Foundations of Mathematics from the Perspective of Computer Verification (original) (raw)

Philosophy of Mathematics: Making a Fresh Start

Studies in History and Philosophy of Science, vol. 44 (2013), pp. 32-42., 2013

The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment of the method of mathematics, the notion of demonstration, the questions of discovery and justification, the nature of mathematical objects, the character of mathematical definition, the role of intuition, the role of diagrams in mathematics, and the effectiveness of mathematics in natural science.

Reflections on the Foundations of Mathematics

2019

We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (section 3). On the way our odyssey from the foundations to the "horizon" of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander.

Modernizing the philosophy of mathematics

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.

Proofs and Programs

Synthese, 2003

2 fondational problem. The goal was to give rigor or logico-formal foundations to all mathematical activities. In spite of the failures of the Hilbert's initial project, the ideas of logicians and mathematicians like Turing, Kleene, Church, Gödel, Herbrand... set the bases for Computer Science. Turing Machines, later developed also by Von Neumann, have formed the paradigm of the first computers and of languages known as imperative programming (based on orders like 'do', 'go to'...). The formal systems for computing we will talk about had above all an influence on functional programming and logical programming, which are recent styles of programming and differ from the imperative style. Moreover, they are linked even more directly to the developments of mathematical logic since the '30s. Mathematical logic, in its 'metamathematical' analysis aspect, has Mathematics themselves as its object of study, its languages and its deductive methods, like geometry, to take a purely mathematical discipline as an example, has as its object of study figures and structures of space. One can therefore imagine rather artificial but convenient a three-level stratification, which has organised, with the passing of this century, the 'mathematical discourse': the geometric-algebraic structures, the mathematical theories which study them (algebra, geometry...) and, finally, metatheories that deal with mathematical theories and where one may develop a 'theory of proofs' (one will also try to see the limits of this 'organisation of mathematical discourse'). In other words, from the point of view of computations and languages, linear algebra and analytic geometry, for instance, study the expressions that represent lines on a plane or surfaces in space; λ−calculus, as a language of Proof Theory, manipulates words or expressions that represent formal proofs. In fact, expressions of this language codify abstract mathematical proofs and, therefore computations carried out on them correspond to formal operations on proofs, rather than on the lines or on the surfaces. The fact that λ-calculus is programmable, and that in fact it is a paradigmatic programming language, allows to describe the passage from Proof Theory, as an abstract theory in mathematical logic, to automated proofs and symbolic calculus, as mathematical methods in computing. Although we will speak further on of Proof Theory, we will underline here the 'constructivist' approach in order to present 'proofs as lambda-terms' and study the computer version of provability. While using λ−calculus, we will mention the role of Category Theory in the mathematical semantics of deduction and of formal programming languages (see below). Indeed, the results that link the different sectors of Mathematics and Computer Science place λ−calculus and Combinatory Logic, an equivalent system, at the meeting point of vast sectors of Logic and their applications, providing these theories with an importance that goes beyond their origins as a system for calculability or effective provability. As we have said, the two theories, which have as a common base the 'algebraic calculus without variables' of Shoenfinkel dating from the early Twenties and that are owed to Church and Curry (1928-1936), essentially proposed to 'formalise'

Some consequences of defining mathematical objects constructively and mathematical truth effectively

Arxiv preprint math/0210078, 2002

Standard interpretations of classical first order theory - rooted primarily in the works of Cantor, Goedel, Tarski, and Turing - argue that the truth of the propositions of a formal mathematical language, under an interpretation, is non-algorithmic and essentially unverifiable constructively. In this essay we consider some arguments for, and consequences of, an interpretation of classical foundational concepts in which such truth is defined effectively.