Diversity Reading List

Abstract:
According to the received view, genuine mathematical justification derives from proofs. In this article, I challenge this view. First, I sketch a notion of proof that cannot be reduced to deduction from the axioms but rather is tailored to human agents. Secondly, I identify a tension between the received view and mathematical practice. In some cases, cognitively diligent, well-functioning mathematicians go wrong. In these cases, it is plausible to think that proof sets the bar for justification too high. I then propose a fallibilist account of mathematical justification. I show that the main function of mathematical justification is to guarantee that the mathematical community can correct the errors that inevitably arise from our fallible practices.

Abstract:
In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice. In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice covering topics including the distinction between formal and informal proofs, visualization and artefacts, mathematical explanation and understanding, value judgments, and mathematical design. We conclude with some remarks on the potential connections between the philosophy of mathematical practice and mathematics education.

Abstract:
Some personal thoughts and opinions on what “good quality mathematics” is and whether one should try to define this term rigorously. As a case study, the story of Szemer´edi’s theorem is presented.

Abstract:

A source of tension between Philosophers of Mathematics and Mathematicians is the fact that each group feels ignored by the other; daily mathematical practice seems barely affected by the questions the Philosophers are considering. In this talk I will describe an issue that does have an impact on mathematical practice, and a philosophical stance on mathematics that is detectable in the work of practising mathematicians. No doubt controversially, I will call this issue ‘morality’, but the term is not of my coining: there are mathematicians across the world who use the word ‘morally’ to great effect in private, and I propose that there should be a public theory of what they mean by this. The issue arises because proofs, despite being revered as the backbone of mathematical truth, often contribute very little to a mathematician’s understanding. ‘Moral’ considerations, however, contribute a great deal. I will first describe what these ‘moral’ considerations might be, and why mathematicians have appropriated the word ‘morality’ for this notion. However, not all mathematicians are concerned with such notions, and I will give a characterisation of ‘moralist’ mathematics and ‘moralist’ mathematicians, and discuss the development of ‘morality’ in individuals and in mathematics as a whole. Finally, I will propose a theory for standardising or universalising a system of mathematical morality, and discuss how this might help in the development of good mathematics.

Abstract: According to an adequacy-for-purpose view, models should be assessed with respect to their adequacy or fitness for particular purposes. Such a view has been advocated by scientists and philosophers alike. Important details, however, have yet to be spelled out. This article attempts to make progress by addressing three key questions: What does it mean for a model to be adequate-for-purpose? What makes a model adequate-for-purpose? How does assessing a model’s adequacy-for-purpose differ from assessing its representational accuracy? In addition, responses are given to some objections that might be raised against an adequacy-for-purpose view.

Abstract: The Prisoner’s Dilemma game is one of the classic games discussed in game theory,  the  study  of  strategic  decision  making  in  situations  of conflict,  which  stretches  between  mathematics  and  the  social  sciences. Game theory was  primarily developed  during  the  late  1940s  and  into  the  1960s  at  a number of research sites funded by various arms of the U.S. military establishment as part of their Cold War research.

Abstract: The chapter addresses the philosophical issues raised by the use of hypothetical modeling in the social sciences. Hypothetical modeling involves the construction and analysis of simple hypothetical systems to represent complex social phenomena for the purpose of understanding those social phenomena. To highlight its main features hypothetical modeling is compared both to laboratory experimentation and to computer simulation. In analogy with laboratory experiments, hypothetical models can be conceived of as scientific representations that attempt to isolate, theoretically, the working of causal mechanisms or capacities from disturbing factors. However, unlike experiments, hypothetical models need to deal with the epistemic uncertainty due to the inevitable presence of unrealistic assumptions introduced for purposes of analytical tractability. Computer simulations have been claimed to be able to overcome some of the strictures of analytical tractability. Still they differ from hypothetical models in how they derive conclusions and in the kind of understanding they provide. The inevitable presence of unrealistic assumptions makes the legitimacy of the use of hypothetical modeling to learn about the world a particularly pressing problem in the social sciences. A review of the contemporary philosophical debate shows that there is still little agreement on what social scientific models are and what they are for. This suggests that there might not be a single answer to the question of what is the epistemic value of hypothetical models in the social sciences.

Abstract:

Abstract: Empirical research into moral decision-making is often taken to have normative implications. For instance, in his recent book, Greene (2013) relies on empirical findings to establish utilitarianism as a superior normative ethical theory. Kantian ethics, and deontological ethics more generally, is a rival view that Greene attacks. At the heart of Greene’s argument against deontology is the claim that deontological moral judgments are the product of certain emotions and not of reason. Deontological ethics is a mere rationalization of these emotions. Accordingly Greene maintains that deontology should be abandoned. This paper is a defense of deontological ethical theory. It argues that Greene’s argument against deontology needs further support. Greene’s empirical evidence is open to alternative interpretations. In particular, it is not clear that Greene’s characterization of alarm-like emotions that are relative to culture and personal experience is empirically tenable. Moreover, it is implausible that such emotions produce specifically deontological judgments. A rival sentimentalist view, according to which all moral judgments are determined by emotion, is at least as plausible given the empirical evidence and independently supported by philosophical theory. I therefore call for an improvement of Greene’s argument.

Abstract: I have always thought that the most sacred duty of men was to give their children an education that prevented them at a more advanced age from regretting their youth, the only time when one can truly gain instruction. You are, my dear son, in this happy age when the mind begins to think, and when the heart has passions not yet lively enough to disturb it.
Now is perhaps the only time of your life that you will devote to the study of nature. Soon the passions and pleasures of your age will occupy all your moments; and when this youthful enthusiasm has passed, and you have paid to the intoxication of the world the tribute of your age and rank, ambition will take possession of your soul; and even if in this more advanced age, which often is not any more mature, you wanted to apply yourself to the study of the true Sciences, your mind then no longer having the flexibility characteristic of its best years, it would be necessary for you to purchase with painful study what you can learn today with extreme facility. So, I want you to make the most of the dawn of your reason; I want to try to protect you from the ignorance that is still only too common among those of your rank, and which is one more fault, and one less merit.
You must early on accustom your mind to think, and to be self-sufficient. You will perceive at all the times in your life what resources and what consolations one finds in study, and you will see that it can even furnish pleasure and delight.

Abstract: Gene–environment (G–E) covariance is the phenomenon whereby genetic differences bias variation in developmental environment, and is particularly problematic for assigning genetic and environmental causation in a heritability analysis. The interpretation of these cases has differed amongst biologists and philosophers, leading some to reject the utility of heritability estimates altogether. This paper examines the factors that influence causal reasoning when G–E covariance is present, leading to interpretive disagreement between scholars. It argues that the causal intuitions elicited are influenced by concepts of agency and blame-worthiness, and are intimately tied with the conceptual understanding of the phenotype under investigation. By considering a phenotype-specific approach, I provide an account as to why causal ascriptions can differ depending on the interpreter. Phenotypes like intelligence, which have been the primary focus of this debate, are more likely to spark disagreement for the interpretation of G–E covariance cases because the concept and ideas about its ‘normal development’ relatively ill-defined and are a subject of debate. I contend that philosophical disagreement about causal attributions in G–E covariance cases are in essence disagreements regarding how a phenotype should be defined and understood. This moves the debate from one of an ontological flavour concerning objective causal claims, to one concerning the conceptual, normative and semantic dependencies.