Chemicals have penetrated everyday lives of men who have sex with men as never before, along with new online and mobile technologies used to seek pleasures and connections. Poststructuralist (including queer) explorations of these new intensities show how bodies exist in the form of (political) surfaces able to connect with other bodies and with other objects where they may find/create a function (e.g., reproduce or disrupt hegemonies). This federally funded netnographic study explored how a variety of chemicals such as recreational (...) drugs, pharmaceuticals and steroids are contributing to the construction of gay, bisexual and other men having sex with men (GBMSM) communities and their interactions with idealized masculinities in the age of increasing technology. Five major thematic categories emerged from our analysis: (1) assembling bodies and technologies, (2) becoming orgiastic, (3) experiencing stigma, (4) becoming machinic and (5) negotiating practices. Our analysis explores how and why GBMSM pursue excesses of pleasure and connection through the assemblages they make with sexualized drug use, online platforms and other men. (shrink)

The paper presents a numerical interpretation of the quantifiers of traditional categorical propositions and then offers a generalization to accommodate all other numerical values. Next, it considers the implications possible on the basis of both minimum and maximum existential presuppositions; and finally, it shows that every pair of categorical premises yields multiple conclusions when appropriate minimum and maximum presuppositions are made for the terms of the premises.

Drawn from the Anguttara Nikaya, Numerical Discourses of the Buddha brings together teachings of the Buddha ranging from basic ethical observances recommended to the busy man or woman of the world, to the more rigorous instructions on mental training prescribed for the monks and nuns. The Anguttara Nikaya is a part of the Pali Canon, the authorized recension of the Buddha's Word for followers of Theravada Buddhism, the form of Buddhism prevailing in the Buddhist countries of southern Asia. These discourses (...) are called numerical because they retain the structure of the original Anguttara Nikaya. Sayings are organized not by topic, but by numbers mentioned in the texts. This organizational scheme, common in ancient Indian literature, can give the reader a haphazard view of the Buddha's teachings. To balance this tendency, Bhikku Bodhi provides a systematic introduction to the Buddha's teachings. To balance this tendency, Bhikku Bodhi provides a systematic introduction to the Buddha's Teaching in the Anguttara Nikaya. The translators also provide notes, a glossary, and another introduction placing the Anguttara Nikaya in the context of the larger Theravada Buddhist Canon. This readable butt precise translation will be welcomed by both students of Theravada Buddhism as well as anyone wishing to learn from the Buddha's teachings. (shrink)

This paper is an attempt to show that my work to establish numerically flexible quantifiers for the syllogism can be aptly combined with the term logic advanced by Sommers, Englebretsen, and others.

I argue that numerical and causal accuracy arguments can be successful only if: (1) the theories in use are known to be true, (2) computational difficulties do not exist, and (3) the experimental data are stable and resolved. When any one or more of these assumptions are not satisfied, additional justificational considerations must be invoked. I illustrate the need for range of validity and intelligibility claims with examples drawn from chemical kinetics. My arguments suggest that the realist and anti-realist accounts (...) of justification are incomplete. Finally, I sketch some reasons why additional justificatory criteria are needed. (shrink)

Humans and other animals have an evolved ability to detect discrete magnitudes in their environment. Does this observation support evolutionary debunking arguments against mathematical realism, as has been recently argued by Clarke-Doane, or does it bolster mathematical realism, as authors such as Joyce and Sinnott-Armstrong have assumed? To find out, we need to pay closer attention to the features of evolved numerical cognition. I provide a detailed examination of the functional properties of evolved numerical cognition, and propose that they prima (...) facie favor a realist account of numbers. (shrink)

The ‘bat and ball’ is one of the problems most frequently employed as a testbed for research on the dual-system hypothesis of reasoning. Frederick (J Econ Perspect 19:25–42, 2005) is the first to envisage the possibility that different numerical arrangements of the ‘bat and ball’ problem could lead to different dynamics of activation of the dual-system, and so to different performances of subjects in task accomplishment. This possibility has triggered a strand of research oriented to accomplish ‘sensitivity analyses’ of the (...) ‘bat and ball’ problem. The scope of this paper is to test experimentally the specific hypothesis that numerals are responsible for the selective activation of the two systems of reasoning in this task. In particular, we argue that their role goes beyond and cannot be reduced to that of numbers conceived as magnitudes. To test our hypothesis, we devise an experimental setting in which the role of numbers (as magnitudes) is rendered irrelevant. We find experimental results consistent with our hypothesis. We further provide a link between the literature on mathematical problem-solving and that on mathematical cognition research, in particular that branch labeled embodied mathematical cognition. (shrink)

This paper studies general numerical networks with support and attack. Our starting point is argumentation networks with the Caminada labelling of three values 1=in, 0=out and ½=undecided. This is generalised to arbitrary values in [01], which enables us to compare with other numerical networks such as predator?prey ecological networks, flow networks, logical modal networks and more. This new point of view allows us to see the place of argumentation networks in the overall landscape of networks and import and export ideas (...) to and from argumentation networks. We make a special effort to make clear how general concepts in general networks relate to the special case of argumentation networks. We pay special attention to the handling of loops and to the special features of numerical support. We find surprising connections with the Dempster?Shafer rule and with the cross-ratio in projective geometry. This paper is an expansion of our 2005 paper and so we also consider higher level features such as numerical attacks on attacks, and propagation of numerical values.We conclude with a brief view of temporal numerical argumentation and with a detailed comparison with related papers published since 2005. (shrink)

ArgumentJohannes Kepler published hisAstronomia novain 1609, based upon a huge amount of computations. The aim of this paper is to show that Kepler's new astronomy was grounded on methods from numerical analysis. In his research he applied and improved methods that required iterative calculations, and he developed precompiled mathematical tables to solve the problems, including a transcendental equation. Kepler was aware of the shortcomings of his novel methods, and called for a new Apollonius to offer a formal mathematical deduction. He (...) was also in great need of computational power, and his friend and colleague, Wilhelm Schickard, constructed the first prototype of a true mechanical calculator, although it never came into regular use. The article concludes that Kepler's new astronomy was clearly backed up by numerical methods and embedded concepts and challenges of great importance for the future development of numerical analysis. (shrink)

Two different definitions of numerical identity occur in Aristotle's works, namely: (i) "A" and "B" are both names of one thing; (ii) A and B constitute unity. These definitions can be traced back respectively to the following theories of predication: (i)' the sentences whose subjects are accidents are actually ill-formed; (ii)' in some cases the accidents are not eliminable subjects. Since (i)' and (ii)' are irreparably inconsistent, the theory of identity is inconsistent too; in this paper are explored the consequences (...) of such an inconsistence, mainly as far as the fallacies depending both on combination and division and on accident are concerned. (shrink)

The thesis of this important new direction in classical logic is that the syllogism is an inherently numerical form of entailment - a position in direct opposition to the perennial view. In the book the author shows both that the propositions of classical logic make implicit numerical claims, and also that alternative numerical values can be substituted for these implicit ones to yield a logic whose quantifiers are infinitely flexible. A significant aspect of this work is the development of a (...) «schematic» which clearly exhibits the validity or invalidity of each categorical argument, whether it be classically or numerically extended. (shrink)

The use of numerical arguments has become part and parcel of evidence-based policymaking, serving increasingly as scientific evidence which is used to back up policy decisions and to convince citizens of the acceptability of those decisions. But numerical arguments and their quality and potential persuasive role in the specific institutional context of policymaking have received little treatment within argumentation theory. This paper endeavours to explain the forms, functions, and quality of numerical arguments in policymaking.

A Theory of Magnitude suggests that space, time, and quantities are processed through a generalized magnitude system. ATOM posits that task-irrelevant magnitudes interfere with the processing of task-relevant magnitudes as all the magnitudes are processed by a common system. Many behavioral and neuroimaging studies have found support in favor of a common magnitude processing system. However, it is largely unknown whether such cross-domain monotonic mapping arises from a change in the accuracy of the magnitude judgments or results from changes in (...) precision of the processing of magnitude. Therefore, in the present study, we examined whether large numerical magnitude affects temporal accuracy or temporal precision, or both. In other words, whether numerical magnitudes change our temporal experience or simply bias duration judgments. The temporal discrimination paradigm was used to present numerical magnitudes across varied durations. We estimated temporal accuracy and precision for each numerical magnitude. The results revealed that temporal accuracy for large numerical magnitude was significantly lower than that of small and identical magnitudes. This implies that the temporal duration was overestimated for large numerical magnitude compared to small and identical numerical magnitude, in line with ATOM’s prediction. However, no influence of numerical magnitude was observed on temporal precision. The findings of the present study suggest that task-irrelevant numerical magnitude selectively affects the accuracy of processing of duration but not duration discrimination itself. Further, we argue that numerical magnitude may not directly affect temporal processing but could influence via attentional mechanisms. (shrink)

Building on conceptual insights from the history and sociology of numbers, media and surveillance studies, and theories of governance and risk, this article analyzes the forms of transparency produced by the use of numbers in social life. It examines what it is about numbers that often makes their ‘truth claims’ so powerful, investigates the role that numerical operations play in the production of retrospective, real-time and anticipatory forms of transparency in contemporary politics and economic transactions, and discusses some of the (...) implications resulting from the increasingly abstract and machine-driven use of numbers. It argues that the forms of transparency generated by machine-driven numerical operations open up for individual and collective practices in ways that are intimately linked to precautionary and pre-emptive aspirations and interventions characteristic of contemporary governance. As such, these numerical operations raise important political and ethical questions that deserve further conceptual and empirical scrutiny. (shrink)

This paper is concerned with the combinator representation of numeral systems with logarithmic space complexity of symbols. The principle used is based on the lexicographic ordering of words over a finite alphabet.

Menary OpenMIND, MIND Group, Frankfurt am Main, 2015) has argued that the development of our capacities for mathematical cognition can be explained in terms of enculturation. Our ancient systems for perceptually estimating numerical quantities are augmented and transformed by interacting with a culturally-enriched environment that provides scaffolds for the acquisition of cognitive practices, leading to the development of a discrete number system for representing number precisely. Numerals and the practices associated with numeral systems play a significant role in this (...) process. However, the details of the relationship between the ancient number system and the discrete number system remain unclear. This lack of clarity is exacerbated by the problem of symbolic estrangement and the fact that unique features of how numeral systems represent require our ancient number system to play a dual role. These issues highlight that Dehaene’s From monkey brain to human brain, MIT Press, Cambridge, pp 133–157, 2005) neuronal recycling hypothesis may be insufficient to explain the neural mechanisms underlying the process of enculturation. In order to explain mathematical enculturation, and enculturation more generally, it may be necessary to adopt Anderson’s :245–266, 2010; After phrenology: neural reuse and the interactive brain, MIT Press, Cambridge, 2014) theory of neural reuse. (shrink)

The idea that there is a “Number Sense” (Dehaene, 1997) or “Core Knowledge” of number ensconced in a modular processing system (Carey, 2009) has gained popularity as the study of numerical cognition has matured. However, these claims are generally made with little, if any, detailed examination of which modular properties are instantiated in numerical processing. In this article, I aim to rectify this situation by detailing the modular properties on display in numerical cognitive processing. In the process, I review literature (...) from across the cognitive sciences and describe how the evidence reported in these works supports the hypothesis that numerical cognitive processing is modular. I outline the properties that would suffice for deeming a certain processing system a modular processing system. Subsequently, I use behavioral, neuropsychological, philosophical, and anthropological evidence to show that the number module is domain specific, informationally encapsulated, neurally localizable, subject to specific pathological breakdowns, mandatory, fast, and inaccessible at the person level; in other words, I use the evidence to demonstrate that some of our numerical capacity is housed in modular casing. (shrink)

Human languages vary in terms of which meanings they lexicalize, but this variation is constrained. It has been argued that languages are under two competing pressures: the pressure to be simple (e.g., to have a small lexicon) and to allow for informative (i.e., precise) communication, and that which meanings get lexicalized may be explained by languages finding a good way to trade off between these two pressures. However, in certain semantic domains, languages can reach very high levels of informativeness even (...) if they lexicalize very few meanings in that domain. This is due to productive morphosyntax and compositional semantics, which may allow for construction of meanings which are not lexicalized. Consider the semantic domain of natural numbers: many languages lexicalize few natural number meanings as monomorphemic expressions, but can precisely convey very many natural number meanings using morphosyntactically complex numerals. In such semantic domains, lexicon size is not in direct competition with informativeness. What explains which meanings are lexicalized in such semantic domains? We will propose that in such cases, languages need to solve a different kind of trade‐off problem: the trade‐off between the pressure to lexicalize as few meanings as possible (i.e, to minimize lexicon size) and the pressure to produce as morphosyntactically simple utterances as possible (i.e, to minimize average morphosyntactic complexity of utterances). To support this claim, we will present a case study of 128 natural languages' numeral systems, and show computationally that they achieve a near‐optimal trade‐off between lexicon size and average morphosyntactic complexity of numerals. This study in conjunction with previous work on communicative efficiency suggests that languages' lexicons are shaped by a trade‐off between not two but three pressures: be simple, be informative, and minimize average morphosyntactic complexity of utterances. (shrink)

We define here a notion of internal copy and of weak internal copy of a category. We will then determine some families of categories having an internal copy or a weak internal copy. We will consider categories of definable classes of first-order theories and we will see that the notion of internal copy is related to the notion of numerical existence property.

There is a category of objects such that for any two occurrences of an object in that category, Establishing the highest degree of their qualitative identity will not be sufficient to establish that the object involved is one and the same. It is first argued that objects in this category occupy positions in a spatio-Temporal continuum and obey certain principles of conservation. And then two criteria for the numerical identity of these objects are developed: (a) that there are scientific laws (...) which can, In principle, Predict or retrodict a continuous path for the object; (b) that the parts of the object and their arrangement are preserved in a certain specified way. (shrink)

Kant's philosophy must be understood nonnaturalistically and anti-psychologistically. Self-consciousness must be interpreted as preceding the distinction between different persons. Kant departs from the traditional idea that I thoughts are always mediated by a certain specific I sense or conceptualization of oneself. At the same time the so-called paradoxes of self-consciousness are resolved. The possibility of a pre-personal self-consciousness is what links the way all objects are given to finite beings to the way they are conceptualized by those beings. It serves (...) as the unifying basis for intuitions and concepts. Intuitions and concepts of objects are connected by their relation to a numerically identical self. ;The subject construed impersonally provides the frame for any possible experience and is one in the same in all possible experience. As such it can never be experienced as a distinct entity. It is the condition under which it is possible to have a point of view at all. One must be in a position to become conscious of the numerical identity of this logical subject of all representation if is to be possible to communicate indeed to compare and contrast representational contents at all. ;Self-consciousness is inherently empty and yet also necessarily in relation to contents of consciousness. This means that its numerical identity can only be sustained if all potential contents of consciousness present themselves to the self in such a way that they sustain the possibility of consciousness of the self as having a discrete empirical identity. Kant's argument in the First Analogy and the Refutation of Idealism fulfills this demand for a correlate to the numerical identity of pre-personal consciousness in the data presented to self-consciousness spatio-temporally. We must view self-construction and construction of objects existing outside of us as part of a single process which presupposes an impersonal self-consciousness. (shrink)

In this article, we describe and attempt to solve a puzzle arising from the interpretation of modified numerals like less than five and between two and five. The puzzle is this: such modified numerals seem to mean different things depending on whether they combine with distributive or non-distributive predicates. When they combine with distributive predicates, they intuitively impose a kind of upper bound, whereas when they combine with non-distributive predicates, they do not. We propose and explore in detail (...) four solutions to this puzzle, each involving some notion of maximality, but differing in the type of maximality involved and in the source of maximality. While the full range of data we consider do not conclusively favor one theory over the other three, we do argue that overall the evidence goes against the view that modified numerals lexically encode a ‘standard’ maximality operator, and suggests the need for a pragmatic blocking mechanism that filters out readings of sentences that are generated by the grammar but intuitively unavailable. (shrink)

This paper investigates the problem of numerical continuity in thomistic metaphysics and attempts to point out the principle of identity in material substances. it has three parts: the first clarifies the issue and presents the possible alternatives; the second rejects various solutions which have been proposed by interpreters of thomas aquinas such as matter, form, accidents, and substance; and the third part argues that within thomistic metaphysics it is only existence ("esse") that may be considered as an acceptable candidate for (...) this ontological role. (shrink)

The progress of Mathematics during the nineteenth century was characterised both by an enormous acquisition of new knowledge and by the attempts to introduce rigour in reasoning patterns and mathematical writing. Cauchy’s presentation of Mathematical Analysis was not immediately accepted, and many writers, though aware of that new style, did not use it in their own mathematical production. This paper is devoted to an episode of this sort that took place in Spain during the first half of the century: It (...) deals with the presentation of a method for numerically solving algebraic equations by José Mariano Vallejo, a late Spanish follower of the Enlightenment ideas, politician, writer, and mathematician who published it in the fourth (1840) edition of his book Compendio de Matemáticas Puras y Mistas, claiming to have discovered it on his own. Vallejo’s main achievement was to write down the whole procedure in a very careful way taking into account the different types of roots, although he paid little attention to questions such as convergence checks and the fulfilment of the hypotheses of Rolle’s Theorem. For sure this lack of mathematical care prevented Vallejo to occupy a place among the forerunners of Computational Algebra. (shrink)

Given an argumentation network with initial values to the arguments, we look for algorithms which can yield extensions compatible with such initial values. We find that the best way of tackling this problem is to offer an iteration formula that takes the initial values and the attack relation and iterates a sequence of intermediate values that eventually converges leading to an extension. The properties surrounding the application of the iteration formula and its connection with other numerical and non-numerical techniques proposed (...) by others are thoroughly investigated in this paper. (shrink)

The dual-code proposal of number representation put forward by Cohen Kadosh & Walsh (CK&W) accounts for only a fraction of the many modes of numerical abstraction. Contrary to their proposal, robust data from human infants and nonhuman animals indicate that abstract numerical representations are psychologically primitive. Additionally, much of the behavioral and neural data cited to support CK&W's proposal is, in fact, neutral on the issue of numerical abstraction.

David Lewis (1969) introduced sender-receiver games as a way of investigating how meaningful language might evolve from initially random signals. In this report I investigate the conditions under which Lewis signaling games evolve to perfect signaling systems under various learning dynamics. While the 2-state/2- term Lewis signaling game with basic urn learning always approaches a signaling system, I will show that with more than two states suboptimal pooling equilibria can evolve. Inhomogeneous state distributions increase the likelihood of pooling equilibria, but (...) learning strategies with negative reinforcement or certain sorts of mutation can decrease the likelihood of, and even eliminate, pooling equilibria. Both Moran and APR learning strategies (Bereby-Meyer and Erev 1998) are shown to promote successful convergence to signaling systems. A model is presented that illustrates how a language that codes state-act pairs in an order-based grammar might evolve in the context of a Lewis signaling game. The terms, grammar, and the corresponding partitions of the state space co-evolve to generate a language whose structure appears to reflect canonical natural kinds. The evolution of these apparent natural kinds, however, is entirely in service of the rewards that accompany successful distinctions between the sender and receiver. Any metaphysics grounded on the structure of a natural language that evolved in this way would track arbitrary, but pragmatically useful distinctions. (shrink)

We present an elementary three-pass algorithm for computing addition in Ostrowski numeration systems. When a is quadratic, addition in the Ostrowski numeration system based on a is recognizable by a finite automaton. We deduce that a subset of X⊆Nn is definable in, where Va is the function that maps a natural number x to the smallest denominator of a convergent of a that appears in the Ostrowski representation based on a of x with a nonzero coefficient if and only if (...) the set of Ostrowski representations of elements of X is recognizable by a finite automaton. The decidability of the theory of follows. (shrink)

Four perspectives on numerical origins are examined. The nativist model sees numbers as an aspect of numerosity, the biologically endowed ability to appreciate quantity that humans share with other species. The linguistic model sees numbers as a function of language. The embodied model sees numbers as conceptual metaphors informed by physical experience and expressed in language. Finally, the extended model sees numbers as conceptual outcomes of a cognitive system that includes material forms as constitutive components. If numerical origins are to (...) be found, each perspective must address one or more critical questions that will require working across discipline boundaries. (shrink)

Recentemente diversi studi hanno mostrato come la distanza tra i Grundlagen e le precedenti pubblicazioni di Hilbert non sia tanto abissale come ritenuto in passato, ma vi sia una significativa consequenzialità con la teoria dei campi numerici. Nel ribadire questa visione, si intende mostrare come i risultati ottenuti da Hilbert, in particolare sui teoremi di Pappo e di Desargues, siano conseguenza di una ricerca più ampia sulla possibilità di introdurre all’interno della geometria dei sistemi numerici atti a coordinatizzare il piano (...) o lo spazio. Nello sviluppo della ricerca hanno avuto un ruolo determinante la scoperta di Hurwitz delle uniche quattro algebre di divisione normate e le idee maturate, già agli inizi della sua carriera, sulla finitezza degli strumenti da utilizzare in geometria, in particolare la rinuncia al concetto di continuità e la limitazione a sistemi numerici ordinati e numerabili. Nelle conclusioni dei Grundlagen, Hilbert rimarca l’importanza di dimostrare l’impossibilità delle assunzioni, tra cui l’impossibilità di determinare, attraverso solo relazioni geometriche di incidenza, dei sistemi numerici ordinati che abbiano le stesse proprietà dei quaternioni e degli ottonioni rispetto alle operazioni dell’aritmetica. (shrink)

This article addresses the physical chemical processes underlying biological self-organisation by which a homogenous solution of reacting chemicals spontaneously self-organises. Theoreticians have predicted that self-organisation can arise from a coupling of reactive processes with molecular diffusion. In addition, the presence of an external field, such as gravity, at a critical moment early in the process may determine the morphology that subsequently develops. The formation, in-vitro, of microtubules, a constituent of the cellular skeleton, shows this type of behaviour. The preparations spontaneously (...) self-organise by reaction-diffusion and the morphology that develops depends upon the presence of gravity at a critical bifurcation time early in the process. Here, we present numerical simulations of a population of microtubules that reproduce this behaviour. Microtubules can grow from one end whilst shrinking from the other. The shrinking end leaves behind a chemical trail of high tubulin concentration. Neighbouring microtubules preferentially grow into these regions, whilst avoiding regions of low tubulin concentration. The chemical trails produced by individual microtubules thus activate and inhibit the formation of neighbouring microtubules and this progressively leads to self-organisation. Gravity acts by way of its directional interaction with the macroscopic density fluctuations present in the solution arising from microtubule disassembly. (shrink)

Mathematical cognition has become an interesting case study for wider theories of cognition. Menary :1–20, 2015) argues that arithmetical cognition not only shows that internalist theories of cognition are wrong, but that it also shows that the Hypothesis of Extended Cognition is right. I examine this argument in more detail, to see if arithmetical cognition can support such conclusions. Specifically, I look at how the use of numerals extends our arithmetical abilities from quantity-related innate systems to systems that can (...) deal with exact numbers of arbitrary size. I then argue that the system underlying our grasp of small numbers is an unhelpful case study for Menary; it doesn’t support an argument for externalism over internalism. The system for large numbers, on the other hand, clearly displays important interactions between public numeral systems and our cognitive processes. I argue that the large number system supports an argument against internalist theories of arithmetical cognition, but that one cannot conclude that the Hypothesis of Extended Cognition is correct. In other words, the large number case doesn’t decide between the Hypothesis of Extended Cognition and the Hypothesis of EMbedded Cognition. (shrink)

We present a number of puzzles arising for the interpretation of modified numerals. Following Büring and others we assume that the main difference between comparative and superlative modifiers is that only the latter convey disjunctive meanings. We further argue that the inference patterns triggered by disjunction and superlative modifiers are hard to capture in existing semantic and pragmatic analyses of these phenomena (neo-Gricean or grammatical alike), and we propose a novel account of these inferences in the framework of bilateral (...) state-based modal logic defining a first order extension of Aloni (Semant Pragmat 15:5-EA, 2022. https://doi.org/10.3765/sp.15.5)’s. (shrink)

The goal of the present commentary is to show that past results on automatic numerical processing in different notations are consistent with the idea of an abstract numerical representation. This is done by reviewing the relevant studies and giving alternative explanations to the ones proposed in the target article.

The consideration of deliberate versus automatic processing of numeric representations is important to math education, memory for numbers, and decision-making. In this commentary, we address the possible roles for numeric representations in such higher-level cognitive processes. Current evidence is consistent with important roles for both automatic and deliberative processing of the representations.

Many researchers, including Clarke and Beck, describe the human numerical system as unitary. We offer an alternative view – the coexistence of several systems; namely, multiple systems existing in parallel, ready to be activated depending on the task/need. Based on this alternative view, we present an account for the representation of rational numbers.

This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a nonstandard cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.

I will examine three claims made by Ackerman and Kripke. First, they claim that not any arithmetical terms is eligible for universal instantiation and existential generalisation in doxastic or epistemic contexts. Second, Ackerman claims that Peano numerals are eligible for universal instantiation and existential generalisation in doxastic or epistemic contexts. Kripke's position is a bit more subtle. Third, they claim that the successor relation and the smaller-than relation must be effectively calculable. These three claims will be examined from the (...) framework of modal-epistemic arithmetic, i.e. arithmetic extended with certain modal, epistemic and modal-epistemic principles. I will present theorems that give support to the claims made by Ackerman and Kripke. (shrink)

Melisa Vivanco objects to my theory of the Arabic numerals in Roads to Reference that the reference fixing procedure that I postulate doesn’t exploit the morphological structure of the Arabic numerals, but it should. Against Vivanco, I argue that the procedure in question does exploit the morphological structure of the numerals in an essential way.