A notion of feasible function of finite type based on the typed lambda calculus is introduced which generalizes the familiar type 1 polynomial-time functions. An intuitionistic theory IPVω is presented for reasoning about these functions. Interpretations for IPVω are developed both in the style of Kreisel's modified realizability and Gödel's Dialectica interpretation. Applications include alternative proofs for Buss's results concerning the classical first-order system S12 and its intuitionistic counterpart IS12 as well as proofs of some of Buss's conjectures concerning IS12, (...) and a proof that IS12 cannot prove that extended Frege systems are not polynomially bounded. (shrink)

Presents wide-ranging and up-to-date interpretations of the Zhuangzi, the Daoist classic and one of the most elusive works ever written.

A beneficial effect of gesture on learning has been demonstrated in multiple domains, including mathematics, science, and foreign language vocabulary. However, because gesture is known to co‐vary with other non‐verbal behaviors, including eye gaze and prosody along with face, lip, and body movements, it is possible the beneficial effect of gesture is instead attributable to these other behaviors. We used a computer‐generated animated pedagogical agent to control both verbal and non‐verbal behavior. Children viewed lessons on mathematical equivalence in which an (...) avatar either gestured or did not gesture, while eye gaze, head position, and lip movements remained identical across gesture conditions. Children who observed the gesturing avatar learned more, and they solved problems more quickly. Moreover, those children who learned were more likely to transfer and generalize their knowledge. These findings provide converging evidence that gesture facilitates math learning, and they reveal the potential for using technology to study non‐verbal behavior in controlled experiments. (shrink)

This study renders the complex corpus of the Guodian texts into a more easily manageable form, incorporating the past several years of scholarly activity on these texts and providing them with a comprehensive introduction along with a complete and well-annotated translation into English.

A beneficial effect of gesture on learning has been demonstrated in multiple domains, including mathematics, science, and foreign language vocabulary. However, because gesture is known to co-vary with other non-verbal behaviors, including eye gaze and prosody along with face, lip, and body movements, it is possible the beneficial effect of gesture is instead attributable to these other behaviors. We used a computer-generated animated pedagogical agent to control both verbal and non-verbal behavior. Children viewed lessons on mathematical equivalence in which an (...) avatar either gestured or did not gesture, while eye gaze, head position, and lip movements remained identical across gesture conditions. Children who observed the gesturing avatar learned more, and they solved problems more quickly. Moreover, those children who learned were more likely to transfer and generalize their knowledge. These findings provide converging evidence that gesture facilitates math learning, and they reveal the potential for using technology to study non-verbal behavior in controlled experiments. (shrink)

Zhuang Zi's relation to the Confucian school is reexamined. It is argued that although Zhuang Zi was fond of highlighting the absurdities of the Confucian enterprise, we can nonetheless detect in his writings a great admiration for much of what constituted the central core of the Confucian vision. This essay analyzes Confucius' image of "musical perfection," representing the total concordance of ritual restraints and harmonious freedom; traces the Confucian notion of self-cultivation through Mencius' passage on the "full-flowing energy"; and concludes (...) with an examination of Zhuang Zi's "Butcher Ding" story, showing that while Zhuang Zi's whole concept of self-nurturing is approached from a diametrically opposite angle, he nonetheless ends up celebrating a state of mastery and freedom in many ways identical to Confucius' notion of musical perfection. (shrink)

We prove the following results: (i) PV proves NP ⊆ P/poly iff PV proves coNP ⊆ NP/O(1). (ii) If PV proves NP ⊆ P/poly then PV proves that the Polynomial Hierarchy collapses to the Boolean Hierarchy. (iii) $S_{2}^{1}$ proves NP ⊆ P/poly iff $S_{2}^{1}$ proves coNP ⊆ NP/O(log n). (iv) If $S_{2}^{1}$ proves NP ⊆ P/poly then $S_{2}^{1}$ proves that the Polynomial Hierarchy collapses to PNP[log n]. (v) If $S_{2}^{2}$ proves NP ⊆ P/poly then $S_{2}^{2}$ proves that the Polynomial Hierarchy (...) collapses to PNP. Motivated by these results we introduce a new concept in proof complexity: proof systems with advice, and we make some initial observations about them. (shrink)

Let H be a proof system for quantified propositional calculus (QPC). We define the Σqj-witnessing problem for H to be: given a prenex Σqj-formula A, an H-proof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the Σq1-witnessing problems for the systems G*1and G1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce and study the systems G*0 and G0, (...) in which cuts are restricted to quantifier-free formulas, and prove that the Σqj-witnessing problem for each is complete for NC1. Our proof involves proving a polynomial time version of Gentzen’s midsequent theorem for G*0 and proving that G0-proofs are TC0-recognizable. We also introduce QPC systems for TC0 and prove witnessing theorems for them. We introduce a finitely axiomatizable second-order system VNC1 of bounded arithmetic which we prove isomorphic to Arai’s first order theory AID + Σb0-CA for uniform NC1. We describe simple translations of VNC1 proofs of all bounded theorems to polynomial size families of G*0 proofs. From this and the above theorem we get alternative proofs of the NC1 witnessing theorems for VNC1 and AID. (shrink)

We introduce three formal theories of increasing strength for linear algebra in order to study the complexity of the concepts needed to prove the basic theorems of the subject. We give what is apparently the first feasible proofs of the Cayley–Hamilton theorem and other properties of the determinant, and study the propositional proof complexity of matrix identities such as AB=I→BA=I.

Although children learn more when teachers gesture, it is not clear how gesture supports learning. Here, we sought to investigate the nature of the memory processes that underlie the observed benefits of gesture on lasting learning. We hypothesized that instruction with gesture might create memory representations that are particularly resistant to interference. We investigated this possibility in a classroom study with 402 second‐ and third‐grade children. Participants received classroom‐level instruction in mathematical equivalence using videos with or without accompanying gesture. After (...) instruction, children solved problems that were either visually similar to the problems that were taught, and consistent with an operational interpretation of the equal sign (interference), or visually distinct from equivalence problems and without an equal sign (control) in order to assess the role of gesture in resisting interference after learning. Gesture facilitated learning, but the effects of gesture and interference varied depending on type of problem being solved and the strategies that children used to solve problems prior to instruction. Some children benefitted from gesture, while others did not. These findings have implications for understanding the mechanisms underlying the beneficial effect of gesture on mathematical learning, revealing that gesture does not work via a general mechanism like enhancing attention or engagement that would apply to children with all forms of prior knowledge. (shrink)

Today the economic diagram is employed universally in teaching and research by professional economists. Yet the history of its construction shows that much that has been regarded as distinctive of twentieth-century visual culture was prefigured in the nineteenth. This paper will place the construction of the first economic diagrams by Alfred Marshall in the context both of contemporary visual technologies developed in other moral sciences, and of his wider theory of industrial production. The paper will argue that an understanding of (...) the background to the historical graphs, constructed by the Cambridge political economist from the late 1860s onwards, will necessitate a revision of the narrative of modern visual reasoning, as recently advanced by Lev Manovich. (shrink)

We introduce a second-order system V1-Horn of bounded arithmetic formalizing polynomial-time reasoning, based on Grädel's 35) second-order Horn characterization of P. Our system has comprehension over P predicates , and only finitely many function symbols. Other systems of polynomial-time reasoning either allow induction on NP predicates , and hence are more powerful than our system , or use Cobham's theorem to introduce function symbols for all polynomial-time functions . We prove that our system is equivalent to QPV and Zambella's P-def. (...) Using our techniques, we also show that V1-Horn is finitely axiomatizable, and, as a corollary, that the class of Σ1b consequences of S21 is finitely axiomatizable as well, thus answering an open question. (shrink)

Traditional sources for understanding of the thought of Confucius, such as the Lunyu 論語 (Analects) and Li ji 禮記 (Book of Ritual), are fraught with uncertainty in terms of dating and reliability. Recently excavated manuscripts from Warring States China, however, have begun to shed new light on the development of Confucian thought and the shaping of Confucius as a narrative figure during the two centuries following his death. This paper surveys and briefly assesses the significance of the relevant texts from (...) two major unearthed corpuses: the Guodian Bamboo Manuscripts of Chu 郭店楚墓竹簡 and the Shanghai Museum Manuscripts of Chu 上海博物館藏戰國楚竹書. These include: “Ziyi,” “Zun deyi,” “Min zhi fumu,” “Zhonggong,” “Zigao,” “Lu bang da han,” “Xiang bang zhi dao,” “Yan Yuan wen yu Kong Zi,” “Junzi wei li,” “Dizi wen,” “Kongzi jian Ji Huanzi,” “Ji Kangzi wen yu Kong Zi,” “Kong Zi shilun,” and “Cong zheng.”. (shrink)

Aside from the Yue Ji 樂記, the “Xiu Wen 修文” chapter of the Shuoyuan 說苑 is perhaps our richest source of pre-Qin and early Han 漢 Confucian musical thought. Though woven together from earlier sources, “Xiu Wen” nonetheless manages to present its own distinctive expression of the role of music and its relationship to the greater system of ritual institutions. This article undertakes an examination of the chapter’s philosophies of ritual and music, focusing especially on the latter, and in the (...) process attempts to demonstrate what is unique about the text in relation to its forerunners in the realm of musical thought. (shrink)

In the 1860s and 1870s the logic of Boole and the calculating machines of Babbage were key resources in W. S. Jevons’s attempt to construct a mechanical model of the mind, and both therefore played an important role in Jevons’s attempted revolution in economic theory. In this same period both Boole and Babbage were studied within the Cambridge Moral Sciences Tripos, but the Cambridge reading of Boole and Babbage was much more circumspect. Implicitly following the division of the moral sciences (...) into material and ‘real’ as established by the Rev. Grote, John Venn treated Boole’s logic as a purely formal science, while Alfred Marshall based his psychological model of the mechanical part of the human mind upon Babbage’s two-level machine. From the different perspectives of logic and psychology, Venn and Marshall did not simply incorporate their readings of Boole and Babbage, but also attempted to establish the limits to any mechanical explanation of the mind. This comparison of the attitudes to mental science of Jevons and Marshall provides a foundation from which the differing conceptions of economic theory of the two men can be established.Keywords: Alfred Marshall; Charles Babbage; George Boole; John Venn; Logic; Psychology. (shrink)

Joseph C. Pitt, based on his understanding of trust and of technology, makes the provocative argument that trusting technology is actually a matter of trusting people. I agree with Pitt’s conclusion but differ with him on the nature of trust. I contend, nonetheless, that my understanding of trust actually reinforces Pitt’s characterization of technology as “humanity at work.”.

SummaryFrom the early 1880s the Cambridge-trained classicist William Ridgeway had applied cutting-edge anthropological theory to his reading of ancient Greek literature in order to develop an evolutionary account of the continuous development of early Greek social institutions. Then, at the turn of the century, he began to argue that archaeological evidence demonstrated that the Achaean warriors described by Homer were in origin Germanic tribesmen from north of the Alps who had but recently conquered Mycenaean Greece. The present paper inquires as (...) to how Ridgeway reconciled these seemingly opposed visions of early Greek society. A fairly comprehensive survey of his writings leads to the suggestion that, in Ridgeway's opinion, Achaean invasion had left little lasting impact upon most early Greek social institutions, but that it had been responsible for a fundamental shift from matriarchy to patriarchy, and that this shift was the key to the subsequent greatness of Greek—and so ultimately Western—civilisation. (shrink)

The debate over what constituted the most effective means of political control—governance through moral suasion or through coercive measures—was one that came to define the main battle lines between the Confucians and their “Legalist” rivals over the course of Warring States period China. While the importance of this debate is by no means new, recently unearthed Warring States manuscripts have done much to help shed new light upon the emergence of this debate: in particular the Chu-region bamboo manuscripts of Guodian (...) 郭店楚墓竹簡, which reveal the extent to which this issue was already at the forefront of Confucian thought by the mid-to-late fourth century bce. This paper discusses how these manuscript texts go about making the argument against coercive order and for rulership through ritual and musical education, which they do, in short, by staking the claim that the latter derive ultimately from human nature itself and, because of this, are the only true and natural means of successfully guiding and controlling it. (shrink)

Since the middle of the 20th century there has been a significant debate about the attribution of capacities of living systems, particularly humans, to technological artefacts, especially computers—from Turing’s opening gambit, to subsequent considerations of artificial intelligence, to recent claims about artificial life. Some now argue that the capacities of future technologies will ultimately make it impossible to draw any meaningful distinctions between humans and machines. Such issues center on what sense, if any, it makes to claim that gadgets can (...) actually think, feel, act, live, etc. I outline this debate and offer a critique of its persistent polarization. I characterize two of the debate’s major camps (associated roughly with Turing and Searle); argue that the debate’s structure (including key assumptions inherent to each camp) precludes resolution; and, contend that some central clashes within the debate actually stem from an inadequately drawn distinction between claims about the capacities of artifacts and claims about the proper criteria for assessing such attributions. I offer a different perspective in which I: challenge some central elements of the debate that contribute to its perennially irresolvable state; hold that the debate needs to be placed more squarely in sync with how we in fact treat the attribution of such capacities to humans themselves; and, offer (unlike the other two camps) a foothold for making moral assessments of such proposed technologies. (shrink)

Since the middle of the 20th century there has been a significant debate about the attribution of capacities of living systems, particularly humans, to technological artefacts, especially computers—from Turing’s opening gambit, to subsequent considerations of artificial intelligence, to recent claims about artificial life. Some now argue that the capacities of future technologies will ultimately make it impossible to draw any meaningful distinctions between humans and machines. Such issues center on what sense, if any, it makes to claim that gadgets can (...) actually think, feel, act, live, etc. I outline this debate and offer a critique of its persistent polarization. I characterize two of the debate’s major camps (associated roughly with Turing and Searle); argue that the debate’s structure (including key assumptions inherent to each camp) precludes resolution; and, contend that some central clashes within the debate actually stem from an inadequately drawn distinction between claims about the capacities of artifacts and claims about the proper criteria for assessing such attributions. I offer a different perspective in which I: challenge some central elements of the debate that contribute to its perennially irresolvable state; hold that the debate needs to be placed more squarely in sync with how we in fact treat the attribution of such capacities to humans themselves; and, offer (unlike the other two camps) a foothold for making moral assessments of such proposed technologies. (shrink)

SUMMARYThe essay identifies and explores the intellectual formation of a hitherto overlooked constellation of ‘anthropologists’ in Edwardian Cambridge. Three core members of this group were William Ridgeway, Hector Munro Chadwick, and William H. R. Rivers, who today are more normally associated with Classics, Anglo-Saxon studies, and Anthropology. However, in the decade before World War I all three were active members of the new Board of Anthropology, and each, in his particular field of study, began to turn away from established evolutionary (...) explanation to investigate social phenomena as arising out of the contact of different peoples. The essay first shows the connections between the work of Chadwick and Rivers, and then suggests that both were following a path recently beaten down by Ridgeway who, in his disputes with the so-called Cambridge Ritualists, advanced an account of ancient Greek tragedy as arising out of a fusion of native and intrusive performances, both relating to the commemoration of the dead. (shrink)