The "Artificial Mathematician" Objection: Exploring the (Im)possibility of Automating Mathematical Understanding

Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer

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

Towards mathematical AI via a model of the content and process of mathematical question and answer dialogues

This paper outlines a strategy for building semantically meaningful representations and carrying out effective reasoning in technical knowledge domains such as mathematics. Our central assertion is that the semi-structured Q and A format, as used on the popular Stack Exchange network of websites, exposes domain knowledge in a form that is already reasonably close to the structured knowledge formats that computers can reason about. The knowledge in question is not only facts - but discursive, dialectical, argument for purposes of proof and pedagogy. We therefore assert that modelling the Q and A process computationally provides a route to domain understanding that is compatible with the day-to-day practices of mathematicians and students. This position is supported by a small case study that analyses one question from Mathoverflow in detail, using concepts from argumentation theory. A programme of future work, including a rigorous evaluation strategy, is then advanced

Bronchopulmonary dysplasia: clinical aspects and preventive and therapeutic strategies

Abstract Background Bronchopulmonary dysplasia (BPD) is the result of a complex process in which several prenatal and/or postnatal factors interfere with lower respiratory tract development, leading to a severe, lifelong disease. In this review, what is presently known regarding BPD pathogenesis, its impact on long-term pulmonary morbidity and mortality and the available preventive and therapeutic strategies are discussed. Main body Bronchopulmonary dysplasia is associated with persistent lung impairment later in life, significantly impacting health services because subjects with BPD have, in most cases, frequent respiratory diseases and reductions in quality of life and life expectancy. Prematurity per se is associated with an increased risk of long-term lung problems. However, in children with BPD, impairment of pulmonary structures and function is even greater, although the characterization of long-term outcomes of BPD is difficult because the adults presently available to study have received outdated treatment. Prenatal and postnatal preventive measures are extremely important to reduce the risk of BPD. Conclusion Bronchopulmonary dysplasia is a respiratory condition that presently occurs in preterm neonates and can lead to chronic respiratory problems. Although knowledge about BPD pathogenesis has significantly increased in recent years, not all of the mechanisms that lead to lung damage are completely understood, which explains why therapeutic approaches that are theoretically effective have been only partly satisfactory or useless and, in some cases, potentially negative. However, prevention of prematurity, systematic use of nonaggressive ventilator measures, avoiding supraphysiologic oxygen exposure and administration of surfactant, caffeine and vitamin A can significantly reduce the risk of BPD development. Cell therapy is the most fascinating new measure to address the lung damage due to BPD. It is desirable that ongoing studies yield positive results to definitively solve a major clinical, social and economic problem