In his Remarks on the Foundations of Mathematics, Wittgenstein claims, puzzlingly, that ‘the proof creates a new concept’ (RFM III-41). This paper aims to contribute to clarifying this idea, and to showing how it marks a major break with the traditional conception of proof. Moreover, since the most natural way to understand his claim is open to criticism, a secondary goal of what follows is to offer an interpretation of it that neutralizes the objection. The discussion proceeds by analysing a (...) well-known geometrical proof. (shrink)

We discuss conceptual change and progress within mathematics, in particular how tools, structural concepts and representations are transferred between fields that appear to be unconnected or remote from each other. The theoretical background is provided by the frame concept, which is used in linguistics, cognitive science and artificial intelligence to model how explicitly given information is combined with expectations deriving from background knowledge. In mathematical proofs, we distinguish two kinds of frames, namely structural frames and ontological frames. The interaction between (...) both kinds of frames can drive mathematical interpretation. We first discuss two examples where structural frames (formulaic notation) drive ontological development (the discovery or exploration of mathematical objects). The development of Boole’s Boolean algebra may at first appear as a metaphorical treatment of the (then) new area of logic. In the analysis, we discuss how different (aspects of) certain algebraic frames change in the transfer, how arising difficulties are solved and overall argue that Boole uses the numerical algebra frame as a research template for the discovery of a system for calculations in logic. Following Ifrah, we analyse the discovery of zero as an extension to the number ontology as driven by the development of notation. Both structural and ontological frames are extended and simplified as notation progresses. Finally, we discuss two examples from infinite combinatorics, viz. topological graph theory, and one foundational issue. In both examples, the two simultaneous frames about one object are maintained independently. They motivate different research questions, but may also fruitfully interact: shifting between multiple synchronously maintained perspectives acts as a motor of innovation. The analysis shows how a frame-based approach allows to model how different perspectives drive mathematical innovation because they highlight different aspects, questions and heuristics. (shrink)

This work explores the later Wittgenstein’s philosophy of mathematics in relation to Lakatos’ philosophy of mathematics and the philosophy of mathematical practice. I argue that, while the philosophy of mathematical practice typically identifies Lakatos as its earliest of predecessors, the later Wittgenstein already developed key ideas for this community a few decades before. However, for a variety of reasons, most of this work on philosophy of mathematics has gone relatively unnoticed. Some of these ideas and their significance as precursors for (...) the philosophy of mathematical practice will be presented here, including a brief reconstruction of Lakatos’ considerations on Euler’s conjecture for polyhedra from the lens of late Wittgensteinian philosophy. Overall, this article aims to challenge the received view of the history of the philosophy of mathematical practice and inspire further work in this community drawing from Wittgenstein’s late philosophy. (shrink)

Open texture is a kind of semantic indeterminacy first systematically studied by Waismann. In this paper, extant definitions of open texture will be compared and contrasted, with a view towards the consequences of open-textured concepts in mathematics. It has been suggested that these would threaten the traditional virtues of proof, primarily the certainty bestowed by proof-possession, and this suggestion will be critically investigated using recent work on informal proof. It will be argued that informal proofs have virtues that mitigate the (...) danger posed by open texture. Moreover, it will be argued that while rigor in the guise of formalisation and axiomatisation might banish open texture from mathematical theories through implicit definition, it can do so only at the cost of restricting the tamed concepts in certain ways. (shrink)

In this work we argue that there is no strong demarcation between pure and applied mathematics. We show this first by stressing non-deductive components within pure mathematics, like axiomatization and theory-building in general. We also stress the “purer” components of applied mathematics, like the theory of the models that are concerned with practical purposes. We further show that some mathematical theories can be viewed through either a pure or applied lens. These different lenses are tied to different communities, which endorse (...) different evaluative standards for theories. We evaluate the distinction between pure and applied mathematics from a late Wittgensteinian perspective. We note that the classical exegesis of the later Wittgenstein’s philosophy of mathematics, due to Maddy, leads to a clear-cut but misguided demarcation. We then turn our attention to a more niche interpretation of Wittgenstein by Dawson, which captures aspects of the aforementioned distinction more accurately. Building on this newer, maverick interpretation of the later Wittgenstein’s philosophy of mathematics, and endorsing an extended notion of meaning as use which includes social, mundane uses, we elaborate a fuzzy, but more realistic, demarcation. This demarcation, relying on family resemblance, is based on how direct and intended technical applications are, the kind of evaluative standards featured, and the range of rhetorical purposes at stake. (shrink)

The frame concept from linguistics, cognitive science and artificial intelligence is a theoretical tool to model how explicitly given information is combined with expectations deriving from background knowledge. In this paper, we show how the frame concept can be fruitfully applied to analyze the notion of mathematical understanding. Our analysis additionally integrates insights from the hermeneutic tradition of philosophy as well as Schmid’s ideal genetic model of narrative constitution. We illustrate the practical applicability of our theoretical analysis through a case (...) study on extremal proofs. Based on this case study, we compare our analysis of proof understanding to Avigad’s ability-based analysis of proof understanding. (shrink)

Often machine learning programs inherit social patterns reflected in their training data without any directed effort by programmers to include such biases. Computer scientists call this algorithmic bias. This paper explores the relationship between machine bias and human cognitive bias. In it, I argue similarities between algorithmic and cognitive biases indicate a disconcerting sense in which sources of bias emerge out of seemingly innocuous patterns of information processing. The emergent nature of this bias obscures the existence of the bias itself, (...) making it difficult to identify, mitigate, or evaluate using standard resources in epistemology and ethics. I demonstrate these points in the case of mitigation techniques by presenting what I call ‘the Proxy Problem’. One reason biases resist revision is that they rely on proxy attributes, seemingly innocuous attributes that correlate with socially-sensitive attributes, serving as proxies for the socially-sensitive attributes themselves. I argue that in both human and algorithmic domains, this problem presents a common dilemma for mitigation: attempts to discourage reliance on proxy attributes risk a tradeoff with judgement accuracy. This problem, I contend, admits of no purely algorithmic solution. (shrink)

ABSTRACTIn this paper I investigate how conceptual engineering applies to mathematical concepts in particular. I begin with a discussion of Waismann’s notion of open texture, and compare it to Shapiro’s modern usage of the term. Next I set out the position taken by Lakatos which sees mathematical concepts as dynamic and open to improvement and development, arguing that Waismann’s open texture applies to mathematical concepts too. With the perspective of mathematics as open-textured, I make the case that this allows us (...) to deploy the tools of conceptual engineering in mathematics. I will examine Cappelen’s recent argument that there are no conceptual safe spaces and consider whether mathematics constitutes a counterexample. I argue that it does not, drawing on Haslanger’s distinction between manifest and operative concepts, and applying this in a novel way to set-theoretic foundations. I then set out some of the questions that need to be engaged with to establish mathematics as involving a kind of conceptual engineering. I finish with a case study of how the tools of conceptual engineering will give us a way to progress in the debate between advocates of the Universe view and the Multiverse view in set theory. (shrink)

Recent studies of Wittgenstein's later writing have made clear that they stand as a defence of two main ideas: that scepticism about the possibility of interpersonal discussions about our subjective feelings is misplaced and, as a seemingly startling corollary; that a mind state account of most 'mental activities' is incoherent. This leads to a great emphasis on skills and practices which, a fortiori, are definable only relationally, by reference to targets. In this paper I try to show that the 'computer' (...) analogue for the mind f ails on both of Wittgenstein's dimensions. There are no physiognomic language games in the computer centre, while the 'target' aspect of skill and practice concepts ties them in to a wholly human world. (shrink)

This paper defends a reading of Wittgenstein’s philosophy of mathematics in the Lectures on the Foundation of Mathematics as a radical conventionalist one, whereby our agreement about the particular case is constitutive of our mathematical practice and ‘the logical necessity of any statement is a direct expression of a convention’ (Dummett 1959, p. 329). -/- On this view, mathematical truths are conceptual truths and our practices determine directly for each mathematical proposition individually whether it is true or false. Mathematical truths (...) are thus not consequences of a prior adoption of a convention or rules as orthodox conventionalism has it. -/- The goal of the paper is not merely exegetical, however, and argues that radical conventionalism can withstand some of the most difficult objections that have been brought forward against it, including those of Dummett himself, and thus that radical conventionalism has been prematurely excluded from consideration by philosophers of mathematics. (shrink)

One of the distinguishing features of mathematics is the exceptional level of consensus among mathematicians. However, an analysis of what mathematicians agree on, how they achieve this agreement, and the relevant historical conditions is lacking. This paper is a programmatic intervention providing a preliminary analysis and outlining a research program in this direction.First, I review the process of ‘negotiation’ that yields agreement about the validity of proofs. This process most often does generate consensus, however, it may give rise to another (...) kind of disagreement: whether the original and new proof are effectively the same, leading to potential disagreement about the validity of the original proof.Second, I historicize the phenomenon of consensus. I show that in earlier European mathematics and other mathematical cultures, consensus about the validity of arguments was substantially weaker or conceived differently than it is today. This means that contemporary consensus about the validity of mathematical proofs should be explained by historical changes in mathematical practice.Finally, I conjecture what brought about the contemporary form of mathematical consensus. Since a sharp rise in consensus occurs around the turn of the 20th century, it makes sense to explain this consensus by the concurrent formalization of mathematics. However, this explanation has a major flaw: it explains a really existing phenomenon (consensus) by something that hardly ever happens (writing proofs in formal languages). I will therefore explain the ways in which formalization does enter mathematical practice so as to account for contemporary forms of mathematical consensus. (shrink)

Data‐driven algorithms are widely used to make or assist decisions in sensitive domains, including healthcare, social services, education, hiring, and criminal justice. In various cases, such algorithms have preserved or even exacerbated biases against vulnerable communities, sparking a vibrant field of research focused on so‐called algorithmic biases. This research includes work on identification, diagnosis, and response to biases in algorithm‐based decision‐making. This paper aims to facilitate the application of philosophical analysis to these contested issues by providing an overview of three (...) key topics: What is algorithmic bias? Why and how can it occur? What can and should be done about it? Throughout, we highlight connections—both actual and potential—with philosophical ideas and concerns. (shrink)

The potential for biases being built into algorithms has been known for some time, yet literature has only recently demonstrated the ways algorithmic profiling can result in social sorting and harm marginalised groups. We contend that with increased algorithmic complexity, biases will become more sophisticated and difficult to identify, control for, or contest. Our argument has four steps: first, we show how harnessing algorithms means that data gathered at a particular place and time relating to specific persons, can be used (...) to build group models applied in different contexts to different persons. Thus, privacy and data protection rights, with their focus on individuals, do not protect from the discriminatory potential of algorithmic profiling. Second, we explore the idea that anti-discrimination regulation may be more promising, but acknowledge limitations. Third, we argue that in order to harness anti-discrimination regulation, it needs to confront emergent forms of discrimination or risk creating new invisibilities, including invisibility from existing safeguards. Finally, we outline suggestions to address emergent forms of discrimination and exclusionary invisibilities via intersectional and post-colonial analysis. (shrink)

The conditions involved in the applicability of mathematics in science are the subject of ongoing debates. One of the best‐received approaches is the inferential account, which involves structural mappings and pragmatic considerations in a three‐step model. According to the inferential account, these pragmatic considerations happen in the immersion and interpretation stages, but not during derivation (symbol‐pushing in a mathematical formalism). In this work, I draw inspiration from the later Wittgenstein and make the case that the applicability of mathematics also rests (...) on pragmatic considerations at the mathematical derivation level. I make the further case that pragmatic considerations at the mathematical derivation level may substitute pragmatic considerations in the other two stages, thus showing how they are holistically interrelated. I illustrate these two points with the solution of a simple geometrical problem. (shrink)

How can metaphors account for the formation of mathematical concepts, for changes in mathematical practices, or for the handling of mathematical problems? Following Hans Blumenberg’s thought, this paper aims to unfold a possible answer to these questions by viewing the metaphorical frameworks accompanying these changes as essential for an understanding of how changes in mathematical practices have been accounted for. I will focus especially on cases in which these changes were caused by encounters with a mathematical object which did not (...) yet have a well-defined concept, but also show that such indeterminacy remains with the mathematical concept even after it is considered ‘well-defined’. As the paper will show, this ‘forefield’ [Vorfeld] of the concept is addressed by Blumenberg’s account of metaphorology on the one hand, and accompanied by his later account of nonconceptuality [Unbegrifflichkeit] on the other hand. While Blumenberg himself did not develop a full-fledged philosophy of mathematics or of mathematical practices, I aim to show that one can nevertheless extract from his writings a unique position concerning the role metaphors play in mathematics. To this end, Blumenberg’s account of nautical and oceanic metaphors and Alexandre Grothendieck’s philosophy of mathematical practice provide fruitful starting points. (shrink)

The association of Wittgenstein’s name with the notion of artificial intelligence is bound to cause some surprise both to Wittgensteinians and to people interested in artificial intelligence. After all, Wittgenstein died in 1951 and the term artificial intelligence didn’t come into use until 1956 so that it seems unlikely that one could have anything to do with the other. However, establishing a connection between Wittgenstein and artificial intelligence is not as insuperable a problem as it might appear at first glance. (...) While it is true that artificial intelligence as a quasi-distinct discipline is of recent vintage, some of its concerns, especially those of a philosophical nature, have been around for quite some time. At the birth of modern philosophy we find Descartes wondering whether it would be possible to create a machine that would be phenomenologically indistinguishable from.. (shrink)