WHO’S AFRAID OF MATHEMATICAL DIAGRAMS?

Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions.   Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs.  In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that diagrams form genuine notational systems, and I argue that this explains why they can play a role in the inferential structure of proofs without undermining their reliability.  I then consider whether diagrams can be essential to the proofs in which they appear.@font-face{font-family:"Cambria Math";panose-1:2 4 5 3 5 4 6 3 2 4;mso-font-charset:0;mso-generic-font-family:roman;mso-font-pitch:variable;mso-font-signature:-536870145 1107305727 0 0 415 0;}@font-face{font-family:Calibri;panose-1:2 15 5 2 2 2 4 3 2 4;mso-font-charset:0;mso-generic-font-family:swiss;mso-font-pitch:variable;mso-font-signature:-536859905 -1073697537 9 0 511 0;}p.MsoNormal, li.MsoNormal, div.MsoNormal{mso-style-unhide:no;mso-style-qformat:yes;mso-style-parent:"";margin:0in;line-height:200%;mso-pagination:widow-orphan;font-size:12.0pt;font-family:"Calibri",sans-serif;mso-fareast-font-family:Calibri;}.MsoChpDefault{mso-style-type:export-only;mso-default-props:yes;font-family:"Calibri",sans-serif;mso-ascii-font-family:Calibri;mso-fareast-font-family:Calibri;mso-hansi-font-family:Calibri;mso-bidi-font-family:Calibri;}.MsoPapDefault{mso-style-type:export-only;line-height:200%;}div.WordSection1{page:WordSection1;}

Aristotle on Intelligent Perception

Aristotle presents perception as a potentially intelligent form of cognition—a form of cognition that allows us to respond in discerning, knowing ways to a range of different situations, and develop certain theoretical insights relevant to some inquiry. But it’s not clear how we should understand the interaction between our rational and perceptual powers in these cases, or how widespread we should take their interaction to be. In this paper I argue against interpretations on which human perception would be an inherently rational power. I then develop an alternative view of intelligent perception that seeks to do justice to the role Aristotle assigns nonrational uses of perception in his account of our learning, and to his emphasis on the continuity between animal and human forms of cognition. On the view I develop, our rational powers allow us to develop a reflective understanding of the knowledge we acquire by purely perceptual means—a reflective understanding that can then affect what we recognize perceptually, and thereby yield forms of action and insight unavailable to other animals.

Unfinished Business

According to an intriguing though somewhat enigmatic line of thought first proposed by Jonathan Bennett, if humanity went extinct any time soon this would be unfortunate because important business would be left unfinished. This line of thought remains largely unexplored. I offer an interpretation of the idea that captures its intuitive appeal, is consistent with plausible constraints, and makes it non-redundant to other views in the literature. The resulting view contrasts with a welfare-promotion perspective, according to which extinction would be bad because it would come at a massive opportunity cost in terms of feasibly achievable welfare over the lifetime of our species.

Remembering objects

Conscious recollection, of the kind characterised by sensory mental imagery, is often thought to involve ‘episodically’ recalling experienced events in one’s personal past. One might wonder whether this overlooks distinctive ways in which we sometimes recall ordinary, persisting objects. Of course, one can recall an object by remembering an event in which one encountered it. But are there acts of recall which are distinctively objectual in that they are not about objects in this mediated way (i.e., by way of being about events in which they featured)? This question has broad implications, not least for understanding the nature and role of imagery in remembering, the requirements of memory-based singular thought about objects, and the sense in which remembering involves ‘mental time travel’ through which one ‘relives’ past events. In this paper, I argue that we sometimes do recall objects from our past without remembering events in which they featured. The positive view of such cases I go on to propose draws on a wide body of empirical work in its support and accommodates a more nuanced picture of the role of imagery in remembering. Succinctly, remembering might essentially involve a kind of ‘re-experiencing’, but it need not involve ‘reliving’.

Outrage and the Bounds of Empathy

Often, when we are angry, we are angry at someone who has hurt us. Our anger is a protest against a perceived mistreatment, and its function is to hold the person accountable for their offense. The anger involves a demand for some sort of change or response: that the hurt be acknowledged, that the relationship be repaired, that the offending party reform in some way. Call this “reform” anger. A different sort of attitude, often contrasted with reform anger, is hatred. Hatred is also a response to a perceived mistreatment, and it also demands some sort of change. Unlike reform anger, however, its goal is not to repair the relationship. Instead, its goal is destruction, to remove the offending party. In this paper, I develop and defend an account of a third sort of attitude, which I call “outrage” anger, that is distinct from both reform anger and hatred. I argue that outrage anger has an important role to play in the context of political injustice, but that it also comes with significant costs. In §1, I introduce outrage anger, and contrast it with reform anger. In §2, drawing on the work of Maria Lugones, I develop an account of outrage anger as a second-order attitude directed at the state of affairs in which a violation is not fully intelligible as the violation it is. I argue its central function is a kind of psychological boundary setting: it closes off the victim’s ability to feel empathy for their abuser. In §3, I show that the benefits of outrage come with serious costs, both epistemic and prudential. In §4, I make some suggestions about when, and for whom, the benefits of outrage outweigh the costs.

Market Exchange and Alienation

Efficient markets are alienating if they inhibit us from caring about one another inour productive activities. I argue that efficient market behaviour is bothexclusionary and fetishistic. As exclusionary, the efficient marketeer cannotmanifest care alongside their market behaviour. As fetishistic, the efficientmarketeer cannot manifest care in their market behaviour. The conjunction entailsthat efficient market behavior inhibits care. It doesn’t follow that efficient marketbehavior is vicious: individuals might justifiably commit to efficiency becausedoing so serves the common good. But efficient market systems nevertheless havesignificant opportunity costs. This serves as a corrective to the prevailingassumption amongst welfare state capitalists, liberal egalitarians and marketsocialists that resolving distributive objections to markets will resolve thisrelational objection. 

Taking the Straight Path. P.F. Strawson's Later Work on Freedom and Responsibility

I highlight three features of P.F. Strawson’s later, neglected work on freedom and responsibility. First, in response to a criticism by Rajendra Prasad, Strawson explicitly rejects an argument put forward in ‘Freedom and Resentment’ against the relevance of determinism to moral responsibility. Second, his remarkable acceptance of Prasad’s criticism motivates him to take the ‘straight path’, that is, to be straightforward about the relation between determinism, freedom, the ability to do otherwise and the conditions of responsibility. He claims that the ability to do otherwise is a necessary condition of responsibility and provides a list of additional conditions, including a knowledge condition. Third, he clarifies the relation between responsibility, quality of will and the reactive attitudes. The latter do not figure essentially in his answer to the question, ‘What are the conditions of responsibility?’, but they do play an essential role in his answer to the question, ‘Why do we have the concept of responsibility?’ We have it, Strawson suggests, because of our natural concern about the quality of will with which people act, a concern expressed in our reactive attitudes. I argue that, although Strawson’s later work definitely involves a shift of emphasis when compared to ‘Freedom and Resentment’, his overall account of freedom and responsibility is coherent. The later work helps to better understand the nature and significance of Strawson’s contribution, and to identify problems with common interpretations of and objections to ‘Freedom and Resentment’. 

Idealizations and ideal policing

Political philosophy often focuses on “major institutions” that make up the “basic structure” of society. These include political, economic, and social institutions. In this paper I argue first that policing plays a substantial role in generating the kinds of inequalities and problems that are concerns of social or structural justice, and therefore that police agencies qualify as a major institution. When we abandon full compliance or similar idealizations, it is clear that policing is not a concern secondary to, e.g., the electoral system or the scheme of property rights in a society. Nor, I argue, does maintaining full compliance or moral character idealizations obviate an active enforcement role. Eliminating that role from an idealized model society requires engaging in a methodologically and substantively unattractive amount of abstraction. The result is that an active enforcement role is at the core of a complete theory of justice rather than something that is significant only “downstream” from more fundamental issues.

For Better And Worse: Taking phenomenalist accounts of intuition seriously

The historically-influential perceptual analogy states that intuitions and perceptual experiences are alike in many important respects. Phenomenalists defend a particular reading of this analogy according to which intuitions and perceptual experiences share a common phenomenal character. The phenomenalist thesis has proven highly influential in recent years. However, insufficient attention has been given to the challenges that the phenomenalist thesis raises for theories of intuitions. In this paper, I first develop one such challenge. I argue that if we take seriously the idea that intuitions and perceptual experiences have a common phenomenal character, then an analogous version of the familiar problem of perceptual presence arises for intuitions. I call this the 'problem of intuitive presence'. In the second part of the paper I sketch a novel enactivist solution to this problem.

Assertion is weak

Recent work has argued that belief is weak: the level of rational credence required for belief is relatively low. That literature has  contrasted belief with assertion, arguing that the latter requires an epistemic state much stronger than (weak) belief---perhaps knowledge or even certainty.  We argue that this is wrong: assertion is just as weak as belief.  We first present a variety of new arguments for this, and then show that the standard arguments for stronger norms are not convincing.  Finally, we sketch an alternative picture on which the fundamental norm of assertion is to say what you believe, but both belief and assertion are weak. To help make sense of this, we propose that both belief and assertion involve navigating a tradeoff between accuracy and informativity, and so it can makes sense to believe/say something you only have weak evidence for, if it is informative enough.