not restrict the consideration to the range of a physical theory but aimed at Nature as a whole. Still basing on his his own views and radical changes in the fundamental views of theoretical physics, Weyl hoped for an insight into the true group of automorphisms of Nature without any further specifications.

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Yin & Yang Logarithmic Spirals

The figure depicts the well-known black and white symbol of Yin and Yang. The dots of different color in the area delimited by each force symbolize the fact that each force bears the seed of its counterpart within itself. According to the principle of Yin and Yang outlined above, neither Yin nor Yang can be observed directly. Both Yin and Yang are intertwined forces always occurring in pairs, rather than being isolated forces independent from each other. In Chinese philosophy, Yin and Yang assume the form of spirals. Let us now show that the net force in

K = −p(K ) ∗ ln(1 − p(K )/p(K))

the performance of a given confidence value K always matches exactly the expectation, i.e. in other words E = p(K), is a spiral too. In order to do so, let us introduce the general definition of the logarithmic spiral before illustrating the similarity to the famous Yin/Yang symbol.

A logarithmic spiral is a special type of spiral curve, which plays an important role in nature. It occurs in all different kinds of objects and processes, such as mollusk shells, hurricanes, galaxies, etc. In polar coordinates (r, θ), the general definition of a logarithmic spiral is

r = ae^bθ

Parameter a is a scale factor determining the size of the spiral, while parameter b con- trols the direction and tightness of the wrapping. For a logarithmic spiral, the distances between the turnings increase. This distinguishes the logarithmic spiral from the Archimedean spiral, which features constant distances between turnings. The figure below  depicts a typical example of a logarithmic spiral.

Resolving r = ae^bθ for θ leads to the following general form of logarithmic spirals:

θ = 1/b ln (r/a)

In order to show that the net force in

K = −p(K ) ∗ ln(1 − p(K )/p(K))

defines a logarithmic spiral, and for the sake of easier illustration, let us look at the negative version of the net force in the above equation for net force and look at the polar coordinates (r, θ) it defines, namely:

θ = −p(K) ∗ ln(p(K)/(1−p(K)))

and

r = (1−p(K)) ∗ e^−θ/p(K)

A comparison of θ = −p(K) ∗ ln(p(K)/(1−p(K))), r = (1−p(K)) ∗ e^−θ/p(K) with the general form of logarithmic spirals in θ = 1/b ln (r/a) shows that the net force does indeed describe a spiral. Both of these equations match when we set the parameters a and b to the following values:

a = 1−p(K)

and

b = − 1/p(K)

In particular, we can check that a and b are identical when p(K) equals the golden ratio, which happens to be

φ ≈ 1.618 ∨ −0.618

If we let p(K) run from 0 to 1, and mirror the resulting spiral along both axes, we receive two spirals. Figure 8 shows both spirals plotted in a Cartesian coordinate system. Both spirals are, of course, symmetrical and their turnings approach the unit circle. A comparison of the Yin/Yang symbol with the spirals in the figure below shows the strong similarities between both figures.

A simple mirror operation transforms the spirals in the above figure into the Yin/Yang symbol. The addition of a time dimension to the above figure generates a three-dimensional object.

The above is an informational universe. Note that the use of performance as time is reasonable because the exponential distribution is typically used to model dynamic time processes and the expectation value is thus typically associated with time.

Yin and Yang with deep roots in Chinese philosophy, stand for two principles that are opposites of each other, and which are constantly trying to gain the upper hand over each other. However, neither one will ever succeed in doing so, though one principle may temporarily dominate the other one. Both principles cannot exist without each other. It is rather the constant struggle between both principles that defines our world and produces the rhythm of life. According to Chinese philosophy, Yin and Yang are the foundation of our entire universe. They flow through, and thus affect, every being. Typical examples of Yin/Yang opposites are, for example, night/day, cold/hot, rest/activity, etc. Chinese philosophy does not confine itself to a mere description of Yin and Yang. It also provides guidelines on how to live in accordance with Yin and Yang. The central statement is that Yin and Yang need to be in harmony. Any imbalance of an economical, biological, physical, or chemical system can be directly attributed to a distorted equilibrium between Yin and Yang.