Wittgenstein on Pseudo-Irrationals, Diagonal Numbers and Decidability

In his early philosophy as well as in his middle period, Wittgenstein holds a purely syntactic view of logic and mathematics. However, his syntactic foundation of logic and mathematics is opposed to the axiomatic approach of modern mathematical logic. The object of Wittgenstein’s approach is not the representation of mathematical properties within a logical axiomatic system, but their representation by a symbolism that identifies the properties in question by its syntactic features. It rests on his distinction of descriptions and operations; its aim is to reduce mathematics to operations. This paper illustrates Wittgenstein’s approach by examining his discussion of irrational numbers

A BIBLIOGRAPHY: JOHN CORCORAN’S PUBLICATIONS ON ARISTOTLE 1972–2015

This presentation includes a complete bibliography of John Corcoran’s publications devoted at least in part to Aristotle’s logic. Sections I–IV list 20 articles, 43 abstracts, 3 books, and 10 reviews. It starts with two watershed articles published in 1972: the Philosophy & Phenomenological Research article that antedates Corcoran’s Aristotle’s studies and the Journal of Symbolic Logic article first reporting his original results; it ends with works published in 2015. A few of the items are annotated with endnotes connecting them with other work. In addition, Section V “Discussions” is a nearly complete secondary bibliography of works describing, interpreting, extending, improving, supporting, and criticizing Corcoran’s work: 8 items published in the 1970s, 22 in the 1980s, 39 in the 1990s, 56 in the 2000s, and 65 in the current decade. The secondary bibliography is annotated with endnotes: some simply quoting from the cited item, but several answering criticisms and identifying errors. As is evident from the Acknowledgements sections, all of Corcoran’s publications benefited from correspondence with other scholars, most notably Timothy Smiley, Michael Scanlan, and Kevin Tracy. All of Corcoran’s Greek translations were done in consultation with two or more classicists. Corcoran never published a sentence without discussing it with his colleagues and students. REQUEST: Please send errors, omissions, and suggestions. I am especially interested in citations made in non-English publications

Theory-related translation research. Some thoughts on methodology

The article illustrates some methodological principles for translation research and for writing theory-related research papers. A number of postulates are formulated with respect to basic structural dimensions of organizing research papers (working hypothesis, ‘state of the art’ in research, solution proposal and validity test). Examples for norm-compliant behavior are given. Since the problem of writing the ‘state of the art’ of  research seems to be one of the major problems facing young scholars, particular emphasis is given to this aspect

Scenario-based portfolio model for building robust and proactive strategies

In order to address major changes in the operational environment, companies can (i) define scenarios that characterize different alternatives for this environment, (ii) assign probabilities to these scenarios, (iii) evaluate the performance of strategic actions across the scenarios, and (iv) choose those actions that are expected to perform best. In this paper, we develop a portfolio model to support the selection of such strategic actions when the information about scenario probabilities is possibly incomplete and may depend on the selected actions. This model helps build a strategy that is robust in that it performs relatively well in view of all available probability information, and proactive in that it can help steer the future as reflected by the scenarios toward the desired direction. We also report a case study in which the model helped a group of Nordic, globally operating steel and engineering companies build a platform ecosystem strategy that accounts for uncertainties related to markets, politics, and technological development

A learning theory approach to students' misconceptions in calculus

Bibliography: leaves 129-138.This study analyses students' errors in calculus through the lens of learning theories. The subjects in this study were 117 students enrolled in a calculus course for students from disadvantaged educational backgrounds at the University of Cape Town. A coding scheme to categorise the errors that these students made in the final examination was developed. This categorisation was supported by error data generated through the administration of a conceptual test and follow-up interviews. The pattern of errors in the coding scheme suggests that the students' perception of algebra is largely that of a "game of letters". As a result of this their construction of calculus knowledge is based on the rehearsal of algorithmic procedures. Their errors indicate that they develop linking and extending mechanisms to deal with the multiplicity of rules that are generated from this process of rehearsal

Formal hardware verification of digital circuits

The use of formal methods to verify the correctness of digital circuits is less constrained by the growing complexity of digital circuits than conventional methods based on exhaustive simulation. This paper briefly outlines three main approaches to formal hardware verification: symbolic simulation, state machine analysis, and theorem-proving

Mill’s Fallacies: Theory and Practice

In noting contemporary neglect of Mill\u27s work on fallacy, Hansen and Pinto say that his account is tied more closely to scientific methodology than to problems of public discourse and everyday argumentation. This paper re-examines Mill\u27s fallacies from a rhetorical perspective, assessing the extent to which his examples—drawn from the domain of popular superstition, science, philosophy, and public discussion—fits his theoretical structure. In articulating the relationship between Mill\u27s philosophical assumptions and the discursive practices of the fields from which he draws his examples, it will suggest the ambiguities in Mill\u27s mentalistic, rationalistic, inductivist approach and the inescapable rhetoricity of his examples

Constructing sample space with combinatorial reasoning: a mixed methods study

Recent curricular developments suggest that students at all levels need to be statistically literate and able to efficiently and accurately make probabilistic decisions. Furthermore, statistical literacy is a requirement to being a well-informed citizen of society. Research also recognizes that the ability to reason probabilistically is supported and enabled by other forms of mathematical reasoning and concepts. One of these supporting concepts is sample space, the enumeration of all outcomes of a probability experiment. As a concept, sample space supports the construction of probability distributions, which in turn enables statistical inference, a form of probabilistic reasoning. This mixed methods study investigated how undergraduate pre-service elementary teachers construct and generalize their understanding about sample space. One hundred fifty students participated in a series of three tasks designed to investigate the ways in which they enumerate sample space and the associations between their enumeration strategies and their generalization rules. A subset of eight participants engaged in follow- up interviews designed to explore their understandings of sample space enumeration and generalization. Findings from the study suggest that there was growth across tasks in the sophistication of the enumeration strategies used and that participants attempted to find explicit and formalized generalizations. However, in spite of this growth in the sophistication of enumeration, there was little association between the enumeration strategies participants used and the generalizations that they constructed. Students compartmentalized their understanding of generalization rules, often looking for a numeric formula that had little do to with their enumerated solutions