From Euclidean Geometry to Knots and Nets

This document is the Accepted Manuscript of an article accepted for publication in Synthese. Under embargo until 19 September 2018. The final publication is available at Springer via https://doi.org/10.1007/s11229-017-1558-x.This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if (a) it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, (b) the information thus displayed is not metrical and (c) it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions.Peer reviewe

On a Partial Decision Method for Dynamic Proofs

This paper concerns a goal directed proof procedure for the propositional fragment of the adaptive logic ACLuN1. At the propositional level, it forms an algorithm for final derivability. If extended to the predicative level, it provides a criterion for final derivability. This is essential in view of the absence of a positive test. The procedure may be generalized to all flat adaptive logics.Comment: 18 pages. Originally published in proc. PCL 2002, a FLoC workshop; eds. Hendrik Decker, Dina Goldin, Jorgen Villadsen, Toshiharu Waragai (http://floc02.diku.dk/PCL/

Aristotle on Geometrical Potentialities

This paper examines Aristotle's discussion of the priority of actuality to potentiality in geometry at Metaphysics Θ9, 1051a21–33. Many scholars have assumed what I call the "geometrical construction" interpretation, according to which his point here concerns the relation between an inquirer's thinking and a geometrical figure. In contrast, I defend what I call the "geometrical analysis" interpretation, according to which it concerns the asymmetrical relation between geometrical propositions in which one is proved by means of the other. His argument as so construed is ultimately based on the asymmetrical relation between the corresponding geometrical facts. Then I explore this ontological priority in geometry by drawing attention to a parallel passage, Posterior Analytics II.11, 94a24–35, where Aristotle explains the relation between the same geometrical propositions in connection to material causation

Evaluating novel pedagogy in higher education: a case study of e-proofs

This thesis is a single case study of the introduction and evaluation of new resources and new technologies in higher education; in which e-Proof was chosen as a single case. E-proofs are a multimedia representation of proofs, were created by Alcock (2009), and aimed to help undergraduates to read proofs for better proof comprehension. My thesis aimed to investigate whether the impact of reading such technology-based resource, e-Proofs, on undergraduates proof comprehension was better compared to reading written textbook proofs and if so, then why (or why not). To evaluate the effectiveness of e-Proofs, I used both qualitative and quantitative methods. First I measured undergraduates satisfaction, which is a most common research practice in evaluation studies, by using self-reporting methods such as web-based survey and interviews. A web-based survey and focus-group interviews showed that undergraduates liked to have e-Proofs and they believed that e-Proofs had positive impact on their proof comprehension. However, their positive views on e-Proofs did not evidence the educational impact of e-Proofs. I conducted an interview with Alcock for better understanding of her intentions of creating e-Proof and her expectations from it. Next, I conducted the first experiment which compared the impact of reading an e-Proof with a written textbook proof on undergraduates proof comprehension. Their comprehension was measured with an open-ended comprehension test twice immediately after reading the proof and after two weeks. I found that the immediate impact of reading an e-Proof and a textbook proof were essentially the same, however the long term impact of reading an e-Proof was worse than reading a textbook proof (for both high and low achieving undergraduates). This leads to the second experiment in which I investigated how undergraduates read e-Proofs and textbook proofs. In the second experiment, participants eye-movements were recorded while read- ing proofs, to explore their reading comprehension processes. This eye-tracking experiment showed that undergraduates had a sense of understanding of how to read a proof without any additional help. Rather, additional help allowed them to take a back seat and to devote less cognitive effort than they would otherwise. Moreover, e-Proofs altered undergraduates reading behaviours in a way which can harm learning. In sum, this thesis contributes knowledge into the area of reading and compre- hending proofs at undergraduate level and presents a methodology for evaluation studies of new pedagogical tools

Factorizing FF-matrices and the XXZ spin-1/2 chain: A diagrammatic perspective

Using notation inherited from the six-vertex model, we construct diagrams that represent the action of the factorizing $F$-matrices associated to the finite length XXZ spin-1/2 chain. We prove that these $F$-matrices factorize the tensor $R^{\sigma}_{1... n}$ corresponding with elements of the permutation group. We consider in particular the diagram for the tensor $R^{\sigma_c}_{1... n}$, which cyclically permutes the spin chain. This leads us to a diagrammatic construction of the local spin operators $S_i^{\pm}$ and $S_i^{z}$ in terms of the monodromy matrix operators.Comment: 26 pages, extra references added, typographical errors correcte