'Algebraic' Approaches to Mathematics

2010

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.

Synthese, 1991

The distinction between analytic and synthetic propositions, and with that the distinction between a priori and a posteriori truth, is being abandoned in much of analytic philosophy and the philosophy of most of the sciences. These distinctions should also be abandoned in the philosophy of mathematics. In particular, we must recognize the strong empirical component in our mathematical knowledge. The traditional distinction between logic and mathematics, on the one hand, and the natural sciences, on the other, should be dropped. Abstract mathematical objects, like transcendental numbers or Hilbert spaces, are theoretical entities on a par with electromagnetic fields or quarks. Mathematicai theories are not primarily logical deductions from axioms obtained by reflection on concepts but, rather, are constructions chosen to solve some collection of problems while fitting smoothly into the other theoretical commitments of the mathematician who formulates them. In other words, a mathematical theory is a scientific theory like any other, no more certain but also no more devoid of content.

This was a talk at "Theology in Mathematics?" (Kraków, Poland, June 8-10, 2014). The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern in the second part of this paper is the early-twentieth-century foundational crisis of mathematics. The hypothesis that pure mathematics partially fulfilled the functions of theology at that time is tested on the views of the leading figures of the three main foundationalist programs: Russell, Hilbert and Brouwer.

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.

I am not the autor, it is a collection of essays edited by Hilary Putnam and Paul Benacerraf. The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field.

2009

We shall assume throughout this essay that platonism1 regarding the subject matter of mathematics is correct. This is not to say that arguments for platonism–that is, arguments for the existence of abstract objects such as numbers, sets, Hilbert spaces, and so on that comprise the subject matter of mathematics–are either uninteresting or unneeded. Nevertheless, I am not personally interested in such arguments.