Steps towards a Dynamic Constructivism

Prologue. The philosophy of mathematics and logic has been an object of my reeections ever since I rst undertook serious study of these subjects. Over many years I have slowly developed a global way of thinking about the foundations, not only of mathematics, but of abstract concepts in general (see 7], 8], 9] and the introduction of 12]). Recently, the fertility of such a global approach, at which I have arrived independently of the usual schools, has been connrmed by two speciic and novel technical developments in logic and in mathematics. This fact has nally convinced me to address a wider audience, and, moreover, in English. 1 Since I cannot convey everything in a few pages, this paper is necessarily an overview. I hope in the future to nd the energy to write up a complete version, which will include all the details, arguments, connections and references omitted here. If my exposition is somewhat ideological and my language nontechnical, it is because my goal is to take the initial steps towards a new conception. 1 A general claim. Right from the start, the philosophy of mathematics of Brouwer { the founder of intuitionism { was rejected by most because of its links with mysticism. Granted. Certainly it is not a good idea to base scientiic knowledge on any form of religious belief. Now the real problem is that, if by religion we mean any form of unprovable belief in something beyond our perceptions, then apparently also all other existing philosophies of mathematics are based no less on religion. Logicism, which claims that mathematics can be reduced to logic, gives no explanation of why logic should stand by itself, given a priori. Formalism purposely deprives mathematics of all intentionality and thus accepts it, without any discussion, as given in its totality, since its aim is to justify all of it by reducing it to itsòbjective' external appearance, that is, to signs. In general, all of the present day approaches seem still to be permeated by some form of platonism: something, be it called`truth', `existence', `objectivity', or the like, is assumed to exist independently of our perceptions and of our minds. Assumptions of this sort may be understandable from a psychological point of view, since they relieve the anxieties of emptiness and uncertainty as well as furnish an aim for our trials and errors, but it must be emphasized that they have …