Mathematical structuralism and the Identity of Indiscernibles

Analysis ◽  
2005 ◽  
Vol 65 (3) ◽  
pp. 218-221 ◽  
Author(s):  
J. Ladyman
Keyword(s):  
Identity Of Indiscernibles ◽  
Mathematical Structuralism

Indiscernibility

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Infinitary Logic ◽  
Identity Of Indiscernibles ◽  
Sheer Number

This chapter explores Leibniz's principle of the Identity of Indiscernibles. Model theory supplies us with the resources to distinguish between many different notions of indiscernibility; we can vary: (a) the primitive ideology (b) the background logic and (c) the grade of discernibility. We use these distinctions to discuss the possibility of singling-out “indiscernibles”. And we then use these to distinctions to explicate Leibniz's famous principle. While model theory allows us to make this principle precise, the sheer number of different precise versions of this principle made available by model theory can serve to mitigate some of the initial excitement of this principle. We round out the chapter with two technical topics: indiscernibility in infinitary logic, and the relation between indiscernibility, orders, and stability.

Indistinguishable elements and mathematical structuralism

Analysis ◽  
2007 ◽  
Vol 67 (294) ◽  
pp. 112-116 ◽  
Author(s):  
José Luis Bermúdez
Keyword(s):  
Mathematical Structuralism

I.—THE IDENTITY OF INDISCERNIBLES

Mind ◽  
1952 ◽  
Vol LXI (242) ◽  
pp. 153-164 ◽  
Author(s):  
MAX BLACK
Keyword(s):  
Identity Of Indiscernibles

scholarly journals Metaphysical Rationalism

2019 ◽  
pp. 164-181
Author(s):  
Martin Lin
Keyword(s):  
Sufficient Reason ◽  
Existence Of God ◽  
Identity Of Indiscernibles ◽  
Principle Of Sufficient Reason ◽  
Limited Scope

This chapter explores the meaning of Spinoza’s Principle of Sufficient Reason (the PSR) and the role it plays in his system. Some commentators have argued that Spinoza’s PSR applies to every truth and that Spinoza relies on it in deriving a great deal of his system. Against such interpretations, this chapter argues that Spinoza’s PSR is restricted to existential truths and is applied only once by Spinoza, to the case of the existence of God. In making this case, it considers Spinoza’s arguments for necessitarianism, causal and conceptual dependence, and the identity of indiscernibles, and it concludes that none of them rely on the PSR. It further argues that the limited scope of Spinoza’s PSR is a philosophical advantage because a fully unrestricted PSR is an unattractive doctrine that creates demands for explanation that cannot be met.

Analysis and the identity of indiscernibles

Analysis ◽  
1982 ◽  
Vol 42 (2) ◽  
pp. 80-82
Author(s):  
W. Godwin
Keyword(s):  
Identity Of Indiscernibles

scholarly journals Methodological Frames: Paul Bernays, Mathematical Structuralism, and Proof Theory

2020 ◽  
pp. 352-382
Author(s):  
Wilfried Sieg
Keyword(s):  
Proof Theory ◽  
Incompleteness Theorem ◽  
Mathematical Structures ◽  
General Characterization ◽  
Mathematical Structuralism ◽  
Mathematical Experience

Mathematical structuralism is deeply connected with Hilbert and Bernays’s proof theory and its programmatic aim to ensure the consistency of all of mathematics. That aim was to be reached on the basis of finitist mathematics. Gödel’s second incompleteness theorem forced a step from absolute finitist to relative constructivist proof-theoretic reductions. This mathematical step was accompanied by philosophical arguments for the special nature of the grounding constructivist frameworks. Against that background, this chapter examines Bernays’s reflections on proof-theoretic reductions of mathematical structures to methodological frames via projections. However, these reflections are focused on narrowly arithmetic features of frames. Drawing on broadened meta-mathematical experience, this chapter proposes a more general characterization of frames that has ontological and epistemological significance. The characterization is given in terms of accessibility: domains of objects are accessible if their elements are inductively generated, and principles for such domains are accessible if they are grounded in our understanding of the generating processes.

scholarly journals Carnap’s Structuralist Thesis

2020 ◽  
pp. 383-420
Author(s):  
Georg Schiemer
Keyword(s):  
Present Article ◽  
Structural Properties ◽  
Philosophy Of Mathematics ◽  
Mathematical Structure ◽  
Mathematical Structuralism ◽  
Definition Of ◽  
Philosophical Debates ◽  
Implicit Definitions

This chapter investigates Carnap’s structuralism in his philosophy of mathematics of the 1920s and early 1930s. His approach to mathematics is based on a genuinely structuralist thesis, namely that axiomatic theories describe abstract structures or the structural properties of their objects. The aim in the present article is twofold: first, to show that Carnap, in his contributions to mathematics from the time, proposed three different (but interrelated) ways to characterize the notion of mathematical structure, namely in terms of (i) implicit definitions, (ii) logical constructions, and (iii) definitions by abstraction. The second aim is to re-evaluate Carnap’s early contributions to the philosophy of mathematics in light of contemporary mathematical structuralism. Specifically, the chapter discusses two connections between his structuralist thesis and current philosophical debates on structural abstraction and the on the definition of structural properties.

Sign in / Sign up

Export Citation Format

Share Document