Recent zbMATH articles in MSC 00A30https://www.zbmath.org/atom/cc/00A302021-04-16T16:22:00+00:00WerkzeugBook review of: E. Casari, Bolzano's logical system.https://www.zbmath.org/1456.000392021-04-16T16:22:00+00:00"Sebestik, Jan"https://www.zbmath.org/authors/?q=ai:sebestik.janReview of [Zbl 1370.01001].The slowdown theorem: a lower bound for computational irreducibility in physical systems.https://www.zbmath.org/1456.680572021-04-16T16:22:00+00:00"Gorard, Jonathan"https://www.zbmath.org/authors/?q=ai:gorard.jonathanSummary: Following from the work of Beggs and Tucker on the computational complexity of physical oracles, a simple diagonalization argument is presented to show that generic physical systems, consisting of a Turing machine and a deterministic physical oracle, permit computational irreducibility. To illustrate this general result, a specific analysis is provided for such a system (namely a scatter machine experiment (SME) in which a classical particle is scattered by a sharp wedge) and proves that it must be computationally irreducible. Finally, some philosophical implications of these results are discussed; in particular, it is shown that the slowdown theorem implies the existence of classical physics experiments with undecidable observables, as well as the existence of a definite lower bound for the computational irreducibility of the laws of physics. Therefore, it is argued that the hypothesis that ``the universe is a computer simulation'' has no predictive (i.e., only retrodictive) power.Philosophy of mathematics in ancient and modern times.https://www.zbmath.org/1456.010012021-04-16T16:22:00+00:00"Felgner, Ulrich"https://www.zbmath.org/authors/?q=ai:felgner.ulrichThe book under review is based on the author's lectures on the philosophy of mathematics given at the University of Tübingen, Germany. It is centered around two main problems: (1) what is the nature of mathematical objects, in which sense do they exist, (2) which sources are used when mathematicians are proving mathematical theorems, what is the epistemological status of mathematical theorems. The aim of the author (who is a mathematician working mainly in set theory) is to present and analyse (on the basis of texts from antiquity till modern times) answers that have been given to those questions.
The material is divided into three parts. Part I ``Philosophy of mathematics in antiquity'' starts with the description of the very beginnings of mathematics in ancient Greece -- it is told about the discovery of incommensurable quantities and about the appearance of ontological problems. Further conceptions of Plato and Aristotle as well as the \textit{Elements} by Euclid are presented. The next two chapters concern problems with the infinity (distinction between potential and actual infinity, the method of exhaustion, paradoxes of Zeno). Part II is devoted to the philosophy of mathematics in the 17th, 18th and 19th centuries. Here are presented and discussed conceptions of René Descartes (nativism), John Locke (who was against nativism), Thomas Hobbes and Gottfried Wilhelm Leibniz (rationalism), David Hume, George Berkeley, John Stuart Mill (empiricism) as well as Immanuel Kant (critical philosophy). Part III presents the philosophy of mathematics in the 19th and early 20th centuries. The following conceptions and problems are discussed: psychologism, logicism, the concept of a set and set theory, Platonism, constructivism, structuralism, formalism. Every part of the book begins with a summary of the presented conceptions and ends with a discussion and concluding remarks. The entire book is closed by final conclusions in which the author goes back to the two main questions and problems of the philosophy of mathematics indicated at the beginning of the book and attempts to show how modern tools such as axiomatic method, set theory, logic and the idea of a mathematical structure help us to answer those questions.
The book is well written. The language is clear and precise. The description of particular conceptions is illustrated by numerous quotations of the original classical texts. The conceptions are presented in a critical way indicating positive and negative elements of them. The whole book discloses the deep erudition of the author.
Reviewer: Roman Murawski (Poznań)Should mathematicians play dice?https://www.zbmath.org/1456.000492021-04-16T16:22:00+00:00"Berry, Don"https://www.zbmath.org/authors/?q=ai:berry.donald-arthurSummary: It is an established part of mathematical practice that mathematicians demand deductive proof before accepting a new result as a theorem. However, a wide variety of probabilistic methods of justification are also available. Though such procedures may endorse a false conclusion even if carried out perfectly, their robust structure may mean they are actually more reliable in practice once implementation errors are taken into account. Can mathmaticians be rational in continuing to reject these probabilistic methods as a means of establishing a mathematical claim? In this paper, I give reasons in favour of their doing so. Rather than appealing directly to individual epistemological considerations, the discussion offers a normative constraint on what constitutes a good mathematical argument. This I call `Univocality': the requirement that the underlying concepts all have clear defining conditions.