- Submit
-
Browse
- All Categories
- Metaphysics and Epistemology
- Value Theory
- Science, Logic, and Mathematics
- Science, Logic, and Mathematics
- Logic and Philosophy of Logic
- Philosophy of Biology
- Philosophy of Cognitive Science
- Philosophy of Computing and Information
- Philosophy of Mathematics
- Philosophy of Physical Science
- Philosophy of Social Science
- Philosophy of Probability
- General Philosophy of Science
- Philosophy of Science, Misc
- History of Western Philosophy
- Philosophical Traditions
- Philosophy, Misc
- Other Academic Areas
- More
Switch to: Citations
References in:
Inferentialism and Structuralism: A Tale of Two Theories
Logique Et Analyse 61 (244):489-512 (2018)
Add references
You must login to add references.
|
|
|
The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify (...) |
|
|
Such a conception, says Dummett, will form "a base camp for an assault on the metaphysical peaks: I have no greater ambition in this book than to set up a base ... |
|
|
A precise notion of ‘mathematical structure’ other than that given by model theory may prove fruitful in the philosophy of mathematics. It is shown how the language and methods of category theory provide such a notion, having developed out of a structural approach in modern mathematical practice. As an example, it is then shown how the categorical notion of a topos provides a characterization of ‘logical structure’, and an alternative to the Pregean approach to logic which is continuous with the (...) |
|
|
|
|
|
In “The Runabout Inference Ticket” AN Prior (1960) examines the idea that logical connectives can be given a meaning solely in virtue of the stipulation of a set of rules governing them, and thus that logical truth/consequence. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
In this study two strands of inferentialism are brought together: the philosophical doctrine of Brandom, according to which meanings are generally inferential roles, and the logical doctrine prioritizing proof-theory over model theory and approaching meaning in logical, especially proof-theoretical terms. |
|
|
After a brief flirtation with logicism around 1917, David Hilbertproposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays andWilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for everstronger and more comprehensive areas of mathematics, and finitisticproofs of consistency of these systems. Early advances in these areaswere made by Hilbert (and Bernays) in a series of lecture courses atthe (...) |
|
|
"Robert Brandom" is one of the most significant philosophers writing today, yet paradoxically philosophers have found it difficult to get to grips with the details and implications of his work. This book aims to facilitate critical engagement with Brandom's ideas by providing an accessible overview of Brandom's project and the context for an initial assessment. Jeremy Wanderer's examination focuses on Brandom's inferentialist conception of rationality, and the core part of this conception that aims to specify the structure that a set (...) |
|
|
|
|
|
Some authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a non-trivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one of them is true of (...) |
|
|
There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or is it an exception to the (...) |
|
|
1. It seems plausible to say that a language is a system of expressions the use of which is subject to certain rules. It would seem, thus, that learning to use a language is learning to obey the rules for the use of its expressions. However, taken as it stands, this thesis is subject to an obvious and devastating refutation. After formulating this refutation, I shall turn to the constructive task of attempting to restate the thesis in a way which (...) |
|
|
|
|
|
I raise an objection to Stewart Shapiro's version of ante rem structuralism: I show that it is in conflict with mathematical practice. Shapiro introduced so-called ‘finite cardinal structures’ to illustrate features of ante rem structuralism. I establish that these structures have a well-known counterpart in mathematics, but this counterpart is incompatible with ante rem structuralism. Furthermore, there is a good reason why, according to mathematical practice, these structures do not behave as conceived by Shapiro's ante rem structuralism. |
|
|
|
|
|
|
|
|
This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics (...) |
|
|
Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...) |
|
|
Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...) |
|
|
This volume examines the notion of an analytic proof as a natural deduction, suggesting that the proof's value may be understood as its normal form--a concept with significant implications to proof-theoretic semantics. |
|
|
|
|
|
The perennial question – What is meaning? – receives many answers. In this paper I present and discuss inferentialism – a recent approach to semantics based on the thesis that to have ( such and such ) a meaning is to be governed by ( such and such ) a cluster of inferential rules . I point out that this thesis presupposes that looking for meaning requires seeing language as a social institution (rather than, say, a psychological reality). I also (...) |
|
|
|
|
|
Basic to Robert Brandom’s project in Making It Explicit is the demarcation of singular terms according to the structure of their inferential roles---rather than, as is usual, according to the kinds of things they purport to denote. But the demarcational effort founders on the need to distinguish extensional and nonextensional occurrences of expressions in terms of inferential roles; the closest that an inferentialist can come to drawing that distinction is to discern degrees of extensionality, and that is not close enough. (...) |
|
|
Basic to Robert Brandom’s project in Making It Explicit is the demarcation of singular terms according to the structure of their inferential roles—rather than, as is usual, according to the kinds of things they purport to denote. But the demarcational effort founders on the need to distinguish extensional and nonextensional occurrences of expressions in terms of inferential roles; the closest that an inferentialist can come to drawing that distinction is to discern degrees of extensionality, and that is not close enough. (...) |
|
|
|
|
|
This paper has two goals. The first goal is to show that the structuralists’ claims about dependence are more significant to their view than is generally recognized. I argue that these dependence claims play an essential role in the most interesting and plausible characterization of this brand of structuralism. The second goal is to defend a compromise view concerning the dependence relations that obtain between mathematical objects. Two extreme views have tended to dominate the debate, namely the view that all (...) |
|
|
|
|
|
|
|
|
Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ... |
|
|
|
|
|
|
|
|
The later Wittgenstein’s perspective on mathematical sentences as norms is motivated for sentences belonging to Hilbertian axiomatic systems where the axioms are treated as implicit definitions. It is shown that in this approach the axioms are employed as norms in that they function as standards of what counts as using the concepts involved. This normative dimension of their mode of use, it is argued, is inherited by the theorems derived from them. Having been motivated along these lines, Wittgenstein’s perspective on (...) |
|
|
|
|
|
Inferentialism is the conviction that to be meaningful in the distinctively human way, or to have a 'conceptual content', is to be governed by a certain kind of inferential rules. The term was coined by Robert Brandom as a label for his theory of language; however, it is also naturally applicable (and is growing increasingly common) within the philosophy of logic. |
|
|
|
|
|
|
|
|
With the rise of multiple geometries in the nineteenth century, and in the last century the rise of abstract algebra, of the axiomatic method, the set-theoretic foundations of mathematics, and the influential work of the Bourbaki, certain views called “structuralist” have become commonplace. Mathematics is seen as the investigation, by more or less rigorous deductive means, of “abstract structures”, systems of objects fulfilling certain structural relations among themselves and in relation to other systems, without regard to the particular nature of (...) |
|
|
|
|
|
|
PhilPapers logo by Andrea Andrews and Meghan Driscoll.
This site uses cookies and Google Analytics (see our terms & conditions for details regarding the privacy implications). Use of this site is subject to terms & conditions.
All rights reserved by The PhilPapers Foundation
Server: philpapers-web-5ff6c685cb-rskd8 uwo