scholarly journals Carnap and Beth on the Limits of Tolerance

2021 ◽  
pp. 1-19
Author(s):  
Benjamin Marschall
Keyword(s):  
Philosophy Of Mathematics ◽  
Mathematical Truth ◽  
Inference Rules ◽  
Linguistic Rules ◽  
New Interpretation

Abstract Rudolf Carnap’s principle of tolerance states that there is no need to justify the adoption of a logic by philosophical means. Carnap uses the freedom provided by this principle in his philosophy of mathematics: he wants to capture the idea that mathematical truth is a matter of linguistic rules by relying on a strong metalanguage with infinitary inference rules. In this paper, I give a new interpretation of an argument by E. W. Beth, which shows that the principle of tolerance does not suffice to remove all obstacles to the employment of infinitary rules.

Naturalism in the Philosophy of Mathematics

Philosophy ◽  
2015 ◽  
Author(s):  
Mary Leng
Keyword(s):  
Natural Science ◽  
Philosophy Of Mathematics ◽  
Sole Source ◽  
Indispensability Argument ◽  
Mathematical Truth ◽  
Methodological Naturalism ◽  
Mathematical Objects ◽  
Ontological Naturalism ◽  
Epistemological Challenge ◽  
Genuine Problem

In the context of the philosophy of mathematics, the term “naturalism” has a number of uses, covering approaches that look to be fundamentally at odds with one another. In one use, the “natural” in naturalism is contrasted with non-natural, in the sense of supernatural; in this sense, naturalism in the philosophy of mathematics appears in opposition to Platonism (the view that mathematical truths are truths about a body of abstract mathematical objects). Naturalism thus construed takes seriously the epistemological challenge to Platonism presented by Paul Benacerraf in his paper “Mathematical Truth” (cited under Ontological Naturalism). Benacerraf points out that a view of mathematics as a body of truths about a realm of abstract objects appears to rule out any (non-mystical) account of how we, as physically located embodied beings, could come to know such truths. The naturalism that falls out of acceptance of Benacerraf’s challenge as presenting a genuine problem for our claims to be able to know truths about abstract mathematical objects is sometimes referred to as “ontological naturalism,” and suggests a physicalist ontology. In a second use, the “natural” in naturalism is a reference specifically to natural science and its methods. Naturalism here, sometimes called methodological naturalism, is the Quinean doctrine that philosophy is continuous with natural science. Quine and Putnam’s indispensability argument for the existence of mathematical objects places methodological naturalism in conflict with ontological naturalism, since it is argued that the success of our scientific theories confirms the existence of the abstract mathematical objects apparently referred to in formulating those theories, suggesting that methodological naturalism requires Platonism. A final use of “naturalism” in the philosophy of mathematics is distinctive to mathematics, and arises out of consideration of the proper extent of methodological naturalism. According to Quine’s naturalism, the natural sciences provide us with the proper methods of inquiry. But, as Penelope Maddy has noted, mathematics has its own internal methods and standards, which differ from the methods of the empirical sciences, and naturalistic respect for the methodologies of successful fields requires that we should accept those methods and standards. This places Maddy’s methodological naturalism in tension with the original Quinean version of the doctrine, because, Maddy argues, letting natural science be the sole source of confirmation for mathematical theories fails to respect the autonomy of mathematics.

Shadows of Syntax

2020 ◽  
Author(s):  
Jared Warren
Keyword(s):  
Logical Pluralism ◽  
Mathematical Truth ◽  
Ontological Commitments ◽  
Linguistic Rules ◽  
Linguistic Conventions ◽  
Popular View ◽  
Big Picture ◽  
Epistemology Of Logic ◽  
And Mathematics ◽  
Algorithmic Process

What is the source of logical and mathematical truth? This volume revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. In Shadows of Syntax, Jared Warren offers the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. He argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims. In Part I, Warren explains exactly what conventionalism amounts to and what linguistic conventions are. Part II develops an unrestricted inferentialist theory of the meanings of logical constants that leads to logical conventionalism. This conventionalist theory is elaborated in discussions of logical pluralism, the epistemology of logic, and of the influential objections that led to the historical demise of conventionalism. Part III aims to extend conventionalism from logic to mathematics. Unlike logic, mathematics involves both ontological commitments and a rich notion of truth that cannot be generated by any algorithmic process. To address these issues Warren develops conventionalist-friendly but independently plausible theories of both metaontology and mathematical truth. Finally, Part IV steps back to address big picture worries and meta-worries about conventionalism. This book develops and defends a unified theory of logic and mathematics according to which logical and mathematical truths are reflections of our linguistic rules, mere shadows of syntax.

Realism in the philosophy of mathematics

2018 ◽  
Author(s):  
Patricia A. Blanchette
Keyword(s):  
Human Activities ◽  
Philosophy Of Mathematics ◽  
Mathematical Truth ◽  
Mathematical Realism ◽  
Mathematical Objects ◽  
Natural Position ◽  
Potential Problems ◽  
The Subject ◽  
Important Form ◽  
Clear Relation

Mathematical realism is the view that the truths of mathematics are objective, which is to say that they are true independently of any human activities, beliefs or capacities. As the realist sees it, mathematics is the study of a body of necessary and unchanging facts, which it is the mathematician’s task to discover, not to create. These form the subject matter of mathematical discourse: a mathematical statement is true just in case it accurately describes the mathematical facts. An important form of mathematical realism is mathematical Platonism, the view that mathematics is about a collection of independently existing mathematical objects. Platonism is to be distinguished from the more general thesis of realism, since the objectivity of mathematical truth does not, at least not obviously, require the existence of distinctively mathematical objects. Realism is in a fairly clear sense the ‘natural’ position in the philosophy of mathematics, since ordinary mathematical statements make no explicit reference to human activities, beliefs or capacities. Because of the naturalness of mathematical realism, reasons for embracing antirealism typically stem from perceived problems with realism. These potential problems concern our knowledge of mathematical truth, and the connection between mathematical truth and practice. The antirealist argues that the kinds of objective facts posited by the realist would be inaccessible to us, and would bear no clear relation to the procedures we have for determining the truth of mathematical statements. If this is right, then realism implies that mathematical knowledge is inexplicable. The challenge to the realist is to show that the objectivity of mathematical facts does not conflict with our knowledge of them, and to show in particular how our ordinary proof-procedures can inform us about these facts.

Taking the History of Mathematics Seriously

Conceptus ◽  
2009 ◽  
Vol 38 (94) ◽  
Author(s):  
Adrian Frey
Keyword(s):  
Historical Development ◽  
Mathematical Knowledge ◽  
Philosophy Of Mathematics ◽  
History Of Mathematics ◽  
Mathematical Truth ◽  
Historical Account ◽  
Historical Turn ◽  
History Of

SummaryKitcher’s philosophy of mathematics rests on the idea that a philosopher who tries to understand mathematical knowledge ought to take its historical development into consideration. In this paper, I take a closer look at Kitcher’s reasons for proposing such a historical turn. I argue that, whereas a historical account is indeed an essential part of the standpoint advanced in The Nature of Mathematical Knowledge, this is no longer the case for the position defended in the manuscript Mathematical Truth? The Wittgensteinian account of mathematics advocated in that manuscript does not force us to take a historical turn.

scholarly journals Ontological Indifference of Theories and Semantic Primacy of Sentences

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Dirk Greimann
Keyword(s):  
Empirical Evidence ◽  
Scientific Theory ◽  
Ontological Commitment ◽  
Philosophy Of Mathematics ◽  
Ontological Commitments ◽  
Ontological Reduction ◽  
Structuralist View ◽  
New Interpretation

Abstract In his late philosophy, Quine generalized the structuralist view in the philosophy of mathematics that mathematical theories are indifferent to the ontology we choose for them. According to his ‘global structuralism’, the choice of objects does not matter to any scientific theory. In the literature, this doctrine is mainly understood as an epistemological thesis claiming that the empirical evidence for a theory does not depend on the choice of its objects. The present paper proposes a new interpretation suggested by Quine’s recently published Kant Lectures from 1980 according to which his global structuralism is a semantic thesis that belongs to his theory of ontological reduction. It claims that a theory can always be reformulated in such a way that its truth does not presuppose the existence of the original objects, but only of some objects that can be considered as their proxies. Quine derives this claim from the principle of the semantic primacy of sentences, which is supposed to license the ontological reductions he uses to establish his global structuralism. It is argued that these reductions do not actually work because they do not account for some hidden ontological commitments that are not detected by his criterion of ontological commitment.

scholarly journals On Gödel's Way In: The Influence of Rudolf Carnap

2005 ◽  
Vol 11 (2) ◽  
pp. 185-193 ◽  
Author(s):  
Warren Goldfarb
Keyword(s):  
Philosophy Of Mathematics ◽  
Vienna Circle ◽  
Early Summer ◽  
Mathematical Truth ◽  
Rudolf Carnap ◽  
Modern Logic ◽  
Kurt Gödel ◽  
Representational Capacity ◽  
The Subject ◽  
The University

The philosopher Rudolf Carnap (1891–1970), although not himself an originator of mathematical advances in logic, was much involved in the development of the subject. He was the most important and deepest philosopher of the Vienna Circle of logical positivists, or, to use the label Carnap later preferred, logical empiricists. It was Carnap who gave the most fully developed and sophisticated form to the linguistic doctrine of logical and mathematical truth: the view that the truths of mathematics and logic do not describe some Platonistic realm, but rather are artifacts of the way we establish a language in which to speak of the factual, empirical world, fallouts of the representational capacity of language. (This view has its roots in Wittgenstein's Tractatus, but Wittgenstein's remarks on mathematics beyond first-order logic are notoriously sparse and cryptic.) Carnap was also the thinker who, after Russell, most emphasized the importance of modern logic, and the distinctive advances it enables in the foundations of mathematics, to contemporary philosophy. It was through Carnap's urgings, abetted by Hans Hahn, once Carnap arrived in Vienna as Privatdozent in philosophy in 1926, that the Vienna Circle began to take logic seriously and that positivist philosophy began to grapple with the question of how an account of mathematics compatible with empiricism can be given (see Goldfarb 1996).A particular facet of Carnap's influence is not widely appreciated: it was Carnap who introduced Kurt Gödel to logic, in the serious sense. Although Gödel seems to have attended a course of Schlick's on philosophy of mathematics in 1925–26, his second year at the University, he did not at that time pursue logic further, nor did the seminar leave much of a trace on him. In the early summer of 1928, however, Carnap gave two lectures to the Circle which Gödel attended, or so I surmise. At these occasions, Carnap presented material from his manuscript treatise, Untersuchungen zur allgemeinen Axiomatik, that is, “Investigations into general axiomatics”, which dealt with questions of consistency, completeness and categoricity. Carnap later circulated this material to various people including Gödel.

Linguistic Competence in Aphasia

2008 ◽  
Vol 17 (3) ◽  
pp. 87-92
Author(s):  
Leonard L. LaPointe
Keyword(s):  
Language Use ◽  
Linguistic Competence ◽  
Careful Study ◽  
Feral Children ◽  
Performance Factors ◽  
Performance Loss ◽  
Linguistic Rules ◽  
Communication Goals ◽  
Aphasia Treatment ◽  
And Performance

Abstract Loss of implicit linguistic competence assumes a loss of linguistic rules, necessary linguistic computations, or representations. In aphasia, the inherent neurological damage is frequently assumed by some to be a loss of implicit linguistic competence that has damaged or wiped out neural centers or pathways that are necessary for maintenance of the language rules and representations needed to communicate. Not everyone agrees with this view of language use in aphasia. The measurement of implicit language competence, although apparently necessary and satisfying for theoretic linguistics, is complexly interwoven with performance factors. Transience, stimulability, and variability in aphasia language use provide evidence for an access deficit model that supports performance loss. Advances in understanding linguistic competence and performance may be informed by careful study of bilingual language acquisition and loss, the language of savants, the language of feral children, and advances in neuroimaging. Social models of aphasia treatment, coupled with an access deficit view of aphasia, can salve our restless minds and allow pursuit of maximum interactive communication goals even without a comfortable explanation of implicit linguistic competence in aphasia.

Explanatory Power for Medical Expert Systems: Studies in the Representation of Causal Relationships for Clinical Consultations

1982 ◽  
Vol 21 (03) ◽  
pp. 127-136 ◽  
Author(s):  
J. W. Wallis ◽  
E. H. Shortliffe
Keyword(s):  
Explanatory Power ◽  
Prototype System ◽  
Inference Rules ◽  
Medical Expert ◽  
Comprehensive Knowledge ◽  
Different Types ◽  
Medical Expert Systems ◽  
Explanation System ◽  
Systems Studies

This paper reports on experiments designed to identify and implement mechanisms for enhancing the explanation capabilities of reasoning programs for medical consultation. The goals of an explanation system are discussed, as is the additional knowledge needed to meet these goals in a medical domain. We have focussed on the generation of explanations that are appropriate for different types of system users. This task requires a knowledge of what is complex and what is important; it is further strengthened by a classification of the associations or causal mechanisms inherent in the inference rules. A causal representation can also be used to aid in refining a comprehensive knowledge base so that the reasoning and explanations are more adequate. We describe a prototype system which reasons from causal inference rules and generates explanations that are appropriate for the user.

Esotericism and the “Coded Word” in Mormonism

2011 ◽  
Vol 2 (1) ◽  
pp. 29-54
Author(s):  
Clyde Forsberg Jr.
Keyword(s):  
Popular Religion ◽  
Latter Day Saints ◽  
Social Commentary ◽  
Joseph Smith ◽  
Playing Field ◽  
Book Of Mormon ◽  
Divine Inspiration ◽  
History Of ◽  
New Interpretation ◽  
New Religion

In the history of American popular religion, the Latter-day Saints, or Mormons, have undergone a series of paradigmatic shifts in order to join the Christian mainstream, abandoning such controversial core doctrines and institutions as polygamy and the political kingdom of God. Mormon historians have played an important role in this metamorphosis, employing a version (if not perversion) of the Church-Sect Dichotomy to change the past in order to control the future, arguing, in effect, that founder Joseph Smith Jr’s erstwhile magical beliefs and practices gave way to a more “mature” and bible-based self-understanding which is then said to best describe the religion that he founded in 1830. However, an “esoteric approach” as Faivre and Hanegraaff understand the term has much to offer the study of Mormonism as an old, new religion and the basis for a more even methodological playing field and new interpretation of Mormonism as equally magical (Masonic) and biblical (Evangelical) despite appearances. This article will focus on early Mormonism’s fascination with and employment of ciphers, or “the coded word,” essential to such foundation texts as the Book of Mormon and “Book of Abraham,” as well as the somewhat contradictory, albeit colonial understanding of African character and destiny in these two hermetic works of divine inspiration and social commentary in the Latter-day Saint canonical tradition.

Mathematical Knowledge and the Interplay of Practices

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Mathematical Knowledge ◽  
A Priori ◽  
Philosophy Of Mathematics ◽  
Advanced Mathematics ◽  
New Approach ◽  
Agent Based ◽  
Epistemology Of Mathematics ◽  
Actual Experience ◽  
Elementary Math ◽  
Mathematical Tradition

This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction in the philosophy of mathematics, the book uses the crucial idea of a continuum to provide an account of the development of mathematical knowledge that reflects the actual experience of doing math and makes sense of the perceived objectivity of mathematical results. Describing a historically oriented, agent-based philosophy of mathematics, the book shows how the mathematical tradition evolved from Euclidean geometry to the real numbers and set-theoretic structures. It argues for the need to take into account a whole web of mathematical and other practices that are learned and linked by agents, and whose interplay acts as a constraint. It demonstrates how advanced mathematics, far from being a priori, is based on hypotheses, in contrast to elementary math, which has strong cognitive and practical roots and therefore enjoys certainty. Offering a wealth of philosophical and historical insights, the book challenges us to rethink some of our most basic assumptions about mathematics, its objectivity, and its relationship to culture and science.

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