Spinors, Twistors and Ontologies of SpaceTime

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Penrose’s conception of spacetime based on the complex analysis as well as Manin’s stem from the notions of spaces of spinors and twistors. Manin’s construction is complex space of spinors, which is a base by means of which 3-dimensional Euclidean space of the classical mechanics and 4-dimensional Minkowski’s space can be defined. The similarity to Plato’s world can be seen through the essential issue of the philosophy of nature, the issue of a relationship between a mathematical model described by a physical theory and the world. In contemporary physics, this relationship poses some subtle problems due to the fact that mathematical models themselves are constructions with abstract, intricate and many level structures. That is the case when we take into consideration the relationship between unitary space of spinors and the structure of physical spacetime. Here the relationship is indirect to a large extent. The Euclidean space and Minkowski’s space are the intermediate structures between fundamental mathematical structure — complex space of spinors — and physical spacetime. That the unitary complex space is fundamental means that it allows to define both, the Euclidean and Minkowski’s spaces. On the other hand, the indispensable intermediary role of those classical structures is played through their relation with experiments and measurement, that can be made only in their categories. Here, we find an analogy with Timaeus’ ontology. Plato’s triangles and bodies, geometrical substratum of the world, correspond to deep structure of spacetime — abstract, complex mathematical structure that allows to define models of spacetime of the classical mechanics. The significance of the latter models is not weakened since they describe adequately — to use the expression taken from the domain of linguistics — surface structure of the physical spacetime, as they enable us make concrete measurements, that serve as a base of verification of a physical theory. Therefore, they connect the ideal Plato’s world with the world of phenomena, similarly like in “Timaeus” a description of this surface structure of nature, i. e. concrete events, was made by means of the four elements, the frame of which was the actual geometrical substratum of nature. It is remarkable that Penrose, whose contribution to the examination of complex spaces of spinors and twistors was the most valuable in our times, shares the view of the strong mathematical Platonism concerning ontology as well as epistemology.

Relation: local-global is of great importance not only in ontology, but also in entire science and philosophy. On the one hand, ontology tends to be defined as knowledge concerning the notion of the whole — the notion of global nature, indeed. On the other hand, the contrast: local-global is often used to define and contrast scientific knowledge and philosophy. Such a view is expressed by René Thom, who thinks that the basic feature of a scientific theory is its locality expressed as the possibility to geometrize it. This view is also shared by Maurin, who states that a category of the whole is specifically philosophical, strictly religious. In the domain of the ontology of spacetime the latest mathematical models that use the methods of global analysis on complex manifolds let us obtain important results concerning the connection between local homogeneity of spacetime and its global homogeneity. The former one, well proved by the whole classical physics and through the Noether’s theorem connected with the principles of conservation in the classical mechanics, has purely scientific nature, the latter, on the other hand, left without any justification would be only an arbitrary metaphysical postulate commonly assumed, since it provides “comfortable” universality of physical laws in the whole Universe. And here, contemporary mathematics can give a kind of solution. The crucial significance for a demonstration of the global homogeneity of spacetime resulting from the local homogeneity of spacetime has Penrose’ postulate which defines spacetime as a 2-dimensional complex, i. e. 4-dimensional real, holomorphic manifold. For such manifolds the principle of identity binds. According to this principle, for any two holomorphic or meromorphic functions, if they are identical in optionally small neighbourhood, they are also identical on the whole manifold. Such a geometrical model of spacetime lessens remarkably the arbitrariness of the metaphysical postulate of global homogeneity of spacetime. What follows, is the connection between what is local, so scientific, and what is global, so ontological, — the means that enables this connection is mathematics, strictly speaking, global analysis.

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