Transreal limits expose category errors in IEEE 754 floating-point arithmetic and in mathematics

The IEEE 754 standard for oating-point arithmetic is widely used in computing. It is based on real arithmetic and is made total by adding both a positive and a negative infinity, a negative zero, and many Not-a-Number (NaN) states. The IEEE infinities are said to have the behaviour of limits. Transreal arithmetic is total. It also has a positive and a negative infinity but no negative zero, and it has a single, unordered number, nullity. We elucidate the transreal tangent and extend real limits to transreal limits. Arguing from this firm foundation, we maintain that there are three category errors in the IEEE 754 standard. Firstly the claim that IEEE infinities are limits of real arithmetic confuses limiting processes with arithmetic. Secondly a defence of IEEE negative zero confuses the limit of a function with the value of a function. Thirdly the definition of IEEE NaNs confuses undefined with unordered. Furthermore we prove that the tangent function, with the infinities given by geometrical con- struction, has a period of an entire rotation, not half a rotation as is commonly understood. This illustrates a category error, confusing the limit with the value of a function, in an important area of applied mathe- matics { trigonometry. We brie y consider the wider implications of this category error. Another paper proposes transreal arithmetic as a basis for floating- point arithmetic; here we take the profound step of proposing transreal arithmetic as a replacement for real arithmetic to remove the possibility of certain category errors in mathematics. Thus we propose both theo- retical and practical advantages of transmathematics. In particular we argue that implementing transreal analysis in trans- floating-point arith- metic would extend the coverage, accuracy and reliability of almost all computer programs that exploit real analysis { essentially all programs in science and engineering and many in finance, medicine and other socially beneficial applications.

National interest to global reform: patterns of reasoning in British foreign policy discourse

Discussion of the national interest often focuses on how Britain's influence can be maximized, rather than on the goals that influence serves. Yet what gives content to claims about the national interest is the means-ends reasoning which links interests to deeper goals. In ideal-typical terms, this can take two forms. The first, and more common, approach is conservative: it infers national interests and the goals they advance from existing policies and commitments. The second is reformist: it starts by specifying national goals and then asks how they are best advanced under particular conditions. New Labour's foreign policy discourse is notable for its explicit use of a reformist approach. Indeed, Gordon Brown's vision of a 'new global society' not only identifies global reform as a key means of fulfilling national goals, but also thereby extends the concept of the national interest well beyond a narrow concern with national security.

Challenging climate change and migration discourse: different understandings of time-scale and temporality in the Maldives

This article draws on ongoing research in the Maldives to explore differences between elite and non-elite perceptions of climate change and migration. It argues that, in addition to variations in perceptions based on diverse knowledge, priorities and agendas, there exists a more fundamental divergence based upon different understandings of the time-scale of climate change and related ideas of urgency and crisis. Specifically, elites tend to focus on a distant future which is generally abstracted from people’s everyday lived realities, as well as utilise the language of a climate change-induced migration ‘crisis’ in their discussions about impacts in a manner not envisaged by non-elites. The article concludes that, rather than unproblematically mapping global, external facing narratives wholesale onto ordinary people’s lives and experiences, there needs to be more dialogue between elites and non-elites on climate change and migration issues. These perspectives should be integrated more effectively in the development of policy interventions designed to help people adapt to the impacts of global environmental change.

Half-remembered quotations from mostly forgotten speeches: the limits of Labour’s European policy discourse

This article uses discourse analysis to study the continuities in British foreign policy thinking within the Labour party from the 1960s to the present day. Using representative extracts from speeches by Hugh Gaitskell, Harold Wilson, Tony Blair and Gordon Brown, it identifies the ideational consis- tencies in the leaders’ attitudes to: Empire; federalism in the EEC/EU; and laying down conditions that have to be met before any constructive engagement with ‘Europe’ can be countenanced. We argue that these consistencies, spanning a 50-year period, exemplify a certain stagnation both within Labour’s European discourses and within British foreign policy thinking more widely. We develop the idea that Labour party thinking has been crucially framed by both small ‘c’ conser- vative and upper-case Conservative ideology, popularised by Winston Churchill in his ‘three circles’ model of British foreign policy.

Teaching and learning Chinese: heritage language classroom discourse in Montreal

This paper explores issues of teaching and learning Chinese as a heritage language in a Chinese heritage language school, the Zhonguo Saturday School, in Montreal, Quebec. With a student population of more than 1000, this school is the largest of the eight Chinese Heritage Language schools in Montreal. Students participating in this study were from seven different classes (grade K, two, three, four, five, six, and special class), their ages ranging from 4 to 13 years. The study took place over a period of two years between 2000 and 2002. Focusing on primary level classroom discourse and drawing on the works of Vygotsky and Bakhtin, I examine how teachers and students use language to communicate, and how their communication mediates teaching, learning and heritage language acquisition. Data sources include classroom observations, interviews with students and their teachers, students’ writings, and video and audio taping of classroom activities. Implications for heritage language development and maintenance are discussed with reference to the findings of this study.

A comparative corpus-assisted discourse study of the representations of hosts in promotional tourism discourse

Most studies concerned with the representations of local people in tourism discourse point to the prevalence of stereotypic images asserting that contemporary tourism perpetuates colonial legacy and gendered discursive practices. This claim has been, to some extent, contested in research that explores representations of hosts in local tourism materials claiming that tourism can also discursively resist the dominant Western imagery. While the evidence for the existence of hegemonic and diverging discourses about the local ‘Other’ seems compelling, the empirical basis of this research is rather small and often limited to one geographic context. The present study addresses these shortcomings by examining representations of hosts in a larger corpus of promotional tourism materials including texts produced by Western and local tourism industries. The data is investigated using the methodology of Corpus-Assisted Discourse Studies (CADS). By comparing external with internal (self) representations, this study verifies and refines some of the claims on the subject and offers a much more nuanced picture of representations that defies the black and white scenarios proposed in previous research

Discourse-sensitive clitic-doubled dislocations in heritage Spanish

This experimental study tests the predictions of the Interface Hypothesis (Sorace, 2011, 2012) using two constructions whose appropriateness depends on monitoring discourse information: Clitic Left Dislocation and Fronted Focus. Clitic Left Dislocation relates a dislocated and clitic-doubled object to an antecedent activated in previous discourse, while Fronted Focus does not relate the fronted constituent to a discourse antecedent. The Interface Hypothesis argues that speakers in language contact situations experience difficulties when they have to integrate syntactic with discourse information. We tested four groups of native speakers on these constructions: Spanish monolinguals, bilinguals with more than 7 years residence in the US, intermediate and advanced proficiency heritage speakers. Our findings suggest that attrition has not set in the adult L2 bilingual speakers, and that the heritage speakers perform similarly to the monolingual and the adult sequential bilingual natives.

A rare structure at the syntax-discourse interface: heritage and Spanish-dominant native speakers weigh in

The present study examines knowledge of the discourse-appropriateness of Clitic Right Dislocation (CLRD) in a population of Heritage (HS) and Spanish-dominant Native Speakers in order to test the predictions of the Interface Hypothesis (IH; Sorace 2011). The IH predicts that speakers in language contact situations will experience difficulties with integrating information involving the interface of syntax and discourse modules. CLRD relates a dislocated constituent to a discourse antecedent, requiring integration of syntax and pragmatics. Results from an acceptability judgment task did not support the predictions of the IH. No statistical differences between the HSs’ performance and that of L1-dominant native speakers were evidenced when participants were presented with an offline task. Thus, our study did not find any evidence of “incomplete acquisition” (Montrul 2008) as it pertains to this specific linguistic structure.

Pragmatic solutions for syntactic problems: Understanding some L2 syntactic errors in terms of discourse-pragmatic deficits

Contemporary research in generative second language (L2) acquisition has attempted to address observable target-deviant aspects of L2 grammars within a UG-continuity framework (e.g. Lardiere 2000; Schwartz 2003; Sprouse 2004; Prévost & White 1999, 2000). With the aforementioned in mind, the independence of pragmatic and syntactic development, independently observed elsewhere (e.g. Grodzinsky & Reinhart 1993; Lust et al. 1986; Pacheco & Flynn 2005; Serratrice, Sorace & Paoli 2004), becomes particularly interesting. In what follows, I examine the resetting of the Null-Subject Parameter (NSP) for English learners of L2 Spanish. I argue that insensitivity to associated discoursepragmatic constraints on the discursive distribution of overt/null subjects accounts for what appear to be particular errors as a result of syntactic deficits. It is demonstrated that despite target-deviant performance, the majority must have native-like syntactic competence given their knowledge of the Overt Pronoun Constraint (Montalbetti 1984), a principle associated with the Spanish-type setting of the NSP.