In his beautiful paper “Physical reality” Max Born proposes a far-reaching theory of objectivity for contemporary physics based on an analogy with perception. When we perceive, for instance, a glass, we see approximately a truncated cone even if the shape of the stimulus is different in accordance with the point of observation from which we look at the glass. The truncated cone has always the same dimensions from every point of view. Born emphasizes that in a certain sense our brain makes almost instantaneously a very complex calculation having as input the particular projection of the truncated cone and as output the invariant and real truncated cone. Since I was a child, my father Guido Fano, a theoretical physicist, reminded me that the brain of a cat must have a similar computer implemented as well in its brain, else it would be not able to catch mice. Moving from this consideration we can attempt a more accurate investigation of the notion of objectivity in contemporary physics.

Consider a set A of objects: for instance all possible geometrical three-dimensional regular and connected shapes. A Transformation is a function T_{i} from A to A. For instance “rotate p/2 around the x axis”. Moreover we assume that A is a model of a certain scientific theory S. Let us consider now the set of all transformations on A and call it T. Let us take a subset of T and call it t_{i}. Consider now the operation ● of composition of transformations, intended as composition of functions. Sometime the structure (t_{i},●) is a group, that is ● has the null element (call it I) and the inverse (call it ^{-1}), is associative and is closed in t_{i}. Take now an element of A, for instance a. Say that A is characterized by the predicates p, that is all sentences of the form pa are true in S. For each element a_{i} of A and each term p_{i} of T the sentence p_{i}a_{i} could be either true or false. Consider the set t_{i}p_{i} of x belonging to A such that if p_{i}x is true and for each transformation l belonging to t_{i} ly=x, then p_{i}y holds. Consider now the relation R_{i} “to be the result of a transformation belonging to t_{i}” on the set t_{i}p_{i}. Since t_{i} is a group, R_{i} is an equivalence relation. Indeed xR_{i}x for each x, because t_{i} has a null element; if xR_{i}y then yR_{i}x, because in t_{i} there is the inverse of every transformation; to prove that R_{i} is transitive is a little more difficult. Suppose that xR_{i}y and yR_{i}z but not xR_{i}z. Then there exist l belonging to t_{i} that brings form x to y and m that brings from y to z, but no transformation in t_{i} brings from x to z. Therefore l●m does not belong to t_{i}; but this is impossible because t_{i} is closed with respect to the composition of transformations (●). It follows that R_{i} defines a certain equivalence class. In other terms equivalence relation R determines equivalence classes.

From these premises it follows that one can state that to the predicates p_{i} corresponds an *objective* properties or relations P_{i} holding for a with respect to t_{i}.

In other words we can ascribe objectivity to certain properties and relations of an object a on the basis of a theory S, which we consider true for a and with respect to a determinate group of transformations t_{i}.

In this perspective there are two important differences with respect to the Neo-positivist point of view magisterially exposed by Carnap in the seminal paper “Empiricism, Semantics and Ontology”. 1. We ascribe objectivity to a certain characteristic of a on the basis of S, but this objectivity is not only internal to S. 2. Nonetheless this objectivity is relative to a certain group of transformations. In other terms, 1. enhances scientific realism with respect to Carnap, whereas 2. involves a different and more adequate concept of objectivity in our ontology.

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