Poincaré Paradox, Similarities and Point-free Geometry. (brief Version of a Paper Named " Similarities, Approximate Objects and Paradoxes " Submitted to Fuzzy Sets and Systems)

1. Introduction Recall that a fuzzy equivalence relation in a nonempty set S is a fuzzy relation eq : S×S → [0,1] such that eq(x,x) = 1 (reflexive); eq(x,y) = eq(y,x) (symmetric) ; eq(x,y)≈eq(y,z) ≤ eq(x,z) (≈-transitive) where ≈ is a t-norm. Now, as observed by De Cock and Kerre in [3], this notion is not suitable to model approximate equality and therefore to solve paradoxes like the so called Poincaré " paradox " (see [19]). This famous paradox refers to the indistinguishability by emphasizing that it is not transitive in spite of the common intuition. In fact, it is possible that we are not able to distinguish d 1 from d 2 , d 2 from d 3 , ...,d m from d m+1 and, nevertheless, that we have no difficulty to distinguish d 1 from d m+1. The argumentation of De Cock and Kerre is the following one. Consider the interval S = [1.00,2.50] of possible heights a man can have and assume that the notion of " approximately equal heights " is modelled by a fuzzy equivalence eq. Also, in accordance with the fact that we cannot distinguish a difference of less than 0.01 in the heights, assume that eq(1.50,1.51) = 1, eq(1.51,1.52) = 1, … eq(2.49,2.50) = 1. Then, by the ≈-transitivity and the fact that 1≈1=1, we obtain that eq(1.50,2.50) = 1, i.e. the height 1.50 is approximately equal to the height 2.50, that is an absurdity. As observed in [2], this argument is based on the fact that kernel(eq) = {(x,y)∈S×S : eq(x,y) = 1} is a transitive relation. Obviously, we can avoid such an argumentation simply by claiming that the values eq(1.50,1.51), eq(1.151,1.152) … are different from 1, but this looks unconvincing. As an example Bodenhofer observed in [2] that,