Results 11  20
of
41
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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
Amending Frege’s Grundgesetze der Arithmetik
, 2002
"... Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, secondorder logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative secondorder fragment of the Grundgeset ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, secondorder logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative secondorder fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in this extended system.
What is Frege’s Theory of Descriptions?
"... When prompted to consider Frege’s views about definite descriptions, many philosophers think about the meaning of proper names, and some of them can cite the following quotation taken from a footnote Frege’s 1892 article “ Über Sinn und Bedeutung.”2 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
When prompted to consider Frege’s views about definite descriptions, many philosophers think about the meaning of proper names, and some of them can cite the following quotation taken from a footnote Frege’s 1892 article “ Über Sinn und Bedeutung.”2
Introduction to Absolute Generality
, 2006
"... absolutely everything there is. A cursory look at philosophical practice reveals numerous instances ..."
Abstract
 Add to MetaCart
absolutely everything there is. A cursory look at philosophical practice reveals numerous instances
(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 ..."
Abstract
 Add to MetaCart
(Show Context)
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
BENACERRAF’S DILEMMA AND INFORMAL MATHEMATICS
"... Abstract. This paper puts forward and defends an account of mathematical truth, and in particular an account of the truth of mathematical axioms. The proposal attempts to be completely nonrevisionist. In this connection, it seeks to satisfy simultaneously both horns of Benacerraf’s dilemma. The acco ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. This paper puts forward and defends an account of mathematical truth, and in particular an account of the truth of mathematical axioms. The proposal attempts to be completely nonrevisionist. In this connection, it seeks to satisfy simultaneously both horns of Benacerraf’s dilemma. The account builds upon Georg Kreisel’s work on informal rigour. Kreisel defends the view that axioms are arrived at by a rigorous examination of our informal notions, as opposed to being stipulated or arrived at by trial and error. This view is then supplemented by a Fregean account of the objectivity and our knowledge of abstract objects. It is then argued that the resulting view faces no insurmountable metaphysical or epistemic obstacles. §1. Introduction. Benacerraf’s (1973) paper ‘Mathematical truth ’ presents a problem with which any position in the philosophy of mathematics must come to terms. Benacerraf’s paper is often seen as presenting a dilemma where common sense seems to pull in opposite directions. Common sense with respect to the truth and the syntactical form of mathematical statements leads us to conclude that mathematical propositions concern
From Lagrange to Frege: Functions and Expressions
, 2011
"... Part 1 I of Frege’s Grundgesetze is devoted to the “exposition [Darlegung] ” of his formal system. It opens with the following claim ([34], § I.1, p. 5; [40], p. 33) 2: When one is concerned with specifying the original reference [Bedeutung] of the word ‘function’ in its mathematical usage, it is e ..."
Abstract
 Add to MetaCart
(Show Context)
Part 1 I of Frege’s Grundgesetze is devoted to the “exposition [Darlegung] ” of his formal system. It opens with the following claim ([34], § I.1, p. 5; [40], p. 33) 2: When one is concerned with specifying the original reference [Bedeutung] of the word ‘function’ in its mathematical usage, it is easy to fall into calling function of x an expression [Ausdruck] formed from ‘x ’ and particular numbers by means of the notations [Bezeichnungen] for sum, product, power, difference, and so on. This is inappropriate [unzutreffend] because in this way a function is depicted [hingestellt] as an expression—that is, as a concatenation of signs [Verbindung von Zeichen]—not as what is designated [Bezeichnete] thereby. Hence, instead of ‘expression’, one should say ‘reference of an expression’. Frege does not explicitly ascribe this inappropriateness to anyone, though he could have ascribed it to many 3. One is Lagrange, which, a little less than one century later, defined functions as follows, both in the Théorie des fonctions analytiques and in the Leçons sur le calcul des fonctions ([58], § 1, p. 1;
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 ..."
Abstract
 Add to MetaCart
(Show Context)
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