The debate about the ethical implications of Artificial Intelligence dates from the 1960s :741–742, 1960; Wiener in Cybernetics: or control and communication in the animal and the machine, MIT Press, New York, 1961). However, in recent years symbolic AI has been complemented and sometimes replaced by Neural Networks and Machine Learning techniques. This has vastly increased its potential utility and impact on society, with the consequence that the ethical debate has gone mainstream. Such a debate has primarily focused on principles—the (...) ‘what’ of AI ethics —rather than on practices, the ‘how.’ Awareness of the potential issues is increasing at a fast rate, but the AI community’s ability to take action to mitigate the associated risks is still at its infancy. Our intention in presenting this research is to contribute to closing the gap between principles and practices by constructing a typology that may help practically-minded developers apply ethics at each stage of the Machine Learning development pipeline, and to signal to researchers where further work is needed. The focus is exclusively on Machine Learning, but it is hoped that the results of this research may be easily applicable to other branches of AI. The article outlines the research method for creating this typology, the initial findings, and provides a summary of future research needs. (shrink)

Knowledge ascriptions are a central topic of research in both philosophy and science. In this collection of new essays on knowledge ascriptions, world class philosophers offer novel approaches to this long standing topic.

Technologies to rapidly alert people when they have been in contact with someone carrying the coronavirus SARS-CoV-2 are part of a strategy to bring the pandemic under control. Currently, at least 47 contact-tracing apps are available globally. They are already in use in Australia, South Korea and Singapore, for instance. And many other governments are testing or considering them. Here we set out 16 questions to assess whether — and to what extent — a contact-tracing app is ethically justifiable.

As the range of potential uses for Artificial Intelligence, in particular machine learning, has increased, so has awareness of the associated ethical issues. This increased awareness has led to the realisation that existing legislation and regulation provides insufficient protection to individuals, groups, society, and the environment from AI harms. In response to this realisation, there has been a proliferation of principle-based ethics codes, guidelines and frameworks. However, it has become increasingly clear that a significant gap exists between the theory of (...) AI ethics principles and the practical design of AI systems. In previous work, we analysed whether it is possible to close this gap between the ‘what’ and the ‘how’ of AI ethics through the use of tools and methods designed to help AI developers, engineers, and designers translate principles into practice. We concluded that this method of closure is currently ineffective as almost all existing translational tools and methods are either too flexible or too strict. This raised the question: if, even with technical guidance, AI ethics is challenging to embed in the process of algorithmic design, is the entire pro-ethical design endeavour rendered futile? And, if no, then how can AI ethics be made useful for AI practitioners? This is the question we seek to address here by exploring why principles and technical translational tools are still needed even if they are limited, and how these limitations can be potentially overcome by providing theoretical grounding of a concept that has been termed ‘Ethics as a Service.’. (shrink)

It has become recently popular to suggest that knowledge is the epistemic norm of practical reasoning and that this provides an important constraint on the correct account of knowledge, one which favours subject-sensitive invariantism over contextualism and classic invariantism. I argue that there are putative counterexamples to both directions of the knowledge norm. Even if the knowledge norm can be defended against these counterexamples, I argue that it is a delicate issue whether it is true, one which relies on fine (...) distinctions among a variety of relevant notions of propriety which our intuitions may reflect. These notions variously apply to the agent herself, her character traits, her beliefs, her reasoning and any resultant action. Given the delicacy of these issues, I argue that the knowledge norm is not a fixed point from which to defend substantive and controversial views in epistemology. Rather, these views need to be defended on other grounds. (shrink)

Introduction : reading Nietzsche skeptically -- Nietzsche and the Pyrrhonian tradition -- Skepticism in Nietzsche's early work : the case of "on truth and lie" -- The question of Nietzsche's "naturalism" -- Perspectivism and Ephexis in interpretation -- Skepticism and health -- Skepticism as immoralism.

Stanley and Williamson have defended the intellectualist thesis that knowing-how is a subspecies of knowing-that by appeal to the syntax and semantics of ascriptions of knowing-how. Critics have objected that this way of defending intellectualism places undue weight on linguistic considerations and fails to give sufficient attention to empirical considerations from the scientific study of the mind. In this paper, I examine and reject Stanley's recent attempt to answer the critics.

In this paper, I defend McKinsey's argument (Analysis 1991) that Burge's antiindividualist position is incompatible with privileged access, viz. the claim that each subject can know his own thought contents just by reflection and without having undertaken an empirical investigation. I argue that Burge thinks that there are certain necessary conditions for a subject to have thoughts involving certain sorts of concepts; these conditions are appropriately different for thoughts involving natural kind concepts and thoughts involving non-natural kind concepts. I use (...) Burge's commitment to these entailments to show that his antiindividualist position is incompatible with privileged access. (shrink)

did plato and aristotle have anything to say about belief? The answer to this question might seem blindingly obvious: of course they did. Plato distinguishes belief from knowledge in the Meno, Republic, and Theaetetus, and Aristotle does so in the Posterior Analytics. Plato distinguishes belief from perception in the Theaetetus, and Aristotle does so in the De anima. They talk about the distinction between true and false beliefs, and the ways in which belief can mislead and the ways in which (...) it can steer us aright. Indeed, they make belief a central component of their epistemologies.The view underlying these claims—one so widespread these days as to remain largely unquestioned—is that when Plato and Aristotle talk... (shrink)

I consider but reject one broad strategy for answering the threshold problem for fallibilist accounts of knowledge, namely what fixes the degree of probability required for one to know? According to the impurist strategy to be considered, the required degree of probability is fixed by one's practical reasoning situation. I distinguish two different ways to implement the suggested impurist strategy. According to the Relevance Approach, the threshold for a subject to know a proposition at a time is determined by the (...) practical reasoning situations she is then in to which that particular proposition is relevant. According to the Unity Approach, the threshold for a subject to know any proposition whatsoever at a time is determined by a privileged practical reasoning situation she then faces, most plausibly the highest stakes practical reasoning situation she is then in. I argue that neither way of implementing the impurist strategy succeeds and so impurism does not offer a satisfactory response to the threshold problem. (shrink)

By mid-2019 there were more than 80 AI ethics guides available in the public domain. Despite this, 2020 saw numerous news stories break related to ethically questionable uses of AI. In part, this is because AI ethics theory remains highly abstract, and of limited practical applicability to those actually responsible for designing algorithms and AI systems. Our previous research sought to start closing this gap between the ‘what’ and the ‘how’ of AI ethics through the creation of a searchable typology (...) of tools and methods designed to translate between the five most common AI ethics principles and implementable design practices. Whilst a useful starting point, that research rested on the assumption that _all_ AI practitioners are aware of the ethical implications of AI, understand their importance, and are actively seeking to respond to them. In reality, it is unclear whether this is the case. It is this limitation that we seek to overcome here by conducting a mixed-methods qualitative analysis to answer the following four questions: what do AI practitioners understand about the need to translate ethical principles into practice? What motivates AI practitioners to embed ethical principles into design practices? What barriers do AI practitioners face when attempting to translate ethical principles into practice? And finally, what assistance do AI practitioners want and need when translating ethical principles into practice? (shrink)

The main contribution of this paper is a new account of how a community may introduce a term for a natural kind in advance of knowing the correct scientific account of that kind. The account is motivated by the inadequacy of the currently dominant accounts of how a community may do this, namely those proposed by Kripke and by Putman. Their accounts fail to deal satisfactorily with the facts that (1) typically, an item that instantiates one natural kind instantiates several (...) - 'the higher-level natural kinds problem', and (2) natural kinds often occur in nature in impure form - 'the composition problem' .On the account I propose, a term for a natural kind gains its reference by being associated with a recognitional capacity for that kind. I show how members of a scientifically ignorant community could have a recognitional capacity for a natural kind, say gold, as opposed to a certain kind of appearance, for instance the appearance that gold actually has. I argue that members of such a community can have recognitional capacities for particular natural kinds despite the actual or possible existence of duplicate kinds, e.g. water. After developing the account in detail, I show how it can deal with the two problems faced by Kripke's and Putnam's problem. The case of natural kind terms is crucial to the central debate in the philosophy of language and the philosophy of mind about whether we can refer non-descriptively to objects and kinds in the world. I take the account I propose to be a non-descriptive account of linguistic reference to natural kinds that can be used to support externalism in the philosophy of mind. (shrink)

I will ask the conditional question: if folk attributions of "know" are not sensitive to the stakes and/or the salience of error, does this cast doubt on contextualism or subject-sensitive invariantism (SSI)? I argue that if it should turn out that folk attributions of knowledge are insensitive to such factors, then this undermines contextualism, but not SSI. That is not to say that SSI is invulnerable to empirical work of any kind. Rather, I defend the more modest claim that leading (...) versions of SSI are not undermined by one particular kind of experimental result, namely the recent suggestion that knowledge attributions are insensitive to the stakes. (shrink)

This article highlights the limitations of the tendency to frame health- and wellbeing-related digital tools (mHealth technologies) as empowering devices, especially as they play an increasingly important role in the National Health Service (NHS) in the UK. It argues that mHealth technologies should instead be framed as digital companions. This shift from empowerment to companionship is advocated by showing the conceptual, ethical, and methodological issues challenging the narrative of empowerment, and by arguing that such challenges, as well as the risk (...) of medical paternalism, can be overcome by focusing on the potential for mHealth tools to mediate the relationship between recipients of clinical advice and givers of clinical advice, in ways that allow for contextual flexibility in the balance between patiency and agency. The article concludes by stressing that reframing the narrative cannot be the only means for avoiding harm caused to the NHS as a healthcare system by the introduction of mHealth tools. Future discussion will be needed on the overarching role of responsible design. (shrink)

According to one contemporary formulation of infallibilism, probability 1 infallibilism, if a subject knows that p, then the probability of p on her evidence is 1. To avoid an implausible scepticism about knowledge, probability 1 infallibilism needs to allow that, in a wide range of cases, a proposition can be evidence for itself. However, such infallibilism needs to explain why it is typically infelicitous to cite p as evidence for p itself. I argue that probability 1 infallibilism has no explanation (...) of this infelicity and should be rejected. (shrink)

As the range of potential uses for Artificial Intelligence (AI), in particular machine learning (ML), has increased, so has awareness of the associated ethical issues. This increased awareness has led to the realisation that existing legislation and regulation provides insufficient protection to individuals, groups, society, and the environment from AI harms. In response to this realisation, there has been a proliferation of principle-based ethics codes, guidelines and frameworks. However, it has become increasingly clear that a significant gap exists between the (...) theory of AI ethics principles and the practical design of AI systems. In previous work , we analysed whether it is possible to close this gap between the ‘what’ and the ‘how’ of AI ethics through the use of tools and methods designed to help AI developers, engineers, and designers translate principles into practice. We concluded that this method of closure is currently ineffective as almost all existing translational tools and methods are either too flexible (and thus vulnerable to ethics washing) or too strict (unresponsive to context). This raised the question: if, even with technical guidance, AI ethics is challenging to embed in the process of algorithmic design, is the entire pro-ethical design endeavour rendered futile? And, if no, then how can AI ethics be made useful for AI practitioners? This is the question we seek to address here by exploring why principles and technical translational tools are still needed even if they are limited, and how these limitations can be potentially overcome by providing theoretical grounding of a concept that has been termed ‘Ethics as a Service’. (shrink)

This article discusses the role of diagrams in mathematical reasoning in the light of a case study in analysis. In the example presented certain combinatorial expressions were first found by using diagrams. In the published proofs the pictures were replaced by reasoning about permutation groups. This article argues that, even though the diagrams are not present in the published papers, they still play a role in the formulation of the proofs. It is shown that they play a role in concept (...) formation as well as representations of proofs. In addition we note that 'visualization' is used in two different ways. In the first sense 'visualization' denotes our inner mental pictures, which enable us to see that a certain fact holds, whereas in the other sense 'visualization' denotes a diagram or representation of something. (shrink)

The concept of homology arose from classical studies of comprative morphology, and took on a new signficance with the advent of evolutionary theory. It is currentlyl undergoing antoher metamorphosis: many developmental geneticists now dfine homology as shared patterns of gene expression. However, this ne usage conflaes difinition with criteri, and fails to recognize the meaninful asignments of homology must speify a biologcal level. We argue the although developmental genetic data can help identify homologus structures. they are niether necessary nor sufficient, (...) and do not in any case jutify a new definition of homology. (shrink)

The computer revolution has had an enormous effect on all aspects of the practice of medicine, yet little thought has been given to the role of social media in identifying treatment choices for incompetent patients. We are currently living in the ?Internet age? and many people have integrated social media into all aspects of their lives. As use becomes more prevalent, and as users age, social media are more likely to be viewed as a source of information regarding medical care (...) preferences. This article explores the ethical and legal issues raised by the use of social media in surrogate decision making. (shrink)

What is the nature of the evidence provided by thought experiments in philosophy? For instance, what evidence is provided by the Gettier thought experiment against the JTB theory of knowledge? According to one view, it provides as evidence only a certain psychological proposition, e.g. that it seems to one that the subject in the Gettier case lacks knowledge. On an alternative, nonpsychological view, the Gettier thought experiment provides as evidence the nonpsychological proposition that the subject in the Gettier case lacks (...) knowledge (e.g., Williamson 2007). Given the centrality of thought experiments to philosophical enquiry, the correct account of thought experiment evidence is important for understanding the nature of philosophical methodology. Further, Williamson argues that a misguided adherence to the psychological view of thought experiment evidence encourages scepticism about philosophy since it opens a gap between our evidence and the nonpsychological subject matter of philosophy. The main aim of this paper is to defend the psychological view against recent objections. In particular, I argue that even if thought experiment evidence is psychological, it can still provide justification for non-psychological claims which are the subject matter of philosophy. (shrink)

It is a longstanding problem for theorists of justice that many victims of injustice seem to prefer mistreatment, and perpetuate their own oppression. One possible response is to simply ignore such preferences as unreliable ‘adaptive preferences’. Capability theorists have taken this approach, arguing that individuals should be entitled to certain capabilities regardless of their satisfaction without them. Although this initially seems plausible, worries have been raised that undermining the reliability of individuals' strongly-held preferences impugns their rationality, and further excludes already (...) marginalised groups. I argue that such criticisms trade on an ambiguity between two uses of the term ‘adaptive preference’. An adaptive preference is often assumed to be irrational, and an unreliable guide to its possessor's best interests. However, I suggest a preference may also be adaptive in the sense that it is an unreliable guide to our distributive entitlements, and that this does not require an assessment of individuals' rationality. I consider this distinction in relation to disability, arguing that this clarification allows us to justifiably ignore some disabled individuals' preferences, in the context of theorising about distributive justice, without disrespecting or undermining their rationality or culture. (shrink)

The World Health Organisation declared COVID-19 a global pandemic on 11th March 2020, recognising that the underlying SARS-CoV-2 has caused the greatest global crisis since World War II. In this chapter, we present a framework to evaluate whether and to what extent the use of digital systems that track and/or trace potentially infected individuals is not only legal but also ethical.

Research groups around the world are currently busy trying to invent new life in the laboratory, looking for extraterrestrial life, or making machines increasingly more life-like. In the case of astrobiology, any newly discovered life would likely be very old, but when discovered it would be new to us. In the case of synthetic organic life or life-like machines, humans will have invented life that did not exist before. Together, these endeavors amount to what we call the emerging plurality of (...) lives because we see a future where we are surrounded by, and interact with, life of many different origins, as well as entities that we have not traditionally seen as being alive. In the future these entities will likely possess properties that we so far have only associated with living beings, thereby transcending and perhaps blurring the borders between life and non-life. (shrink)

This paper compares the statement ‘Mathematics is the study of structure’ with the actual practice of mathematics. We present two examples from contemporary mathematical practice where the notion of structure plays different roles. In the first case a structure is defined over a certain set. It is argued firstly that this set may not be regarded as a structure and secondly that what is important to mathematical practice is the relation that exists between the structure and the set. In the (...) second case, from algebraic topology, one point is that an object can be a place in different structures. Which structure one chooses to place the object in depends on what one wishes to do with it. Overall the paper argues that mathematics certainly deals with structures, but that structures may not be all there is to mathematics. (shrink)

Plato links pleasure with illusion, and this link explains his rejection of the view that all desires are rational desires for the good. The Protagoras and Gorgias show connections between pleasure and illusion; the Republic develops these into a psychological theory. One part of the soul is not only prone to illusions, but also incapable of the kind of reasoning that can dispel them. Pleasure appears good; therefore this part of the soul (the appetitive part) desires pleasures qua good but (...) ignores reasoning about what is really good. Hence the new moral psychology of the Republic: not all desires are rational, and thus virtue depends on bringing one's non-rational desires under the control of reason. (shrink)

If there is any consensus about knowledge in contemporary epistemology, it is that there is one primary kind: knowledge-that. I put forth a view, one I find in the works of Aristotle, on which knowledge-of – construed in a fairly demanding sense, as being well-acquainted with things – is the primary, fundamental kind of knowledge. As to knowledge-that, it is not distinct from knowledge-of, let alone more fundamental, but instead a species of it. To know that such-and-such, just like to (...) know a person or place, is to be well-acquainted with a portion of reality – in this case a fact. In part by comparing classic Gettier cases to cases in which one has true impressions of but fails to know a person, I argue that this account not only respects our intuitions about knowledge-that – in particular that it is or entails non-accidentally true justified belief – but also explains them, providing a compelling analysis. (shrink)

It has been suggested that to overcome the challenges facing the UK’s National Health Service (NHS) of an ageing population and reduced available funding, the NHS should be transformed into a more informationally mature and heterogeneous organisation, reliant on data-based and algorithmically-driven interactions between human, artificial, and hybrid (semi-artificial) agents. This transformation process would offer significant benefit to patients, clinicians, and the overall system, but it would also rely on a fundamental transformation of the healthcare system in a way that (...) poses significant governance challenges. In this article, we argue that a fruitful way to overcome these challenges is by adopting a pro-ethical approach to design that analyses the system as a whole, keeps society-in-the-loop throughout the process, and distributes responsibility evenly across all nodes in the system. (shrink)

In this paper I propose a position in the ontology of mathematics which is inspired mainly by a case study in the mathematical discipline if-theory. The main theses of this position are that mathematical objects are introduced by mathematicians and that after mathematical objects have been introduced, they exist as objectively accessible abstract objects.

Healthcare systems across the globe are struggling with increasing costs and worsening outcomes. This presents those responsible for overseeing healthcare with a challenge. Increasingly, policymakers, politicians, clinical entrepreneurs and computer and data scientists argue that a key part of the solution will be ‘Artificial Intelligence’ (AI) – particularly Machine Learning (ML). This argument stems not from the belief that all healthcare needs will soon be taken care of by “robot doctors.” Instead, it is an argument that rests on the classic (...) counterfactual definition of AI as an umbrella term for a range of techniques that can be used to make machines complete tasks in a way that would be considered intelligent were they to be completed by a human. Automation of this nature could offer great opportunities for the improvement of healthcare services and ultimately patients’ health by significantly improving human clinical capabilities in diagnosis, drug discovery, epidemiology, personalised medicine, and operational efficiency. However, if these AI solutions are to be embedded in clinical practice, then at least three issues need to be considered: the technical possibilities and limitations; the ethical, regulatory and legal framework; and the governance framework. In this article, we report on the results of a systematic analysis designed to provide a clear overview of the second of these elements: the ethical, regulatory and legal framework. We find that ethical issues arise at six levels of abstraction (individual, interpersonal, group, institutional, sectoral, and societal) and can be categorised as epistemic, normative, or overarching. We conclude by stressing how important it is that the ethical challenges raised by implementing AI in healthcare settings are tackled proactively rather than reactively and map the key considerations for policymakers to each of the ethical concerns highlighted. (shrink)

This article explores notions about Nietzsche’s career as a philologist and his fascination with the Greeks. It considers his interest in Homer and the Greek philosophers—in particular, Heraclitus and Pyrrho. For Nietzsche, ancient Greeks such as Heraclitus and Homer were interesting not because of their doctrines, but because of the example they themselves provided of certain psychological types. Like the ancient skeptics following Pyrrho, Nietzsche was generally more interested in the psychological consequences of philosophical doctrines than in their content, and (...) like those skeptics he often rejected any ambitions to limn the true nature of reality. (shrink)

The suggestion that individuals should be considered disadvantaged, and consequently entitled to compensation, only if they consider themselves disadvantaged (Dworkin’s ‘continuity test’) is initially appealing. However, it also faces problems. First, if individuals are routinely mistaken, then we routinely fail to assist the deserving. Second, if individuals assess their circumstances differently then the state will provide different levels of assistance to people in identical situations. Thus, should we instead ignore individuals’ convictions and provide assistance that some, at least, do not (...) feel they need? One set of cases where this dilemma is salient are those in which disabled individuals disagree over whether they are disadvantaged. Focussing on these, I argue that despite objections, individuals should have a voice in determining whether they are disadvantaged. However, I contend that our goal should not be ensuring continuity with individuals’ ethical convictions (concerning the pursuits they deem worthwhile or valuable), but their convictions regarding whether they are relevantly disadvantaged (whether they have what they are entitled to). (shrink)

I årtusenden har människan försökt definiera livet – hur levande djur och växter skiljer sig från död materia. Problemet är dock att livet är mångfacetterat, och varje regel har sitt undantag. Vi försöker möta kommande utmaningar med nya livsformer, genom att lyfta fram en ny definition av liv.

This paper discusses the notion of necessity in the light of results from contemporary mathematical practice. Two descriptions of necessity are considered. According to the first, necessarily true statements are true because they describe ‘unchangeable properties of unchangeable objects’. The result that I present is argued to provide a counterexample to this description, as it concerns a case where objects are moved from one category to another in order to change the properties of these objects. The second description concerns necessary (...) ‘structural properties’. Although I grant that mathematical statements could be considered as necessarily true in this sense, I question whether this justifies the claim that mathematics as a whole is necessary. (shrink)

In this paper, I criticise one main strategy for supporting anti-intellectualism, the view that whether a subject knows may depend on the stakes. This strategy appeals to difficulties with developing contextualist and pragmatic treatments of the shiftiness of our talk about knowledge to motivate anti-intellectualism. I criticise this strategy by drawing an analogy between debates about causation and knowledge. In each case, talk about a phenomenon is shifty and contextualist and pragmatic explanations of the shifty talk face the same objections. (...) However, in the case of causation it would be implausible to argue that difficulties with the relevant contextualist and pragmatic accounts motivate a revisionary metaphysics of causation. I conclude that the defender of anti-intellectualism needs to employ a different strategy to defend her view. (shrink)

Noninvasive prenatal screening using cell-free DNA, which analyzes placental DNA circulating in maternal blood to provide information about fetal chromosomal disorders early in pregnancy and without risk to the fetus, has been hailed as a potential “paradigm shift” in prenatal genetic screening. Commercial provision of cell-free DNA screening has contributed to a rapid expansion of the tests included in the screening panels. The tests can include screening for sex chromosome anomalies, rare subchromosomal microdeletions and aneuploidies, and most recently, the entire (...) fetal genome. The benefits of this screening tool are generally framed, by both providers and commercial laboratories, as enhancing reproductive autonomy and choice by providing an earlier, simpler, and more accurate screening while potentially reducing the need for invasive follow-up testing. The majority of the literature has explored these issues empirically or conceptually from a European or North American vantage point, one that assumes normative priorities such as individual reproductive autonomy and the clinical availability of maternal health care or prenatal screening programs within which cell-free DNA screening is offered. While its implementation has raised both challenges and opportunities, very little is known about real-world experiences and the implications of the rapid introduction of cell-free DNA screening outside of North America and Europe, especially in low- and middle-income countries. To begin addressing this gap in knowledge, we organized a four-day international workshop to explore the ethical, legal, social, economic, clinical, and practical implications of the global expansion of cell-free DNA screening. We describe eight key insights that arose from the workshop. (shrink)

Aristotle's De Anima discusses the psychological causes of what he calls locomotion – i.e, roughly, purpose-driven behavior. One cause is desire. The other is cognition, which falls into two kinds: thought (nous) and imagination (phantasia). Aristotle’s discussion is dense and confusing, but I argue that we can extract from it an account that is coherent, compelling, and that in many ways closely anticipates modern psychological theories, in particular Dual Processing theory. Animals and humans are driven to pursue objects that attract (...) us. Objects take on that power when we cognize them as valuable. If we rely on imagistic, automatic, uncontrolled processing mechanisms – Aristotle’s phantasia, which closely anticipates the modern notion of Type 1 processing – our resulting desires and actions will be impulsive. If we rely instead on rational, critical, deliberative capacities – Aristotle’s thought, which closely anticipates the modern notion of Type 2 processing – our resulting desires and actions will be reflective. Animals are capable only of the first kind of behavior; the human psyche is constituted of an animal psyche united with an intellectual one, so we are capable of both. (shrink)

This article takes as a starting point the current popular anti realist position, Fictionalism, with the intent to compare it with actual mathematical practice. Fictionalism claims that mathematical statements do purport to be about mathematical objects, and that mathematical statements are not true. Considering these claims in the light of mathematical practice leads to questions about how mathematical objects are handled, and how we prove that certain statements hold. Based on a case study on Riemann’s work on complex functions, I (...) propose that mathematicians deal with systems of representations and that truth—or what we can prove—depends on available representations in some context where the problem can be solved. (shrink)

The words in children's language learning environments are strongly predictive of cognitive development and school achievement. But how do we measure language environments and do so at the scale of the many words that children hear day in, day out? The quantity and quality of words in a child's input are typically measured in terms of total amount of talk and the lexical diversity in that talk. There are disagreements in the literature whether amount or diversity is the more critical (...) measure of the input. Here we analyze the properties of a large corpus of speech to children and simulate learning environments that differ in amount of talk per unit time, lexical diversity, and the contexts of talk. The central conclusion is that what researchers need to theoretically understand, measure, and change is not the total amount of words, or the diversity of words, but the function that relates total words to the diversity of words, and how that function changes across different contexts of talk. (shrink)