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

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

Lithosphere tearing along STEP faults and synkinematic formation of lherzolite and wehrlite in the shallow subcontinental mantle

Subduction-transform edge propagator (STEP) faults are the locus of continual lithospheric tearing at slab edges, resulting in sharp changes in the lithospheric and crustal thickness and triggering lateral and/or near-vertical mantle flow. However, the mechanisms at the lithospheric mantle scale are still poorly understood. Here, we present the microstructural study of olivine-rich lherzolite, harzburgite and wehrlite mantle xenoliths from the Oran volcanic field (Tell Atlas, northwest Algeria). This alkali volcanic field occurs along a major STEP fault responsible for the Miocene westward slab retreat in the westernmost Mediterranean. Mantle xenoliths provide a unique opportunity to investigate the microstructures in the mantle section of a STEP fault system. The microstructures of mantle xenoliths show a variable grain size ranging from coarse granular to fine-grained equigranular textures uncorrelated with lithology. The major element composition of the mantle peridotites provides temperature estimates in a wide range (790–1165 ºC) but in general, the coarse-grained and fine-grained peridotites suggest deeper and shallower provenance depth, respectively. Olivine grain size in the fine-grained peridotites depends on the size and volume fraction of the pyroxene grains, which is consistent with pinning of olivine grain growth by pyroxenes as second-phase particles. In the coarse-grained peridotites, well-developed olivine crystal-preferred orientation (CPO) is characterized by orthorhombic and [100]-fiber symmetries, and orthopyroxene has a coherent CPO with that of olivine, suggesting their coeval deformation by dislocation creep at high temperature. In the fine-grained microstructures, along with the weakening of the fabric strength, olivine CPO symmetry exhibits a shift towards [010] fiber and the [010] and [001] axes of orthopyroxene are generally distributed subparallel to those of olivine. These data are consistent with deformation of olivine in the presence of low amounts of melts and the precipitation of orthopyroxenes from a melt phase. The bulk CPO of clinopyroxene mimics that of orthopyroxene via a topotaxial relationship of the two pyroxenes. This observation points to a melt-related origin of most clinopyroxenes in the Oran mantle xenoliths.This research has been supported by the Agencia Estatal de Investigación (grant nos. CGL2016-75224-R, CGL2016-81085-R and CGL2015-67130-C2-1-R), the Junta de Andalucía research groups RNM-131 and RNM-148, and the International Lithosphere Program (grant no. CC4-MEDYNA)

Biological interaction of living cells with COSAN-based synthetic vesicles

Cobaltabisdicarbollide (COSAN) [3,3′-Co(1,2-C2B9H11)2]−, is a complex boron-based anion that has the unusual property of self-assembly into membranes and vesicles. These membranes have similar dimensions to biological membranes found in cells, and previously COSAN has been shown to pass through synthetic lipid membranes and those of living cells without causing breakdown of membrane barrier properties. Here, we investigate the interaction of this inorganic membrane system with living cells. We show that COSAN has no immediate effect on cell viability, and cells fully recover when COSAN is removed following exposure for hours to days. COSAN elicits a range of cell biological effects, including altered cell morphology, inhibition of cell growth and, in some cases, apoptosis. These observations reveal a new biology at the interface between inorganic, synthetic COSAN membranes and naturally occurring biological membranes

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