What Mathematical Logic Says about the Foundations of Mathematics

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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.

This paper brings up some important points about logic, e.g., mathematical logic, and also an inconsistence in logic as per Gödel's incompleteness theorems which state that there are mathematical truths that are not decidable or provable. These incompleteness theorems have shaken the solid foundation of mathematics where innumerable proofs and theorems have a place of pride. The great mathematician David Hilbert had been much disturbed by them. There are much long unsolved famous conjectures in mathematics, e.g., the twin primes conjecture, the Goldbach conjecture, the Riemann hypothesis, etc. Perhaps, by Gödel's incompleteness theorems the proofs for these famous conjectures will not be possible and the numerous mathematicians attempting to find the solutions for these conjectures are simply banging their heads against the metaphorical wall. Besides mathematics, Gödel's incompleteness theorems will have ramifications in other areas involving logic. This paper looks at the ramifications of the incompleteness theorems, which pose the serious problem of inconsistency, and offers a solution to this dilemma. The paper also looks into the apparent inconsistence of the axiomatic method in mathematics. [Published in international mathematics journal. Acknowledgments: The author expresses his gratitude to the referees and the Editor-in-Chief for their valuable comments in strengthening the contents of this paper.]

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.

In P. Odifreddi (ed.), Kreiseliana. About and around Kreisel, pp. 365-388, A K Peters, 1996

The aim of this paper is not to provide any systematic reconstruction of Kreisel’s views but only to discuss some claims concerning the relationship between mathematical logic and the philosophy of mathematics that repeatedly occur in his writings. Although I do not know to what extent they are representative of his present position, they correspond to widespread views of the logical community and so seem worth discussing anyhow. Such claims will be used as reference to make some remarks about the present state of relations between mathematical logic and the philosophy of mathematics.

Corcoran and Shapiro review “What is Mathematical Logic?” updated PDF. 1976. Crossley, J. N. What is Mathematical Logic? OxfordUP 1972. Philosophy of Science 43, 301–302 (Co-author: S. Shapiro). J This book—still in print—pretends to tell the clueless neophyte what mathematical logic is—and in 82 small pages. The idea, we do not say fact, that someone knowledgeable in the subject thinks this possible is astounding. As we show in this review and in another much longer review, it is unlikely that Oxford University Press even copy-edited the book much less had it vetted by a competent logician. People who think that Oxford University Press has recently lowered its standards should read this book—or at least the reviews. 1978. Crossley on Mathematical Logic (essay review, Co-author: Stewart Shapiro), Philosophia 8, 79–94. 1988. Ensayo-Resenas: Introduciendo La Logica Matematica, Mathesis X, 133–150. Spanish translation by A. Garciadiego of revised version of “Crossley on Mathematical Logic”, Philosophia 8 (1978) 79–94. Co-author: S. Shapiro.