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

Shoe-stiffening inserts for first metatarsophalangeal joint osteoarthritis (the SIMPLE trial): study protocol for a randomised controlled trial

BACKGROUND: This article describes the design of a parallel-group, participant- and assessor-blinded randomised controlled trial comparing the effectiveness of shoe-stiffening inserts versus sham shoe insert(s) for reducing pain associated with first metatarsophalangeal joint (MTPJ) osteoarthritis (OA). METHODS: Ninety participants with first MTPJ OA will be randomised to receive full-length shoe-stiffening insert(s) (Carbon Fibre Spring Plate, Paris Orthotics, Vancouver, BC, Canada) plus rehabilitation therapy or sham shoe insert(s) plus rehabilitation therapy. Outcome measures will be obtained at baseline, 4, 12, 24 and 52 weeks; the primary endpoint for assessing effectiveness being 12 weeks. The primary outcome measure will be the foot pain domain of the Foot Health Status Questionnaire (FHSQ). Secondary outcome measures will include the function domain of the FHSQ, severity of first MTPJ pain (using a 100-mm Visual Analogue Scale), global change in symptoms (using a 15-point Likert scale), health status (using the Short-Form-12® Version 2.0 and EuroQol (EQ-5D-5L™) questionnaires), use of rescue medication and co-interventions, self-reported adverse events and physical activity levels (using the Incidental and Planned Activity Questionnaire). Data will be analysed using the intention-to-treat principle. Economic analysis (cost-effectiveness and cost-utility) will also be performed. In addition, the kinematic effects of the interventions will be examined at 1 week using a three-dimensional motion analysis system and multisegment foot model. DISCUSSION: This study will determine whether shoe-stiffening inserts are a cost-effective intervention for relieving pain associated with first MTPJ OA. The biomechanical analysis will provide useful insights into the mechanism of action of the shoe-stiffening inserts. TRIAL REGISTRATION: Australian New Zealand Clinical Trials Registry, identifier: ACTRN12616000552482 . Registered on 28 April 2016

Centre of pressure characteristics in normal, planus and cavus feet

Background The aim of this study was to compare centre of pressure (COP) characteristics between healthy adults with normal, planus or cavus feet who were allocated to groups based on reliable foot posture measurement techniques. Methods Ninety-two healthy adult participants (aged 18 to 45) were recruited and classified as either normal (n = 35), pes planus (n = 31) or pes cavus (n = 26) based on Foot Posture Index, Arch Index and normalised navicular height truncated measurements. Barefoot walking trials were conducted using an emed®-x 400 plantar pressure system (Novel GmbH, Munich, Germany). Average, maximum, minimum and range (difference between maximum and minimum) values were calculated for COP velocity and lateral-medial force index during loading response, midstance, terminal stance and pre-swing phases of stance. The COP excursion index was also calculated. One-way analyses of variance were used to compare the three foot posture groups. Results The cavus foot exhibited the slowest average and minimum COP velocity during terminal stance, but this pattern was reversed during pre-swing, when the cavus foot exhibited the fastest maximum COP velocity. The planus foot exhibited the smallest lateral medial force index range during terminal stance. There were no differences between the groups for COP excursion index. Conclusion These findings indicate that there are differences in COP characteristics between foot postures, which may represent different mechanisms for generating force to facilitate forward progression of the body during the propulsive phases of gait

How to think about informal proofs

This document is the Accepted Manuscript version of the following article: Brendan Larvor, ‘How to think about informal proofs’, Synthese, Vol. 187(2): 715-730, first published online 9 September 2011. The final publication is available at Springer via doi:10.1007/s11229-011-0007-5It is argued in this study that (i) progress in the philosophy of mathematical practice requires a general positive account of informal proof; (ii) the best candidate is to think of informal proofs as arguments that depend on their matter as well as their logical form; (iii) articulating the dependency of informal inferences on their content requires a redefinition of logic as the general study of inferential actions; (iv) it is a decisive advantage of this conception of logic that it accommodates the many mathematical proofs that include actions on objects other than propositions; (v) this conception of logic permits the articulation of project-sized tasks for the philosophy of mathematical practice, thereby supplying a partial characterisation of normal research in the fieldPeer reviewedFinal Accepted Versio

A formally verified abstract account of Gödel's incompleteness theorems

We present an abstract development of Gödel’s incompleteness theorems, performed with the help of the Isabelle/HOL theorem prover. We analyze sufficient conditions for the theorems’ applicability to a partially specified logic. In addition to the usual benefits of generality, our abstract perspective enables a comparison between alternative approaches from the literature. These include Rosser’s variation of the first theorem, Jeroslow’s variation of the second theorem, and the S ́wierczkowski–Paulson semantics-based approach. As part of our framework’s validation, we upgrade Paulson’s Isabelle proof to produce a mech- anization of the second theorem that does not assume soundness in the standard model, and in fact does not rely on any notion of model or semantic interpretation

Force-induced conformational change of bacteriorhodopsin

The cytoplasmic surface topography of purple membranes imaged by the atomic force microscope depends mainly on the force applied to the stylus. Imaged at forces of 300 pN, individual bacteriorhodopsin molecules reveal two domains. The resulting donut-shaped trimers reversibly transform into structures exhibiting three prominent protrusions when scanned at 100 pN. In parallel, the height of the protein moiety above the lipid layer increases from 4 A to 6 A. From the known structure of bacteriorhodopsin it appears that this change may be related to a bending of the most prominent cytoplasmic loop