This chapter argues that the distinction between ambitious and modest transcendental arguments, developed and deployed by various authors in the wake of Stroud’s influential critique of transcendental reasoning, may be pointless when applied to transcendental arguments from performative inconsistency that have moral statements as their conclusions. If moral truth is assertorically constrained, then any modest moral transcendental argument from performative inconsistency can be converted into an ambitious moral transcendental argument. The chapter provides an account of performative inconsistency and suggests an (...) alternative to the widespread reading of transcendental conditionals in terms of an ‘if-then’-sentences whose antecedents express a proposition to the effect that some x is possible and whose consequents express a statement to the effect that some y is actual. (shrink)

This article informally presents a solution to the paradoxes of truth and shows how the solution solves classical paradoxes (such as the original Liar) as well as the paradoxes that were invented as counterarguments for various proposed solutions (“the revenge of the Liar”). This solution complements the classical procedure of determining the truth values of sentences by its own failure and, when the procedure fails, through an appropriate semantic shift allows us to express the failure in a classical two-valued language. (...) Formally speaking, the solution is a language with one meaning of symbols and two valuations of the truth values of sentences. The primary valuation is a classical valuation that is partial in the presence of the truth predicate. It enables us to determine the classical truth value of a sentence or leads to the failure of that determination. The language with the primary valuation is precisely the largest intrinsic fixed point of the strong Kleene three-valued semantics (LIFPSK3). The semantic shift that allows us to express the failure of the primary valuation is precisely the classical closure of LIFPSK3: it extends LIFPSK3 to a classical language in parts where LIFPSK3 is undetermined. Thus, this article provides an argumentation, which has not been present in contemporary debates so far, for the choice of LIFPSK3 and its classical closure as the right model for the truth predicate. In the end, an erroneous critique of Kripke-Feferman axiomatic theory of truth, which is present in contemporary literature, is pointed out. (shrink)

The aim of Modality and Explanatory Reasoning (MER) is to shed light on metaphysical necessity and the broader class of modal properties to which it belongs. This topic is approached with two goals: to develop a new and reductive analysis of modality, and to understand the purpose and origin of modal thought. I argue that a proper understanding of modality requires us to reconceptualize its relationship to causation and other forms of explanation such as grounding, a relation that connects metaphysically (...) fundamental facts to non-fundamental ones. While many philosophers have tried to give modal analyses of causation and explanation, often in counterfactual terms, I argue that we obtain a more plausible, explanatorily powerful and unified theory if we regard explanation as more fundamental than modality. The function of modal thought is to facilitate a common type of thought experiment – counterfactual reasoning – that allows us to investigate explanatory connections and which is closely related to the controlled experiments of empirical science. Necessity is defined in terms of explanation, and modal facts often reflect underlying facts about explanatory relationships. The study of modal facts is important for philosophy not because these facts are of much metaphysical interest in their own right, but largely because they provide evidence about explanatory connections. (shrink)

In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us, all directions are the same to us and all units of length we use to create geometric figures are the same to us. On the other hand, through the process of algebraic simplification, this system of axioms directly provides the Weyl’s (...) system of axioms for Euclidean geometry. The system of axioms, together with its a priori interpretation, offers new views to philosophy and pedagogy of mathematics: it supports the thesis that Euclidean geometry is a priori, it supports the thesis that in modern mathematics the Weyl’s system of axioms is dominant to the Euclid’s system because it reflects the a priori underlying symmetries, it gives a new and promising approach to learn geometry which, through the Weyl’s system of axioms, leads from the essential geometric symmetry principles of the mathematical nature directly to modern mathematics. (shrink)

In this handbook, I put into practice my philosophical views on children's mathematics. The handbook contains brief instructions and examples of mathematical activities. In the INSTRUCTIONS section, instructions are given on how, and in part why that way, to help preschool children in their mathematical development. In the ACTIVITIES section, there are examples of activities through which the child develops her mathematical abilities.

Boris Kment takes a new approach to the study of modality that emphasises the origin of modal notions in everyday thought. He argues that the concepts of necessity and possibility originate in counterfactual reasoning, which allows us to investigate explanatory connections. Contrary to accepted views, explanation is more fundamental than modality.

This paper analyzes the historical context and systematic importance of Kant's hypothetical use of reason. It does so by investigating the role of hypotheses in Kant's philosophy of science. We first situate Kant’s account of hypotheses in the context of eighteenth-century German philosophy of science, focusing on the works of Wolff, Meier, and Crusius. We contrast different conceptions of hypotheses of these authors and elucidate the different theories of probability informing them. We then adopt a more systematic perspective to discuss (...) Kant's idea that scientific hypotheses must articulate real possibilities. We argue that Kant's views on the intelligibility of scientific hypotheses constitute a valuable perspective on scientific understanding and the constraints it imposes on scientific rationality. (shrink)

I propose a general alethic theory of epistemic risk according to which the riskiness of an agent’s credence function encodes her relative sensitivity to different types of graded error. After motivating and mathematically developing this approach, I show that the epistemic risk function is a scaled reflection of expected inaccuracy. This duality between risk and information enables us to explore the relationship between attitudes to epistemic risk, the choice of scoring rules in epistemic utility theory, and the selection of priors (...) in Bayesian epistemology more generally. (shrink)

EveryK-trivial set is computable from an incomplete Martin-Löf random set, i.e., a Martin-Löf random set that does not compute the halting problem.

We consider effective versions of two classical theorems, the Lebesgue density theorem and the Denjoy–Young–Saks theorem. For the first, we show that a Martin-Löf random real z ∈ [0, 1] is Turing incomplete if and only if every effectively closed class.

We further investigate a divisibility relation on the set of BN ultrafilters on the set of natural numbers. We single out prime ultrafilters (divisible only by 1 and themselves) and establish a hierarchy in which a position of every ultrafilter depends on the set of prime ultrafilters it is divisible by. We also construct ultrafilters with many immediate successors in this hierarchy and find positions of products of ultrafilters.

We investigate the role of continuous reductions and continuous relativization in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with respect to van Lambalgen’s theorem and the Miller–Yu/Levin theorem. We study lowness for continuous relativization of randomness, and show the equivalence of the higher analogues of the different characterizations of lowness for Martin-Löf randomness. We also characterize computing higher [Formula: see text]-trivial sets by higher (...) random sequences. We give a separation between higher notions of randomness, in particular between higher weak 2-randomness and [Formula: see text]-randomness. To do so we investigate classes of functions computable from Kleene’s [Formula: see text] based on strong forms of the higher limit lemma. (shrink)

Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are our internally imagined objects, some of which, at least approximately, we can realize or represent; (ii) mathematical truths are not (...) truths about the external world but specifications (formulations) of mathematical conceptions; (iii) mathematics is first and foremost our imagined tool by which, with certain assumptions about its applicability, we explore nature and synthesize our rational cognition of it. (shrink)

While there are satisfactory answers to the question “How should we teach children mathematics?”, there are no satisfactory answers to the question “What mathematics should we teach children?”. This paper provides an answer to the last question for preschool children (early childhood), although the answer is also applicable to older children. This answer, together with an appropriate methodology on how to teach mathematics, gives a clear conception of the place of mathematics in the children’s world and our role in helping (...) children develop their mathematical abilities. Briefly, children’s mathematics consists of the world of children’s internal activities that they eventually purposefully organize in order to understand and control the outside world and organize their overall activities in it. We need to support a child in mathematical activities that she does spontaneously and in which she shows interest, and we need to teach her mathematics that she is interested in developing through these activities. In doing so, we must be fully aware that the child’s mathematics is part of the child’s world of internal activities and is not outside of it. We help the child develop mathematical abilities by developing them in the context of her world and not outside of it. From the point of view of this conception, the standards established today are limiting and too focused on numbers and geometric figures: these topics are too prominent and elaborated, and other mathematical contents are subordinated to them. Adhering to the standards, we drastically limit the mathematics of the child’s world, hamper the correct mathematical development of a child, and we can turn her away from mathematics. (shrink)

We continue the research of an extension of the divisibility relation to the Stone‐Čech compactification. First we prove that ultrafilters we call prime actually possess the algebraic property of primality. Several questions concerning the connection between divisibilities in and nonstandard extensions of are answered, providing a few more equivalent conditions for divisibility in. Results on uncountable chains in are proved and used in a construction of a well‐ordered chain of maximal cardinality. Probably the most interesting result is the existence of (...) a chain of type in. Finally, we consider ultrafilters without divisors in and among them find the greatest class. (shrink)

In this paper, I make a contribution to a naturalistically-minded theory of truthmakers by proposing a solution to the nasty problem of truthmakers for negative truths. After formulating the difficulty, I consider and reject a number of solutions to the problem, including Armstrong's states of affairs of totality, incompatibility accounts, and JC Beall 's polarity view. I then defend the position that absences of truthmakers are real and are responsible for making negative truths true. According to the positive account of (...) absences I offer, absences of contingent states of affairs are causally relevant mind-independent features of the physical world, located within space and time, and capable of being discovered by scientific inquiry. Recognition of the reality of absences strengthens truthmaker theory as a naturalistic metaphysics, as truth and falsity of each and every contingent proposition finds an ontological grounding in some region of the physical universe. (shrink)

We construct and study structures imitating the field of complex numbers with exponentiation. We give a natural, albeit non first-order, axiomatisation for the corresponding class of structures and prove that the class has a unique model in every uncountable cardinality. This gives grounds to conjecture that the unique model of cardinality continuum is isomorphic to the field of complex numbers with exponentiation.

In this paper we develop a general theory of algorithmic fairness. Drawing on Johnson King and Babic’s work on moral encroachment, on Gary Becker’s work on labor market discrimination, and on Strawson’s idea of resentment and indignation as responses to violations of the demand for goodwill toward oneself and others, we locate attitudes to fairness in an agent’s utility function. In particular, we first argue that fairness is a matter of a decision-maker’s relative concern for the plight of people from (...) different groups, rather than of the outcomes produced for different groups. We then show how an agent’s preferences, including in particular their attitudes to error, give rise to their decision thresholds. Tying these points together, we argue that the agent’s relative degrees of concern for different groups manifest in a difference in decision thresholds applied to these groups. (shrink)

This paper explores the use of new scientific techniques to examine collections of historic scientific apparatus and other technological artefacts. One project under discussion uses interferometry to examine the history of lens development, while another uses X-ray fluorescence to discover the kinds of materials used to make early mathematical and astronomical instruments. These methods lead to surprising findings: instruments turn out to be fake, and lens makers turn out to have been adept at solving the riddle of aperture. Although exciting, (...) in some ways this is neither novel nor particularly unusual. After all, lab techniques have been used in art and archaeological collections for a very long time. In fact, scientific instruments themselves have been examined in this way since at least the 1950s. What, then, is special about the use of new instruments to examine old instruments? We argue that the answer has less to do with measuring historical innovation or establishing priority, and more to do with networks of craft know-how that, typically, have left no other historical traces than those embodied in surviving instruments themselves. We show, in particular, how collections of objects can be mobilised within wider histories of knowledge, placing instruments within a dynamic interplay of craft knowledge, expertise, labour, commerce, and material exchange, over the longue durée. Finally, we suggest that these kinds of lab analyses can be given an extra dimension through the use of computational modelling, and we introduce the “Tools of Knowledge” project, which is designed to bring together XRF with techniques from the digital humanities, in order to tell a new story about the development of scientific instruments from the 16th to the 20th century. (shrink)

A central object of study in the field of algorithmic randomness are notions of randomness for sequences, i.e., infinite sequences of zeros and ones. These notions are usually defined with respect to the uniform measure on the set of all sequences, but extend canonically to other computable probability measures. This way each notion of randomness induces an equivalence relation on the computable probability measures where two measures are equivalent if they have the same set of random sequences. In what follows, (...) we study the equivalence relations induced by Martin-Löf randomness, computable randomness, Schnorr randomness and Kurtz randomness, together with the relations of equivalence and consistency from probability theory. We show that all these relations coincide when restricted to the class of computable strongly positive generalized Bernoulli measures. For the case of arbitrary computable measures, we obtain a complete and somewhat surprising picture of the implications between these relations that hold in general. (shrink)

Since the mid-1980s, there has been a gradual ethical drift in the provision of maternal care in African health facilities in general, and in Cameroon in particular, despite government efforts. In fact, in Cameroon, an increasing number of caregivers are reportedly not providing compassionate care in maternity services. Consequently, many women, particularly the financially vulnerable, experience numerous difficulties in accessing these health services. In this article, we highlight the unequal access to care in public maternity services in Cameroon in general (...) and the Noun Division in particular. For this study, in addition to documentary review, two qualitative data collection techniques were used: direct observation and individual interviews. Following the field work, the observation data were categorized and analyzed to assess their relevance and significance in relation to the topics listed in the observation checklist. Interviews were recorded using a dictaphone; they were subsequently transcribed and the data categorized and coded. After this stage, an analysis grid was constructed for content analysis of the transcripts, to study the frequency of topics addressed during the interviews, as well as divergences and convergences among the respondents. The results of this data analysis showed that money has become the driving force in service provision. As such, it is the patient’s economic capital that counts. Considered “clients”, pregnant women without sufficient financial resources wait long hours in corridors; some die in pain under the indifferent gaze of the professionals who are supposed to take care of them. In sharp contrast, the findings revealed that financially privileged patients are able to bribe caregivers to attract their favour and obtain prompt, careful, and effective care. These ethical abuses observed in the Noun public health facilities drive women to use, from the beginning of their pregnancies to the delivery, only healthcare delivered by traditional health attendants. (shrink)

We continue the exploration of various aspects of divisibility of ultrafilters, adding one more relation to the picture: multiplicative finite embeddability. We show that it lies between divisibility relations \ and \. The set of its minimal elements proves to be very rich, and the \-hierarchy is used to get a better intuition of this richness. We find the place of the set of \-maximal ultrafilters among some known families of ultrafilters. Finally, we introduce new notions of largeness of subsets (...) of \, and compare it to other such notions, important for infinite combinatorics and topological dynamics. (shrink)

The essential role of language in rational cognition is analysed. The approach is functional: only the results of the connection between language, reality, and thinking are considered. Scientific language is analysed as an extension and improvement of everyday language. The analysis gives a uniform view of language and rational cognition. The consequences for the nature of ontology, truth, logic, thinking, scientific theories, and mathematics are derived.

In this paper, we reflect on the connection between the notions of organism and organisation, with a specific interest in how this bears upon the issue of the reality of the organism. We do this by presenting the case of Buffon, who developed complex views about the relation between the notions of “organised” and “organic” matter. We argue that, contrary to what some interpreters have suggested, these notions are not orthogonal in his thought. Also, we argue that Buffon has a (...) view in which organisation is not just ubiquitous, but basic and fundamental in nature, and hence also fully natural. We suggest that he can hold this view because of his anti-mathematicism. Buffon’s case is interesting, in our view, because he can regard organisation, and organisms, as perfectly natural, and can admit their reality without invoking problematic supernaturalist views, and because he allows organisation and the organismal to come in kinds and degrees. Thus, his view tries to do justice to two cautionary notes for the debate on the reality of the organism: the need for a commitment to a broadly naturalist perspective, and the need to acknowledge the interesting features of organisms through which we make sense of them. (shrink)

We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15]. We fully characterize lowness for Demuth randomness using an appropriate notion of traceability. Central to this characterization is a partial relativization of Demuth randomness, which may be more natural than the fully relativized version. We also show that an oracle is low for weak Demuth randomness if and only if it is computable.

In this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence and every 2-random sequence forms a minimal pair in the Turing degrees with every 2-generic sequence. We strengthen these results by showing that every Demuth random sequence computes a 1-generic sequence and that every Demuth random sequence forms a minimal pair with every pb-generic sequence. Moreover, we prove that for every (...) comeager${\cal G} \subseteq {2^\omega }$, there is some weakly 2-random sequenceXthat computes some$Y \in {\cal G}$, a result that allows us to provide a fairly complete classification as to how various notions of effective randomness interact in the Turing degrees with various notions of effective genericity. (shrink)

British social survey movement ‘Mass-Observation’ (M-O) was founded in 1937 by a poet, a film-maker and an ornithologist. It purported to offer a new kind of sociology – one informed by surrealism and working with a ‘mass’ of Observers recording day-to-day interactions. Various commentators have debated the importance and precise identity of M-O in its first phase, especially in light of its combination of social science and surrealism. This article draws on new archival research, in particular into the ‘paperwork’ practices (...) of Charles Madge, arguing that M-O is best understood as an attempt to define a new relationship between the survey subject and information organiser. The latter – as sociologist, planner or artist – was a distinctive interwar persona, central to ‘scientific humanism’. Bathos, as a formal strategy in modernist aesthetics, is introduced as an explanation of the failure of this particular part of the M-O project. Questions of subjectivity and data link M-O to a longer history of heterodox sociological inquiry. This analysis resolves some of the apparent paradoxes that have been prominent in studies of M-O, and draws attention to the unfulfilled promise of a vast archive of social data. (shrink)

We investigate two questions, the relevance of memory for evaluative conditioning effects based on smell-taste pairings, and the potential preparedness of smell-taste combinations for...

We continue the research of the relation $\hspace {1mm}\widetilde {\mid }\hspace {1mm}$ on the set $\beta \mathbb {N}$ of ultrafilters on $\mathbb {N}$, defined as an extension of the divisibility relation. It is a quasiorder, so we see it as an order on the set of $=_{\sim }$ -equivalence classes, where $\mathcal {F}=_{\sim }\mathcal {G}$ means that $\mathcal {F}$ and $\mathcal {G}$ are mutually $\hspace {1mm}\widetilde {\mid }$ -divisible. Here we introduce a new tool: a relation of congruence modulo an (...) ultrafilter. We first recall the congruence of ultrafilters modulo an integer and show that $=_{\sim }$ -equivalent ultrafilters do not necessarily have the same residue modulo $m\in \mathbb {N}$. Then we generalize this relation to congruence modulo an ultrafilter in a natural way. After that, using iterated nonstandard extensions, we introduce a stronger relation, which has nicer properties with respect to addition and multiplication of ultrafilters. Finally, we introduce a strengthening of $\hspace {1mm}\widetilde {\mid }\hspace {1mm}$ and show that it also behaves well with respect to the congruence relation. (shrink)

In this paper, I argue that Kant adopted, throughout his career, a position that is much more akin to classical accounts of epigenesis, although he does reject the more radical forms of epigenesis proposed in his own time, and does make use of preformationist sounding terms. I argue that this is because Kant thinks of what is pre-formed as a species, not an individual or a part of an individual; has no qualm with the idea of a specific, teleological principle (...) or force underlying generation, and conceives of germs and predispositions as specific constraints on such a principle or force. Neither of these conceptions of what is “preformed”, I argue, is in strict opposition to classical epigenesis. I further suggest that Kant’s lingering use of preformationist terminology is due to his belief that this is required to account for the specificity of the specific generative force; his resistance towards the unrestricted plasticity of the generative force in radical epigenesis, which violates species-fixism; and his insistence on the internal, organic basis of developmental plasticity and variation within species. I conclude by suggesting that this terminological and interpretative peculiarity is partly due to a larger shift in the natural philosophical concerns surrounding the debate on epigenesis and preformation. Specifically, it is a sign that the original reasons for resisting epigenesis, namely its use of specific, teleological principles and its commitment to the natural production of biological structure, became less of a concern, whereas unrestricted plasticity and its undermining of fixism became a real issue, thereby also becoming the focal point of the debate. (shrink)

If definitive evidence concerning treatment effectiveness becomes available from an ongoing randomized clinical trial, then the trial could be stopped early, with the public release of results benefiting current and future patients. However, stopping an ongoing trial based on accruing outcome data requires methodological rigor to preserve validity of the trial conclusions. This has led to the use of formal interim monitoring procedures, which include inefficacy monitoring that will stop a trial early when the experimental treatment appears not to be (...) working. For participants, inefficacy monitoring is especially important as it ensures that they are not being treated worse than if they had not enrolled on the trial. We discuss the importance of reporting with trial results the formal interim inefficacy monitoring guidelines that were utilized, and, if none were used, the reasons for their absence. A survey of two leading medical journals suggests that this is not current practice. (shrink)

In this article, logical concepts are defined using the internal syntactic and semantic structure of language. For a first-order language, it has been shown that its logical constants are connectives and a certain type of quantifiers for which the universal and existential quantifiers form a functionally complete set of quantifiers. Neither equality nor cardinal quantifiers belong to the logical constants of a first-order language.

The complete system of knowledge is a standard trope of science fiction, a techno-utopian dream and an aesthetic ideal. It is Solomon’s House, the Encyclopaedia and the Museum. It is also an ideology – of Enlightenment, High Modernism and absolute governance. Far from ending the dream of a total archive, 20th-century positivist rationality brought it ever closer. From Paul Otlet’s ‘Mundaneum’ to Mass-Observation, from the Unity of Science movement to Wikipedia, the dream of universal knowledge dies hard. As a political (...) tool, the total archive encompasses population statistics, gross domestic product, indices of the Standard of Living and the international ideology of the United Nations Educational, Scientific and Cultural Organization, the World Health Organization, the free market and, most recently, Big Data. Questions of the total archive engage key issues in the philosophy of classification, the poetics of the universal, the ideology of surveillance and the technologies of information retrieval. What are the social structures and political dynamics required to sustain total archives, and what are the temporalities implied by such projects? This introduction and the articles that follow describe and place in historical context a series of concrete instances of totality. Our analysis is arranged according to four central themes: the relationship between the Archive (singular) and archives (plural); the image of the archive and the aesthetics of totality; pathologies of accumulation; and the specific historical trajectory of the total archive in the 19th and 20th centuries. (shrink)