When we speak of similarity between two individuals, we mean that some of the properties that are attributed to one can be attributed to the other. When we speak of similarity between two systems, the expression 'what is true of one is equally true of the other' can receive a more subtle meaning. We mean that there exists a projection f which makes it possible to replace the individuals x of the first system by individuals f(x) of the second system, in such a way that true statements about the first system are replaced by true statements about the second system. Such a projection is called in mathematics a morphism, or an isomorphism if it is bijective, to say that the two systems have the same form, or the same structure. The current use of the concept of structure is ambiguous. The structure designates sometimes the object, the system, sometimes its property. Structures have a structure. From a logical point of view, a structure as an object is a logically possible world or part of such a world. A structure as a property can be defined from the equivalence relation x has the same structure as y. This equivalence relation can be defined with the concept of isomorphism: Two structures (or two systems) have the same structure if and only if they are isomorphic. An isomorphism between two structures E and F is a bijective function f which replaces the individuals of E by individuals of F so that all fundamental properties and relations are retained. Formally: If P is a fundamental property, for all x in E, x has the property P if and only if f(x) has the property P. If R is a fundamental binary relation, for all x and all y in E, xRy if and only if f(x)Rf(y) The same goes for the fundamental relations between more terms. (A relation between the elements of E and the elements of F defines an application of E into F when each element of E is connected to a single element of F. An application of E into F is bijective when each element of F is connected to a single element of E. In other words, a bijective function is an application whose inverse is also an application.) An isomorphism between two structures makes it possible to transform all true statements about one into true statements about the other, by replacing everywhere x by f(x). When two structures are isomorphic, they are models of the same theories. Any system of axioms of true one is necessarily true of the other. A complex natural being is a natural structure, defined with natural properties and relations. Two isomorphic complex natural beings are essentially similar, naturally indiscernible. They have the same natural properties. Everything that is naturally possible with one is naturally possible with the other. The nature of a complex natural being is its structure. Two isomorphic complex natural beings have the same nature. The concept of isomorphism is often defined in a more general way. The bijective function f is allowed to replace not only individuals but also properties and relations, always in such a way that true statements on one system are replaced by true statements on another system. When the similarity between systems is defined in this way, it is commonly said that similar systems are analogous and that the projection f is an analogy. An isomorphism can be defined as a bijective analogy. We can also define the concept of structure in a more general way: Two structures have the same structure if and only if they are models of the same theory. With this second definition, a structure as a property is determined by the axioms of a theory. More precisely, systems of different axioms define the same structure when they have the same models, when any model of one is a model of the other. A theory is categorical when all its models are isomorphic. The fundamental structures of mathematics, the set of natural numbers and the set of real numbers in particular, are determined with categorical theories. A categorical theory forbids any contingency. There is essentially one logically possible world that obeys its principles. The laws of Nature do not determine a categorical theory of Nature. They leave room for contingency. When a theory is not categorical, different, non-isomorphic, structures or systems may have the same structure, as defined by the theory. For example, we can say of all vector spaces that they have a vector space structure. An automorphism of a structure E is an internal isomorphism, an isomorphism from E to E. Every structure has a trivial automorphism, the identity-function defined by id(x)=x. A structure is symmetrical when it has at least one non-trivial automorphism. A non-trivial automorphism is a symmetry of a structure. The automorphisms of a structure form a group, in the algebraic sense, because the inverse of an automorphism is an automorphism and because the composite of two automorphisms is also an automorphism. The group of all the automorphisms of a structure is also called the group of its symmetries. For example, the group of symmetries of a circle, or a disc, is the group of rotations around their center and reflections with respect to a diameter. When there is an automorphism g such that y=g(x), x and y are essentially indistinguishable within the structure, in the sense that any truth on one can be transformed into an equivalent truth on the other. . The equivalence class, or orbit, of an element x of a symmetric structure is the set of y such that y=g(x) where g is an automorphism of the structure. An equivalence class is a set of elements that are essentially indistinguishable within the structure. For example, all the points of a circle are in the same equivalence class because there is nothing on the circle to distinguish them. All the points of a disc at the same distance from the center are also in the same equivalence class, but different concentric circles are different equivalence classes, because the points are distinguished by their distance to the center. A structure is symmetrical when it contains distinct but essentially indistinguishable elements, because their properties and their relations within the structure determine distinct but equivalent places. A natural structure is perfectly symmetrical when it contains naturally indiscernible elements such that their relations within the structure give them equivalent places. A natural structure is imperfectly symmetrical when it contains naturally very similar elements such that their relations within the structure give them equivalent or almost equivalent places. When a structure contains many constituents, the more symmetrical it is, the easier it is to know it, because we know all the symmetrical parts as soon as we know one. To develop empirical science, we must assume that all experimenters are equivalent in the sense that any observation made by one can be made again by another. Experiments must be reproducible. If an experiment is not so, then it is not well controlled. For experiments to be reproducible, it is necessary in particular that their results do not depend on the place or time. Experimental conditions can be reproduced always and everywhere, and have to lead always to the same result. By postulating the principle of equivalence of observers, we thus assume at the same time that the laws of physics are true always and everywhere. This leads to define the symmetry group of space-time. All points of space-time are necessarily similar, they are all in the same equivalence class. When we know one, we know them all. It is the same for all directions in space and more generally for all frames of reference. There is no center of space-time, no preferred direction in space (isotropy, no up and down) and no absolute state of rest (Galileo-Einstein relativity principle). Like the Knights of the Round Table, but in a much larger space, space-time observers never have a privileged position. The group of symmetries of space-time (Poincaré's group) is a mathematical translation of the principle of equivalence of all observers, just as the group of symmetries of a round table is a mathematical translation of the principle of equivalence of all knights. The principle of equivalence of all observers is not only a foundation of theoretical physics but also of all sciences, because reason requires that knowledge be universal, that all that is known by one can be known by all others.
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