Computing systems are everywhere today. Even the brain is thought to be a sort of computing system. But what does it mean to say that a given organ or system computes? What is it about laptops, ...smartphones, and nervous systems that they are deemed to compute, and why does it seldom occur to us to describe stomachs, hurricanes, rocks, or chairs that way? The book provides an extended argument for the semantic view of computation, which states that semantic properties are involved in the nature of computing systems. Laptops, smartphones, and nervous systems compute because they are accompanied by representations. Stomachs, hurricanes, and rocks, for instance, which do not have semantic properties, do not compute. The first part of the book argues that the linkage between the mathematical theory of computability and the notion of physical computation is weak. Theoretical notions such as algorithms, effective procedure, program, and automaton play only a minor role in identifying physical computation. The second part of the book reviews three influential accounts of physical computation and argues that while none of these accounts is satisfactory, each of them highlights certain key features of physical computation. The final part of the book develops and argues for a semantic account of physical computation and offers a characterization of computational explanations.
Abstract In this essay, we seek to problematize the modes of enunciation and validation of truth from discursive practices that support mathematical rigor. We question the aspects and circumstances ...by which a truth is constituted as such, seeking to intend its operation within mathematics. To this end, the text is divided into three acts that aim to develop the hypothesis that, being rigor a practice, it is an event.
Resumo Neste ensaio, buscamos problematizar os modos de enunciação e validação da verdade a partir de práticas discursivas que sustentam o rigor matemático. Questionamos os aspectos e circunstâncias pelos quais uma verdade se constitui como tal, visando entender seu funcionamento no interior da Matemática. Para tanto, o texto é dividido em três atos, que buscam desenvolver a hipótese de que, sendo o rigor uma prática, ele é um evento.
This open access book is a superb collection of some fifteen chapters inspired by Schroeder-Heister's groundbreaking work, written by leading experts in the field, plus an extensive autobiography and ...comments on the various contributions by Schroeder-Heister himself. For several decades, Peter Schroeder-Heister has been a central figure in proof-theoretic semantics, a field of study situated at the interface of logic, theoretical computer science, natural-language semantics, and the philosophy of language. The chapters of which this book is composed discuss the subject from a rich variety of angles, including the history of logic, the proper interpretation of logical validity, natural deduction rules, the notions of harmony and of synonymy, the structure of proofs, the logical status of equality, intentional phenomena, and the proof theory of second-order arithmetic. All chapters relate directly to questions that have driven Schroeder-Heister's own research agenda and to which he has made seminal contributions. The extensive autobiographical chapter not only provides a fascinating overview of Schroeder-Heister's career and the evolution of his academic interests but also constitutes a contribution to the recent history of logic in its own right, painting an intriguing picture of the philosophical, logical, and mathematical institutional landscape in Germany and elsewhere since the early 1970s. The papers collected in this book are illuminatingly put into a unified perspective by Schroeder-Heister's comments at the end of the book. Both graduate students and established researchers in the field will find this book an excellent resource for future work in proof-theoretic semantics and related areas.
George Berkeley is known as "The Good Bishop," a name celebrating his faith, pastoral ministry and earnest commitment to his philosophical views. To mathematicians, he is known for his agitated ...performance in his 1734 critique of fluxions, The Analyst. That work and its petulant tone were occasioned by (i) his "philo-mathematical" opponents' alleged admonitions on religious mysteries' lack of logical respectability and (ii) what Berkeley saw as a related public appetite for reformist and deist religious movements. This paper questions Berkeley's saintly reputation by outlining the hostility and flippancy in his handling of Jacob Walton's contributions to the debate. Berkeley's interactions with Walton bear many of the hallmarks of contemporary "trolling" culture and this paper emphasises and explains these parallels. Finally, given Berkeley's theological and political agenda in his mathematical interactions, I argue that we should not read his mathematical rhetoric as straightforwardly representing his philosophical views.
If numbers were objects, how could there be human knowledge of number? Numbers are not physical objects: must we conclude that we have a mysterious power of perceiving the abstract realm? Or should ...we instead conclude that numbers are fictions? This book argues that numbers are not objects: they are magnitude properties. Properties are not fictions and we certainly have scientific knowledge of them. Much is already known about magnitude properties such as inertial mass and electric charge, and much continues to be discovered. The book says the same is true of numbers. In the theory of magnitudes, the categorial distinction between quantity and individual is of central importance, for magnitudes are properties of quantities, not properties of individuals. Quantity entails divisibility, so the logic of quantity needs mereology, the a priori logic of part and whole. The three species of quantity are pluralities, continua and series, and the book presents three variants of mereology, one for each species of quantity. Given Euclid’s axioms of equality, it is possible without the use of set theory to deduce the axioms of the natural, real and ordinal numbers from the respective mereologies of pluralities, continua and series. Knowledge and the Philosophy of Number carries out these deductions, arriving at a metaphysics of number that makes room for our a priori knowledge of mathematical reality.
In 1977, the first computer-assisted proof of a mathematical theorem was presented by K. Appel and W. Haken. The proof was met with a lot of criticism from both mathematicians and philosophers. In ...this paper, I present some examples of computer-assisted proofs, including Appel and Haken’s work. Then, I analyze the most famous arguments against the equal acceptance of computer-based and human-based proofs in mathematics and examine the philosophical assumptions behind the presented criticism. In the conclusion, I talk about whether the philosophical assumptions are justified as they are, or one needs to take a specific philosophical position to accept them.
Mathematical proofs are both paradigms of certainty and some of the most explicitly-justified arguments that we have in the cultural record. Their very explicitness, however, leads to a paradox, ...because the probability of error grows exponentially as the argument expands. When a mathematician encounters a proof, how does she come to believe it? Here we show that, under a cognitively-plausible belief formation mechanism combining deductive and abductive reasoning, belief in mathematical arguments can undergo what we call an epistemic phase transition: a dramatic and rapidly-propagating jump from uncertainty to near-complete confidence at reasonable levels of claim-to-claim error rates. To show this, we analyze an unusual dataset of forty-eight machine-aided proofs from the formalized reasoning system Coq, including major theorems ranging from ancient to 21st Century mathematics, along with five hand-constructed cases including Euclid, Apollonius, Hernstein's Topics in Algebra, and Andrew Wiles's proof of Fermat's Last Theorem. Our results bear both on recent work in the history and philosophy of mathematics on how we understand proofs, and on a question, basic to cognitive science, of how we justify complex beliefs.
