What mathematical knowledge could not be

This survey paper will critically discuss four different strategies to explain our knowledge of mathematics. In the first section I will outline Benacerraf ’s dilemma as put forth in Benacerraf ’s famous paper “Mathematical Truth”1 – a dilemma faced by any account of mathematical knowledge. The aim of this section is to clarify and discuss the semantic and epistemic constraints that Benacerraf (explicitly and implicitly) imposes, and show how they give rise to his well-known dilemma. In the second section I will review four strategies to overcome this dilemma as they occur in the philosophical literature. The first two platonistic strategies comply with the semantic constraint but, I will argue, provide insufficient answers to the epistemic constraint, while the other two, nominalistic strategies either reject the idea of mathematical knowledge altogether or fail the semantic constraint. In the last section, I will elicit, on the basis of my discussion of the four conceptions, what I label the fundamental assumption. I will argue that it is presupposed by all four strategies and suggest that a rejection of this assumption will give rise to a different type of platonistic response. A thorough discussion of this fifth alternative will, however, be postponed to another occasion

The context principle and implicit definitions : towards an account of our a priori knowledge of arithmetic

This thesis is concerned with explaining how a subject can acquire a priori knowledge of arithmetic. Every account for arithmetical, and in general mathematical knowledge faces Benacerraf's well-known challenge, i.e. how to reconcile the truths of mathematics with what can be known by ordinary human thinkers. I suggest four requirements that jointly make up this challenge and discuss and reject four distinct solutions to it. This will motivate a broadly Fregean approach to our knowledge of arithmetic and mathematics in general. Pursuing this strategy appeals to the context principle which, it is proposed, underwrites a form of Platonism and explains how reference to and object-directed thought about abstract entities is, in principle, possible. I discuss this principle and defend it against different criticisms as put forth in recent literature. Moreover, I will offer a general framework for implicit definitions by means of which - without an appeal to a faculty of intuition or purely pragmatic considerations - a priori and non-inferential knowledge of basic mathematical principles can be acquired. In the course of this discussion, I will argue against various types of opposition to this general approach. Also, I will highlight crucial shortcomings in the explanation of how implicit definitions may underwrite a priori knowledge of basic principles in broadly similar conceptions. In the final part, I will offer a general account of how non-inferential mathematical knowledge resulting from implicit definitions is best conceived which avoids these shortcomings

The Convenience of the Typesetter; Notation and Typography in Frege’s Grundgesetze der Arithmetik

We discuss the typography of the notation used by Gottlob Frege in his Grundgesetze der Arithmetik

Methodological and conceptual challenges in rare and severe event forecast verification

There are distinctive methodological and conceptual challenges in rare and severe event (RSE) forecast verification, that is, in the assessment of the quality of forecasts of rare but severe natural hazards such as avalanches, landslides or tornadoes. While some of these challenges have been discussed since the inception of the discipline in the 1880s, there is no consensus about how to assess RSE forecasts. This article offers a comprehensive and critical overview of the many different measures used to capture the quality of categorical, binary RSE forecasts – forecasts of occurrence and non-occurrence – and argues that of skill scores in the literature there is only one adequate for RSE forecasting. We do so by first focusing on the relationship between accuracy and skill and showing why skill is more important than accuracy in the case of RSE forecast verification. We then motivate three adequacy constraints for a measure of skill in RSE forecasting. We argue that of skill scores in the literature only the Peirce skill score meets all three constraints. We then outline how our theoretical investigation has important practical implications for avalanche forecasting, basing our discussion on a study in avalanche forecast verification using the nearest-neighbour method (Heierli et al., 2004). Lastly, we raise what we call the “scope challenge”; this affects all forms of RSE forecasting and highlights how and why working with the right measure of skill is important not only for local binary RSE forecasts but also for the assessment of different diagnostic tests widely used in avalanche risk management and related operations, including the design of methods to assess the quality of regional multi-categorical avalanche forecasts

Safety in numbers: how social choice theory can inform avalanche risk management

Avalanche studies have undergone a transition in recent years. Early research focused mainly on environmental factors. More recently, attention has turned to human factors in decision making, such as behavioural and cognitive biases. This article adds a social component to this human turn in avalanche studies. It identifies lessons for decision making by groups of skiers from the perspective of social choice theory, a sub-field of economics, decision theory, philosophy and political science that investigates voting methods and other forms of collective decision making. In the first part, we outline the phenomenon of wisdom of crowds, where groups make better decisions than their individual members. Drawing on the conceptual apparatus of social choice theory and using idealised scenarios, we identify conditions under which wisdom of crowds arises and also explain how and when deciding together can instead result in worse decisions than may be expected from individual group members. In the second part, we use this theoretical understanding to offer practical suggestions for decision making in avalanche terrain. Finally, we make several suggestions for risk management in other outdoor and adventure sports and for outdoor sports education

Gratuitous risk: danger and recklessness perception of adventure sports participants

Since the 1970’s there has been a major increase in adventure sports participation but it seems that engagement in such sports comes with a stigma: adventure sports participants are often regarded as reckless ‘daredevils’. We approach the questions about people’s perception of risk and recklessness in adventure sports by combining empirical research with philosophical analysis. First, we provide empirical evidence that suggests that laypeople tend to assess the danger of adventure sports as greater than more mundane sports and judge adventure sports participants as more reckless than participants in non-adventure sports. We contextualise these findings within existing psychological risk perception paradigms and outline new philosophical explanations of the identified pattern in laypeople’s risk perception

A Plea for Risk

Mountaineering is a dangerous activity. For many mountaineers, part of its very attraction is the risk, the thrill of danger. Yet mountaineers are often regarded as reckless or even irresponsible for risking their lives. In this paper, we offer a defence of risk-taking in mountaineering. Our discussion is organised around the fact that mountaineers and non-mountaineers often disagree about how risky mountaineering really is. We hope to cast some light on the nature of this disagreement – and to argue that mountaineering may actually be worthwhile because of the risks it involves. Section 1 introduces the disagreement and, in doing so, separates out several different notions of risk. Sections 2–4 then consider some explanations of the disagreement, showing how a variety of phenomena can skew people’s risk judgements. Section 5 then surveys some recent statistics, to see whether these illuminate how risky mountaineering is. In light of these considerations, however, we suggest that the disagreement is best framed not simply in terms of how risky mountaineering is but whether the risks it does involve are justified. The remainder of the paper, sections 6–9, argues that risk-taking in mountaineering often is justified – and, moreover, that mountaineering can itself be justified (in part) by and because of the risks it involves