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)

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 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)

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)

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.

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.

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)

In recent decades, non-representational approaches to mental phenomena and cognition have been gaining traction in cognitive science and philosophy of mind. In these alternative approach, mental representations either lose their central status or, in its most radical form, are banned completely. While there is growing agreement that non-representational accounts may succeed in explaining some cognitive capacities, there is widespread skepticism about the possibility of giving non-representational accounts of cognitive capacities such as memory, imagination or abstract thought. In this paper, I (...) will critically examine the view that there are fundamental limitations to non-representational explanations of cognition. Rather than challenging these arguments on general grounds, I will examine a set of human cognitive capacities that are generally thought to fall outside the scope of non-representational accounts, i.e. numerical cognition. After criticizing standard representational accounts of numerical cognition for their lack of explanatory power, I will argue that a non-representational approach that is inspired by radical enactivism offers the best hope for developing a genuine naturalistic explanatory account for these cognitive capacities. (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)

In a recent paper in Astrophysics and Space Science Vol. 364 no. 11 (2019), S. Ershkov & D. Leschenko presented a new solving procedure for Euler-Poisson equations for solving momentum equations of the CR3BP near libration points for uniformly rotating planets having inclined orbits in the solar system with respect to the orbit of the Earth. The system of equations of the CR3BP has been explored with regard to the existence of an analytic way of presentation of the approximated solution (...) in the vicinity of libration points. A new and elegant ansatz has been suggested in their publication whereby, in solving, the momentum equation is reduced to a system two coupled Riccati ODEs. In this paper, we presented a numerical solution of such coupled Riccati ODEs using Mathematica software package. (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)

It has been known for long time that most of the existing cosmology models have singularity problem. Cosmological singularity has been a consequence of excessive symmetry of flow, such as “Hubble’s law”. More realistic one is suggested, based on Newtonian cosmology model but here we include the vertical-rotational effect of the whole Universe. We review a Riccati-type equation obtained by Nurgaliev, and solve the equation numerically with Mathematica. It is our hope that the new proposed method can be verified with (...) observation data. (shrink)

A numerical classification was performed on 69 elements with 54 chemicaland physicochemical properties. The elements fell into clusters in closeaccord with the electron shell s-, p- andd-blocks. The f-block elements were not included forlack of sufficiently complete data. The successive periods ofs- and p-block elements appeared in an ovalconfiguration, with d-block elements lying to one side. Morethan three axes were required to give good representation of thevariation, although the interpretation of the higher axes is difficult.Only 15 properties were scorable for (...) the noble gases, but despite thepaucity of properties reflecting chemical reactivity, analysis of the 69elements on these properties still showed the major features seen fromthe full set. (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.

Aquinas, Ockham, and Burdan all claim that a person can be numerically identical over time, despite changes in size, shape, and color. How can we reconcile this with the Indiscernibility of Identicals, the principle that numerical identity implies indiscernibility across time? Almost all contemporary metaphysicians regard the Indiscernibility of Identicals as axiomatic. But I will argue that Aquinas, Ockham, and Burdan would reject it, perhaps in favor of a principle restricted to indiscernibility at a time.

Arithmetic is one of the foundations of our educational systems, but what exactly is it? Numbers are everywhere in our modern societies, but what is our knowledge of numbers really about? This book provides a philosophical account of arithmetical knowledge that is based on the state-of-the-art empirical studies of numerical cognition. It explains how humans have developed arithmetic from humble origins to its modern status as an almost universally possessed knowledge and skill. Central to the account is the realisation that, (...) while arithmetic is a human creation, the development of arithmetic is constrained by our evolutionarily developed cognitive architecture. Arithmetic is a sophisticated cultural development, but it is ultimately based on abilities with numerosities that we already possess as infants and share with many non-human animals. Therefore, arithmetic is not purely conventional, an arbitrary game akin to chess. Instead, arithmetic is deeply connected to our basic cognitive capacities. (shrink)

The first aim of the paper is to show that under certain conditions generative syntax can be made suitable for Montague semantics, based on his type logic. One of the conditions is to make branching in the so-called X-bar syntax strictly binary, This makes it possible to provide an adequate semantics for Noun Phrases by taking them as referring to sets of collections of sets of entities ( type <ett,t>) rather than to sets of sets of entities (ett).

Numerical identity is the non-relational sameness of an object to itself. It is concerned with understanding how entities undergo change and maintain their identity. In substance metaphysics, an entity is considered a substance with an essence and such an essence is the source of its power. However, such a framework fails to explain the sense in which an entity is still the entity it was, amidst changes. Those who claim that essence is unaffected by existence are faced with challenge of (...) exploring the epistemic access to such an essence, which is questionable at best. Process metaphysics is a strong candidate for a theory that can ontologically explain regularity and change without appeal to essence. Process and its interactions is the main category. Every process is an emergent organization of constitutive interactions and is individuated on the basis of its interactive powers, that is, the ways in which it interacts with the world around it. Interactions are situated adaptation to changes. In this way, changes are crucial within process metaphysics and are included in the starting point of its investigation. What seems to the naked eyes as one-ness/singularity is a complex process where an organization of interactions is emerging from moment to moment by continually adapting to the changes around and within it. The question of numerical identity over time becomes valid only within substance metaphysics which has no space to accommodate change, due to its allegiance to essence. (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)

_ Source: _Volume 46, Issue 2, pp 205 - 220 I investigate the phenomenological significance of Husserl’s appeal to the “numerical identity” of _irreality_ as it appears in recollected manifolds of lived-experience in his mature account of the transcendental constitution of transcendence and find it wanting. I show that what is at stake for Husserl in this appeal is the descriptive mark that exhibits the distinction between a unit of meaning as it is constituted in psychologically determined lived-experience and as (...) it is constituted in lived-experience that is determined transcendentally. In other words, I show that numerical identity functions for Husserl as the criterion that signals transcendental psychologism has been overcome. I then present the argument that it has not been overcome in Husserl’s investigations, because the collective unity characteristic of numerical unity is presupposed by those investigations rather than made evidentially manifest and articulated. (shrink)

The study of neuronal specialisation in different cognitive and perceptual domains is important for our understanding of the human brain, its typical and atypical development, and the evolutionary precursors of cognition. Central to this understanding is the issue of numerical representation, and the question of whether numbers are represented in an abstract fashion. Here we discuss and challenge the claim that numerical representation is abstract. We discuss the principles of cortical organisation with special reference to number and also discuss methodological (...) and theoretical limitations that apply to numerical cognition and also to the field of cognitive neuroscience in general. We argue that numerical representation is primarily non-abstract and is supported by different neuronal populations residing in the parietal cortex. (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)

Following a hypothesis by Marciak-Kozlowska, 2011, we consider one-dimensional Schumann wave transfer phenomena. Numerical solution of that equation was obtained by the help of Mathematica.

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)

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)

An interpretive system for the IBM 704 computer permitting interpretation of the genetic pattern of a numeric symbioorganism as a game strategy has been developed. Selection for best performance in a simple game has been applied in a preliminary experiment. An objective method to measure the quality of a game played is described. The results presented in the article show a small but significant improvement of game quality during a period of 2300 generations.The general characteristics of the phenomena observed are (...) similar to those of preceding evolution experiments . However, a startling infection process caused by a parasite whose behaviour was influenced by the game competition is described. The parasite never developed an independent game strategy to any degree of efficiency and was entirely dependent on its host organism for game competitions with, and transmission of the infection to, uninfected hosts.The consequences of substituting the reproduction and mutation rules used in the preceding evolution experiments are discussed. Particularly, the use of rules applying the reproduction pattern ofDNA-molecules is considered, and the theoretic aspects of symbiogenetic evolution experiments based onDNA-norm are discussed. A few inferences concerning the origin and history of terrestrial life suggested by the results of the symbiogenesis experiments are presented.Ein interpretierendes System für die IBM 704 Rechenmaschine, dass Interpretation des genetischen Modells von einem numerischen Symbio-organismus wie eine Spiel Strategie, erlaubt, wurde entwickelt. Eine Forführung für Fähigkeit in einem einfachen Spiel wurde angewandt in einem preliminarisches Experiment. Eine objektive Methode zur Messung der Qualität des betreffenden Spiels wurde beschrieben. Die in dem Artikel angegebenen Resultate zeigen eine kleine, aber bedeutende Verbesserung der Spiel-Qualität während einr Periode von 2300 Generationen.Die allgemeinen Kennziechen von den beobachteten Phänomenen sind denen von vorhergehenden Evolutions-Experimenten ähnlich . Jedoch, ein überraschender Infektions-Prozess, verursacht durch einen Parasit, wessen Verhalten beinflusst wurde durch die Spiel-Konkurrenz, wurde beschrieben. Der Parasit entwickelte niemals eine unabhängige Spiel-Strategie zu irgendeinem Masze von Fähigkeit und war ganz abhängig von seinem Gastherr-Organismus für Spiel-Konkurrenz mit, und Überbringung von der Infektion auf, uninfektierte Gäste.Die Folgerungen von dem Ersetzen der Fortpflanzung und Mutations-Regeln , angewandt in vorhergehenden Evolutions-Experimente wurden besprochen. Besonders der Gebrauch von Regeln welche das FortpflanzungsModell von DNA-Molekülen anwenden wurde betrachtet, und die theorethischen Aspekte von Symbiogenetischen Evolutions-Experimenten basiert auf DNA-Norm wurden besprochen. Einige Schlussfolgerungen in Bezug auf die Entstehung und Geschichte von dem Erdenleben suggeriert durch die Resultate von den Symbiogenese-Experimenten wurden gezeigt. (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)

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)

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)

Predicates like gather and ones like be numerous have both been described as ‘collective predicates,’ since they predicate something of a plurality. The two classes of predicates differ, however, with respect to plural quantifiers, which are grammatical with gather-type predicates but ungrammatical with numerous-type predicates. Here, I show that the gather/numerous opposition derives from mereological properties that are familiar from the domains of telicity and mass/count. I address problems of undergeneration and overgeneration with two technical innovations: first, I weaken the (...) property of divisibility to Champollion’s concept of stratified reference; second, I provide mechanisms to rule out accidental satisfaction of the logical property. More broadly, I place collective predication in a larger context by building empirical connections to mass/count and collectivity across semantic domains. (shrink)

Modified numerals, such as at least three and more than five, are known to sometimes give rise to ignorance inferences. However, there is disagreement in the literature regarding the nature of these inferences, their context dependence, and differences between at least and more than. We present a series of experiments which sheds new light on these issues. Our results show that the ignorance inferences of at least are more robust than those of more than, the presence and strength of the (...) ignorance inferences triggered by both at least and more than depends on the question under discussion, and whether ignorance inferences are detected in a given experimental setting depends partly on the task that participants are asked to perform. We offer an Optimality Theoretic account of these findings. In particular, the task effect is captured by assuming that in performing an acceptability task, participants take the speaker’s perspective in order to determine whether an expression is optimal given a certain epistemic state, while in performing an inference task they take the addressee’s perspective in order to determine what the most likely epistemic state of the speaker is given a certain expression. To execute the latter task in a fully rational manner, participants have to perform higher-order reasoning about alternative expressions the speaker could have used. Under the assumption that participants do not always perform such higher-order reasoning but also often resort to so-called unidirectional optimization, the task effect finds a natural explanation. This also allows us to relate our finding to asymmetries between comprehension and production that have been found in language acquisition. (shrink)

The central topic of this article is (the possibility of) de re knowledge about natural numbers and its relation with names for numbers. It is held by several prominent philosophers that (Peano) numerals are eligible for existential quantification in epistemic contexts (‘canonical’), whereas other names for natural numbers are not. In other words, (Peano) numerals are intimately linked with de re knowledge about natural numbers, whereas the other names for natural numbers are not. In this article I am looking for (...) an explanation of this phenomenon. It is argued that the standard induction scheme plays a key role. (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)

Quantifier expressions like “many” and “at least” are part of a rich repository of words in language representing magnitude information. The role of numerical processing in comprehending quantifiers was studied in a semantic truth value judgment task, asking adults to quickly verify sentences about visual displays using numerical or proportional quantifiers. The visual displays were composed of systematically varied proportions of yellow and blue circles. The results demonstrated that numerical estimation and numerical reference information are fundamental in encoding the meaning (...) of quantifiers in terms of response times and acceptability judgments. However, a difference emerges in the comparison strategies when a fixed external reference numerosity is used for numerical quantifiers, whereas an internal numerical criterion is invoked for proportional quantifiers. Moreover, for both quantifier types, quantifier semantics and its polarity biased the response direction. Overall, our results indicate that quantifier comprehension involves core numerical and lexical semantic properties, demonstrating integrated processing of language and numbers. (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.

The target article undermines the existence of a shared unitary numerical format, illustrating a variety of representations. The / dichotomy does not capture their specific features. These representations are with respect to the sensory modality of the stimulus, and independent of its specific notation, with a main role of spatial codes, both related and unrelated to the mental number line.

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.

In Science Without Numbers (1980), Hartry Field defends a theory of quantity that, he claims, is able to provide both i) an intrinsic explanation of the structure of space, spacetime, and other quantitative properties, and ii) an intrinsic explanation of why certain numerical representations of quantities (distances, lengths, mass, temperature, etc.) are appropriate or acceptable while others are not. But several philosophers have argued otherwise. In this paper I focus on arguments from Ellis and Milne to the effect that one (...) cannot provide an account of quantity in ''purely intrinsic'' terms. I show, first, that these arguments are confused. Second, I show that Field's treatment of quantity can provide an intrinsic explanation of the structure of quantitative properties; what it cannot do is provide an intrinsic explanation of why certain numerical representations are more appropriate than others. Third, I show that one could provide an intrinsic explanation of this sort if one modified Field's account in certain ways. (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)

A new class of polynomials investigates the numerical solution of the fractional pantograph delay ordinary differential equations. These polynomials are equipped with an auxiliary unknown parameter a, which is obtained using the collocation and least-squares methods. In this study, the numerical solution of the fractional pantograph delay differential equation is displayed in the truncated series form. The upper bound of the solution as well as the error analysis and the rate of convergence theorem are also investigated in this study. In (...) five examples, the numerical results of the present method have been compared with other methods. For the first time, a -polynomials are used in this study to numerically solve delay equations, and accurate approximations have been displayed. (shrink)

What does it look like when a group of scientists set out to re-envision an entire field of biology in symbolic and formal terms? I analyze the founding and articulation of Numerical Taxonomy between 1950 and 1970, the period when it set out a radical new approach to classification and founded a tradition of mathematics in systematic biology. I argue that introducing mathematics in a comprehensive way also requires re-organizing the daily work of scientists in the field. Numerical taxonomists sought (...) to establish a mathematical method for classification that was universal to every type of organism, and I argue this intrinsically implicated them in a qualitative re-organization of the work of all systematists. I also discuss how Numerical Taxonomy’s re-organization of practice became entrenched across systematic biology even as opposing schools produced their own competing mathematical methods. In this way, the structure of the work process became more fundamental than the methodological theories that motivated it. (shrink)

Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept (...) from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas. (shrink)

Phenomenological psychology has emphasized that experience as it is immediately "given" to the experiencing individual is an appropriate subject matter for psychological investigation. Consideration of the methodological implications of this stance suggests that certain text analytic and cluster analytic methods could be used to discern the identifying properties of different types of experience. We present results of a study in which textual analysis was used to identify recurrent properties of participants' verbal accounts of their experience, cluster analysis was used to (...) classify participants' accounts according to the similarity of their profiles of properties, and the resulting clusters were examined for their more or less characteristic prpoerties. Using these methods, three distinct types of experience of a Renaissance painting were identified and described. This demonstration of numerically aided phenomenological methods indicates the compatibility of rigorous and sensitive descriptions of experiential accounts. (shrink)

Our epistemic access to mathematical objects, like numbers, is mediated through our external representations of them, like numerals. Nevertheless, the role of formal notations and, in particular, of the internal structure of these notations has not received much attention in philosophy of mathematics and cognitive science. While systems of number words and of numerals are often treated alike, I argue that they have crucial structural differences, and that one has to understand how the external representation works in order to form (...) an advanced conception of numbers. The relation between external representations and mathematical cognition will be discussed by drawing on experimental results from cognitive science and mathematics education, and I argue for the constitutive role of external representations for the development of an advanced conception of numbers. (shrink)

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)