Review of In The Light of Logic by Solomon Feferman

arXiv: History and Overview, 2014

My purpose is to examine some concepts of mathematical logic, which have been studied by Carlo Cellucci. Today the aim of classical mathematical logic is not to guarantee the certainty of mathematics, but I will argue that logic can help us to explain mathematical activity; the point is to discuss what and in which sense logic can "explain". For example, let's consider the basic concept of an axiomatic system: an axiomatic system can be very useful to organize, to present, and to clarify mathematical knowledge. And, more importantly, logic is a science with its own results: so, axiomatic systems are interesting also because we know several revealing theorems about them. Similarly, I will discuss other topics such as mathematical definitions, and some relationships between mathematical logic and computer science. I will also consider these subjects from an educational point of view: can logical concepts be useful in teaching and learning elementary mathematics?

haverford.edu

Catalyzed by the failure of Frege's logicist program, logic and the foundations of mathematics first became a philosophical topic in Europe in the early years of the twentieth century. Frege had aimed to show that logic constitutes the foundation for mathematics in the sense of providing both the primitive concepts in terms of which mathematical concepts were to be defined and the primitive truths on the basis of which mathematical truths were to be proved. 1 Russell's paradox showed that the project could not be completed, at least as envisaged by Frege. It nevertheless seemed clear to many that mathematics must be founded on something, and over the first few decades of the twentieth century four proposals emerged: two species of logicism, namely ramified type theory as developed in Russell and Whitehead's Principia and Zermelo-Frankel set theory, Hilbert's finitist program (a species of formalism), and finally Brouwerian intuitionism. 2 Across the Atlantic, already by the time Russell had discovered his famous paradox, the great American pragmatist Charles Sanders Peirce was developing a radically new non-foundationalist picture of mathematics, one that, through the later influence of Quine, Putnam, and Benacerraf, would profoundly shape the course of the philosophy of mathematics in the United States.

Among sciences, mathematics has a unique relation to philosophy. Many philosophers have taken mathematics to be the paradigm of knowledge, and the reasoning employed in some mathematical proofs are often regarded as rational thought, however mathematics is also a rich source of philosophical problems which have been at the centre of epistemology and metaphysics. Since antiquity 1 , philosophers have envied their ideas as the model of mathematical perfection because of the clarity of its concepts and the certainty of its conclusions. Many efforts should be done if we strive to elaborate philosophy and foundation of mathematics. Philosophy of mathematics covers the discussion of ontology of mathematics, epistemology of mathematics, mathematical truth, and mathematical objectivity. While the foundation of mathematics engages with the discussion of ontological foundation, epistemological foundation which covers the schools of philosophy such as platonism, logicism, intuitionism, formalism, and structuralism.

Mathematics starts by introducing some simple concepts (in the founding cycle), which may seem self-sufficient. It is natural to start with a set theory that is not fully formalized as an axiomatic theory. It is usually presented as a popularized or implicit version of ZFC, admitting its axioms as necessary or obvious.

2010

The Philosophical Quarterly, 2004

Since virtually every mathematical theory can be interpreted in set theory, the latter is a foundation for mathematics. Whether set theory, as opposed to any of its rivals, is the right foundation for mathematics depends on what a foundation is for. One purpose is philosophical, to provide the metaphysical basis for mathematics. Another is epistemic, to provide the basis of all mathematical knowledge. Another is to serve mathematics, by lending insight into the various fields. Another is to provide an arena for exploring relations and interactions between mathematical fields, their relative strengths, etc. Given the different goals, there is little point to determining a single foundation for all of mathematics.

Reflections on the Foundations of Mathematics, 2019

This note opens with brief evaluations of classical foundationalist endeavors – those of Frege, Russell, Brouwer and Hilbert. From there we proceed to some pluralist approaches to foundations, focusing on Putnam and Wittgenstein, making a note of what enables their pluralism. Then, I bring up approaches that find foundations potentially harmful, as expressed by Rav and Lakatos. I conclude with a brief discussion of a late medieval Indian case study (Śaṅkara’s and Nārāyaṇa’s Kriyākramakarī) in order to show what an “unfounded” mathematics could look like. The general purpose is to re-evaluate the desiderata of foundational programs in mathematics.

Metascience, 2012

Essays on the foundations of mathematics and logic, 2005

Is there still a Sense in which Mathematics can have Foundations? Jody Azzouni Abstract. An analysis of traditional mathematical proof is undertaken, with an im-plicit contrast to formal derivations. The semantic interpretation of mathematical terms plays a role in the former ...

Springer eBooks, 2021

Logic, Epistemology, and the Unity of Science aims to reconsider the question of the unity of science in light of recent developments in logic. At present, no single logical, semantical or methodological framework dominates the philosophy of science. However, the editors of this series believe that formal frameworks, for example, constructive type theory, deontic logics, dialogical logics, epistemic logics, modal logics, and proof-theoretical semantics, have the potential to cast new light on basic issues in the discussion of the unity of science. This series provides a venue where philosophers and logicians can apply specific systematic and historic insights to fundamental philosophical problems. While the series is open to a wide variety of perspectives, including the study and analysis of argumentation and the critical discussion of the relationship between logic and philosophy of science, the aim is to provide an integrated picture of the scientific enterprise in all its diversity. This book series is indexed in SCOPUS.