Claus Beisbart

Kant famously thought that mathematics contains synthetic a priori truths. In his book, Wille defends a version of the Kantian thesis on not-so-Kantian grounds. Wille calls his account neo-Kantian, because it makes sense of Kantian tenets by using a methodology that takes the linguistic and pragmatic turns seriously.Wille's work forms part of a larger project in which the statuses of mathematics and proof theory are investigated. The official purpose of the present book is to answer the question: what is mathematics. Wille sets himself the task of finding a definition that enables him to distinguish between mathematics and proof theory. His solution reads roughly as follows. Mathematics is about how to generate synthetic a priori knowledge by acting within some calculus. This definition does not seem to be a promising starting point, because in this way a very controversial claim becomes true by definition. However, Wille's strategy is to make sense of his definition during the course of his argument and to show that the definition is appropriate regarding research that is commonly taken to be mathematics.The second part of the book explains what ‘acting within some calculus’ amounts to. In Wille's view, mathematics should be characterized in …

The voting power of a voter—the extent to which she can affect the outcome of a collective decision—is often quantified in terms of the probability that she is critical. This measure is extended to a series of power measures of different ranks. The measures quantify the extent to which a voter can be part of a group that can jointly make a difference as to whether a bill passes or not. It is argued that the series of these measures allow for a more appropriate assessment of voting power, particularly of a posteriori voting power in case the votes are stochastically dependent. Also, the new measures discriminate between voting games that cannot be distinguished in terms of the probability of only criticality

Monte Carlo simulations arrive at their results by introducing randomness, sometimes derived from a physical randomizing device. Nonetheless, we argue, they open no new epistemic channels beyond that already employed by traditional simulations: the inference by ordinary argumentation of conclusions from assumptions built into the simulations. We show that Monte Carlo simulations cannot produce knowledge other than by inference, and that they resemble other computer simulations in the manner in which they derive their conclusions. Simple examples of Monte Carlo simulations are analysed to identify the underlying inferences.