## Russell’s Absolutism vs.(?)

### BibTeX

@MISC{Hellman_russell’sabsolutism,

author = {Geoffrey Hellman},

title = {Russell’s Absolutism vs.(?)},

year = {}

}

### OpenURL

### Abstract

Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And Dedekind’s “structuralist” views come in for criticism in the Principles. But, on reflection, Russell also flirted with views very close to a (different) version of structuralism. Main varieties of modern structuralism and their challenges are reviewed, taking account of Russell’s insights. Problems of absolutism plague some versions, and, interestingly, Russell’s critique of Dedekind can be extended to one of them, ante rem structuralism. This leaves modal-structuralism and a category theoretic approach as remaining non-absolutist

### Citations

50 |
Philosophy of Mathematics: Structure and Ontology
- Shapiro
- 1997
(Show Context)
Citation Context ...Four main varieties of modern structuralism are readily identified: (1) set-theoretic (“STS”), based on model theory, (2) structures as sui generis universals (the “ante rem” structuralism of Shapiro =-=[18]-=- and Resnik [15]) (“SGS”), (3) modal-structuralism (“MS”), and (4) an approach based on category theory (“CTS”). These have been described in some detail elsewhere. (See, e.g., [10] and [11].) Here we... |

38 |
Mathematics: Form and Function
- Lane, S
- 1986
(Show Context)
Citation Context ...s the idea of a totality...After each careful delimitation, bigger totalities appear. No set theory and no category theory can encompass them all—and they are needed to grasp what Mathematics does.” (=-=[14]-=-, 390.) That STS violates this “open-endedness” is familiar from its commitment to a fixed “real world” of sets. But SGS also appears to violate it in it commitment to a totality of “all positions in ... |

24 |
1966] “The category of categories as a foundation for mathematics
- Lawvere
- 1965
(Show Context)
Citation Context ...overn categories and topoi themselves?”, “Are modal notions involved?”, “Is an extendability principle recognized, as in MS, or should we take seriously an all-embracing category of categories?” (see =-=[13]-=-), etc. A full-fledged CT version of structuralism should address such questions. On one proposal, ordinary mathematics can be carried out relative to any number of topoi as universes of discourse, an... |

22 |
Mathematics without Numbers: Towards a Modal-Structural Interpretation
- Hellman
- 1989
(Show Context)
Citation Context ...t of a modal-structuralist interpretation, which simply prefixes the above with a necessity operator, as governed by a suitable modal logic (naturally chosen to be S-5, with certain restrictions, see =-=[8]-=-, Ch. 1, and [9]). The same procedure generalizes to analysis and many extensions, including set theories. Furthermore, as Russell clearly recognized, the axiom of infinity, however it is stated preci... |

19 |
Stetigkeit und irrationale Zahlen
- Dedekind
- 1872
(Show Context)
Citation Context ... built out of pure sets, or properties, etc., if recognized, I have called SGS a “hyperplatonist” view of mathematical structure.) This procedure has its most distinguished antecedent in Dedekind 10s(=-=[5]-=- and [6]), as is made clear from correspondence with Weber. 4 The natural numbers (or the reals, the topic of the Weber correspondence) form a unique, particular system over and above all other simply... |

12 |
The consistency of Frege’s foundations of arithmetic
- Boolos
- 1987
(Show Context)
Citation Context ...ust number theory and classical analysis, the advantages of the Frege-Russell definition can be realized in a demonstrably consistent second-order system invented by Boolos called “Frege Arithmetic”. =-=[3]-=- (The demonstration is relative to the consistency of second-order Peano Arithmetic, also called “classical analysis” in a formal sense.) 3sWe naturally think that the class of couples (for example) i... |

12 | 2003] “Does Category Theory Provide a Framework for
- Hellman
(Show Context)
Citation Context ...f Shapiro [18] and Resnik [15]) (“SGS”), (3) modal-structuralism (“MS”), and (4) an approach based on category theory (“CTS”). These have been described in some detail elsewhere. (See, e.g., [10] and =-=[11]-=-.) Here we recall briefly their leading characteristics. STS goes back to the Bourbaki and today would appeal to model theory, with ZF as the background, as providing general concepts of mathematical ... |

11 |
From absolute to local mathematics, Synthese
- Bell
- 1986
(Show Context)
Citation Context ...ld address such questions. On one proposal, ordinary mathematics can be carried out relative to any number of topoi as universes of discourse, and this can be done without a set-theoretic background. =-=[1]-=- The result is a kind of relativity of ordinary mathematical concepts, and a distinction then arises between invariant mathematics (e.g. obeying intuitionistic logic, arising naturally inside a topos)... |

11 |
What Numbers Could Not Be”, Philosophical Review 74
- Benacerraf
- 1965
(Show Context)
Citation Context ...umbers cannot really be sets, 14ssince many progressions of sets are equally available to serve as natural numbers, and it would be absurd to say we are really speaking of one as opposed to any other.=-=[2]-=- Indeed, he generalized the argument to conclude that natural numbers cannot “really be” objects at all, and here, with ante rem structures, we have another example of why not. Hyperplatonist abstract... |

7 |
Structuralism without structures
- Hellman
- 1996
(Show Context)
Citation Context ...ucturalist interpretation, which simply prefixes the above with a necessity operator, as governed by a suitable modal logic (naturally chosen to be S-5, with certain restrictions, see [8], Ch. 1, and =-=[9]-=-). The same procedure generalizes to analysis and many extensions, including set theories. Furthermore, as Russell clearly recognized, the axiom of infinity, however it is stated precisely, while form... |

7 |
The Identity Problem for Realist Structuralism
- Keränen
- 2001
(Show Context)
Citation Context ...” should be individuated sufficiently by intra-structural relations (including functions) alone, without help “from outside” or from individual constants. (The reasons for this are given in detail in =-=[12]-=-.) This yields a version of Leibniz’ “identity of indiscernibles”: any items bearing exactly the same intra-structural relations to other items must be not many but one. But this immediately implies t... |

5 |
Truth and proof: the Platonism of mathematics, Synthese 69
- Tait
- 1986
(Show Context)
Citation Context ...s as defined by structural relationships specified in the mathematics itself. (Tait calls the passage from particular realizations, e.g. set theoretic, to such pure archetypes “Dedekind abstraction”. =-=[19]-=-) Now one may immediately question the idea of “purity”. How can any objects fail to have mathematically irrelevant properties, such as arise from adventitious relations to external things, e.g. “bein... |

3 |
The Principles of Mathematics (London: Allen and Unwin
- Russell
- 1903
(Show Context)
Citation Context ...any assigned progression is what all progressions have in common...His demonstrations nowhere–not even when he comes to cardinals–involve any property distinguishing numbers from other progressions. (=-=[16]-=-, 249.) Nevertheless, ante rem structures are posited in SGS, in apparent defiance of these strictures. But perhaps it can be conceded that places in structures “must be intrinsically something”, but ... |

2 |
2001] “Three Varieties of Mathematical Structuralism”, Philosophia Mathematica 9: 184-211. Hellman, G. [2003] “Does Category Theory Provide a Framework for Mathematical Structuralism?” Philosophia Mathematica 11
- Hellman
(Show Context)
Citation Context ...uralism of Shapiro [18] and Resnik [15]) (“SGS”), (3) modal-structuralism (“MS”), and (4) an approach based on category theory (“CTS”). These have been described in some detail elsewhere. (See, e.g., =-=[10]-=- and [11].) Here we recall briefly their leading characteristics. STS goes back to the Bourbaki and today would appeal to model theory, with ZF as the background, as providing general concepts of math... |

2 |
MathematicsasaScienceofPatterns(Oxford
- Resnik
- 1997
(Show Context)
Citation Context ...ies of modern structuralism are readily identified: (1) set-theoretic (“STS”), based on model theory, (2) structures as sui generis universals (the “ante rem” structuralism of Shapiro [18] and Resnik =-=[15]-=-) (“SGS”), (3) modal-structuralism (“MS”), and (4) an approach based on category theory (“CTS”). These have been described in some detail elsewhere. (See, e.g., [10] and [11].) Here we recall briefly ... |

1 | Review of Stewart Shapiro - Burgess - 1997 |

1 |
Gesammelte mathematische Werke 3
- Dedekind
- 1932
(Show Context)
Citation Context ...e rem structures are posited in SGS, in apparent defiance of these strictures. But perhaps it can be conceded that places in structures “must be intrinsically something”, but that all that need 4 See =-=[7]-=-, vol. 3, 489-90. There is also, however, a letter to Lipschitz (dated June 10, 1876 [7] §65, cf. [18], 173) saying, “if one does not want to introduce new numbers, I have nothing against this...” Whi... |

1 |
Introduction to Mathematical Philosophy (New
- Russell
- 1919
(Show Context)
Citation Context ...tes, The question, ‘What is number?’, is one which has been often asked but has only been correctly answered in our own time. The answer was given by Frege in 1884, in his Grundlagen der Arithmetik. (=-=[17]-=-, 11.) Here we have a good expression of an absolutist stance: There is such a thing as the correct answer to the question, “What is number?”, and, moreover, it is (essentially) the one Frege gave. (C... |