Wittgenstein and the infinity. The overpassing of metaphysics by linguistic analysis of logic

Our principal goal in the article is to highlight the relation of Ludwig Wittgenstein (in the period of Philosophical Investigations) with what he calls metaphysics. His strategy of overcoming (Uberwindung) is quite different from the one undertaken by the Vienna Circle. The second Wittgenstein develops an original philosophy which, being considerably different from the Tractatus, remains in continuity with his first philosophy. The aim is no more the logical analysis of language that strives to create an ideal logical calculus, but a kind of parsing of our linguistic habits, including the mathematical logic. The latter with the concept of infinity that haunted the research of the foundations of mathematics during the first half of the twentieth century, will be the subject of our investigation. Wittgenstein continues to reflect on the philosophy of mathematics because it reveals the philosophical problems in all their acuity. Nowadays, in our opinion, there is no another place where Platonism with all its problematical diversity is still seriously at stake. In other words, there is no another intellectual domain where we can still feel the old taste for metaphysics. This is one of the main reasons why Wittgenstein devoted half of his notes from his second philosophy to the problems of mathematics and mathematical logic: the Platonism inherent to the mathematical logic fools us with a "infinity of shadowy worlds."