Abstraction and Identity

Cook, Roy T.; Ebert, Philip A.

doi:10.1111/j.1746-8361.2005.1023.x

Cited by 24 publications

(16 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If we assume, in addition, that there are infinitely many non-sets we can prove infinity (thanks to the fact that all urelemente are now in the basis). Note that obtaining infinitely many non-sets may be straightforward for the abstractionist, if the cardinal numbers governed by, say, Finite Hume,46 are not identical to extensions-that is, if we can suppose a favorable solution regarding the question of the identity of abstracta governed by di¯erent abstraction principles (Cook (2003b, §10); see also Fine (2002), Cook and Ebert (2005), Mancosu (2015b)). Moreover, once again, foundation holds for pure sets.…”

Section: Abstractionism and Neo-fregeanismmentioning

confidence: 99%

Introduction to Abstractionism

Ebert¹,

Rossberg²

2016

Abstractionism

Self Cite

View full text Add to dashboard Cite

fi r s t , u n c o r r e c t e d p r o o f-f o r c o p y-e d i t i n g o n l y what we would call its graph today. In the special case of concepts, the value-range is the extension of the concept. Concepts F and G have the same extensions if and only they are co-extensional (i.e., the same objects fall under them):-"F " =-"G" $ 8x(F x $ G x) 3 As we now know, if embedded in a weaker logic-e.g., predicative second-order logic-Basic Law V does not entail a contradiction. This has generated some very interesting research on identifying consistent fragments of Frege's Grundgesetze logic that retain Basic Law V-for further details see §1.3 below. For all we know, Frege never considered a weakening of the logic as a way out of the paradox. Indeed, it might seem to go against Frege's general conception of logic. 4 In his excellent critical survey, MacBride (2003) distinguishes "neo-logicism" from "neo-Fregeanism". Neo-logicism stands for "the doctrine that Frege's judgement was premature [.. .] Frege should not have abandoned (HP)" (p. 106) while "neo-Fregeanism" stands for the general conception of the relation between language and reality that Hale and Wright are interpreted to have adopted. We here use the term "neo-Fregeanism" to stand for Hale and Wright's version of Abstractionism generally. 5 HP may be glossed as: the cardinal number belonging to the concept F is identical to the cardinal number belonging to the concept G if and only if there is a one-to-one correspondence between the objects falling under F and those falling under G. The equivalence relation of equinumerosity (one-to-one correspondence, bijection) can be formulated in purely (second-order) logical vocabulary. In full detail, HP is the following statement: N x : F x = N x : G x $ 9R 8x[F x 9y(Gy^Rxy^8z(Gz^Rx z z = y))]8 y[Gy 9x(F x^Rxy^8z(F z^Rzy z = x))] 42 Union also holds for hereditary sets (see fn. 40 above), but foundation does not. 43 Let us call the non-sets urelemente. It is tempting to think of the basis as the collection of urelemente, but there is in fact no guarantee that it contains all or even only urelemente. 44 Interestingly, in this setting, SOAP can be dispensed with: instead of the ordinals provided by SOAP, the stages can be ordered according to the more "conventional" ordinals (transitive pure sets, well-ordered by '2') supplied by New V, but some complicating adjustments in New V and Newer V are required; see Cook (2003b, p. 90, n. 30).

show abstract

Section: Abstractionism and Neo-fregeanismmentioning

confidence: 99%

Introduction to Abstractionism

Ebert¹,

Rossberg²

2016

Abstractionism

Self Cite

View full text Add to dashboard Cite

show abstract

“…Cook's system, therefore, no longer seems to count as a form of neologicism. The second concern is that the system as a whole is subject to the form of the Julius Caesar problem that Cook and Ebert [2005] call the 'C-R' problem. The C-R problem is that when more than one abstraction principle is added to second-order logic, it is not clear how to prove that the abstractions introduced by one principle are identical with those of another.…”

Section: ])mentioning

confidence: 99%

What is Neologicism?

Linsky¹,

Zalta²

2006

Bull. symb. log.

View full text Add to dashboard Cite

Logicism is a thesis about the foundations of mathematics, roughly, that mathematics is derivable from logic alone. It is now widely accepted that the thesis is false and that the logicist program of the early 20th century was unsuccessful. Frege's [1893/1903] system was inconsistent and the Whitehead and Russell [1910–1913] system was not thought to be logic, given its axioms of infinity, reducibility, and choice. Moreover, both forms of logicism are in some sense non-starters, since each asserts the existence of objects (courses of values, propositional functions, etc.), something which many philosophers think logic is not supposed to do. Indeed, the tension in the idea underlying logicism, that the axioms and theorems of mathematics can be derived as theorems of logic, is obvious: on the one hand, there are numerous existence claims among the theorems of mathematics, while on the other, it is thought to be impossible to prove the existence of anything from logic alone. According to one well-received view, logicism was replaced by a very different account of the foundations of mathematics, in which mathematics was seen as the study of axioms and their consequences in models consisting of the sets described by Zermelo-Fraenkel set theory (ZF). Mathematics, on this view, is just applied set theory.

show abstract

“…The neo-logicist, on the other hand, in rejecting the recipe-and a single domain of primitive objects generally-in favor of a multitude of distinct abstraction principles describing distinct (yet possibly overlapping) domains of mathematical objects, 44 The name is a play on the familiar phrase "the Caesar problem", and refers to the specific case of determining whether the real numbers R generated by one abstraction principle are identical to a sub-collection of the complex numbers C given by a distinct abstraction principle. For a fuller discussion of this problem, see (Cook and Ebert, 2005) and more recently (Mancosu, 2015). 45 We do not mean to imply that settling whether two extensions in a non-well-founded theory of extensions such as that found within Grundgesetze (or consistent sub fragments of Grundgesetze is trivial, effective, etc.…”

Section: The Definitional Strategy and Neo-logicismmentioning

confidence: 99%

Frege's Recipe

Cook

Ebert

2016

Journal of Philosophy

Self Cite

View full text Add to dashboard Cite

In this paper, we present a formal recipe that Frege followed in his magnum opus “Grundgesetze der Arithmetik” when formulating his definitions. This recipe is not explicitly mentioned as such by Frege, but we will offer strong reasons to believe that Frege applied it in developing the formal material of Grundgesetze. We then show that a version of Basic Law V plays a fundamental role in Frege’s recipe and, in what follows, we will explicate what exactly this role is and explain how it differs from the role played by extensions in his earlier book “Die Grundlagen der Arithmetik”. Lastly, we will demonstrate that this hitherto neglected yet foundational aspect of Frege’s use of Basic Law V helps to resolve a number of important interpretative challenges in recent Frege scholarship, while also shedding light on some important differences between Frege’s logicism and recent neo-logicist approaches to the foundations of mathematics

show abstract

Abstraction and Identity

Cited by 24 publications

References 20 publications

Introduction to Abstractionism

Introduction to Abstractionism

What is Neologicism?

Frege's Recipe

Contact Info

Product

Resources

About