On some first-order theories which can be represented by definitions. (Report at the conference) (original) (raw)
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.
The Palgrave Centenary Companion to Principia Mathematica
From 1914, when Behmann first lectured on Principia in Göttingen, to 1930, when Gödel proved the incompleteness of its system, Principia Mathematica played a large role in the development of modern metatheory. ii The Principia system, with its explicit axiomatic approach to the fundamental principles of logic, was just what was needed in the early years of the 20 th century to make possible the precise formulation and treatment of meta-logical questions. One might have thought, then, that at least by the time of finishing his work on Principia, Russell would have been in just the right position to appreciate such straightforward metatheoretical issues as those of the completeness and soundness of a logical system, of the independence of its axioms, and so on. But, notoriously, he seems curiously far removed from anything like modern metatheory. Russell never formulates a completeness theorem, or even raises anything like a modern completeness question about his system. He even seems strangely confused about what we now take to be an entirely straightforward method of proving the independence of logical axioms. In Principles of Mathematics, Russell remarks that [W]e require certain indemonstrable propositions, which hitherto I have not succeeded in reducing to less than ten. Some indemonstrables there must be; and some propositions, such as the syllogism, must be of the number, since no demonstration is possible without them. But concerning others, it may be doubted whether they are indemonstrable or merely undemonstrated; and it should be observed that the method of supposing an axiom false, and deducing the consequences of this assumption, which has been found admirable in such cases as the axiom of parallels, is here not universally available. For all our axioms are principles of deduction; and if they are true, the consequences which appear to follow from the employment of an opposite principle will not really follow, so that arguments from the supposition of the falsity of an axiom are here subject to special fallacies. Thus the number of indemonstrable propositions may be capable of further reduction, and in regard to some of them I know of no grounds for regarding them as indemonstrable except that they have hitherto remained undemonstrated. iii This view, that we can't use standard methods to demonstrate the independence of logical axioms, is one that Russell maintains up to and including the period of writing Principia. iv Why doesn't Russell, apparently well placed to appreciate modern metatheoretical questions and techniques, ever raise, employ, or even appear to understand them? One answer to this question has been proposed by a group of scholars including Burt Dreben and Jean van Heijenoort, Warren Goldfarb, and Tom Ricketts. To quote the first pair: [N]either in the tradition in logic that stemmed from Frege through Russell and Whitehead, that is, logicism, nor in the tradition that stemmed from Boole through Peirce and Schröder, that is, algebra of logic, could the question of the completeness of a formal system arise. For Frege, and then for Russell and Whitehead, logic was universal: within each explicit formulation of logic all deductive reasoning, including all of classical analysis and much of Cantorian set theory, was to be formalized. Hence not only was pure quantification theory never at the center of their attention, but metasystematic questions as such, for example the question of completeness,
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.
Mark Wilson: Which Came First: The Logic or the Math? / Response: Mathematics and Logic
Manuscrito, 2008
Abstract Mark Wilson: Many authors, including Oswaldo Chateaubriand, maintain that "properties" should be structured in logical grades, where the least abstract quantities comprise the lowest ranks of a hierarchy that embraces more abstract and mathematized qualities only at higher levels. But applied mathematicians warns that no quantities can be expected to possess crisp, real world extensions unless they have already been processed with a fair amount of set theoretic machinery beforehand. Abstract response: Mark Wilson argues that in order to make physical first-order properties suitable for inclusion in the bottom levels of a logical hierarchy of properties, their proper treatment must take into account the methods of applied mathematics. I agree that the methods of applied mathematics are essential for studying physical properties, and in my response focus on the nature of the logical hierarchy and on the requirements of classical logic.
Mathematical logic: What has it done for the philosophy of mathematics?
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.
Logic in general and mathematical logic in particular
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.]
Corcoran and Shapiro review “What is Mathematical Logic?” updated PDF.
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.
2010
Middle age cosmology produced demonstrably false predictions applying logical reasoning to experimentally unsubstantiated assumptions. To resolve this problem, empiricism was developed as a protocol to extract general statements, mostly in the form of mathematical equations, from gathered experimental results. In the midst of the Religious Reformation, criticism on middle age Catholic cosmology went too far to dismiss logical reasoning as metaphysics. Ironically it was the further development of physics itself which gave rise to Cantorian Set Theory which brought logic back to the centre of mathematical science. George Cantor, in his attempt to show the uniqueness of the Fourier expansion developed abstract set theory. Fourier calculus was a most powerful mathematical tool for physics and engineering in the late 19Th century. Define a set to be well-founded if there is no infinite descending chain of membership relations starting from any element of the set. Let W be the set of all well-founded sets. Is W well-founded? If yes then W W. So, we have an infinite chain of membership relation. W W W. So, we have an infinite descending chain of membership relation in W. Thus W is not well-founded. If W is not well-founded, then it contains at least one element from which an infinite chain of membership relation descend. So, it is not the set of all well-founded set. This paradox due to Miramanov establishes that Cantorian Set Theory is logically inconsistent. As set theory was considered to be the foundation of mathematics, the discovery of inconsistency of this theory sent a shock wave through mathematics community. This gave rise to the formalism of David Hilbert. His idea was to formalise mathematics, foundation of mathematics in particular, so that we could prove its consistency.
WHAT ARE MATHEMATICAL LOGICS? Let ML= (G, D, S) be a mathematical logic with G for grammar, D, for deductive system and S or semantics. The grammar G is often studied separately from D and S and purely grammatical theorems are proved –e.g., that every sentence is uniquely decomposable according to a certain mode of decomposition, that sentencehood in G is decidable and so on. When the deductive system D is studied apart from the semantics one gets purely proof-theoretic results - e.g., that every deduction is equivalent to a deduction which does not use a given rule of inference, that not all sentences are provable, and so on. When the semantics S is studied apart from the deductive system, one gets purely model-theoretic or semantic results - e.g., that every infinite interpretation is equivalent to a countable one, that certain sets of sentences are or are not satisfiable, that all interpretations satisfying a given set of sentences are isomorphic and so on. Results which relate the deductive system to the semantics have been called bridge results, by Addison and other the members of the Berkeley School (but this terminology has not caught on). Examples of bridge results are completeness and soundness theorems. A deductive system can only be complete or sound relative to a semantics. It is only a logic as a whole that can be complete or sound. Corcoran 1973 [“Gaps”] pages 28-29 TRUNCATIONS Semi-logics DL = (G, D), SL = (G, S), and the relational systems RS studied in “universal logic” cannot be regarded as adequate for most contemporary purpose because they lack bridge phenomena. Consider RS = (P A) where P is a set of propositions and A is a set of premise-conclusion arguments whose component propositions are taken from P. EXPANDSIONS For bridge theorems we need two sets of arguments, e.g. R2S = (P, A, B) where P is a set of propositions , A is the set of DEDUCIBLE premise-conclusion arguments whose component propositions are taken from P, and B is the set of VALID premise-conclusion arguments whose component propositions are taken from P. In fact even the full relational systems derived from mathematical logics are not sufficient for contemporary purposes: three or more sets of arguments are required.
Logic as a Methodological Discipline (penultimate)
Synthese, 2021
This essay offers a conception of logic by which logic may be considered to be exceptional among the sciences on the backdrop of a naturalistic outlook. The conception of logic focused on emphasises the traditional role of logic as a methodology for the sciences, which distinguishes it from other sciences that are not methodological. On the proposed conception, the methodological aims of logic drive its definitions and principles, rather than the description of scientific phenomena. The notion of a methodological discipline is explained as a relation between disciplines or practices. Logic serves as a methodological discipline with respect to any theoretical practice, and this generality, as well as logic's reflexive nature, distinguish it from other methodological disciplines. Finally, the evolution of model theory is taken as a case study, with a focus on its methodological role. Following recent work by John Baldwin and Juliette Kennedy, we look at model theory from its inception in the mid-twentieth century as a foundational endeavour until developments at the end of the century, where the classification of theories has taken centre-stage.
On some first-order theories which can be represented by definitions V.A. Smirnov was interested and aware of the latest advances in logic. The relations between theories was one of his favorite themes. He has dedicated his latest book "The logical methods of analysis of scientific knowledge" to this issue. Currently, the axiomatic approach dominates in the construction of theories at which we postulate a number of non-logical axioms, and then study their properties. Accepted theory postulates restrict the set of its possible interpretations. It is believed that the postulates play the role of implicit definitions of included in them descriptive terms of language. VA Smirnov was interested in a possible redundancy of language of theories, where not all the terms are independent, and some of them may be defined by other descriptive terms. The redundancy of the language is useful for ease of work in theory, but in thinking about what we have actually postulated in theory, it can be misleading. Therefore, the problem of reduction of the original set of descriptive terms of theories is important not only for philosophers but also for scientists working in specific areas of science.
Hilbert, logicism, and mathematical existence
Synthese, 2008
David Hilbert's early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind's footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the shape and evolution of Hilbert's foundational ideas, including his early contributions to the foundations of geometry and the real number system. Most interestingly, the context of Dedekind-style logicism makes it possible to offer a new analysis of the emergence of Hilbert's famous ideas on mathematical existence. And a careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets (and of the dichotomic conception of set theory) in Hilbert's early axiomatics, and detailed analyses of Hilbert's paradox and of his completeness axiom (Vollständigkeitsaxiom). It is well known that in the address 'Axiomatisches Denken' (1918) Hilbert expressed great interest in the work of Frege and Russell, praising their "magnificent enterprise" of the axiomatization of logic, and saluting its "completion … as the crowning achievement of the work of axiomatization as a whole" 1 (1918, 1113). His high praise not only aimed at formal logic, but more particularly at logicism, as the remarkable statement that anteceded the above phrases made clear: since the examination of consistency is a task that cannot be avoided, it appears necessary to axiomatize logic itself and to prove that number theory and set theory are only parts of logic. (Hilbert 1918, 412; emphasis mine) This constitutes obvious endorsement of the logicistic viewpoint, but usually it is regarded as a short-lived outburst of enthusiasm, perhaps arising from study of the Principia Mathematica. 2 In a summer course in 1920, Hilbert had become agnostic with regards to logicism, and in his publications of the following decade he clearly opposed that conception. Indeed, he had already expressed serious doubts in a well-known talk given in August 1904 (published as Hilbert 1905). That this usual reconstruction is inadequate has been made clear by several unpublished documents, especially those that are coming to light thanks to the Hilbert Edition. For one thing, 1 Hilbert immediately went on: "However, this completion will still require new and many-sided work," which makes it clear that the "crowing achievement" was not Russell's work itself but rather its promised completion.