Conjecture, proof, and sense in Wittgenstein's philosophy of mathematics

One of the key tenets in Wittgenstein’s philosophy of mathematics is that a mathematical proposition gets its meaning from its proof. This seems to have the paradoxical consequence that a mathematical conjecture has no meaning, or at least not the same meaning that it will have once a proof has been found. Hence, it would appear that a conjecture can never be proven true: for what is proven true must ipso facto be a different proposition from what was only conjectured. Moreover, it would appear impossible that the same mathematical proposition be proven in different ways. — I will consider some of Wittgenstein’s remarks on these issues, and attempt to reconstruct his position in a way that makes it appear less paradoxical.

The Spanish Ambassador and Samuel Daniel's Vision of the Twelve Goddesses: A New Document [with text]

The performance of Samuel Daniel's masque The Vision of the Twelve Goddesses at court on January 8, 1604 took place in the midst of the preliminary negotiations that would lead to the signing of the Anglo-Spanish peace at Somerset House the following August. Philip III sent a special ambassador to England to congratulate James on his accession, and a series of tussles between Juan de Tassis and his French counterpart ensued. As a recently-discovered document in the Archivo General de Simancas reveals, Anna of Denmark intervened personally to insure that de Tassis, and not the Frenchman, attended the masque. This was a clear signal of James and Anna's peace aims, which de Tassis conveyed to the King of Spain; moreover, he enclosed in his dispatch a text of Daniel's masque which he clearly considered both political intelligence and of interest to the theater-loving Hapsburg monarch. The Simancas text of the Daniel masque is a new version, hitherto unknown, which adds to our knowledge of the circumstances in which the first Stuart masque was performed. Here we present a transcription and annotated translation of both de Tassis' letter and the text of the masque he had compiled for Philip III. (B. C.-E. and M. H.)

Military doctrine, command philosophy and the generation of fighting power: genesis and theory

Military doctrine is one of the conceptual components of war. Its raison d’être is that of a force multiplier. It enables a smaller force to take on and defeat a larger force in battle. This article’s departure point is the aphorism of Sir Julian Corbett, who described doctrine as ‘the soul of warfare’. The second dimension to creating a force multiplier effect is forging doctrine with an appropriate command philosophy. The challenge for commanders is how, in unique circumstances, to formulate, disseminate and apply an appropriate doctrine and combine it with a relevant command philosophy. This can only be achieved by policy-makers and senior commanders successfully answering the Clausewitzian question: what kind of conflict are they involved in? Once an answer has been provided, a synthesis of these two factors can be developed and applied. Doctrine has implications for all three levels of war. Tactically, doctrine does two things: first, it helps to create a tempo of operations; second, it develops a transitory quality that will produce operational effect, and ultimately facilitate the pursuit of strategic objectives. Its function is to provide both training and instruction. At the operational level instruction and understanding are critical functions. Third, at the strategic level it provides understanding and direction. Using John Gooch’s six components of doctrine, it will be argued that there is a lacunae in the theory of doctrine as these components can manifest themselves in very different ways at the three levels of war. They can in turn affect the transitory quality of tactical operations. Doctrine is pivotal to success in war. Without doctrine and the appropriate command philosophy military operations cannot be successfully concluded against an active and determined foe.

Subject-to-subject raising and the syntax of tense in L2 Spanish: a full access approach

This paper investigates the acquisition of syntax in L2 grammars. We tested adult L2 speakers of Spanish (English L1) on the feature specification of T(ense), which is different in English and Spanish in so-called subject-to-subject raising structures. We present experimental results with the verb parecer “to seem/to appear” in different tenses, with and without experiencers, and with Tense Phrase (TP), verb phrase (vP) and Adjectival Phrase (AP) complements. The results show that advanced L2 learners can perform just like native Spanish speakers regarding grammatical knowledge in this domain, although the subtle differences between both languages are not explicitly taught. We argue that these results support Full Access approaches to Universal Grammar (UG) in L2 acquisition, by providing evidence that uninterpretable syntactic features can be learned in adult L2, even when such features are not directly instantiated in the same grammatical domain in the L1 grammar.

Clitic-doubled left dislocation and focus fronting in L2 Spanish: a case of successful acquisition at the syntax-discourse interface

This experimental study tests the Interface Hypothesis by looking into processes at the syntax– discourse interface, teasing apart acquisition of syntactic, semantic and discourse knowledge. Adopting López’s (2009) pragmatic features [±a(naphor)] and [±c(ontrast)], which in combination account for the constructions of dislocation and fronting, we tested clitic left dislocation and fronted focus in the comprehension of English native speakers learning Spanish. Furthermore, we tested knowledge of an additional semantic property: the relationship between the discourse anaphor and the antecedent in clitic left dislocation (CLLD). This relationship is free: it can be subset, superset, part/whole. Syntactic knowledge of clitics was a condition for inclusion in the main test. Our findings indicate that all learners are sensitive to the semantic constraints. While the near-native speakers display native-like discourse knowledge, the advanced speakers demonstrated some discourse knowledge, and intermediate learners did not display any discourse knowledge. The findings support as well as challenge the Interface Hypothesis.

The syntax-semantics of bare and definite plural subjects in the L2 Spanish of English natives

This study investigates the extent to which advanced native-English L2 learners of Spanish come to acquire restrictions on bare plural preverbal subjects in L2 Spanish (e.g. gatos “cats” vs. definite plurals such as los gatos “the cats”). It tests L2 knowledge of available semantic readings of bare plurals and definite plurals in Spanish, where [+specific] and [+generic] interpretations are syntactically represented differently from English. Assuming L1 transfer, and in view of a potential subset/superset relationship of the two grammars, the learning task in this domain is not a straightforward one. Target acquisition requires both grammatical expansion and retraction; Spanish definite plural subjects require the addition of an L1-unavailable [+generic] reading, while a loss of an L1-available [+generic] reading for preverbal subject bare plurals is required. The results and analysis show that advanced L2 learners of Spanish (English L1) can circumvent a superficial subset/superset learnability problem by means of feature resetting in line with the Nominal Mapping Parameter.

On the (un)-ambiguity of adjectival modification in Spanish determiner phrases: informing debates on the mental representations of L2 syntax.

This study contributes to a central debate within contemporary generative second language (L2) theorizing: the extent to which adult learners are (un)able to acquire new functional features that result in a L2 grammar that is mentally structured like the native target (see White, 2003). The adult acquisition of L2 nominal phi-features is explored, with focus on the syntactic and semantic reflexes in the related domain of adjective placement in two experimental groups: English-speaking intermediate (n = 21) and advanced (n = 24) learners of Spanish, as compared to a native-speaker control group (n = 15). Results show that, on some of the tasks, the intermediate L2 learners appear to have acquired the syntactic properties of the Spanish determiner phrase but, on other tasks, to show some delay with the semantic reflexes of prenominal and postnominal adjectives. Crucially, however, our data demonstrate full convergence by all advanced learners and thus provide evidence in contra the predictions of representational deficit accounts (e.g., Hawkins & Chan, 1997; Hawkins & Franceschina, 2004; Hawkins & Hattori, 2006).

Interface vulnerability and knowledge of the subjunctive/indicative distinction with negated epistemic predicates in L2 Spanish.

Much recent research in SLA is guided by the hypothesis of L2 interface vulnerability (see Sorace 2005). This study contributes to this general project by examining the acquisition of two classes of subjunctive complement clauses in L2 Spanish: subjunctive complements of volitional predicates (purely syntactic) and subjunctive vs. indicative complements with negated epistemic matrix predicates, where the mood distinction is discourse dependent (thus involving the syntax-discourse interface). We provide an analysis of the volitional subjunctive in English and Spanish, suggesting that English learners of L2 Spanish need to access the functional projection Mood P and an uninterpretable modal feature on the Force head available to them from their formal English register grammar, and simultaneously must unacquire the structure of English for-to clauses. For negated epistemic predicates, our analysis maintains that they need to revalue the modal feature on the Force head from uninterpretable to interpretable, within the L2 grammar.With others (e.g. Borgonovo & Prévost 2003; Borgonovo, Bruhn de Garavito & Prévost 2005) and in line with Sorace's (2000, 2003, 2005) notion of interface vulnerability, we maintain that the latter case is more difficult for L2 learners, which is borne out in the data we present. However, the data also show that the indicative/subjunctive distinction with negated epistemics can be acquired by advanced stages of acquisition, questioning the notion of obligatory residual optionality for all properties which require the integration of syntactic and discourse information.

Aspect selection in adult L2 Spanish and the competing systems hypothesis: when pedagogical and linguistic rules conflict.

Native-like use of preterit and imperfect morphology in all contexts by English learners of L2 Spanish is the exception rather than the rule, even for successful learners. Nevertheless, recent research has demonstrated that advanced English learners of L2 Spanish attain a native-like morphosyntactic competence for the preterit/imperfect contrast, as evidenced by their native-like knowledge of associated semantic entailments (Goodin-Mayeda and Rothman 2007, Montrul and Slabakova 2003, Slabakova and Montrul 2003, Rothman and Iverson 2007). In addition to an L2 disassociation of morphology and syntax (e.g., Bruhn de Garavito 2003, Lardiere 1998, 2000, 2005, Prévost and White 1999, 2000, Schwartz 2003), I hypothesize that a system of learned pedagogical rules contributes to target-deviant L2 performance in this domain through the most advanced stages of L2 acquisition via its competition with the generative system. I call this hypothesis the Competing Systems Hypothesis. To test its predictions, I compare and contrast the use of the preterit and imperfect in two production tasks by native, tutored (classroom), and naturalistic learners of L2 Spanish.