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.

Crisis. Thought of the Day 66.0


Why do we have a crisis? The system, by being liberal, allowed for the condensation of wealth. This went well as long as there was exponential growth and humans also saw their share of the wealth growing. Now, with the saturation, no longer growth of wealth for humans was possible, and actually decline of wealth occurs since the growth of capital has to continue (by definition). Austerity will accelerate this reduction of wealth, and is thus the most-stupid thing one could do. If debt is paid back, money disappears and economy shrinks. The end point will be zero economy, zero money, and a remaining debt. It is not possible to pay back the money borrowed. The money simply does not exist and cannot be printed by the borrowers in a multi-region single-currency economy.

What will be the outcome? If countries are allowed to go bankrupt, there might be a way that economy recovers. If countries are continuing to be bailed-out, the crisis will continue. It will end in the situation that all countries will have to be bailed-out by each-other, even the strong ones. It is not possible that all countries pay back all the debt, even if it were advisable, without printing money by the borrowing countries. If countries are not allowed to go bankrupt, the ‘heritage’, the capital of the citizens of countries, now belonging to the people, will be confiscated and will belong to the capital, with its seat in fiscal paradises. The people will then pay for using this heritage which belonged to them not so long time ago, and will actually pay for it with money that will be borrowed. This is a modern form of slavery, where people posses nothing, effectively not even their own labor power, which is pawned for generations to come. We will be back to a feudal system.

On the long term, if we insist on pure liberalism without boundaries, it is possible that human production and consumption disappear from this planet, to be substituted by something that is fitter in a Darwinistic way. What we need is something that defends the rights and interests of humans and not of the capital, there where all the measures – all politicians and political lobbies – defend the rights of the capital. It is obvious that the political structures have no remorse in putting humans under more fiscal stress, since the people are inflexible and cannot flee the tax burden. The capital, on the other hand, is completely flexible and any attempt to increase the fiscal pressure makes that it flees the country. Again, the Prisoner’s Dilemma makes that all countries increase tax on people and labor, while reducing the tax on capital and money. We could summarize this as saying that the capital has joined forces – has globalized – while the labor and the people are still not united in the eternal class struggle. This imbalance makes that the people every time draw the short straw. And every time the straw gets shorter.