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 Ti 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 ti. Consider now the operation ● of composition of transformations, intended as composition of functions. Sometime the structure (ti,●) is a group, that is ● has the null element (call it I) and the inverse (call it -1), is associative and is closed in ti. 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 ai of A and each term pi of T the sentence piai could be either true or false. Consider the set tipi of x belonging to A such that if pix is true and for each transformation l belonging to ti ly=x, then piy holds. Consider now the relation Ri “to be the result of a transformation belonging to ti” on the set tipi. Since ti is a group, Ri is an equivalence relation. Indeed xRix for each x, because ti has a null element; if xRiy then yRix, because in ti there is the inverse of every transformation; to prove that Ri is transitive is a little more difficult. Suppose that xRiy and yRiz but not xRiz. Then there exist l belonging to ti that brings form x to y and m that brings from y to z, but no transformation in ti brings from x to z. Therefore l●m does not belong to ti; but this is impossible because ti is closed with respect to the composition of transformations (●). It follows that Ri 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 pi corresponds an objective properties or relations Pi holding for a with respect to ti.
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 ti.
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.