Published Article / Language and Mathematics are both so-called "tools" that are used by other disciplines to explain / describe phenomena in those disciplines, but they are scientific disciplines in their own right. Language is a system of symbols, but so is Mathematics. These symbols carry meaning or value. Both originate in the human mind and are then translated into messages of logic. What is important are the relationships between units that are inherent to both disciplines. In practicing the two disciplines, there are elements that correspond. These are a vocabulary, grammar, a community and meaning. Psycholinguists and psychologists are interested in the role that language might have in enabling other functions in the human cognitive repertoire. Some argue that language is a prerequisite for a whole range of intellectual activities, including mathematics. They claim that mathematical structures are, in a way, parasitic on the human linguistic faculty. Some evidence for the language: maths connection comes from neurology. Functional imaging studies of the brain show increased activation of the language areas as certain mathematical tasks / challenges are performed. Lesions to a certain part of the brain impair both the linguistic as well as the mathematical ability. We are looking at a fundamentally shared enterprise, a deeply interwoven development of numerical and linguistic aspects. This co-evolution of number concepts and number words suggests that it is no accident that the same species that possesses the language faculty as a unique trait, should also be the one that developed a systematic concept of number.

This doctoral thesis addresses one major difficulty in formal proof: removing obstructions to intuition which hamper the proof endeavour. We investigate this in the context of formally verifying geometric algorithms using the theorem prover Isabelle, by first proving the Graham’s Scan algorithm for finding convex hulls, then using the challenges we encountered as motivations for the design of a general, modular framework for combining mathematical tools. We introduce our integration framework — the Prover’s Palette, describing in detail the guiding principles from software engineering and the key differentiator of our approach — emphasising the role of the user. Two integrations are described, using the framework to extend Eclipse Proof General so that the computer algebra systems QEPCAD and Maple are directly available in an Isabelle proof context, capable of running either fully automated or with user customisation. The versatility of the approach is illustrated by showing a variety of ways that these tools can be used to streamline the theorem proving process, enriching the user’s intuition rather than disrupting it. The usefulness of our approach is then demonstrated through the formal verification of an algorithm for computing Delaunay triangulations in the Prover’s Palette.

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Thesis (MA) -- Stellenbosch University, 1949. / INLEIDING: Enige denker wat die geskiedenis van die filosofie bestudeer, gee daarmee te kenne dat die geskiedenis van die filosofie meer is as bloot 'n ry van aaneengeskakelde dog verwerpte sisteme. Die geskiedenis van die filosofie is nie 'n museum van historiese kuriositeite, wat vir die huidige denker van geen waar is nie. In die geskiedenis van die filosofie vind die denker die probleme terug waarmee hy homself ook besig hu alhoewel hierdie probleme steeds in 'n ander verband of vanuit 'n ander gesigspunt gestel is. Die geskiedenis van die filosofie is 'n leerskool vir elke denker. Die feit dat die geskiedenis van die filosofie 'n leerskool vir elke denker is, op wat ter punt van die geskiedenis hy ook staan, hou egter verskillende implikasies in.

Case-Based Reasoning (CBR) is a technique in Artificial Intelligence where a new problem is solved by making use of the solution to a similar past problem situation. People naturally solve problems in this way, without even thinking about it. For example, an occupational therapist (OT) that assesses the needs of a new disabled person may be reminded of a previous person in terms of their disabilities. He may or may not decide to recommend the same devices based on the outcome of an earlier (disabled) person. Case-based reasoning makes use of a collection of past problem-solving experiences thus enabling users to exploit the information of others’ successes and failures to solve their own problem(s). This project has developed a CBR tool to assist in matching SmartHouse technology to the needs of the elderly and people with disabilities. The tool makes suggestions of SmartHouse devices that could assist with given impairments. SmartHouse past problem-solving textual reports have been used to obtain knowledge for the CBR system. Creating a case-based reasoning system from textual sources is challenging because it requires that the text be interpreted in a meaningful way in order to create cases that are effective in problem-solving and to be able to reasonably interpret queries. Effective case retrieval and query interpretation is only possible if a domain-specific conceptual model is available and if the different meanings that a word can take can be recognised in the text. Approaches based on methods in information retrieval require large amounts of data and typically result in knowledge-poor representations. The costs become prohibitive if an expert is engaged to manually craft cases or hand tag documents for learning. Furthermore, hierarchically structured case representations are preferred to flat-structured ones for problem-solving because they allow for comparison at different levels of specificity thus resulting in more effective retrieval than flat structured cases. This project has developed SmartCAT-T, a tool that creates knowledge-rich hierarchically structured cases from semi-structured textual reports. SmartCAT-T highlights important phrases in the textual SmartHouse problem-solving reports and uses the phrases to create a conceptual model of the domain. The model then becomes a standard structure onto which each semi-structured SmartHouse report is mapped in order to obtain the correspondingly structured case. SmartCAT-T also relies on an unsupervised methodology that recognises word synonyms in text. The methodology is used to create a uniform vocabulary for the textual reports and the resulting harmonised text is used to create the standard conceptual model of the domain. The technique is also employed in query interpretation during problem solving. SmartCAT-T does not require large sets of tagged data for learning, and the concepts in the conceptual model are interpretable, allowing for expert refinement of knowledge. Evaluation results show that the created cases contain knowledge that is useful for problem solving. An improvement in results is also observed when the text and queries are harmonised. A further evaluation highlights a high potential for the techniques developed in this research to be useful in domains other than SmartHouse. All this has been implemented in the Smarter case-based reasoning system.

published_or_final_version / Psychology / Master / Master of Philosophy

published_or_final_version / Clinical Psychology / Doctoral / Doctor of Psychology

Students often use study groups to prepare for class or exams; yet to date, we know very little about how these groups actually function. This study looked at the ways in which undergraduate organic chemistry students prepared for exams through self-initiated study groups. We sought to characterize the methods of social regulation, levels of content processing, and types of reasoning processes used by students within their groups. Our analysis showed that groups engaged in predominantly three types of interactions when discussing chemistry content: co-construction, teaching, and tutoring. Although each group engaged in each of these types of interactions at some point, their prevalence varied between groups and group members. Our analysis suggests that the types of interactions that were most common depended on the relative content knowledge of the group members as well as on the difficulty of the tasks in which they were engaged.Additionally, we were interested in characterizing the reasoning methods used by students within their study groups. We found that students used a combination of three content-relevant methods of reasoning: model-based reasoning, case-based reasoning, or rule-based reasoning, in conjunction with one chemically-irrelevant method of reasoning: symbol-based reasoning. The most common way for groups to reason was to use rules, whereas the least common way was for students to work from a model. In general, student reasoning correlated strongly to the subject matter to which students were paying attention, and was only weakly related to student interactions.Overall, results from this study may help instructors to construct appropriate tasks to guide what and how students study outside of the classroom. We found that students had a decidedly strategic approach in their study groups, relying heavily on material provided by their instructors, and using the reasoning strategies that resulted in the lowest levels of content processing. We suggest that instructors create more opportunities for students to explore model-based reasoning, and to create opportunities for students to be able to co-construct in a collaborative manner within the context of their organic chemistry course.

The central thesis of this dissertation is that it is not possible to determine the nature of epistemic justification apart from psychological investigation. Two sub-theses provide the primary support for this claim. The first sub-thesis is that no account of epistemic justification is correct which requires for the possession of justified beliefs a psychological capacity which humans do not have. A different way of stating this view is that the correct account of epistemic justification must be psychologically realistic. The second sub-thesis is that it is not possible to determine whether an account of epistemic justification is psychologically realistic apart from psychological investigation. In sum, there is a meta-theoretical constraint of psychological realism on accounts of epistemic justification which requires appeal to psychological investigation in its employment. After defending these proposals, I illustrate how the constraint of psychological realism has been and can be used both to test candidate accounts of epistemic justification and to guide the construction of such an account which is intuitive and psychologically realistic. These two kinds of applications of the constraint can involve either scientific or non-scientific psychological investigation. I give examples from current epistemological literature of critical employments of the constraint which appeal to both of these kinds of psychological investigation. Finally, in illustrating the role of the constraint of psychological realism in guiding the construction of an account of epistemic justification, I consider both reliabilist views and a variety of positions which feature the notion of cognitive design. I suggest that this latter approach holds out promise for yielding an account of epistemic justification which is both psychologically realistic and intuitive.

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