Arithmetic knowledge from the spontaneous attention to relations

Children's knowledge of arithmetic principles is a key aspect of early mathematics knowledge. Knowledge of arithmetic principles predicts how children approach solving arithmetic problems and the likelihood of their success. Prior work has begun to address how children might learn arithmetic principles in a classroom setting. Understanding of arithmetic principles involves understanding how numbers in arithmetic equations relate to another. For example, the Relation to Operands (RO) principle is that for subtracting natural numbers (A ? B = C), the difference (C) must be smaller than the minuend (A). In the current study we evaluate if individual differences in arithmetic principle knowledge (APK) can be predicted by the learners' spontaneous attention to relations (SAR) and if feedback can increase their attention to relations. Results suggest that participants’ Spontaneous Attention to Number (SAN) does not predict their knowledge of the RO principle for symbolic arithmetic. Feedback regarding the attention to relations did not show a significant effect on SAR or participants’ APK. We also did not find significant relations between reports of parent talk and the home environment with individual differences in SAN. The amount of parent's talk about relations was not significantly associated with learner's SAR and APK. We conclude that children's SAR with non‐symbolic number does not generalize to attention to relations with symbolic arithmetic.     

The Role of Structural Reasoning in the Genesis of Graph Theory

The concept of quantity (Größe) plays a key role in Frege's theory of real numbers. Typically enough, he refers to this theory as ‘theory of quantity’ (‘Größenlehre’) in the second volume of his opus magnum Grundgesetze der Arithmetik (Frege 1903). In this essay, I deal, in a critical way, with Frege's treatment of the concept of quantity and his approach to analysis from the beginning of his academic career until Frege 1903. I begin with a few introductory remarks. In Section 2, I first analyze Frege's use of the term ‘source of knowledge’ (‘Erkenntnisquelle’) with particular emphasis on the logical source of knowledge. The analysis includes a brief comparison between Frege and Kant's conceptions of logic and the logical source of knowledge. In a second step, I examine Frege's theory of quantity in Rechnungsmethoden, die sich auf eine Erweiterung des Größenbegriffes gründen (Frege 1874). Section 3 contains a couple of critical observations on Frege's comments on Hankel's theory of real numbers in Die Grundlagen der Arithmetik (Frege 1884). In Section 4, I consider Frege's discussion of the concept of quantity in Frege 1903. Section 5 is devoted to Cantor's theory of irrational numbers and the critique deployed by Frege. In Section 6, I return to Frege's own constructive treatment of analysis in Frege 1903 and succinctly describe what I take to be the quintessence of his account.     

Numerical predictors of arithmetic success in grades 1–6

Math relies on mastery and integration of a wide range of simpler numerical processes and concepts. Recent work has identified several numerical competencies that predict variation in math ability. We examined the unique relations between eight basic numerical skills and early arithmetic ability in a large sample (N = 1391) of children across grades 1–6. In grades 1–2, children's ability to judge the relative magnitude of numerical symbols was most predictive of early arithmetic skills. The unique contribution of children's ability to assess ordinality in numerical symbols steadily increased across grades, overtaking all other predictors by grade 6. We found no evidence that children's ability to judge the relative magnitude of approximate, nonsymbolic numbers was uniquely predictive of arithmetic ability at any grade. Overall, symbolic number processing was more predictive of arithmetic ability than nonsymbolic number processing, though the relative importance of symbolic number ability appears to shift from cardinal to ordinal processing.     

Reading Kant’s doctrine of schematism algebraically

Kant’s investigations into so-called a priori judgments of pure mathematics in the Critique of Pure Reason (KrV) are mainly confined to geometry and arithmetic both of which are grounded on our pure forms of intuition, space, and time. Nevertheless, as regards notions such as irrational numbers and continuous magnitudes, such a restricted account is crucially problematic. I argue that algebra can play a transcendental role with respect to the two pure intuitive sciences, arithmetic and geometry, as the condition of their possibility. It follows that Kant’s schematism of the concept of magnitude ought to be quantitatively represented in algebraic formulas in general, and also undergo several modifications in order to suit continuities.     

The Sacrifice of Silence: Fear and Trembling and the Secret of Faith

This article examines the important hermeneutical and theological relation between silence and sacrifice in Søren Kierkegaard's (1813–55) divisively enigmatic Fear and Trembling. I contend that this relation becomes clearest when the silence of Abraham is explicated in relation to his esoteric proclamation that ‘God himself will provide a lamb for the burnt offering’. In Abraham's reply to Isaac, the secret of Abraham's faith is concomitantly revealed (as a trust in the notion that ‘with God all things are possible’) and concealed (as a paradoxically ‘impossible’ possibility which cannot be adequately conveyed to ‘the other’). This thereby proposes a qualitative distinction between Abraham's reverent silence before God and his aporetic silence before the other.     

Buddhist functionalism—Instrumentality reaffirmed

This article seeks to determine if Buddhism can best be understood as primarily a functionalist tradition. In pursuing this, some analogies arise with various Western strands—particularly James’ ‘pragmatism’, Dewey's ‘instrumentalism’, Braithwaite's ‘empiricism’, Wittgenstein's ‘language games’, and process thinkers like Hartshorne and Jacobson. Within the Buddhist setting, the traditional Theravāda framework of sila (ethics/precepts), samādhi (meditation) and pañña (wisdom) are examined, together with Theravāda rituals. Despite some ‘correspondence’ approaches with regard to truth claim statements, e.g. vipassanā ’insight’ and Abhidharma analysis, a more profound functionalism seems present. This is even more clear with the Mahāyāna. Apart from the basic and explicit Mahāyāna underpinning of upāya, the Mādhyamika, Tantras and Ch'an (Zen) schools are clearly functionalist. Moreover, despite initially seeming more ‘absolutist’ in their positions, other strands like the Pure Land and Nichiren faith traditions, and Dharmakirti's Vijñānavāda epistemology can also be tied into this functionalist setting.     

Leon Chwistek,The Principles of the Pure Type Theory (1922), translated by Adam Trybus with an Introductory Note by Bernard Linsky

‘The Principles of the Pure Type Theory’ is a translation of Leon Chwistek's 1922 paper ‘Zasady czystej teorii typów’. It summarizes Chwistek's results from a series of studies of the logic of Whitehead and Russell's Principia Mathematica which were published between 1912 and 1924. Chwistek's main argument involves a criticism of the axiom of reducibility. Moreover, ‘The Principles of the Pure Type Theory’ is a source for Chwistek's views on an issue in Whitehead and Russell's ‘no-class theory of classes’ involving the notion of ‘scope’.     

Arithmetic,Mathematical Intuition,and Evidence

Abstract

This paper provides examples in arithmetic of the account of rational intuition and evidence developed in my book After Gödel: Platonism and Rationalism in Mathematics and Logic (Oxford: Oxford University Press, 2011). The paper supplements the book but can be read independently of it. It starts with some simple examples of problem-solving in arithmetic practice and proceeds to general phenomenological conditions that make such problem-solving possible. In proceeding from elementary ‘authentic’ parts of arithmetic to axiomatic formal arithmetic, the paper exhibits some elements of the genetic analysis of arithmetic knowledge that is called for in Husserl’s philosophy. This issues in an elaboration on a number of Gödel’s remarks about the meaning of his incompleteness theorems for the notion of evidence in mathematics.     

From Buddha's speech to Buddha's essence: philosophical discussions of Buddha‐vacana in India and China1

This is a comparative study of the discourses on the nature of sacred language found in Indian Abhidharma texts and those written by 7th century Chinese Buddhist scholars who, unlike the Indian Buddhists, questioned ‘the essence of the Buddha’s teaching'. This issue labeled fo‐chiao t'i lun, the theory of ‘the essence of the Buddha’s teaching', was one of the topics on which Chinese Yogācāra scholars have shown a keen interest and served as the inspiration for extensive intellectual dialogues in their texts. It is in Hsüan‐tsang's massive and organized translation works, begun in 648, that various previous translations of the term buddhavacana from Indian Abhidharma texts were given the unified translation of fo‐chiao. (Fo‐chiao literally means “the Buddha's teachings,” and is the term used in the modern period for “Buddhism.”) By combining fo‐chiao with the term t'i, meaning ‘essence’ or ‘substance’ throughout his translations, Hsüan‐tsang attempted to define ‘the essence of the Buddha’s teaching'. In Indian Abhidharma texts, the nature of the Buddha's word was either ‘sound’ (?abdha), the oral component of speech, or ‘name’ (nāma), the component of language that conveys meaning, or some combination of the two. From the time of Hsüan‐tsang's translation, however, discourse on the nature of sacred language was no longer relegated to the category of language or of epistemological investigation, but became grounded in the Chinese discussion investigating the ‘essence’ or ‘substance’ of the Buddha's teaching, and even of ‘Buddhism’ itself. As such, it sought to transcend the distinction between language and meaning. This gradual but explicit process of inquiry into the nature of ‘the Buddha’s word' was a necessary antecedent to the transition to a ‘Chinese’ Buddhism.     

The End(s) of Philosophy: Rhetoric, Therapy and Wittgenstein's Pyrrhonism

In Culture and Value Wittgenstein remarks: ‘Thoughts that are at peace. That's what someone who philosophizes yearns for’. The desire for such conceptual tranquillity is a recurrent theme in Wittgenstein's work, and especially in his later ‘grammatical‐therapeutic’ philosophy. Some commentators (notably Rush Rhees and C. G. Luckhardt) have cautioned that emphasising this facet of Wittgenstein's work ‘trivialises’ philosophy – something which is at odds with Wittgenstein's own philosophical ‘seriousness’ (in particular his insistence that philosophy demands that one ‘Go the bloody hard way’). Drawing on a number of correlations between Wittgenstein's conception of philosophy and that of the Pyrrhonian Sceptics, in this paper I defend a strong ‘therapeutic’ reading of Wittgenstein, and show how this can be maintained without ‘trivialising’ philosophy.     

Western ‘sincerity’ and Confucian ‘Cheng’

In philology, both ‘sincerity’ and ‘cheng’ primarily mean, ‘to be true to oneself’. As a philosophical term, ‘sincerity’ roots in Aristotle's ‘aletheutikos’. In medieval Europe, it is regarded as a neutral value that may either serve or disserve for ‘truth.’ As for Romantics, it is a positive value, and an individualistic concept whose two elements ‘true’ and ‘self’ refer to a person's ‘true feeling’ and ‘individuality’. In contrast, both ‘self’ and ‘true’ in Confucianism are universalistic concepts, meaning ‘good nature’ common to all humans, and ‘true feeling’ distinguishing them from beasts. Cheng itself means to face one's universal self with universal true feeling.     

Deconstructing the Linearity of Grace: The Risk and Reflexive Paradox of Mary's Immaculate Fiat

Abstract: Beginning from Pope Pius IX's doctrinal definition in Ineffabilis Deus, this article explores the circular paradox of the Virgin Mary's immaculate fiat. Fully contingent on Christ's work of reconciliation (and ‘immaculate’ by virtue of it), Mary's fiat paradoxically precedes that work and consents to it. The article suggests that this circularity is integral to the intimate bond that unites Mary's fiat to the Son's kenosis on the cross. Her fiat thus points the way of redeemed creation into the reflexivity of God's own intra‐trinitarian communication. Mary is hereby read as ‘the way to prayer’, the ‘epiphany’ of the Holy Spirit (as Alexander Schmemann names her) who cries ‘Abba, Father’ on behalf of those who do not know how to pray.     

Frege's Cardinals Do Not Always Obey Hume's Principle

Hume's Principle, dear to neo-Logicists, maintains that equinumerosity is both necessary and sufficient for sameness of cardinal number. All the same, Whitehead demonstrated in Principia Mathematica's logic of relations (where non-homogeneous relations are allowed) that Cantor's power-class theorem entails that Hume's Principle admits of exceptions. Of course, Hume's Principle concerns cardinals and in Principia's ‘no-classes’ theory cardinals are not objects in Frege's sense. But this paper shows that the result applies as well to the theory of cardinal numbers as objects set out in Frege's Grundgesetze. Though Frege did not realize it, Cantor's power-theorem entails that Frege's cardinals as objects do not always obey Hume's Principle.     

‘There is no brute world,only an elaborated world’: Merleau-Ponty on the intersubjective constitution of the world

In his later works, Merleau-Ponty proposes the notion of ‘the flesh’ (la chair) as a new ‘element’, as he put it, in his ontological monism designed to overcome the legacy of Cartesian dualism with its bifurcation of all things into matter or spirit. Most Merleau-Ponty commentators recognise that Merleau-Ponty's notion of ‘flesh’ is inspired by Edmund Husserl's conceptions of ‘lived body’ (Leib) and ‘vivacity’ or ‘liveliness’ (Leiblichkeit). But it is not always recognised that, for Merleau-Ponty, the constitution of the world of perception, the problem of embodiment or incarnation, is at the very same time one with the problem of the experience of others in what Husserl called Einfühlung or Fremderfahung and indeed one with the problem of the constitution of the commonly shared world ‘for all’. As Merleau-Ponty put it in his late essay ‘The Philosopher and His Shadow’ in Signs, ‘the problem of Einfühlung, like that of my incarnation, opens on the meditation of sensible being, or, if you prefer, it betakes itself there’. In other words, the problem of the apprehension of the other is part of the overall apprehension of the transcendent world. In this paper I want to meditate on the relations between embodiment, experience of others, and experience of the world in Merleau-Ponty's philosophy. I will take particular note, as in the title of this presentation, of the claim made by Merleau-Ponty in The Visible and the Invisible that ‘there is no brute world, only an elaborated world’ (il n'y a pas de monde brut, il n'y a qu'un monde élaboré).     

Journal Reviews

Field , Nathan (London). ‘The Primitive and the Pathological’. British Journal of Psychotherapy Fordham , Michael . (London) ‘Ending psychotherapy’ Zinkin , Louis . (London) ‘All's well that ends well, or is it?’Group Analysis Fosshage , J.L. (New York). ‘Towards reconceptualising transference: theoretical and clinical considerations‘. International Journal of Psycho-Analysis Hillman , James . (Thompson, CT, U.S.A.) ‘Once more into the fray (A response to Wolfgang Giegerich's “Killings”)’. Spring Josey , Alden . (Delaware, U.S.A.) Molecules as mandalas: modern chemistry and the quest for the Self.’Psychological Perspectives Phillips , Adam . ‘Contingency for Beginners’. Winnicott Studies Rigsby , Roberta . (U.S.A.).‘Jungians, archetypalists and fear of feminism’. Continuum Samuels , Andrew . (London). “The professionalization of Carl G. Jung's Analytical Psychology Clubs’. Journal of the History of the Behavioural Sciences     

Objectless activity: Marx's ‘theses on feuerbach’

According to Friedrich Engels (Ludwig Feuerbach and the end of classical German philosophy) the so‐called ‘Thesen über Feuerbach’ are ‘the brilliant germ of the new world conception’. For Karl Korsch ('Review of Vernon Venable’, Journal of Philosophy 42 [1945], no. 26) there are ‘magnificently summed up’ in them the ‘texts of Marx and Engels's first (Hegelian and post‐Hegelian) period’. Even given the important distinction between the ‘young’ and the ‘mature’ Marx these two opinions are not incompatible. The present paper's concern, however, is with the relationship of the ‘Thesen’ to the materialist conception of history. Once the ‘Thesen’ are read as a consistent whole it is clear that they are incompatible with any non‐social (non‐human) nature; hence with the ontological independence of nature from man; hence with any materialism, historical or otherwise. Furthermore, taken as a whole the ‘Thesen’ form an attempted solution to the problem of the justification of ideals, a solution both activist and dogmatist. Since the attitude expressed in the ‘Thesen’ underlies both Marx's ‘theory of alienation’ and his ‘critique of political economy’ neither of these can lay claim to the status of knowledge.     

Two cheers for Tanhā

The central character in Sartre's 1938 novel La Nausée, Antoine Roquentin, has lost his sense of things, and now the world appears to him as utterly unstable. Roquentin suffers from what he calls ‘nausea,’ a condition caused by an ontological intuition that the self, as well as the world through which that ‘self’ moves, lacks a substantial nature. The novel portrays Sartre's own philosophical account of the self in La transcendence de l'égo. Here Sartre argues that Husserl's account of consciousness is not radical enough; the ‘I’ or ego is a pseudo-source of activity (and Sartre thus draws very close to a particularly Buddhist account of personal identity). My essay questions Roquentin's response to his ontological insight: why is this the occasion for ‘nausea’? Why doesn't Roquentin (as King Milinda famously does) celebrate and embrace his ‘non-self’? I argue that Sartre's depiction of Roquentin's ailment, and the unsatisfactory solution he provides, misunderstands both the aggregate nature of things as well as authentically rendered consciousness-only (vijñaptimātra).