I argue for the Wittgensteinian thesis that mathematical statements are expressions of norms, rather than descriptions of the world. An expression of a norm is a statement like a promise or a New Year's resolution, which says that someone is committed or entitled to a certain line of action. A expression of a norm is not a mere description of a regularity of human behavior, nor is it merely a descriptive statement which happens to entail a norms. The view can (...) be thought of as a sort of logicism for the logical expressivist---a person who believes that the purpose of logical language is to make explicit commitments and entitlements that are implicit in ordinary practice. The thesis that mathematical statements are expression of norms is a kind of logicism, not because it says that mathematics can be reduced to logic, but because it says that mathematical statements play the same role as logical statements. ;I contrast my position with two sets of views, an empiricist view, which says that mathematical knowledge is acquired and justified through experience, and a cluster of nativist and apriorist views, which say that mathematical knowledge is either hardwired into the human brain, or justified a priori, or both. To develop the empiricist view, I look at the work of Kitcher and Mill, arguing that although their ideas can withstand the criticisms brought against empiricism by Frege and others, they cannot reply to a version of the critique brought by Wittgenstein in the Remarks on the Foundations of Mathematics. To develop the nativist and apriorist views, I look at the work of contemporary developmental psychologists, like Gelman and Gallistel and Karen Wynn, as well as the work of philosophers who advocate the existence of a mathematical intuition, such as Kant, Husserl, and Parsons. After clarifying the definitions of "innate" and "a priori," I argue that the mechanisms proposed by the nativists cannot bring knowledge, and the existence of the mechanisms proposed by the apriorists is not supported by the arguments they give. (shrink)

The article contains the replies to the collection of contributions discussing my research on the philosophy of information.

The World 3 theory outlined by Popper in the 1960s has been diversely received, raising enthusiasms as well as severe criticisms. In this paper, I resituate the theory in the development of Popper’s philosophy. I not only present the three “sources” of W3, but I also show that the first one dominates the two others. Comparing it with a few other evolutionary philosophies, I propose to understand Popper’s metaphysics as part of an evolutionary and spiritualist philosophy, which was, however, always (...) balanced by a strong ethical reminder of the inescapability of human responsibility. (shrink)

Abstract This paper aims to analyse Karl Popper's conception of ?three worlds?, and especially the problem of world 3?the world of objective knowledge. Firstly, I try to explain Popper's turn to ontological questions which I link to his antipsychologism and to issues raised by the development of logic after World War II. I then consider Popper's concept of the autonomy of world 3 and his attempt to introduce world 3 as a world of knowledge without a knowing subject. I conclude (...) that Popper did not succeed in unifying his central idea of autonomy of knowledge with the requirement of the creative role of the critical subject carrying out the evolution of knowledge. I see the core of this contradiction especially in his co?existing ideas of the timeless existence of world 3 and the elimination of the subject from it. The attempt to desubjectivize the realm of objective knowledge leads to a philosophically unbalanced standpoint which presupposes a creative subject and at the same time neglects it. Finally, I question Popper's account of the growth of world 3. Popper considers only cognitive motivations, and excludes a broad range of motivating factors which originate in the problems which we face in our lives, and affect our cognitive interests in world 3. (shrink)