Wittgenstein's stance in mathematics in the transitional period can be characterized genererally as 'constructivist', which means that the concept of iteration of an operation is the key-concept to the definition of number series. This view holds for another central concept of his philosophy of mathematics at that time: the infinite. He defended the view that the infinite is not an actual totality but consists in a rule for the construction of infinite mathematical series. This would not raise any difficulty, had the very notion of the infinite no extra-mathematical meaning. In fact, Wittgenstein's change of mind is strongly related to his discovery in a field that he himself labeled 'theory of knowledge' or 'phenomenology': the infinite is also a feature of the forms of space and time. This poses a problem: if his intensionalist-constructivist stance arose from his investigations outside mathematics, how, then, can he claim that it is only through a reflection upon the properties of a mathematical notation for representing space (any space) that we can solve the problems pertaining to the structure of visual space? This will be our problem in this lecture which will focus on Wittgenstein's Manuscripts 105 and 106 (1929).
Ludovic Soutif is Doctoral Student and Associate Member of the Wittgenstein Research Team at the University of Paris I-Pantheon-Sorbonne.