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

Optimization of coagulant presence in the natural raw water at the Ain Nokbi station in Fez

An optimization study of the coagulation-flocculation chemical treatment process was carried out at the Ain Nokbi station in Fez in order to identify the problems associated with liquid rejects. The purpose of the study involved optimizing three key factors in the coagulation-flocculation process: the pH of the raw water, the coagulant amount (aluminum sulphate) and the flocculent amount (sodium alginate solution or polyelectrolyte at 0.1 mg/l). This demonstrated how acidification can improve treatment performance, particularly in terms of pH. Specifically, reducing the pH to 7 revealed advantages during disinfection, boosting the efficiency of the process. In addition, the optimum dosage of coagulant (aluminum sulphate) has been determined to be around 50 mg/l instead of 60 mg/l, saving 10 mg/l and reducing annual costs by MAD 240,000

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

Infinitesimal Idealization, Easy Road Nominalism, and Fractional Quantum Statistics

It has been recently debated whether there exists a so-called “easy road” to nominalism. In this essay, I attempt to fill a lacuna in the debate by making a connection with the literature on infinite and infinitesimal idealization in science through an example from mathematical physics that has been largely ignored by philosophers. Specifically, by appealing to John Norton’s distinction between idealization and approximation, I argue that the phenomena of fractional quantum statistics bears negatively on Mary Leng’s proposed path to easy road nominalism, thereby partially defending Mark Colyvan’s claim that there is no easy road to nominalism

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

GIZA 11 AND GIZA 12; TWO NEW FLAX DUAL PURPOSE TYPE VARIETIES

Sixteen flax genotypes {13 promising lines and 3 check varieties viz., Giza 8 (oil type), Sakha 1 (dual purpose type) and Sakha 3 (fiber type)} were evaluated for straw, seed, oil yields and their related traits under twelve different environments; four locations (Sakha, Etay El-Baroud, Ismailia and Giza Exp. Stations through three successive seasons (2011/12, 2012/13 and 2013/14). These materials were evaluated in a randomized complete blocks design with three replications at the twelve above-mentioned environments. The analysis of variance revealed highly significant differences among genotypes (G), environments (E) and G x E interaction for all studied traits except straw weight per plant, indicating a wide range of variation among genotypes, environments and these genotypes exhibited differential response to environmental conditions. The significant variance due to residual for all characters except both straw weight per plant and oil yield per fad indicated that genotypes differed with respect to their stability suggesting that prediction would be difficult, which means that mean performance alone would not be appropriate. Interaction component of variance (σ2ge) was less than the genotypic variance (σ2g) for all characters, indicating that genotypes differ in their genetic potential for these traits. This was reflected in high heritability and low discrepancy between phenotypic (PCV) and genotypic (GCV) coefficients of variability values for these traits indicating the possibility of using each of long fiber percentage, plant height and technical stem length as selection indices for improving straw weight per plant, as well as, using 1000-seed weight and capsules number per plant as selection indices for improving seed weight per plant. Yield stability (YSi) statistic indicated that S.541-C/3 and S.541-D/10 gave high mean performance and stability for straw, fiber, seed and oil yields per fad in addition to oil percentage, capsules number per plant and 1000-seed weight. Therefore, the two genotypes well be released under the name Giza 11 and Giza 12, respectively. These newly released varieties are of dual purpose type for straw, fiber, seed and oil yield. They may replace the low yielding cultivars Giza 8, Sakha 1 and Sakha 3