Mathematical knowledge: a case study in empirical philosophy of mathematics

In this paper, we present a paradigm for the philosopher of mathematics who takes mathematical practice seriously: Empirical Philosophy of Mathematics. In this philosophical paradigm, we use empirical methods (from sociology, psychology, cognitive science, didactics) to evaluate empirical questions connected to or derived from philosophical positions about mathematics. The paper presents a concrete case study

Theories of Knowledge and Ignorance

What does it mean to say that an agent only knows a particular fact,i.e., knowing that fact and not more than that? The problem of describing so-called minimal knowledge has been discussed in the literature since 1985, when Halpern and Moses published a paper on Knowledge and Ignorance.The present chapter reviews a considerable part of the most important proposals for only knowing and provides a number of generalizations over these proposals. The focus of this study is on a theoretical understanding of the subject, but several (possible) applications are indicated too. The proposals for solving the problem of minimal knowledge vary along some dimensions. Most of these proposals are restricted to a single agent, whereas a few deal with the multi-agent case. Also, most deal with the problem of minimal knowledge on the level of meta-language, by formulating inferential conditions, semantic constraints on verifying models, or rules for establishing belief sets; a few suggest an explicit operator for only knowing in the object language. Moreover, the majority of proposals employ specific modal systems which, e.g., point out whether the agent is (fully) introspective, i.e., knows that she knows (and knows that and what she does not know); the authors, however, suggest and discuss general modal approaches too. Finally, the advantages of `going partial' (using models that may leave the truth value of certain propositions undefined) are demonstrated

The Proof Is in the Process: A Preamble for a Philosophy of Computer-Assisted Mathematics

According to some well-known mathematicians well-versed in computer-assisted mathematics (CaM), “Computers are changing the way we are doing mathematics”. To what extent this is really true is still an open question. Indeed, even though some philosophers of math have taken up the challenge to think about CaM, it is unclear in what sense exactly a machine (can) affect(s) the so-called “queen of the sciences”. In fact, some have concluded that issues raised by the use of the computer in mathematics are not specific to the use of the computer per se. However, such findings seem precarious since a systematic study of computer-assisted mathematics is still lacking. In this paper I argue that in order to understand the impact of CaM, it is necessary to take more seriously the computer itself and how it is actually used in the process of doing mathematics. Within such an approach, one searches for characteristics that are specific to the use of the computer in mathematics. I will focus on a feature that is beyond any doubt inherently connected to the use of computing machinery, viz. mathematician-computer interactions. I will show how such interactions are fundamentally different from the usual interactions between mathematicians and non-human aids (a piece of paper, a blackboard etc) and how such interactions determine at least two more characteristics of CaM, viz. the significance of time and processes and the steady process of internalization of mathematical tools and knowledge into the machine. I will restrict myself to the use of the computer within so-called experimental mathematics since this is the main object of CaM within the philosophical literature