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Lagrange’s theory of analytical functions and his ideal of purity of method. Archive for History of Exact Sciences. Forthcoming
"... ABSTRACT. We reconstruct essential features of Lagrange’s theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange’s notions of function and algebraic quantity, and concentrate on powerseries expansions, on the algorithm ..."
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ABSTRACT. We reconstruct essential features of Lagrange’s theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange’s notions of function and algebraic quantity, and concentrate on powerseries expansions, on the algorithm for derivative functions, and the remainder theorem—especially the role this theorem has in solving geometric and mechanical problems. We thus aim to provide a better understanding of Enlightenment mathematics and to show that the foundations of mathematics did not, for Lagrange, concern the solidity of its ultimate bases, but rather purity of method—the generality and internal organization of the discipline. hal00614606, version 1 12 Aug 2011 1. PRELIMINARIES AND PROPOSALS Foundation of mathematics was a crucial topic for 18thcentury mathematicians. A pivotal aspect of it was the interpretation of the algoritihms of the calculus. This was often referred to as the question of the “metaphysics of the calculus ” 1 (see Carnot 1797, as an example). Around 1800 Lagrange devoted two large treatises to the matter, both of which went through two editions in Lagrange’s
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legal responsibility for the information which this document contains or the use to which this information is subsequently put. Although every step is taken to ensure that the information is as accurate as possible, it is understood that this material is supplied on the basis that there is no legal responsibility for these materials or resulting from the use to which these can or may be put. Note: the telephone and fax numbers given in this guide for addresses outside the United Kingdom are those to be used if you are in that country. If you are telephoning or faxing from another country, we suggest you contact your local telecommunications provider for details of the country code and area code that you should use. Main contents
Plural Descriptions and . . .
 MIND
, 2005
"... Russell had two theories of definite descriptions: one for singular descriptions, another for plural descriptions. We chart its development, in which ‘On Denoting’ plays a part but not the part one might expect, before explaining why it eventually fails. We go on to consider manyvalued functions, s ..."
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Russell had two theories of definite descriptions: one for singular descriptions, another for plural descriptions. We chart its development, in which ‘On Denoting’ plays a part but not the part one might expect, before explaining why it eventually fails. We go on to consider manyvalued functions, since they too bring in plural terms—terms such as ‘�4 ’ or the descriptive ‘the inhabitants of London ’ which, like plain plural descriptions, stand for more than one thing. Logicians need to take plural reference seriously if only because mathematicians take manyvalued functions seriously. We assess the objection (by Russell, Frege and others) that manyvalued functions are illegitimate because the corresponding terms are ambiguous. We also assess the various methods proposed for getting rid of them. Finding the objection illfounded and the methods ineffective, we introduce a logical framework that admits plural reference, and use it to answer some earlier questions and to raise some more.
In Defense of the Ideal 2nd DRAFT
"... This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size of α. ..."
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This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size of α. The question of the grounds for doing so is perhaps the central question of the workshop. But I want to discuss another question which, as I said, arises at the periphery: How do we know that there are structures 〈Rα, ∈ ∩R2 α〉? How do we know that there exist such things as sets and how do we know that, given such things, the axioms we write down are true of them? These seem very primitive questions, but the skepticism implicit in them has deep (and ancient) roots. In particular, they are questions about ideal objects in general, and not just about the actual infinite. I want to explain why I think the questions (as intended) are empty and the skepticism unfounded. 1 I will be expanding the argument of the first part of my paper “Proof and truth: the Platonism of mathematics”[1986a]. 2 The argument in question
What is Neologicism? 2 What is Neologicism? ∗
"... Logicism is a thesis about the foundations of mathematics, roughly, that mathematics is derivable from logic alone. It is now widely accepted that ..."
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Logicism is a thesis about the foundations of mathematics, roughly, that mathematics is derivable from logic alone. It is now widely accepted that
(HP) (V) (H 2 P)
"... 2 How groundedness might help • What extensions there are depends on what concepts there are. • What concepts there are depends on what extensions there are. A heavyhanded proposal Require that an abstraction principle be predicative in the sense that its RHS not quantify over the sort of objects t ..."
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2 How groundedness might help • What extensions there are depends on what concepts there are. • What concepts there are depends on what extensions there are. A heavyhanded proposal Require that an abstraction principle be predicative in the sense that its RHS not quantify over the sort of objects to which its LHS refers. A more general idea • Abstraction as a process by which more and more abstracts are ‘introduced’. • Each step of the process ‘presupposes ’ only what was available at the previous step Old news? Asymmetry The righthand side of an abstraction principle must not ‘presuppose ’ any of the objects to which the lefthand side refers. Quantification incurs ‘presupposition ’ (QIP) A quantified statement ‘presupposes ’ every object in the range of its quantifiers. Conclusion Impredicative abstraction principles are impermissible. 1 Can we resist QIP by analyzing ‘presupposition’? • Following Kripke, we consider monotonic operators J • J(X) = Y formally represents the philosophical idea that the elements of Y ‘depend on X’, or X ‘suffices for ’ each element of Y, or the elements of Y ‘presuppose ’ at most X. • Need to spell out clearly what philosophical idea J is meant to track. Two key questions Q1. What are the sets on which J operates (e.g. sets of abstracta, (representations of) identity facts, etc.)? Q2. What is the operator J (e.g. strong Kleene, supervaluational, or Leitgebstyle)? 3 Option 1: Leitgeb on grounded abstraction (Leitgeb, ming) Consider an abstraction principle: nx.F x = nx.Gx ↔ Φ[F, G] We wish to restrict this to the instances that are grounded (in a sense to be articulated). Presumably, this means accept only instances nx.φ(x) = nx.ψ(x) ↔ Φ[φ/F, ψ/G] whose RHS is grounded. The ingredients • A domain Dom of preobjects. • A domain Con ⊆ ℘(Dom) of [pre]concepts on these preobjects. • A [pre] abstraction operator N: Con → Dom, which is a bijection. The recipe • Build up identity and distinctness facts on Dom by iterating a Leitgebjump
Introduction to Absolute Generality
, 2006
"... absolutely everything there is. A cursory look at philosophical practice reveals numerous instances ..."
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absolutely everything there is. A cursory look at philosophical practice reveals numerous instances