On some first-order theories which can be represented by definitions. (Report at the conference) (original) (raw)
Related papers
What Mathematical Logic Says about the Foundations of Mathematics
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?
Logic and the Foundations of 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.
Russell's Conception of Logic in the Principles of Mathematics
Word count: 15,657 I hereby declare that the attached piece of written work is my own work and that I have not reproduced, without acknowledgement, the work of another. This work has not been accepted in any previous application for any other degree.
Notre Dame Journal of Formal Logic, 2000
You'll be pleased to know that I don't intend to use these remarks to comment on all of the papers presented at this conference. I won't try to show that one paper was right about this topic, that another was wrong was about that topic, or that several of our conference participants were talking past one another. Nor will I try to adjudicate any of the discussions which took place in between our sessions. Instead, I'll use these remarks to make two simple points: one about logicism in the 20th century and one about neo-logicism here at the start of the 21st.
2021
We argue that logicism, the thesis that mathematics is reducible to logic and analytic truths, is true. We do so by (a) developing a formal framework with comprehension and abstraction principles, (b) giving reasons for thinking that this framework is part of logic, (c) showing how the denotations for terms and predicates of a mathematical theory can be viewed as logical objects that exist in the framework, and (d) showing how each theorem of a mathematical theory can be given a true reading in the logical framework. ∗This paper originated as a presentation that the third author prepared for the 31st Wittgenstein Symposium, in Kirchberg, Austria, August 2008. Discussions between the co-authors after this presentation led to a collaboration on, and further development of, the thesis and the technical material grounding the thesis. The authors would especially like to thank Daniel Kirchner for suggesting important refinements of the technical development. We’d also like to thank . . ....
On First-order Theories Which Can Be Represented by Definitions
In the paper we consider the classical logicism program restricted to first-order logic. The main result of this paper is the proof of the theorem, which contains the necessary and sufficient conditions for a mathematical theory to be reducible to logic. Those and only those theories, which don't impose restrictions on the size of their domains, can be reduced to pure logic. Among such theories we can mention the elementary theory of groups, the theory of combinators (combinatory logic), the elementary theory of topoi and many others. It is interesting to note that the initial formulation of the problem of reduction of mathematics to logic is principally insoluble. As we know all theorems of logic are true in the models with any number of elements. At the same time, many mathematical theories impose restrictions on size of their models. For example, all models of arithmetic have an infinite number of elements. If arithmetic was reducible to logic, it would had finite models, including an one-element model. But this is impossible in view of the axiom 0 ̸ = x ′ .
Innovations in the History of Analytical Philosophy, 2017
One of the most significant events in the emergence of analytic philosophy was Bertrand Russell's rejection of the idealist philosophy of mathematics contained in his An Essay on the Foundations of Geometry [EFG] (1897) in favor of the logicism that he developed and defended in his Principles of Mathematics [POM] (1903). Russell's language in POMand in the sharp and provocative paper, "Recent Work on the Principles of Mathematics" (1901a), which is the first public presentation of many of the doctrines of POM-is deliberately revolutionary. "The proof that all pure mathematics, including Geometry, is nothing but formal logic, is a fatal blow to Kantian philosophy" (1901a, 379)-including, of course, the Kantian philosophy that he defended just four years earlier in EFG. "The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age" (POM, §4), a discovery that will initiate a new era in philosophy: "[T]here is every reason to hope that the near
Mathematics, Logic, and their Philosophies
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.
Russell's reasons for logicism
Journal of the History of Philosophy, 2006
What is at stake for Russell in espousing logicism? I argue that Russell's aims are chiefly epistemological and mathematical in nature. Russell develops logicism in order to give an account of the character of mathematics and of mathematical knowledge that is compatible with what he takes to be the uncontroversial status of this science as true, certain and exact. I argue for this view against the view of Peter Hylton, according to which Russell uses logicism to defend the unconditional truth of mathematics against various Idealist positions that treat mathematics as true only partially or only relative to a particular point of view.