On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system, that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical (...) confounds or exotic substitutes for number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind of number being represented. In response, we propose that the ANS represents not only natural numbers, but also non-natural rational numbers. It does not represent irrational numbers, however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research. (shrink)

On a now orthodox view, humans and many other animals are endowed with a “number sense”, or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques, with critics maintaining either that numerical content is absent altogether, or else that some primitive analog of number (‘numerosity’) is represented as opposed to number itself. We distinguish three arguments for these claims – the arguments from congruency, confounds, and imprecision (...) – and show that none succeed. We then highlight positive reasons for thinking that the ANS genuinely represents numbers. The upshot is that proponents of the orthodox view should not feel troubled by recent critiques of their position. (shrink)

The study determined the influence of innate mathematical characteristics on the number sense competencies of junior high school students in a Philippine public school. The descriptive-correlational research design was used to accomplish the study involving a nonrandom sample of sixty 7th-grade students attending synchronous math sessions. Data obtained from the math-specific Learning Style and Self-Efficacy questionnaires and the modified Number Sense Test (NST) were analyzed and interpreted using descriptive statistics, Pearson’s Chi-Square, and Simple Linear Regression analysis. (...) The research instruments and statistics were all validated and tested for reliability. The analysis revealed that the students are visual learners, had no or slight self-efficacy, and their number sense competency level is poor. They encountered difficulty in all the components and domains of the NST. Moreover, the students’ mathematical self-efficacy is significantly related and may influence their number sense competency level. Building upon the learners’ self-efficacy to further their understanding and skills in number sense is necessary. (shrink)

Many philosophers still countenance senses or meanings in the broadly Fregean vein. However, it is difficult to posit the existence of senses without positing quite a lot of them, including at least one presenting every entity in existence. I discuss a number of Cantorian paradoxes that seem to result from an overly large metaphysics of senses, and various possible solutions. Certain more deflationary and nontraditional understanding of senses, and to what extent they fare better in solving the problems, are (...) also discussed. In the end, it is concluded that one must divide senses into various ramified-orders in order to avoid antinomy, but that the philosophical justification of such orders is, as yet, still somewhat problematic. (shrink)

In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system represents numbers or numerosities, and why the ANS represents rational numbers.

The approximate number system (ANS) enables organisms to represent the approximate number of items in an observed collection, quickly and independently of natural language. Recently, it has been proposed that the ANS goes beyond representing natural numbers by extracting and representing rational numbers (Clarke & Beck, 2021a). Prior work demonstrates that adults and children discriminate ratios in an approximate and ratio-dependent manner, consistent with the hallmarks of the ANS. Here, we use a well-known “connectedness illusion” to provide evidence (...) that these ratio-dependent ratio discriminations are (a) based on the perceived number of items in seen displays (and not just non-numerical confounds), (b) are not dependent on verbal working memory, or explicit counting routines, and (c) involve representations with a part-whole (or subset-superset) format, like a fraction, rather than a part-part format, like a ratio. These results vindicate key predictions of the hypothesis that the ANS represents rational numbers. (shrink)

The present paper deals with the ontological status of numbers and considers Frege ́s proposal in Grundlagen upon the background of the Post-Kantian semantic turn in analytical philosophy. Through a more systematic study of his philosophical premises, it comes to unearth a first level paradox that would unset earlier still than it was exposed by Russell. It then studies an alternative path, that departin1g from Frege’s initial premises, drives to a conception of numbers as synthetic a priori in a more (...) Kantian sense. On this basis, it tentatively explores a possible derivation of basic logical rules on their behalf, suggesting a more rudimentary basis to inferential thinking, which supports reconsidering the difference between logical thinking and AI. Finally, it reflects upon the contributions of this approach to the problem of the a priori. (shrink)

It is often assumed that adaptation — a temporary change in sensitivity to a perceptual dimension following exposure to that dimension — is a litmus test for what is and is not a “primary visual attribute”. Thus, papers purporting to find evidence of number adaptation motivate a claim of great philosophical significance: That number is something that can be seen in much the way that canonical visual features, like color, contrast, size, and speed, can. Fifteen years after its (...) reported discovery, number adaptation’s existence seems to be nearly undisputed, with dozens of papers documenting support for the phenomenon. The aim of this paper is to offer a counterweight — to critically assess the evidence for and against number adaptation. After surveying the many reasons for thinking that number adaptation exists, we introduce several lesser-known reasons to be skeptical. We then advance an alternative account — the old news hypothesis — which can accommodate previously published findings while explaining various (otherwise unexplained) anomalies in the existing literature. Next, we describe the results of eight pre-registered experiments which pit our novel old news hypothesis against the received number adaptation hypothesis. Collectively, the results of these experiments undermine the number adaptation hypothesis on several fronts, whilst consistently supporting the old news hypothesis. More broadly our work raises questions about the status of adaptation itself as a means of discerning what is and is not a visual attribute. (shrink)

Number Nativism is the view that humans innately represent precise natural numbers. Despite a long and venerable history, it is often considered hopelessly out of touch with the empirical record. I argue that this is a mistake. After clarifying Number Nativism and distancing it from related conjectures, I distinguish three arguments which have been seen to refute the view. I argue that, while popular, two of these arguments miss the mark, and fail to place pressure on Number (...) Nativism. Meanwhile, a third argument is best construed as a challenge: rather than refuting Number Nativism, it challenges its proponents to provide positive evidence for their thesis and show that this can be squared with apparent counterevidence. In response, I introduce psycholinguistic work on The Tolerance Principle (not yet considered in this context), propose that it is hard to make sense of without positing precise and innate representations of natural numbers, and argue that there is no obvious reason why these innate representations couldn’t serve as a basis for mature numeric conception. (shrink)

In this paper, I study the kind of questions we can ask about the existence of numbers. In addition to asking whether numbers exist, and how, I argue that there is also a third relevant question: why numbers exist. In platonist and nominalist accounts this question may not make sense, but in the psychologist account I develop, it is as well-placed as the other two questions. In fact, there are two such why-questions: the causal why-question asks what causes numbers (...) to exist and the teleological why-question asks for what purpose numbers exist. I argue that in a psychologist constructivist account, in which numbers are understood to exist as referents of a particular type of culturally shared concepts, both why-questions can get plausible answers. (shrink)

Is the way we use propositions to individuate beliefs and other intentional states analogous to the way we use numbers to measure weights and other physical magnitudes? In an earlier paper [2], I argued that there is an important disanalogy. One and the same weight can be 'related to' different numbers under different units of measurement. Moreover, the choice of a unit of measurement is arbitrary,in the sense that which way we choose doesn't affect the weight attributed to the (...) object. But it makes little sense to say that one and the same belief can be related to different propositions: different proposition means different belief. So there is no analogous arbitrary choice. (shrink)

Visual number adaption is a widely accepted phenomenon. This paper advances an alternative explanation for putative cases of the phenomenon. We propose that such cases may simply reflect observers adapting to the items in perceived displays, rather than their numerical quantity. Three experiments motivate consideration of this novel proposal and call into question the evidential basis for received formulations of the number adaptation hypothesis.

Our objectives in this paper are, first, to identify several puzzling aspects of the “Trilemma Argument” of Section 6 against the Sense Datum Theory; second, to resolve these puzzles by reconstructing the Trilemma Argument; third to point to a distinction Sellars makes between two versions of the Sense Datum Theory, the “nominalist” version and the “realist” version; fourth, to reconstruct Sellars’s arguments against both; and, finally, to find in an earlier paper, “Is There a Synthetic A Priori?” that (...) his argument against the second version, assumed but not actually given in EPM, is against property realism and depends on taking language expressing propositional perception as fundamental and language expressing non-propositional perception as incomplete and derivative from the former. (shrink)

Making sense of modeling: beyond representation Content Type Journal Article Category Original paper in Philosophy of Science Pages 335-352 DOI 10.1007/s13194-011-0032-8 Authors Isabelle Peschard, Philosophy Department, San Francisco State University, 1600 Holloway Ave, San Francisco, CA 94132, USA Journal European Journal for Philosophy of Science Online ISSN 1879-4920 Print ISSN 1879-4912 Journal Volume Volume 1 Journal Issue Volume 1, Number 3.

It is widely accepted that it counts for a metaphysical theory when the theory is in accord with common sense and against a metaphysical theory when the theory clashes with common sense. It is unclear, however, why this should be the case. When engaging in metaphysics, why should we give common sense any weight? This chapter maintains that it is only against the backdrop of a particular metametaphysical stance that questions about metaphysical best practices become tractable. From (...) the perspective of a metaphysics-as-modelling approach, common sense ought to play a significant, though defeasible, role in metaphysical theorizing. According with common sense is one of a number of theoretical virtues that metaphysicians should strive for. Nevertheless, it is important for the metaphysician to be cautious when appealing to common sense. She should distinguish what actually falls within the bounds of common sense as such from what a particular researcher happens to find intuitive. Furthermore, our best scientific theories may undercut the evidence provided by common sense. Finally, the metaphysician should attend to the context in which she invokes common sense. For some topics of inquiry, common sense ought to play a more expansive role in our metaphysical theorizing than for others. (shrink)

The article is centered on the question of how best to understand the logical pluralism/logical monism debate. A number of suggestions are brought up and rejected on the ground that they re...

Moral luck poses a problem for out conception of responsibility because it highlights a tension between morality and lack of control. Michael Slote’s common-sense virtue ethics claims to avoid this problem. However there are a number of objections to this claim. Firstly, it is not clear that Slote fully appreciates the problem posed by moral luck. Secondly, Slote’s move from the moral to the ethical is problematic. Thirdly it is not clear why we should want to abandon judgements (...) of moral blame in favour of judgements of ethical deplorability. Finally this paper defends an alternative solution to the problem of moral luck, which focuses on judgements of probability, but which has been rejected by Slote. (shrink)

Janette Dinishak’s work has helped shed critical light on the scientifically questionable and ethically troubling tendency in psychology and philosophy of mind to theorize autistic people as deficient empathizers. In a recently published reply on the Social Epistemology Review and Reply Collective, Dinishak (2024) brings her important perspective on this topic to bear on our paper “AAC Technology, Autism, and the Empathic Turn” (2022). Dinishak is largely sympathetic to our view while also raising a number of rich and thoughtful (...) philosophical questions. Since each of these questions is capable of jumpstarting a lengthy exchange, we focus our response on those questions that, we hope, are particularly fruitful for advancing further conversation. With this aim in mind, we also engage with some of Dinishak’s own insights from her 2016 chapter “Empathy, Like-mindedness, and Autism.” … [please read below the rest of the article]. (shrink)

What is a number? Answering this will answer questions about its philosophical foundations - rational numbers, the complex numbers, imaginary numbers. If we are to write or talk about something, it is helpful to know whether it exists, how it exists, and why it exists, just from a common-sense point of view [Quine, 1948, p. 6]. Generally, there does not seem to be any disagreement among mathematicians, scientists, and logicians about numbers existing in some way, but currently, in (...) the mainstream arena only definitions, descriptions of properties, and effects are presented as evidence. Enough historical description of numbers in history provides an empirical basis of number, although a case can be made that numbers do not exist by themselves empirically. Correspondingly, numbers exist as abstractions. All the while, though, these "descriptions" beg the question of what numbers are ontologically. Advocates for numbers being the ultimate reality have the problem of wrestling with the nature of reality. I start on the road to discovering the ontology of number by looking at where people have talked about numbers as already existing: history. Of course, we need to know not only what ontology is but the problems of identifying one, leading to the selection between metaphysics and provisional approaches. While we seem to be dimensionally limited, at least we can identify a more suitable bootstrapping ontology than mere definitions, leading us to the unity of opposites. The rest of the paper details how this is done and modifies Peano's Postulates. (shrink)

My dissertation covers a number of different topics in Kant scholarship, but is driven by one central question: how do our sense-based capacities to perceive, desire, and feel relate to our capacity to reason? I take the answer to this question to be key to understanding much about Kant’s philosophical system. For topics as diverse as the role that sensation plays in practical knowledge, the character of moral motivation, the nature of evil, or Kant’s theory that we are (...) morally required to believe in our immortality and God’s existence, I claim that the correct reading turns on a novel approach to the relation between our rational and sensitive capacities. To properly understand this relation, we must overcome the deeply entrenched thought that Kant inherits his conception of reason from the rationalists, and his conception of sense experience from the empiricists. This would be to adopt a fundamentally dualistic philosophy, on which reason and the senses are independently intelligible and sometimes even diametrically opposed. On Kant’s actual view, our capacities to perceive, feel, and desire are metaphysically interdependent with and thus not intelligible in isolation from our capacity to reason. Developing the details of this more unified position leads to satisfying philosophical accounts of a number of controversial issues in Kant scholarship that have suffered under a dualistic framework. (shrink)

Developing earlier studies of the system of numbers in Mundurucu, this paper argues that the Mundurucu numeral system is far more complex than usually assumed. The Mundurucu numeral system provides indirect but insightful arguments for a modular approach to numbers and numerals. It is argued that distinct components must be distinguished, such as a system of representation of numbers in the format of internal magnitudes, a system of representation for individuals and sets, and one-to-one correspondences between the numerosity expressed by (...) the number and its metrics. It is shown that while many-number systems involve a compositionality of units, sets and sets composed of units, few-number languages, such as Mundurucu, do not have access to sets composed of units in the usual way. The nonconfigurational character of the Mundurucu language, which is related to a property for which we coin the term 'low compositionality power', accounts for this and explains the curious fact that Mundurucus make use of marked one-to-one correspondence strategies in order to overcome the limitations of the core system at the perceptual/motor interface of the language faculty. We develop an analysis of a particular construction, parallel numbers, which has not been studied before, elucidating the whole system. This analysis, we argue, sheds new light on classical philosophical, psychological and linguistic debates about numbers and numerals and their relation to language, and more particularly, sheds light on few-number languages. (shrink)

The idea that there are five senses dates back to Aristotle, who was one of the first philosophers to examine them systematically. Though it has become conventional wisdom, many scientists and philosophers would argue that this idea is outdated and inaccurate. Indeed, they have given many different answers to this question, ranging from just three (the number of different kinds of physical energy we can detect) to 33 or more senses. Perhaps surprisingly, the issue remains controversial, partly because it (...) is not clear exactly what should count as a ‘sense’. Should proprioception, balance and kinaesthesis, amongst others, be considered additional senses? And if so, how do we figure out how many senses we really have? -/- (Published in Turkish under the title ‘Çok Duyumlu Algi’). (shrink)

Suppose you can save only one of two groups of people from harm, with one person in one group, and five persons in the other group. Are you obligated to save the greater number? While common sense seems to say ‘yes’, the numbers skeptic says ‘no’. Numbers Skepticism has been partly motivated by the anti-consequentialist thought that the goods, harms and well-being of individual people do not aggregate in any morally significant way. However, even many non-consequentialists think that (...) Numbers Skepticism goes too far in rejecting the claim that you ought to save the greater number. Besides the prima facie implausibility of Numbers Skepticism, Michael Otsuka has developed an intriguing argument against this position. Otsuka argues that Numbers Skepticism, in conjunction with an independently plausible moral principle, leads to inconsistent choices regarding what ought to be done in certain circumstances. This inconsistency in turn provides us with a good reason to reject Numbers Skepticism. Kirsten Meyer offers a notable challenge to Otsuka’s argument. I argue that Meyer’s challenge can be met, and then offer my own reasons for rejecting Otsuka’s argument. In light of these criticisms, I then develop an improved, yet structurally similar argument to Otsuka’s argument. I argue for the slightly different conclusion that the view proposed by John Taurek that ‘the numbers don’t count’ leads to inconsistent choices, which in turn provides us with a good reason to reject Taurek’s position. (shrink)

Research on dual character concepts has explored cases in which people think that a term applies to an object in a superficial sense but does not apply to that same object in a deeper sense. Most of this research has focused on cases of one particular type, namely, cases in which the object fails to embody the characteristic values of a particular category. However, there are also other types of cases in which we would be inclined to say (...) that a term does not apply in a deeper sense. For example, we might also say this if an object has acquired a certain status but it failed to acquire this status through a morally or legally legitimate process. Or we might say it if a person has certain emotions but these emotions are not rooted in the person’s true self. I argue that these apparently unrelated types of cases are surprisingly similar in a number of important respects. Thus, we should not be seeking a separate account for each separate type of case; we should be seeking an account that explains in a more general way what people mean when they say that a term does not apply in a deeper sense. (shrink)

It’s sometimes suggested that we can (in a sense) settle the truth-value of some statements in the language of number theory by stipulation, adopting either φ or ¬φ as an additional axiom. For example, in Clarke-Doane (2020b) and a series of recent APA presentations, Clarke-Doane suggests that any Σ01 sound expansion of our current arithmetical practice would express a truth. In this paper, I’ll argue that (given a certain popular assumption about the model-theoretic representability of languages like ours) (...) we can’t know ourselves to have any such freedom. (shrink)

This article offers an interpretation of a controversial aspect of Frege’s The Foundations of Arithmetic, the so-called Julius Caesar problem. Frege raises the Caesar problem against proposed purely logical definitions for ‘0’, ‘successor’, and ‘number’, and also against a proposed definition for ‘direction’ as applied to lines in geometry. Dummett and other interpreters have seen in Frege’s criticism a demanding requirement on such definitions, often put by saying that such definitions must provide a criterion of identity of a certain (...) kind. These interpretations are criticized and an alternative interpretation is defended. The Caesar problem is that the proposed definitions fail to well-define ‘number’ and ‘direction’. That is, the proposed definitions, even when taken together with the extra-definitional facts, fail to fix unique semantic extensions for ‘number’ and ‘direction’, and thereby fail to fix unique truth-values for sentences like ‘Caesar is a number’ and ‘England is a direction’. A minor modification of the criticized definitions well-defines ‘0’, ‘successor’ and ‘number’, thereby avoiding the Caesar problem as Frege understands it, but without providing any criterion of number identity in the usual sense. (shrink)

Donald Trump and Boris Johnson are notoriously uninterested in truthtelling. They also often appear uninterested even in constructing plausible falsehoods. What stands out above all is the brazenness and frequency with which they repeat known falsehoods. In spite of this, they are not always greeted with incredulity. Indeed, Republicans continue to express trust in Donald Trump in remarkable numbers. The only way to properly make sense of what Trump and Johnson are doing, we argue, is to give a greater (...) role to audience relativity - and in some cases, audience participation - in notions like bullshitting and bald-faced lying. In this paper, we develop a new understanding of bullshitting, one that includes bald-faced lying, and recognizes that different communicative acts may be directed at different audiences with a single utterance. In addition, we argue for recognition of the category of bald-faced bullshitting, a particular speciality of both Trump and Johnson, and one especially useful to authoritarian leaders. (shrink)

Philosophical discussion of human rights has long been monopolised by what might be called the ‘natural-law view’. On this view, human rights are fundamental moral rights which people enjoy solely by virtue of their humanity. In recent years, a number of theorists have started to question the validity of this outlook, advocating instead what they call a ‘political’ view. My aim in this article is to explore the latter view in order to establish whether it constitutes a valuable alternative (...) to the ‘natural-law view’. In particular, I distinguish between three ways in which human rights can be political: in relation to their (1) iudicandum, (2) justification and (3) feasibility constraints. I argue that it makes sense to think of human rights as political in relation to both their iudicandum and their justification but in a way that is not always adequately captured by proponents of the political view. Moreover, I also claim that, if we take the political view seriously, we still need to engage in the sort of abstract moral reasoning that characterises the natural-law approach and which proponents of the political view significantly downplay. (shrink)

In this paper I make a number of points about “political correctness”. Although individually these arguments seem straightforward - and will hopefully be uncontroversial - put together in context they reveal the idea of a “politically correct”, left-wing dominated, media or intelligentsia in Western political culture to be a conservative bogeyman. The rhetoric of “political correctness” is in fact overwhelmingly a right-wing conservative one which itself is used mainly to silence dissenting political viewpoints. However, the same investigation also suggests (...) that a “politics of speech” is an inevitable fact of social life and that some sorts of censorship are likewise inevitable. The question of censorship is therefore revealed as not “Whether we should tolerate all sorts of speech?” but “Which sorts of speech should we tolerate?”. (shrink)

The recent distribution of nude photos of a number of high profile Hong Kong celebrities has provoked intense discussion about the state of Hong Kong's obscenity and indecency laws. In this paper, I argue that Hong Kong's laws prohibiting the transfer of obscene and indecent information and images between consenting adults are both under-inclusive and over-inclusive. The Control of Obscene and Indecent Articles Ordinance is under-inclusive in that it does not adequately criminalise grave violations of privacy. It is also (...) over-inclusive because it is a blanket prohibition against the transfer by all parties (including consenting adults) of all forms of obscene and indecent materials. The laws unnecessarily violate the free expression rights of both the producer and consenting viewer of the offensive materials. The producer/publisher of such materials does not harm his or her audience as they willingly view such materials. The justification for maintaining a blanket prohibition against all transfers of such materials is invalid and utterly and totally out of touch with modern life in Hong Kong. The proponents of such laws have used Victorian positive morality considerations to justify continued criminalisation. These laws should be abrogated and replaced with a new piece of legislation that is narrowly tailored to deal with those types of offensive displays that are wrongful in a critical rather than a mere positive morality sense. Criminalisation should be limited to those offences that target children or use children in the production process, violate the rights of non-consenting adult audiences not to receive certain intimate information in certain public contexts, and violate privacy rights by publishing a person's private and intimate information without consent. If x obtains y's profoundly private information and publishes it without y's consent, then x violates y's privacy rights in a grave way. The violation in the right circumstances will justify a criminal law response rather than a mere civil law response. Similarly, if x and y copulate on a public bus they subject the captive audience to an offensive display which violates the non-consenting audience's right not to receive certain intimate information. I argue below that these types of privacy violations give the lawmaker a legitimate justification for invoking the criminal law. (shrink)

Where do human numerical abilities come from? This article is a commentary on Leibovich et al.’s “From 'sense of number' to 'sense of magnitude' —The role of continuous magnitudes in numerical cognition”. Leibovich et al. argue against nativist views of numerical development by noting limitations in newborns’ vision and limitations regarding newborns’ ability to individuate objects. I argue that these considerations do not undermine competing nativist views and that Leibovich et al.'s model itself presupposes that infant learners (...) have numerical representations. (shrink)

The principal thesis in this review-essay is that the key linguistic terms in Fichte’s Wissenschaftslehre especially have two main meanings that appear at first sight to be almost in contradiction or opposed to each other. The reader of Fichte therefore has to work hard to overcome any apparent conflicts in the “double sense” of his philosophical terminology. Accordingly, I argue that Fichte’s linguistic method and use of language should be seen as part of his chief philosophical method of synthesis, (...) where we have to carry out a similar procedure and attempt to reconcile opposites using the power of the imagination. This thesis is put forward by means of a number of practical examples and in the context of some critical reflections on the recently published Cambridge Companion to Fichte, eds. David James and Günter Zöller. Review essay published in Volume 15 (December, 2017) of the journal Revista de Estud(i)os sobre Fichte (ed. Emiliano Acosta). (shrink)

This articles exposes the methodological errors involved in attempting to operationalize or value-neutralize the concept of 'terrorism.' It defends, instead, an effects-based approach to the taxonomy of 'terrorism' that builds out from a central conceptual connection between the term's negative connotation and a widely shared moral presumption against the killing of innocent non-combatants. Although this approach to the core meaning of 'terrorism' is far from value-neutral, it has a number of virtues to recommend it. First, it has the political (...) virtue of even-handedness in the way it enables competing appraisals of asymmetric conflicts. Second, it is has the ethical virtue of being flexible enough to accommodate nuanced appraisals of various modes and degrees of terrorist violence. And third, it has the empirical virtue of being useful for purposes of rigorous social scientific research. (shrink)

The question of whether the term ”art,” or art as an array of objects, can be defined depends upon the sense of “art” and its extension. The generic sense of “art” is its broadest meaning having its widest extension. I argue that the term is very much like the generic term “science.” Uses of both terms don’t depend upon rigorous definition. Rather, the terms organize an enormous number of varied and sometimes incompatible sub-categories. Most informative topics in (...) art and science are under these sub-categories. As dynamic areas of activity, art and science continue to evolve without an efficacious generic sense. We can dispense with it for these purposes. (shrink)

It is widely agreed by philosophers that the so-called “Frege-Russell definition of natural number” is actually an assertion concerning the nature of the numbers and that it cannot be regarded as a definition in the ordinary mathematical sense. On the basis of the reasoning in this paper it is clear that the Frege-Russell definition contradicts the following three principles (taken together): (1) each number is the same entity in each possible world, (2) each number exists in (...) each possible world, (3) some entities existing in the actual world do not exist in every possible world. Since these principles seem to be true, the paper is a refutation of the Frege-Russell definition. The paper does more. It shows that the contradictory of the Frege-Russell definition follows even when principles 2 and 3 are replaced by one considerably weaker principle. The ideas contained in the paper are related to two earlier objections to the definition. The first, sometimes attributed to the mathematician, C. S. Keyser, is that existence of the numbers as defined implies the existence of infinitely many particulars in each possible world. The second is, in effect, an idea which is said to have led Whitehead to reject the definition of number to which he had subscribed in Principia Mathematica. Whitehead is supposed to have said that he could not believe that the number two changes every “time twins are born”. The mathematician H. Jeffreys expressed similar ideas [Philos. of Sci. 5 (1938), 434–451]. One of the merits of the author’s work is that it refutes the Frege-Russell definition without the need to take sides on controversial points presupposed by the Keyser and Whitehead objections. The objections made by the author are therefore not to be identified with the Keyser and Whitehead objections. Even if the author’s work is to be regarded as a refinement and integration of previous ideas, it is nevertheless a contribution—not only because the basic points are well worth repeating but also because the refinements are logically significant improvements and because the author has stated them clearly and concisely in the idiom of contemporary philosophy. (shrink)

Recent work in contemporary compatibilist theory displays considerable sophistication and subtlety when compared with the earlier theories of classical compatibilism. Two distinct lines of thought have proved especially influential and illuminating. The first developed around the general hypothesis that moral sentiments or reactive attitudes are fundamental for understanding the nature and conditions of moral responsibility. The other important development is found in recent compatibilist accounts of rational self-control or reason responsiveness. Strictly speaking, these two lines of thought have developed independent (...) of each other. However, in the past decade or so they have been fused together in several prominent statements of compatibilist theory. I will refer to theories that combine these two elements in this way as RS theories. RS theories face a number of familiar difficulties that relate to each of their two components. Beyond this, they also face a distinct set of problems concerning how these two main components relate or should be integrated. My concerns in this paper focus primarily on this set of problems. According to one version of RS compatibilism, the role of moral sentiments is limited to explaining what is required for holding an agent responsible. In contrast with this, the role of reason responsiveness is to explain what moral capacities are required for an agent to be responsible, one who is a legitimate or fair target of our moral sentiments. More specifically, according to this view, moral sense is not required for rational selfcontrol or reason responsiveness. There is, therefore, no requirement that the responsible agent has some capacity to feel moral sentiment. Contrary to this view, I argue that a responsible agent must be capable of holding herself and others responsible. Failing this, an agent’s powers of rational self-control will be both limited and impaired. In so far as holding responsible requires moral sense, it follows that being responsible also requires moral sense. (shrink)

This paper argues that Carnap both did not view and should not have viewed Frege's project in the foundations of mathematics as misguided metaphysics. The reason for this is that Frege's project was to give an explication of number in a very Carnapian sense — something that was not lost on Carnap. Furthermore, Frege gives pragmatic justification for the basic features of his system, especially where there are ontological considerations. It will be argued that even on the question (...) of the independent existence of abstract objects, Frege and Carnap held remarkably similar views. I close with a discussion of why, despite all this, Frege would not accept the principle of tolerance. (shrink)

Felix reaches up to catch a high line drive to left field and fires the ball off to Benji at home plate, who then tags the runner trying to score. For Felix to catch the ball and transfer it from his glove to his throwing hand, he needs to have a sense of where his hands are relative to one another and the rest of his body. This sort of information is subconsciously tracked in the body schema (or postural (...) schema), a representation of the current bodily posture that is updated on the basis of proprioceptive inputs. While the existence of the body schema in not in dispute, its origin is. After reviewing the competing proposals, I introduce the conceptual tools needed to move the debate forward and apply them to the question of the extent to which the body schema could be learned from perceptual input in utero. I argue that it could give rise to something recognizable as the body schema, though not quite rising to the level of the mature body schema. After considering the implications for further research on the origins of the body schema, I show how these results apply to other body representations, helping clarify the vexing question of the number, nature, and interactions among body representations in the brain. This theoretical work also promises to advance our understanding and treatment protocols for disorders affecting such body representations (e.g., anorexia nervosa). (shrink)

Daniel Isaacson has advanced an epistemic notion of arithmetical truth according to which the latter is the set of truths that we grasp on the basis of our understanding of the structure of natural numbers alone. Isaacson’s thesis is then the claim that Peano Arithmetic (PA) is the theory of finite mathematics, in the sense that it proves all and only arithmetical truths thus understood. In this paper, we raise a challenge for the thesis and show how it can (...) be overcome. We introduce the concept of purity for theories of arithmetic: a theory of arithmetic is pure when it only proves arithmetical truths. Then, we argue that, under Isaacson’s thesis, some PA-provable truths—including transfinite induction claims for infinite ordinals and some consistency statements—are seemingly not arithmetical in Isaacson’s sense, and hence that Isaacson’s thesis might entail the impurity of PA. Nonetheless, we conjecture that the advocate of Isaacson’s thesis can avoid this undesirable consequence: the arithmetical nature, as understood by Isaacson, of all contentious PA-provable statements can be justified. As a case study, we explore how this is done for transfinite induction claims with infinite ordinals below ε0. To this end, we show that the PA-proof of such claims employs exclusively resources from finite mathematics, and that ordinals below ε0 are finitary objects despite being infinite. (shrink)

Science has always strived for objectivity, for a ‘‘view from nowhere’’ that is not marred by ideology or personal preferences. That is a lofty ideal toward which perhaps it makes sense to strive, but it is hardly the reality. This collection of thirteen essays assembled by Denis R. Alexander and Ronald L. Numbers ought to give much pause to scientists and the public at large, though historians, sociologists and philosophers of science will hardly be surprised by the material covered (...) here. (shrink)

This paper is an attempt to bring together the convergent elements in J.L. Austin’s and Paul Ricoeur’s philosophies of language. Though a number of studies have already claimed that Ricoeur has in some ways been influenced by Austin, to date, not a single study has been made that exclusively focuses on the interrelatedness between Austin’s and Ricoeur’s philosophies of language. Thus, in this paper, I will start with a general exposition of the philosophical connection between Austin and Ricoeur. I (...) will then show how Austin and Ricoeur define and also understand the nature of words, the central component of every language. I will next explore the interplay of meaning and hermeneutics through a detailed discussion of the Speech Acts Theory (Austin) and the Hermeneutics of Symbols (Ricoeur). Afterward, I will argue that in Austin and Ricoeur, words, meaning, and hermeneutics constitute the art of making sense of things and of interpreting certain aspects and features of language. (shrink)

The telepresence experience can be evoked in a number of ways. A well-known example is a player of videogames who reports about a telepresence experience, a subjective experience of being in one place or environment, even when physically situated in another place. In this paper we set the phenomenon of telepresence into a theoretical framework. As people react subjectively to stimuli from telepresence, empirical studies can give more evidence about the phenomenon. Thus, our contribution is to bridge the theoretical (...) with the empirical. We discuss theories of perception with an emphasis on Heidegger, Merleau-Ponty and Gibson, the role of the senses and the Spinozian belief procedure. The aim is to contribute to our understanding of this phenomenon. A telepresence-study that included the affordance concept is used to empirically study how players report sense-reactions to virtual sightseeing in two cities. We investigate and explore the interplay of the philosophical and the empirical. The findings indicate that it is not only the visual sense that plays a role in this experience, but all senses. (shrink)

Albert Einstein once made the following remark about "the world of our sense experiences": "the fact that it is comprehensible is a miracle." (1936, p. 351) A few decades later, another physicist, Eugene Wigner, wondered about the unreasonable effectiveness of mathematics in the natural sciences, concluding his classic article thus: "the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve" (1960, p. (...) 14). At least three factors are involved in Einstein's and Wigner's miracles: the physical world, mathematics, and human cognition. One way to relate these factors is to ask how the universe could possibly be structured in such a way that mathematics would be applicable to it, and we would be able to understand that application. This is roughly Wigner's question. Alternatively, the way of the mathematical naturalist is to argue that we abstract certain properties from the world, perhaps using our bodies and physical tools, thereby articulating basic mathematical concepts, which we continue building into the complex formal structures of mathematics. John Stuart Mill, Penelope Maddy, and Rafael Nuñez teach this strategy of cognitive abstraction, in very different manners. But what if the very concepts and basic principles of mathematics were built into our cognitive structure itself? Given such a cognitive a priori mathematical endowment, would the miracles of the link between world and cognition (Einstein) and mathematics and world (Wigner) not vanish, or at least significantly diminish? This is the stance of Stanislas Deheane and Elizabeth Brannon's 2011 anthology, following a venerable rationalist tradition including Plato and Immanuel Kant. (shrink)

Mary Wollstonecraft argues that women must be independent citizens, but that they cannot be that unless they fulfill certain duties as mothers. This is problematic in a number of ways, as argued by Laura Brace in a 2000 article. However, I argue that if we understand Wollstonecraft's concept of independence in a republican, rather than a liberal context, and at the same time pay close attention to her discussion of motherhood, a feminist reading of Wollstonecraft is not only possible (...) but enriching. I will attempt to show, in particular, that the seeds of a feminist argument for co-parenting are to be found in the Vindication of the Rights of Woman. (shrink)

Is calculation possible without language? Or is the human ability for arithmetic dependent on the language faculty? To clarify the relation between language and arithmetic, we studied numerical cognition in speakers of Mundurukú, an Amazonian language with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their naming range. However, they fail in exact arithmetic with numbers larger than (...) 4 or 5. Our results imply a distinction between a nonverbal system of number approximation and a language-based counting system for exact number and arithmetic. (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)

The ability to solve problems is a prerequisite in preparing mathematics preservice teachers. This study assessed preservice teachers’ problem-solving difficulties and performance, particularly in worded problems on number sense, measurement, geometry, algebra, and probability. Also, academic profile differences in the preservice teacher’s problem-solving performance and common errors were determined. A descriptive-comparative research design was employed with 158 random respondents. Data were gathered face-to-face during the first semester of the school year 2022-2023, and data were analyzed with the aid (...) of jamovi software, ensuring ethical measures. Overall findings revealed that the preservice teachers experienced average difficulty in solving problems. The low performance of the preservice teachers on the given problems was also demonstrated. Further analysis revealed a significant difference between the preservice teachers’ problem-solving performance based on their subject preference and program. Moreover, the error analysis revealed that the preservice teachers incurred comprehension errors in misrepresentation, misinterpretation, and miscalculation. These results will serve as a measure for policymakers and curriculum developers of the teacher education institution concerned to make relevant enhancements to the math courses offered in the elementary and secondary education programs. (shrink)

The study determined the problem-solving performance and skills of prospective elementary teachers (PETs) in the Northern Philippines. Specifically, it defined the PETs’ level of problem-solving performance in number sense, measurement, geometry, algebra, and probability; significant predictors of their problem-solving performance in terms of sex, socio-economic status, parents’ educational attainment, high school graduated from and subject preference; and their problem-solving skills. The PETs’ problem-solving performance was determined by a problem set consisting of word problems with number sense, (...) measurement, geometry, algebra, and probability. A mixed-method research design was employed. Senior PETs purposively served as a sample where they mostly preferred to teach other subjects than mathematics. PETs who preferred math performed satisfactorily, while prospective teachers who opted for other subjects performed unsatisfactorily. The PETs’ unsatisfactory output indicates the need for remediation to advance the mathematical material skills and enrich the problem-solving abilities of these primary schools' potential teachers. Besides, results showed that subject preference strongly affected and predicted the problem-solving success of the PETs. PETs who preferred to teach mathematics performed significantly better than their counterparts; hence, mathematics as a field of specialization in the Bachelor of Elementary Education program may be considered by teacher education institutions. Further, most PETs displayed lack of problem-solving skills; thus, a Problem-Solving course is recommended for them. (shrink)

This inquiry seeks to understand how the original form of writing in Mesopotamia—the small pictures and conventions of protocuneiform—became cuneiform, a script that could not be read without acquiring the neurological and behavioral reorganizations understood today as literacy. The process is described as involving small neurological and behavioral changes realized, accumulated, and distributed to new users through interactions with and concomitant incremental changes in the material form of writing. A related inquiry focuses on why and how numerical notations differ from (...) other written signs. Crucially, numerical signs instantiate their meaning, a representational mode that contrasts with the signification used to represent non-numerical language and which makes numerical notations contiguous with their unwritten precursors, technologies like fingers, tallies, and counters. Instantiation is related to the perceptual system for quantity; this so-called number sense influences the function and form of numerical signs. Reading is then discussed as a cognitive activity that necessarily involves a material form, a plausible example of extended cognition. Because numerical notations share function and often form with precursor technologies, if the former participate in extended cognition, the latter likely do as well. In conjunction with the contiguity between numerical notations and their unwritten precursors, this complicates the idea that (all) writing is (just) language. Finally, potential follow-on research is suggested. (shrink)

Are numbers and other mathematical objects "out there" in some philosophically meaningful sense?