One central aim of science is to provide explanations of natural phenomena. What role(s) does mathematics play in achieving this aim? How does mathematics contribute to the explanatory power of science? Rules to Infinity defends the thesis, common though perhaps inchoate among many members of the Vienna Circle, that mathematics contributes to the explanatory power of science by expressing conceptual rules, rules which allow the transformation of empirical descriptions. Mathematics should not be thought of as describing, in any substantive sense, (...) an abstract realm of eternal mathematical objects, as traditional platonists have thought. In Rules to Infinity, this view, which I call mathematical normativism, is updated with contemporary philosophical tools, and it is argued that normativism is compatible with mainstream semantic theory. This allows the normativist to accept that there are mathematical truths, while resisting the platonistic idea that there exist abstract mathematical objects that explain such truths. Furthermore, Rules to Infinity defends a particular account of the distinction between scientific explanations that are in some sense distinctively mathematical – those that explain natural phenomena in some uniquely mathematical way – and those that are only standardly mathematical, and it lays out desiderata for any account of this distinction. Normativism is compared with other prominent views in the philosophy of mathematics such as neo-Fregeanism, fictionalism, conventionalism, and structuralism. Rules to Infinity serves as an entry point into debates at the forefront of philosophy of science and mathematics, and it defends novel positions in these debates. (shrink)

Annalisa Coliva (Int J Study Skept 10(3–4):346–366, 2020) asks, “Are there mathematical hinges?” I argue here, against Coliva’s own conclusion, that there are. I further claim that this affirmative answer allows a case to be made for taking the concept of a hinge to be a useful and general-purpose tool for studying mathematical practice in its real complexity. Seeing how Wittgenstein can, and why he would, countenance mathematical hinges additionally gives us a deeper understanding of some of his latest thoughts (...) on mathematics. For example, a view of how mathematical hinges relate to Wittgenstein’s well-known river-bed analogy enables us to see how his way of thinking about mathematics can account nicely for a “dynamics of change” within mathematical research—something his philosophy of mathematics has been accused of missing (e.g., by Robert Ackermann (Wittgenstein’s city, The University of Massachusetts Press, Amherst, 1988) and Mark Wilson (Wandering significance: an essay on conceptual behavior, Oxford University Press, Oxford, 2006). Finally, the perspective on mathematical hinges ultimately arrived at will be seen to provide us with illuminating examples of how our conceptual choices and theories can be ungrounded but nevertheless the right ones (in a sense to be explained). (shrink)

Political philosophy has traditionally been defined as a normative discipline with a distinctively ideal component, largely informed by moral philosophy. In this paper, I investigate a prominent critique of ideal theory specifically with the goal of resituating the debate within a larger framework in the philosophy of science. I then mount a novel case for how ideal theory should be viewed in terms of scientific modelling. I close with a discussion of how this view can dissolve apparent paradoxes and provide (...) distinct advantages to the political theorist. (shrink)

This paper aims to unite two seemingly disparate themes in the philosophy of mathematics and language respectively, namely ante rem structuralism and inferentialism. My analysis begins with describing both frameworks in accordance with their genesis in the work of Hilbert. I then draw comparisons between these philosophical views in terms of their similar motivations and similar objections to the referential orthodoxy. I specifically home in on two points of comparison, namely the role of norms and the relation of ontological dependence (...) in both accounts. Lastly, I show that insights from this purported connection can address certain objections to both theories respectively. (shrink)

Wittgenstein's conception of infinity can be seen as continuing the tradition of the potential infinite that begins with Aristotle. Transfinite cardinals in set theory might seem to render the potential infinite defunct with the actual infinite now given mathematical legitimacy. But Wittgenstein's remarks on set theory argue that the philosophical notion of the actual infinite remains philosophical and is not given a mathematical status as a result of set theory. The philosophical notion of the actual infinite is not to be (...) found in the mathematics of set theory, only in a certain associated philosophy – what Wittgenstein calls a certain kind of “prose”. (shrink)

In this paper, it is examined how, if at all, the logical laws can be normative for human reasoning, wherein the notion of normativity is analyzed primarily with respect to Wittgenstein’s philosophy. During the ancient and the medieval periods, logic was being considered in terms of discourse and dialogical practice, but since Descartes and especially Kant, it has been treated as a system of laws with which the process of individual human reasoning has been compared. Therefore, normativity can be investigated (...) in the private sphere and in the public sphere. Wittgenstein discussed both aspects of normativity: in his early philosophy, the focus is on the laws of logic that are primarily normative for the state of affairs in the world, while in his later works the emphasis is on a social aspect of normativity, which is derived from the adopted rules that have been applied in a certain community. Taken that way, logic is certainly normative in the public sphere, but the more difficult issue is whether logic is normative for thinking, regarding the difference between the logical laws and those of thought. (shrink)

An overview of the German philosophy of science community is given for the years 1992–2012, based on a survey in which 159 philosophers of science in Germany participated. To this end, the institutional background of the German philosophy of science community is examined in terms of journals, centers, and associations. Furthermore, a qualitative description and a quantitative analysis of our survey results are presented. Quantitative estimates are given for: (a) academic positions, (b) research foci, (c) philosophers’ of science most important (...) publications, and (d) externally funded projects, where for (c) all survey participants had indicated their five most important publications in philosophy of science. In addition, the survey results for (a)–(c) are also qualitatively described, as they are interesting in their own right. With respect to (a), we estimated the gender distribution among academic positions. Concerning (c), we quantified philosophers’ of science preference for (i) journals and publishers, (ii) publication format, (iii) language, and (iv) coauthorship for their most important publications. With regard to research projects, we determined their (i) prevalence, (ii) length, and (iii) trend (an increase in number?) as well as their most frequent (iv) research foci and (v) funding organizations. We also distinguished between German-based and non-German-based journals, publishers, and funding institutions, making it thereby possible to evaluate the involvement of the German philosophy of science community in the international research landscape. Finally, we discuss some implications of our findings. (shrink)

Additional theorizing about mathematical practice is needed in order to ground appeals to truly useful notions of the virtues in mathematics. This paper aims to contribute to this theorizing, first, by characterizing mathematical practice as being epistemic and “objectual” in the sense of Knorr Cetina The practice turn in contemporary theory, Routledge, London, 2001). Then, it elaborates a MacIntyrean framework for extracting conceptions of the virtues related to mathematical practice so understood. Finally, it makes the case that Wittgenstein’s methodology for (...) examining mathematics and its practice is the most appropriate one to use for the actual investigation of mathematical practice within this MacIntyrean framework. At each stage of thinking through mathematical practice by these means, places where new virtue-theoretic questions are opened up for investigation are noted and briefly explored. (shrink)