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By Breit G.

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A metric space is a set (the points) with a distance function. This is a function d which maps every pair of points into a non-negative real number, and such that Which metric spaces are the Euclidean ones is most easily explained through the representation theorem. The Euclidean space of dimension k is isomorphic to the metric space, whose elements are the k-tuples of real numbers, with the distance d(x, y) being the norm of their difference; that is, . All other geometric notions such as angle and congruence can here be defined in terms of distance.

Each sphere would have some asymmetry, but it does not even make sense to ask whether all the spheres have their heart on the same side. The significant relationship exhibited by the human species must have to do with some other features as well, like feet, head, navel. . All this seems so accidental that we speculate at once that the whole of fundamental physics can be written without reference to this ‘handedness’ of nature. Such a pervasive asymmetry on earth must have been due to some asymmetry in the initial conditions.

In the semantic view a theory typically, in presenting a class of models, describes a kind of system, with an implied classification of sub-kinds. Whether a given real system—such as the world, or you, or I—is deterministic is therefore definitely not a univocal question. This also suggests various options for the interpretation of theories. First, it does not matter how a theory presents its class of models. So it might just say, for example, that the models all have a certain state-space type, and the possible trajectories satisfy certain equations and are deterministic.

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