In this paper which consists of two parts (Teil I and Teil II) we champion Diophantus of Alexandria and Isabella BaÅ¡makova against Norbert Schappacher. In two publications ([Schappacher 1998a] and [Schappacher 1998b]) he puts forward inter alia two propositions: Questioning Diophantus’ originality he considers affirmatively the possibility that the Arithmetica are the joint work of a team of authors like Bourbaki. And he calls BaÅ¡makova’s claim (in [BaÅ¡makova 1972]) that Diophantus uses negative numbers, a nonsense , reproaching her for her (...) thoughtlessness . Teil I: First, we disprove Schappacher’s Bourbaki thesis. Second, we investigate the semantic meaning and historical significance of Diophantus’ keywords $ λvarepsilon tildeιψιζ and ὕπαρξιζ. Next, we discuss Schappacher’s epistemology of the history of mathematics and defend BaÅ¡makova’s methods. Finally we analyse in detail three problems from Diophantus’ Arithmetica (and their solutions) given by Thomas Heath and Helmuth Gericke as proof of the their claim that Diophantus did not use negative numbers. Teil II: In this Part, we give 33 places where Diophantus uses negative quantities as intermediate results; they appear as differences a − b of positive rational numbers, the subtrahend b being bigger than the minuend a; they each represent the (negative) basis ( πλvarepsilonυρacuteα ) of a square number ( τvarepsilonτρacuteαγω ν o ζ ), which is afterwards computed by the formula (a - b) 2 = a 2 + b 2 - 2ab $ . Finally, we report how the topic Diophantus and the negative numbers has been dealt with by translators and commentators from Maximus Planudes onwards. Und er kommt zu dem Ergebnis: ≪Nur ein Traum war das Erlebnis. Weil≫, so schlieβt er messerscharf, ≪nicht sein k a n n, was nicht sein d a r f.≫ CHRISTIAN MORGENSTERN: Palmstrm. (shrink)

People frequently gesture when problem-solving, particularly on tasks that require spatial transformation. Gesture often facilitates task performance by interacting with internal mental representations, but how this process works is not well understood. We investigated this question by exploring the case of mental abacus, a technique in which users not only imagine moving beads on an abacus to compute sums, but also produce movements in gestures that accompany the calculations. Because the content of MA is transparent and readily manipulated, the task (...) offers a unique window onto how gestures interface with mental representations. We find that the size and number of MA gestures reflect the length and difficulty of math problems. Also, by selectively interfering with aspects of gesture, we find that participants perform significantly worse on MA under motor interference, but that perceptual feedback is not critical for success on the task. We conclude that premotor processes involved in the planning of gestures are critical to mental representation in MA. (shrink)

Reasoning about ulterior motives was investigated among children ages 6–10 years (total N = 119). In each of two studies, participants were told about children who offered gifts to peers who needed help. Each giver chose to present a gift in either a public setting, which is consistent with having an ulterior motive to enhance one's reputation, or in a private setting, which is not consistent with having an ulterior motive. In each study, the 6- to 7-year olds showed no (...) evidence of understanding that the public givers might have ulterior motives, but the 8- to 10-year olds rated the private givers more favorably. In , the older children were more likely than the younger children to refer to impression management when explaining their judgments of the givers. The younger children who mentioned impression management did so to justify a preference for public givers (e.g., by explaining that public givers are nicer because more of their peers will know that they are nice). Results from suggest that developmental change in children's reasoning about intentions and social outcomes contributes to their understanding of ulterior motives. (shrink)

To show that discrimination against the terminally ill is a real and worrisome phenomenon Reed presents four examples 1. Here, I focus on the final two: right-to-try and right-to-die laws. I argue that they are not instances of discrimination, because they grant rights. Reed appears to have overlooked that rights differ from obligations in ways that leave his argumentation unsuccessful. According to the most prominent theory of rights, rights function to protect the personal interests of their holders. 2 For that (...) reason, strengthening rights implies weakening discrimination. Granting women the right to vote by itself meant that they were being less discriminated against. Rights also fail to meet standardly accepted definitions of discrimination. As Reed explains, for discrimination to occur, it is necessary that a significant disadvantage is imposed on a socially salient group. Policies can impose disadvantages by being obligations. While rights sometimes enable disadvantages,... (shrink)

In this important study D. M. Armstrong offers a comprehensive system of analytical metaphysics that synthesises but also develops his thinking over the last twenty years. Armstrong's analysis, which acknowledges the 'logical atomism' of Russell and Wittgenstein, makes facts the fundamental constituents of the world, examining properties, relations, numbers, classes, possibility and necessity, dispositions, causes and laws. All these, it is argued, find their place and can be understood inside a scheme of states of affairs. This is a comprehensive and (...) rigorously this-worldly account of the most general features of reality, argued from a distinctive philosophical perspective, and it will appeal to a wide readership in analytical philosophy. (shrink)

Though there are holes in the theory of how children move through stages of numerical competence, the current approach offers the most promising avenue for characterizing changes in competence as children confront new mathematical concepts. Like the science of mathematics, children's discovery of number is rooted in intuitions about sets, and not purely in analytic truths.

In this second part of “Negative Größen bei Diophant?” we start, as announced, by giving 33 places where Diophantus uses negative quantities as intermediate results; they appear as differences a − b of positive rational numbers, the subtrahend b being bigger than the minuend a; they each represent the (negative) basis $(\pi\lambda\varepsilon\upsilon\rho\acute{\alpha})$ of a square number $(\tau\varepsilon\tau\rho\acute{\alpha}\gamma\omega\nu o \zeta)$ , which is afterwards computed by the formula (a - b)2 = a 2 + b 2 - 2ab. Finally, we report (...) how the topic “Diophantus and the negative numbers” has been dealt with by translators and commentators from Maximus Planudes onwards. (shrink)

" -- Africa Today "The excellence of this book lies in the wealth of perspectives that it brings to the discussion on what constitutes philosophy, rationality, ...

Authors philosophical notes on life, conduct of life from the perspective of a life.

A comprehensive introduction to the ways in which meaning is conveyed in language. Alan Cruse covers semantic matters, but also deals with topics that are usually considered to fall under pragmatics. A major aim is to highlight the richness and subtlety of meaning phenomena, rather than to expound any particular theory. Rich in examples and exercises, Meaning in Language provides an invaluable descriptive approach to this area of linguistics for undergraduates and postgraduates alike.

This book: * assumes no mathematical background and keeps the technicalities to a minimum * explains the most important applications of probability theory to ...

A record of the words and teachings of Confucius, _The Analects_ is considered the most reliable expression of Confucian thought. However, the original meaning of Confucius's teachings have been filtered and interpreted by the commentaries of Confucianists of later ages, particularly the Neo-Confucianists of the Song dynasty, not altogether without distortion.In this monumental translation by Professor D. C. Lau, an attempt has been made to interpret the sayings as they stand. The corpus of the sayings is taken as an organic (...) whole and the final test of the interpretation rests on the internal consistency it exhibits. In other words, _The Analects_ is read in the light of _The Analects._This results in a truer understanding of Confucius' thought than the traditional interpretation and paves the way for a re-assessment of its importance in the history of Chinese thought and its relevance to the present day world.This volume also contains an introduction to the life and teachings of Confucius, and three appendices on the events in the life of Confucius, on his disciples, and on the composition of _The Analects._. (shrink)

By around the age of 5½, many children in the United States judge that numbers never end, and that it is always possible to add 1 to a set. These same children also generally perform well when asked to label the quantity of a set after one object is added (e.g., judging that a set labeled “five” should now be “six”). These findings suggest that children have implicit knowledge of the “successor function”: Every natural number, n, has a successor, n (...) + 1. Here, we explored how children discover this recursive function, and whether it might be related to discovering productive morphological rules that govern language‐specific counting routines (e.g., the rules in English that represent base‐10 structure). We tested 4‐ and 5‐year‐old children’s knowledge of counting with three tasks, which we then related to (a) children’s belief that 1 can always be added to any number (the successor function) and (b) their belief that numbers never end (infinity). Children who exhibited knowledge of a productive counting rule were significantly more likely to believe that numbers are infinite (i.e., there is no largest number), though such counting knowledge was not directly linked to knowledge of the successor function, per se. Also, our findings suggest that children as young as 4 years of age are able to implement rules defined over their verbal count list to generate number words beyond their spontaneous counting range, an insight which may support reasoning over their acquired verbal count sequence to infer that numbers never end. (shrink)

Do children understand how different numbers are related before they associate them with specific cardinalities? We explored how children rely on two abstract relations – contrast and entailment – to reason about the meanings of ‘unknown’ number words. Previous studies argue that, because children give variable amounts when asked to give an unknown number, all unknown numbers begin with an existential meaning akin to some. In Experiment 1, we tested an alternative hypothesis, that because numbers belong to a scale of (...) contrasting alternatives, children assign them a meaning distinct from some. In the “Don’t Give-a-Number task”, children were shown three kinds of fruit (apples, bananas, strawberries), and asked to not give either some or a number of one kind (e.g. Give everything, but not [some/five] bananas). While children tended to give zero bananas when asked to not give some, they gave positive amounts when asked to not give numbers. This suggests that contrast – plus knowledge of a number’s membership in a count list – enables children to differentiate the meanings of unknown number words from the meaning of some. Experiment 2 tested whether children’s interpretation of unknown numbers is further constrained by understanding numerical entailment relations – that if someone, e.g. has three, they thereby also have two, but if they do not have three, they also do not have four. On critical trials, children saw two characters with different quantities of fish, two apart (e.g. 2 vs. 4), and were asked about the number inbetween – who either has or doesn’t have, e.g. three. Children picked the larger quantity for the affirmative, and the smaller for the negative prompts even when all the numbers were unknown, suggesting that they understood that, whatever three means, a larger quantity is more likely to contain that many, and a smaller quantity is more likely not to. We conclude by discussing how contrast and entailment could help children scaffold the exact meanings of unknown number words. (shrink)

Word learning depends critically on the use of linguistic context to constrain the likely meanings of words. However, the mechanisms by which children infer word meaning from linguistic context are still poorly understood. In this study, we asked whether adults (n = 58) and 2‐ to 6‐year‐old children (n = 180) use discourse coherence relations (i.e., the meaningful relationships between elements within a discourse) to constrain their interpretation of novel words. Specifically, we showed participants videos of novel animals exchanging objects. (...) These videos were accompanied by a linguistic description of the events in which we manipulated a single word within a sentence (and vs. because) in order to alter the causal and temporal relations between the events in the discourse (e.g., “One animal handed the baby to the other animal [and/because] the baby started crying in the talfa's arms”). We then asked participants which animal (the giver or the receiver) was the referent of the novel word. Across two experiments, we found evidence that young children used the causal and temporal relations in each discourse to constrain their interpretations of novel words. (shrink)

In this entry we provide a critical review of recent work on the nature and ethics of blame, including issues of moral standing.

In identifying that the 'essential tension' is the balance between conservative and innovative approaches in the development of knowledge - tried-and tested or new directions - Kuhn pointed out that these two attitudes are both appropriate. This study adds to this picture the social and psychological dynamics that underpin any such balancing.