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The Search for Certainty : A Philosophical Account of Foundations of Mathematics
The nineteenth century saw a movement to make higher mathematics rigorous. This seemed to be on the brink of success when it was thrown into confusion by the discovery of the class paradoxes. That initiated a period of intense research into the foundations of mathematics, and with it the birth of mathematical logic and a new, sharper debate in the philosophy of mathematics. The Search for Certainty examines this foundational endeavour from the discovery of the paradoxes to the present. Focusing on Russell's logicist programme and Hilbert's finitist programme, Giaquinto investigates how successful they were and how successful they could be. These questions are set in the context of a clear, non-technical exposition and assessment of the most important discoveries in mathematical logic, above all G--ouml--;del's underivability theorems. More than six decades after those discoveries, Giaquinto asks what our present perspective should be on the question of certainty in mathematics. Taking recent developments into account, he gives reasons for a surprisingly positive response.
MANNER OF THE SUNDOG
Nearing the end of the 21st century, the world has been devastated by catastrophic climate change, war, disease and overpopulation. Now the most sought after of all continents to live, Antarctica, has developed into a vastly inhabited and civilized oasis with its own cities, communities, cultures and native population known as the Annortics. Still under the control of the United Nations Consultative Parties, this utopian society established in the exclusive city of Ellsworth is governed by an international coalition of scientists known as ICS, more simply referred to as ICE. The governing laws have been set in accordance with the Antarctic treaty, which has continued to ban any exploitation ...
On Brouwer
This book offers a concise, yet comprehensive, introduction to this philosopher's most important ideas.
A Multi-layered Quantitative in Vivo Expression Atlas of the Podocyte Unravels Kidney Disease Candidate Genes
Abstract: Damage to and loss of glomerular podocytes has been identified as the culprit lesion in progressive kidney diseases. Here, we combine mass spectrometry-based proteomics with mRNA sequencing, bioinformatics, and hypothesis-driven studies to provide a comprehensive and quantitative map of mammalian podocytes that identifies unanticipated signaling pathways. Comparison of the in vivo datasets with proteomics data from podocyte cell cultures showed a limited value of available cell culture models. Moreover, in vivo stable isotope labeling by amino acids uncovered surprisingly rapid synthesis of mitochondrial proteins under steady-state conditions that was perturbed under autophagy-deficient, disease-susceptible conditions. Integration of acquired omics dimensions suggested FARP1 as a candidate essential for podocyte function, which could be substantiated by genetic analysis in humans and knockdown experiments in zebrafish. This work exemplifies how the integration of multi-omics datasets can identify a framework of cell-type-specific features relevant for organ health and disease
A Novel Domain Regulating Degradation of the Glomerular Slit Diaphragm Protein Podocin in Cell Culture
Abstract: Mutations in the gene NPHS2 are the most common cause of hereditary steroid-resistant nephrotic syndrome. Its gene product, the stomatin family member protein podocin represents a core component of the slit diaphragm, a unique structure that bridges the space between adjacent podocyte foot processes in the kidney glomerulus. Dislocation and misexpression of slit diaphragm components have been described in the pathogenesis of acquired and hereditary nephrotic syndrome. However, little is known about mechanisms regulating cellular trafficking and turnover of podocin. Here, we discover a three amino acids-comprising motif regulating intracellular localization of podocin in cell culture systems. Mutations of this motif led to markedly reduced degradation of podocin. These findings give novel insight into the molecular biology of the slit diaphragm protein podocin, enabling future research to establish the biological relevance of podocin turnover and localization
When Einstein Walked with Gödel
From Jim Holt, the New York Times bestselling author of Why Does the World Exist?, comes an entertaining and accessible guide to the most profound scientific and mathematical ideas of recent centuries in When Einstein Walked with Gödel: Excursions to the Edge of Thought. Does time exist? What is infinity? Why do mirrors reverse left and right but not up and down? In this scintillating collection, Holt explores the human mind, the cosmos, and the thinkers who’ve tried to encompass the latter with the former. With his trademark clarity and humor, Holt probes the mysteries of quantum mechanics, the quest for the foundations of mathematics, and the nature of logic and truth. Along the way, he...
From Dedekind to Gödel
Discussions of the foundations of mathematics and their history are frequently restricted to logical issues in a narrow sense, or else to traditional problems of analytic philosophy. From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics illustrates the much greater variety of the actual developments in the foundations during the period covered. The viewpoints that serve this purpose included the foundational ideas of working mathematicians, such as Kronecker, Dedekind, Borel and the early Hilbert, and the development of notions like model and modelling, arbitrary function, completeness, and non-Archimedean structures. The philosophers discussed include not only the household names in logic, but also Husserl, Wittgenstein and Ramsey. Needless to say, such logically-oriented thinkers as Frege, Russell and Gödel are not entirely neglected, either. Audience: Everybody interested in the philosophy and/or history of mathematics will find this book interesting, giving frequently novel insights.
Autonomy Platonism and the Indispensability Argument
Mathematical platonism is the view that mathematical statements are true of real mathematical objects like numbers, shapes, and sets. One central problem with platonism is that numbers, shapes, sets, and the like are not perceivable by our senses. In contemporary philosophy, the most common defense of platonism uses what is known as the indispensability argument. According to the indispensabilist, we can know about mathematics because mathematics is essential to science. Platonism is among the most persistent philosophical views. Our mathematical beliefs are among our most entrenched. They have survived the demise of millennia of failed scientific theories. Once established, mathematical the...
