Why the naïve Derivation Recipe model cannot explain how mathematicians’ proofs secure mathematical knowledge

This is a pre-copyedited, author-produced PDF of an article accepted for publication in Philosophia Mathematica following peer review. Under embargo. Embargo end date: 7 July 2018 The version of record [Lavor, B., 'Why the Naive Derivation Recipe Model Cannot Explain How Mathematician's Proofs Secure Mathematical Knowledge', Philosophia Mathematica (2016) 24(3): 401-404, is available online at: https://doi.org/10.1093/philmat/nkw012. © The Author [2016]. Published by Oxford University Press. All rights reserved.The view that a mathematical proof is a sketch of or recipe for a formal derivation requires the proof to function as an argument that there is a suitable derivation. This is a mathematical conclusion, and to avoid a regress we require some other account of how the proof can establish it.Peer reviewedFinal Accepted Versio

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

An integrated approach to high integrity software verification.

Computer software is developed through software engineering. At its most precise, software engineering involves mathematical rigour as formal methods. High integrity software is associated with safety critical and security critical applications, where failure would bring significant costs. The development of high integrity software is subject to stringent standards, prescribing best practises to increase quality. Typically, these standards will strongly encourage or enforce the application of formal methods. The application of formal methods can entail a significant amount of mathematical reasoning. Thus, the development of automated techniques is an active area of research. The trend is to deliver increased automation through two complementary approaches. Firstly, lightweight formal methods are adopted, sacrificing expressive power, breadth of coverage, or both in favour of tractability. Secondly, integrated solutions are sought, exploiting the strengths of different technologies to increase automation. The objective of this thesis is to support the production of high integrity software by automating an aspect of formal methods. To develop tractable techniques we focus on the niche activity of verifying exception freedom. To increase effectiveness, we integrate the complementary technologies of proof planning and program analysis. Our approach is investigated by enhancing the SPARK Approach, as developed by Altran Praxis Limited. Our approach is implemented and evaluated as the SPADEase system. The key contributions of the thesis are summarised below: • Configurable and Sound - Present a configurable and justifiably sound approach to software verification. • Cooperative Integration - Demonstrate that more targeted and effective automation can be achieved through the cooperative integration of distinct technologies. • Proof Discovery - Present proof plans that support the verification of exception freedom. • Invariant Discovery - Present invariant discovery heuristics that support the verification of exception freedom. • Implementation as SPADEase - Implement our approach as SPADEase. • Industrial Evaluation - Evaluate SPADEase against both textbook and industrial subprograms

Mathematics, cognition, and you!

In what follows I argue for an epistemic bridge principle that allows us to move from real mathematics to ideal mathematics (and back again) without losing anything that is characteristic of either methodological class. Mathematics is a collection of actions performed in the pursuit of mathematical understanding. The actions are the processes of proving claims that come in such forms as lemmas, theorems, or conjectures. These proofs can be accomplished through a variety of means including (though not limited to) logical deduction, geometric intuition, diagramming, or computer assistance. What is common to each of these is the characteristic of being convincing to a sound mathematical mind. Even though what in particular makes each of these methods of proof convincing differs, that they are convincing is enough to usher forth mathematical understanding. The chapters of this dissertation explore (i) a new naturalistic metaphysics for under- standing “where mathematics comes from”, (ii) recent psychological findings in the the nature of mathematical reasoning, (iii) the concatenation of two historical forms of reasoning (real and ideal), (iv) the possibility of an epistemic bridge between real and ideal methods, and (v) the implication of this new bridge principle for the long-standing concern that Go ̈del’s Incompleteness Theorems shake the foundation of modern mathematics

Computerbeweise und ihr Einfluss auf die Philosophie der Mathematik

Der von Appel und Haken im Jahr 1977 veröffentlichte "Computerbeweis" löste eine Kontroverse unter Mathematikern aus: Es wurde über den Einsatz empirischer Methoden und den Stellenwert eines Beweises in der Mathematik diskutiert. Unterschiedliche Standpunkte werden dargestellt. Es wird untersucht, wie und warum sich die Diskussion über spezielle Probleme, die ein Computerbeweis mit sich bringt, auf allgemeine philosophische Grundlagen der Mathematik ausweitete

Proof, rigour and informality : a virtue account of mathematical knowledge

This thesis is about the nature of proofs in mathematics as it is practiced, contrasting the informal proofs found in practice with formal proofs in formal systems. In the first chapter I present a new argument against the Formalist-Reductionist view that informal proofs are justified as rigorous and correct by corresponding to formal counterparts. The second chapter builds on this to reject arguments from Gödel's paradox and incompleteness theorems to the claim that mathematics is inherently inconsistent, basing my objections on the complexities of the process of formalisation. Chapter 3 looks into the relationship between proofs and the development of the mathematical concepts that feature in them. I deploy Waismann's notion of open texture in the case of mathematical concepts, and discuss both Lakatos and Kneebone's dialectical philosophies of mathematics. I then argue that we can apply work from conceptual engineering to the relationship between formal and informal mathematics. The fourth chapter argues for the importance of mathematical knowledge-how and emphasises the primary role of the activity of proving in securing mathematical knowledge. In the final chapter I develop an account of mathematical knowledge based on virtue epistemology, which I argue provides a better view of proofs and mathematical rigour.Funded by the Caroline Elder PG Scholarship and a SASP scholarship, and with travel funded by the Indo-European Research Training Network in Logic and the Arché Travel Fund

Proceedings of the Problem@Web International Conference: technology, creativity and affect in mathematical problem solving

The Problem@Web International Conference is a satellite conference of a research project conducted in Portugal between December 2010 and June 2014, jointly developed by the Institute of Education of the University of Lisbon and the University of Algarve. The project was launched to embrace the opportunity of studying mathematical problem solving beyond the mathematics classroom, by looking at the context of two web-based inclusive mathematical competitions. The research team and the project external consultant (Prof. Keith Jones) took upon themselves the task of promoting a conference aiming to present and disseminate the main results of the research undertaken and to create the opportunity of sharing and discussing neighbouring perspectives and ideas from scholars and researchers in the international field. Technology, creativity, and affect in mathematical problem solving were the three strands of this conference within which pedagogical and research perspectives were offered. The conference proceedings provide an overview of all the research work presented in the form of plenary sessions, research papers and e-posters

Proceedings of the Problem@Web International Conference: technology, creativity and affect in mathematical problem solving

The Problem@Web International Conference is a satellite conference of a research project conducted in Portugal between December 2010 and June 2014, jointly developed by the Institute of Education of the University of Lisbon and the University of Algarve. The project was launched to embrace the opportunity of studying mathematical problem solving beyond the mathematics classroom, by looking at the context of two web-based inclusive mathematical competitions. The research team and the project external consultant (Prof. Keith Jones) took upon themselves the task of promoting a conference aiming to present and disseminate the main results of the research undertaken and to create the opportunity of sharing and discussing neighbouring perspectives and ideas from scholars and researchers in the international field. Technology, creativity, and affect in mathematical problem solving were the three strands of this conference within which pedagogical and research perspectives were offered. The conference proceedings provide an overview of all the research work presented in the form of plenary sessions, research papers and e-posters