At first glance, it might seem an odd pairing: the Analects and Wittgenstein. Comparison between a classical Chinese philosophical text, whose primary topics were the cultivation of xiao and he, and the corpus of an early to mid-twentieth-century Austrian philosopher, whose primary topics had to do with logic, language, and the nature of philosophy, does not obviously recommend itself. Yet, I contend in this article that there is much to be gained from careful comparison between these two very different (...) pictures of philosophy, particularly where it comes to the practices of clarifying that which is confused or obscured. This study will aid primarily in developing Wittgensteinian... (shrink)

In this chapter we take stock of some of the complexities of the sprawling field of AI ethics. We consider questions like "what is the proper scope of AI ethics?" And "who counts as an AI ethicist?" At the same time, we flag several potential uses and abuses of AI ethics. These include challenges for the AI ethicist, including what qualifications they should have; the proper place and extent of futuring and speculation in the field; and the dilemmas concerning how (...) we use our public and academic resources. (shrink)

In this chapter we identify three potentially morally problematic behaviours that are common among users of dating and hook-up apps (DHAs) and provide arguments as to why they may or may not be considered (a) in a category of their own, distinct from similar behaviours outside of DHAs; (b) caused or facilitated by affordances and business logic of DHAs; (c) as indeed morally wrong. We also consider ways in which morally problematic behaviours can be anticipated, mitigated, or even prevented by (...) analysis of the ethical and moral dimensions of technologies and their afforded uses. Finally, we offer some possible directions for future work on these topics in particular and on the ethical consequences of DHAs in general. (shrink)

How could ectogestative technology disrupt gender roles, parenting practices, and concepts such as ‘birth’, ‘body’, or ‘parent’? In this chapter, we situate this emerging technology in the context of the history of reproductive technologies and analyse the potential social and conceptual disruptions to which it could contribute. An ectogestative device, better known as ‘artificial womb’, enables the extra-uterine gestation of a human being, or mammal more generally. It is currently developed with the main goal of improving the survival chances of (...) extremely premature neonates. We argue that the intended use of the technology in neonatal intensive care units, as an alternative to current incubators (’partial-ectogestation’), challenges concepts such as ‘birth’, ‘fetus’, and ‘neonate’, and has several ethico-legal implications. We moreover address a more futuristic scenario where the entire embryological and fetal development could happen within an artificial womb (’full-ectogestation’). Such a scenario reveals the disruption of gender roles, parenting practices, and concepts such as ‘mother’, ‘father’, and ‘parent’. Both full- and partial-ectogestation would have implications for engineering and design, law-making, ethics, and philosophical anthropology. (shrink)

One of the most legendary educational books ever written is Jean-Jacques Rousseau’s “Émile ou de l’Education”. Most obviously Rousseau wrote this book guided by diverse more or less conscious purposes and one of the main problems it presents is paradoxical: Does education have to promote freedom by force? In this article I will, firstly, present several aims that might have triggered Rousseau to write “Émile”. Secondly, I will discuss Rousseau’s view of the so called “educational paradox”. Since this quandary touches (...) the topic of many other of his books, I will discuss “Émile” along with Rousseau’s other works and thus place his educational story in his “great narrative”. (shrink)

In this article, we engage with a theory of management advanced by MacIntyrean scholars of business ethics and organization studies to develop an account of “chronic moral injury” in the workplace. In contrast to what we call “acute moral injury,” which focuses on grave, traumatic events, chronic moral injury results from poor institutional form—when an individual desiring excellence must function within a vicious institution that impedes the acquisition of virtues and marginalizes practices. In other words, chronic moral injury occurs when (...) practitioners who pursue excellence in their practice work within corrupt or malformed organizations. To demonstrate this point, we recount the events associated with the rise and fall of the biotech company, Theranos. This case study advances an empirical contribution to MacIntyrean studies by demonstrating how chronic moral injury can happen under such conditions and what the negative consequences may entail for workers. (shrink)

Kunstmatige intelligentie (AI) en systemen die met machine learning (ML) werken, kunnen veel onderdelen van het medische besluitvormingsproces ondersteunen of vervangen. Ook zouden ze artsen kunnen helpen bij het omgaan met klinische, morele dilemma’s. AI/ML-beslissingen kunnen zo in de plaats komen van professionele beslissingen. We betogen dat dit belangrijke gevolgen heeft voor de relatie tussen een patiënt en de medische professie als instelling, en dat dit onvermijdelijk zal leiden tot uitholling van het institutionele vertrouwen in de geneeskunde.

In this chapter, we consider ethical and philosophical aspects of trust in the practice of medicine. We focus on trust within the patient-physician relationship, trust and professionalism, and trust in Western (allopathic) institutions of medicine and medical research. Philosophical approaches to trust contain important insights into medicine as an ethical and social practice. In what follows we explain several philosophical approaches and discuss their strengths and weaknesses in this context. We also highlight some relevant empirical work in the section on (...) trust in the institutions of medicine. It is hoped that the approaches discussed here can be extended to nursing and other topics in the philosophy of medicine. (shrink)

It is not clear to what the projects of creating an artificial intelligence (AI) that does ethics, is moral, or makes moral judgments amounts. In this paper we discuss some of the extant metaethical theories and debates in moral philosophy by which such projects should be informed, specifically focusing on the project of creating an AI that makes moral judgments. We argue that the scope and aims of that project depend a great deal on antecedent metaethical commitments. Metaethics, therefore, plays (...) the role of an Archimedean fulcrum in this context, very much like the Archimedean role that it is often taken to take in context of normative ethics (Dworkin 1996; Dreier 2002; Fantl 2006; Ehrenberg 2008). (shrink)

Volume 19, Issue 7, July 2019, Page 39-40.

Advocates of moral enhancement through pharmacological, genetic, or other direct interventions sometimes explicitly argue, or assume without argument, that traditional moral education and development is insufficient to bring about moral enhancement. Traditional moral education grounded in a Kohlbergian theory of moral development is indeed unsuitable for that task; however, the psychology of moral development and education has come a long way since then. Recent studies support the view that moral cognition is a higher-order process, unified at a functional level, and (...) that a specific moral faculty does not exist. It is more likely that moral cognition involves a number of different mechanisms, each connected to other cognitive and affective processes. Taking this evidence into account, we propose a novel, empirically informed approach to moral development and education, in children and adults, which is based on a cognitive-affective approach to moral dispositions. This is an interpretative approach that derives from the cognitive-affective personality system (Mischel and Shoda, 1995). This conception individuates moral dispositions by reference to the cognitive and affective processes that realise them. Conceived of in this way, moral dispositions influence an agent's behaviour when they interact with situational factors, such as mood or social context. Understanding moral dispositions in this way lays the groundwork for proposing a range of indirect methods of moral enhancement, techniques that promise similar results as direct interventions whilst posing fewer risks. (shrink)

Objectives: The growing rate of retraction of scientific publications has attracted much attention within the academic community, but there is little knowledge about the nature of such retractions in schizophrenia-related research. This study aimed to analyze the characteristics of retractions of schizophrenia-related publications.

All through the COVID-19-crisis Conspiracy Theories and False Information spread all around the globe. In this article, we want to suggest that the spreading and retainment of disinformation despite counter-evidence is best to be understood in the context of echo chambers as described by Chris Thi Nguyen. Moreover, we want to argue that people active in those echo chambers are at the same time perpetrators as well as victims of epistemic injustice to different amounts. Although this article cannot cover the (...) phenomenon as a whole, we hope to outline a path for further investigation. (shrink)

Engineering an artificial intelligence to play an advisory role in morally charged decision making will inevitably introduce meta-ethical positions into the design. Some of these positions, by informing the design and operation of the AI, will introduce risks. This paper offers an analysis of these potential risks along the realism/anti-realism dimension in metaethics and reveals that realism poses greater risks, but, on the other hand, anti-realism undermines the motivation for engineering a moral AI in the first place.

Ang papel ay paglalagom ng itinuturing na pinakamahalagang yugto ng buhay ni José Rizal—ang kanyang pagkakatapon sa Dapitan, sa lente ng konseptong “bagani” ng ating mga ninuno. Sa pagbanggit sa lang nauna nang pag-aaral ukol kay Rizal, ipapakitang kung papaanong ang yugto sa Dapitan ay ang katuparan ng kanyang mga layunin sa La Liga Filipina, na kanyang isinabuhay kasama ang bayan. Sa lente ng kasalukuyang pandemyang COVID 19, makikita natin na tila naranasan din ni Rizal sa kanyang pagkakatapon ang ilang (...) bagay na ating dinaranas ngayon, na tila siya isang frontliner na nasa lockdown, inilagay sa kuwarentena, nag-homeschooling, nagkaroon ng lockdown sickness at namahagi ng ayudang pangkabuhayan sa mga mamamayan ng isang natutulog na bayan upang ito ay sumiglang muli. Nawa ay maging inspirasyon sa bayan na anuman ang hamon at mga balakid, walang makapipigil sa isang Pilipino na magsilbi sa bayan gamit ang kanyang mga talento at manatiling liwanag ng katotohanan at kaalaman. (shrink)

Ratio, Volume 34, Issue 4, Page 303-311, December 2021.

Dating and hookup apps (DHAs) are now widely used and may be transforming our intimate relationships. The apps are beneficial in fostering intimate connections among those who are lonely, who are members of minority or marginalized groups, or who live nomadic lifestyles because of work or recreational travel. However, the wider social and relational changes that DHAs portend are merely beginning to be seriously discussed by academics (Arias et al., 2017). In this chapter, we employ concepts from the philosophy of (...) technology that inform our analysis of these changes. We also collate some of the morally significant impacts of DHAs and present some of the elements of the architecture of DHAs that facilitate them. Although there are many reasons to be concerned about DHAs, we argue that there is room for cautious optimism that DHAs may be vehicles of positive social and moral change. (shrink)

Chinese international students who studied in the United States received “double stigmatization” from American and Chinese authorities because of the “political othering” tactic during COVID-19. The research used a phenomenological approach to examine why and how specifically the transnational identity of Chinese international students in the United States shifted during the double stigmatization. The researcher conducted a total of three rounds of interviews with 15 Chinese international students who studied in the United States and returned to China between 2018 and (...) 2020, which culminated in 45 interviews through a longitudinal study to probe the transnational identities of this population before and during the double stigmatization; the study also examined how the mindsponge mechanism worked during the identity shifts and the interplay among stigmatization, transnational identity shifts, and the mindsponge mechanism. The study concluded that before COVID-19, Chinese international students had been stigmatized in both China and the United States. (shrink)

This paper provides an analysis of the way in which two foundational principles of medical ethics–the trusted doctor and patient autonomy–can be undermined by the use of machine learning (ML) algorithms and addresses its legal significance. This paper can be a guide to both health care providers and other stakeholders about how to anticipate and in some cases mitigate ethical conflicts caused by the use of ML in healthcare. It can also be read as a road map as to what (...) needs to be done to achieve an acceptable level of explainability in an ML algorithm when it is used in a healthcare context. (shrink)

A combinatorial analysis of intrinsicality takes intrinsic properties to be independent of accompaniment: a property is intrinsic only if it is possible for a lonely or an accompanied thing to have it or lack it (I). Cameron argues that the combinatorial analysis in Langton & Lewis (1998) faces an epistemic circularity, which makes (I) suspicious. In this paper, I examine two approaches to free the combinatorial analyses from the circularity and find them all fail. Then I provide an outline of (...) an argument for (I) by appealing to the grounding analyses of intrinsicality, which shows an advantage of the grounding analyses over the combinatorial ones. (shrink)

The proposition that digital innovations can put people in charge of their health has been accompanied by prolific talk of empowerment. In this paper we consider ethical challenges and opportunities of trying to achieve justice and empowerment using digital health initiatives. The language of empowerment can misleadingly suggest that by using technology, people can control their health and take responsibility for health outcomes to a greater degree than is realistic or fair. Also, digital health empowerment often primarily reaches people who (...) already have high technological and health literacy, leaving others behind. We critically investigate whether the concept of health empowerment could be re-engineered to mean something different, namely the process of improving the health literacy and access of those who least possess it, in line with recent proposals for conceptual engineering in the service of justice. We settle on the weaker conclusion that underlying ethical values including justice should be used to interpret the existing concept of empowerment. To conclude, we take a high-level view of various strategies for achieving the ethical value associated with digital health empowerment. (shrink)

Both motivational internalism and externalism need to explain why sometimes moral judgments tend to motivate us. In this paper, I argue that Dreier’ second-order desire model cannot be a plausible externalist alternative to explain the connection between moral judgments and motivation. I explain that the relevant second-order desire is merely a constitutive requirement of rationality because that desire makes a set of desires more unified and coherent. As a rational agent with the relevant second-order desire is disposed towards coherence, she (...) will have some motivation to act in accordance with her moral judgments. Dreier’s second-order desire model thus collapses into a form of internalism and cannot be a plausible externalist option to explain the connection between moral judgments and motivation. (shrink)

Pain may appear to undermine the radically intentionalist view that the phenomenal character of any experience is entirely constituted by its representational content. That appearance is illusory. After categorizing versions of pain intentionalism along two dimensions, I argue that an 'objectivist' and 'non-mentalist' version is the most promising, if it can withstand two objections concerning what we say when in pain, and the distinctiveness of pain. I rebut these objections, in a way available to both opponents of and adherents to (...) the view that experiential content is entirely conceptual. (shrink)

Lack of a historical sense is the hereditary defect of philosophers. … So what is needed from now on is historical philosophizing, and with it the virtue of modesty.

Philosophers have rightly condemned lookism—that is, discrimination in favor of attractive people or against unattractive people—in education, the justice system, the workplace and elsewhere. Surprisingly, however, the almost universal preference for attractive romantic and sexual partners has rarely received serious ethical scrutiny. On its face, it’s unclear whether this is a form of discrimination we should reject or tolerate. I consider arguments for both views. On the one hand, a strong case can be made that preferring attractive partners is bad. (...) The idea is that choosing partners based on looks seems essentially similar to other objectionable forms of discrimination. (In particular, the preference for attractive partners is arguably both unfair and harmful to a significant degree.) One can try to resist this conclusion in several ways. I consider three possible replies. The first has to do with the possibility of controlling our partner preferences. The second pertains to attractiveness and “good genes”. The last attempts to link certain aspects of attractiveness to a prospective partner’s personality and values. I argue that the first two replies fail conclusively, while the third only amounts to a limited defense of a particular kind of attractiveness preference. So the idea that we should often avoid preferring attractive partners is compelling. (shrink)

This article is a preliminary presentation of conjunctive paraconsistency, the claim that there might be non-explosive true contradictions, but contradictory propositions cannot be considered separately true. In case of true ‘p and not p’, the conjuncts must be held untrue, Simplification fails. The conjunctive approach is dual to non-adjunctive conceptions of inconsistency, informed by the idea that there might be cases in which a proposition is true and its negation is true too, but the conjunction is untrue, Adjunction fails. While (...) non-adjunctivism is a well-known option, the other view is not so much studied nowadays, but it was not unknown in the tradition, and there are some positive suggestions, in recent literature, that the position is plausible and deserves to be developed. The article compares conjunctivism, non-adjunctivism and dialetheism, then focuses on some possible justifications, costs and benefits of the conjunctive view. (shrink)

This paper presents a Least Square Monte Carlo approach for accurately calculating credit value adjustment (CVA). In contrast to previous studies, the model relies on the probability distribution of a default time/jump rather than the default time itself, as the default time is usually inaccessible. As such, the model can achieve a high order of accuracy with a relatively easy implementation. We find that the valuation of a defaultable derivative is normally determined via backward induction when their payoffs could be (...) positive or negative. Moreover, the model can naturally capture wrong or right way risk. (shrink)

In this thesis, I explore motivational internalism and externalism, which concern the relationship between moral judgments and motivation. I first introduce the basic terms and different forms of internalism and externalism, including the externalist objections to internalism based on the famous counterexamples. I then argue against externalism by defending and developing Michael Smith’s fetishism argument. I not only respond to the externalist objections to the fetishism argument but also further argue against different externalist explanations of moral motivation that intend to (...) avoid the fetishism charge. Finally, I re-examine different forms of internalism in order to argue for a new form of internalism that can better preserve our internalist intuitions whilst accommodating the externalist counterexamples. My ultimate conclusion will be that the most plausible form of internalism to accept is constitutional, unconditional, relatively strong, direct internalism that is formulated in terms of dispositional desires. (shrink)

In this paper, I develop and defend a new adverbial theory of perception. I first present a semantics for direct-object perceptual reports that treats their object positions as supplying adverbial modifiers, and I show how this semantics definitively solves the many-property problem for adverbialism. My solution is distinctive in that it articulates adverbialism from within a well-established formal semantic framework and ties adverbialism to a plausible semantics for perceptual reports in English. I then go on to present adverbialism as a (...) theory of the metaphysics of perception. The metaphysics I develop treats adverbial perception as a directed activity: it is an activity with success conditions. When perception is successful, the agent bears a relation to a concrete particular, but perception need not be successful; this allows perception to be fundamentally non-relational. The result is a novel formulation of adverbialism that eliminates the need for representational contents, but also treats successful and unsuccessful perceptual events as having a fundamental common factor. (shrink)

This paper presents a new model for pricing OTC derivatives subject to collateralization. It allows for collateral posting adhering to bankruptcy laws. As such, the model can back out the market price of a collateralized contract. This framework is very useful for valuing outstanding derivatives. Using a unique dataset, we find empirical evidence that credit risk alone is not overly important in determining credit-related spreads. Only accounting for both collateral arrangement and credit risk can sufficiently explain unsecured credit costs. This (...) finding suggests that failure to properly account for collateralization may result in significant mispricing of derivatives. We also empirically gauge the impact of collateral agreements on risk measurements. Our findings indicate that there are important interactions between market and credit risk. (shrink)

We discuss potential biorisks from large language models (LLMs). AI assistants based on LLMs such as ChatGPT have been shown to significantly reduce barriers to entry for actors wishing to synthesize dangerous, potentially novel pathogens and chemical weapons. The harms from deploying such bioagents could be further magnified by AI-assisted misinformation. We endorse several policy responses to these dangers, including prerelease evaluations of biomedical AIs by subject-matter experts, enhanced surveillance and lab screening procedures, restrictions on AI training data, and access (...) controls on biomedical AI. We conclude with a suggestion about future research directions in bioethics. (shrink)

According to a widespread view in metaphysics and philosophy of science, all explanations involve relations of ontic dependence between the items appearing in the explanandum and the items appearing in the explanans. I argue that a family of mathematical cases, which I call “viewing-as explanations”, are incompatible with the Dependence Thesis. These cases, I claim, feature genuine explanations that aren’t supported by ontic dependence relations. Hence the thesis isn’t true in general. The first part of the paper defends this claim (...) and discusses its significance. The second part of the paper considers whether viewing-as explanations occur in the empirical sciences, focusing on the case of so-called fictional models. It’s sometimes suggested that fictional models can be explanatory even though they fail to represent actual worldly dependence relations. Whether or not such models explain, I suggest, depends on whether we think scientific explanations necessarily give information relevant to intervention and control. Finally, I argue that counterfactual approaches to explanation also have trouble accommodating viewing-as cases. (shrink)

The problem of truth in fiction concerns how to tell whether a given proposition is true in a given fiction. Thus far, the nearly universal consensus has been that some propositions are ‘implicitly true’ in some fictions: such propositions are not expressed by any explicit statements in the relevant work, but are nevertheless held to be true in those works on the basis of some other set of criteria. I call this family of views ‘implicitism’. I argue that implicitism faces (...) serious problems, whereas the opposite view is much more plausible than has previously been thought. After mounting a limited defence of explicitism, I explore a difficult problem for the view and discuss some possible responses. (shrink)

In this paper I show that we have strong empirical and theoretical reasons to treat the verbs we use in our semantic theorizing—particularly ‘refers to ’, ‘applies to ’, and ‘is true of ’—as intensional transitive verbs. Stating our semantic theories with intensional vocabulary allows us to partially reconcile two competing approaches to the nature and subject-matter of semantics: the Chomskian approach, on which semantics is non-relational, internalistic, and concerns the psychology of language users, and the Lewisian approach, on which (...) semantics is fully relational, specifies truth-conditions, and has metaphysical implications. ITVs have two readings: an intensional, de dicto reading, and a relational, de re reading. A semantic theory stated with the de dicto readings of our semantic verbs captures the core insights of the Chomskian approach to semantics, in part because it allows us to assign extremely fine-grained semantic values to expressions, even when those expressions are empty. On the other hand, the de re reading yields a theory that is fully relational, and issues in truth-conditions. The resulting theories are related—and compatible—in that they are expressed by two different readings of the very same semantic vocabulary, and plausibly, the distinction between these two readings is one of scope. (shrink)

Equity-linked securities with a guaranteed return become popular in a volatile market environment. This paper presents a new model for valuing guaranteed equity-linked notes. We consider a security whose value depends on the performance of a basket of equities averaged over certain points in time, but that is floored by a guaranteed amount. We show that the security’s price is given by the sum of the guaranteed amount plus the price of an Asian style option on the basket above. The (...) model provides analytical formulas for the security’s price as well as for corresponding hedge ratios; these respective formulas appear to be accurate over a wide range of underlying security parameter values, based on numerical testing against a Monte Carlo benchmark, Finally, we apply our method to value a type of segregated fund with a maturity guarantee. (shrink)

This paper explores the connections between being Chinese in various senses and the study of Chinese philosophy.

In this essay, we will follow Tang Junyi’s lead in exploring issues related to Chinese diaspora and Chinese philosophy. While we largely endorse Tang’s call for overseas Chinese to establish themselves in their adopted lands, we will argue for a more nuanced view on the identity of Chinese people outside China: they are not marginalized individuals scattered out of “homeland” China, rather they are people legitimately established in their own respective countries. In this connection, we will also advance a view (...) of future Chinese philosophy as a world philosophy that takes roots in China as well as in the world at large. In our view, although Chinese immigrant thinkers in the past century have played a major role in promoting Chinese philosophy outside China, the importance of such a role is likely to decline along with the success of Chinese philosophy becoming a world philosophy. (shrink)

Deductive inference is usually regarded as being “tautological” or “analytical”: the information conveyed by the conclusion is contained in the information conveyed by the premises. This idea, however, clashes with the undecidability of first-order logic and with the (likely) intractability of Boolean logic. In this article, we address the problem both from the semantic and the proof-theoretical point of view. We propose a hierarchy of propositional logics that are all tractable (i.e. decidable in polynomial time), although by means of growing (...) computational resources, and converge towards classical propositional logic. The underlying claim is that this hierarchy can be used to represent increasing levels of “depth” or “informativeness” of Boolean reasoning. Special attention is paid to the most basic logic in this hierarchy, the pure “intelim logic”, which satisfies all the requirements of a natural deduction system (allowing both introduction and elimination rules for each logical operator) while admitting of a feasible (quadratic) decision procedure. We argue that this logic is “analytic” in a particularly strict sense, in that it rules out any use of “virtual information”, which is chiefly responsible for the combinatorial explosion of standard classical systems. As a result, analyticity and tractability are reconciled and growing degrees of computational complexity are associated with the depth at which the use of virtual information is allowed. (shrink)

This is the first chapter to our edited collection of essays on the nature and ethics of blame. In this chapter we introduce the reader to contemporary discussions about blame and its relationship to other issues (e.g. free will and moral responsibility), and we situate the essays in this volume with respect to those discussions.

This article presents a new methodology for pricing and hedging commodity derivatives. A generic model calibration is provided. The calibration procedure consists of an offline step where the mean reversion rates, the ratio of the long and short factor volatilities and the correlation between the long and short factors are determined via historical analysis. This offline step is performed relatively infrequently. There’s also an online step of the calibration which happens every time the model is used to price an option (...) or to simulate price paths. Empirical study shows that the model is quite accurate. (shrink)

Appropriate information and consent has been one of the most intensely discussed topics within the context of biobank research. In parallel to the normative debate, many socio-empirical studies have been conducted to gather experiences, preferences and views of patients, healthy research participants and further stakeholders. However, there is scarcity of literature which connects the normative debate about justifications for different consent models with findings gained in empirical research. In this paper we discuss findings of a limited review of socio-empirical research (...) on patients’ and healthy research participants’ experiences and views regarding consent to biobank research in light of ethical principles for appropriate information and consent. (shrink)

Multidimensional adjectives are ubiquitous in natural language. An adjective F is multidimensional just in case whether F applies to an object or pair of objects depends on how those objects stand with respect to multiple underlying dimensions of F-ness. Developing a semantics for multidimensional adjectives requires us to address the problem of dimensional aggregation: how do the application conditions of an adjective F in its positive and comparative forms depend on its underlying dimensions? Here we develop a semantics for multidimensional (...) adjectives that incorporates aggregation functions. We then explore an analogy between dimensional aggregation and preference aggregation, bringing results from social choice theory to bear on the number and kind of aggregation functions that are admissible in a context. These results suggest that, for any given adjective, there will often be multiple aggregation functions admissible, meaning that multidimensional comparatives are often vague. (shrink)

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss (...) and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation. (shrink)