Definition Of Real Numbers Research Articles

FREGE’S CONSTRAINT AND THE NATURE OF FREGE’S FOUNDATIONAL PROGRAM

AbstractRecent discussions on Fregean and neo-Fregean foundations for arithmetic and real analysis pay much attention to what is called either ‘Application Constraint’ ($AC$) or ‘Frege Constraint’ ($FC$), the requirement that a mathematical theory be so outlined that it immediately allows explaining for its applicability. We distinguish between two constraints, which we, respectively, denote by the latter of these two names, by showing how$AC$generalizes Frege’s views while$FC$comes closer to his original conceptions. Different authors diverge on the interpretation of$FC$and on whether it applies to definitions of both natural and real numbers. Our aim is to trace the origins of$FC$and to explore how different understandings of it can be faithful to Frege’s views about such definitions and to his foundational program. After rehearsing the essential elements of the relevant debate (§1), we appropriately distinguish$AC$from$FC$(§2). We discuss six rationales which may motivate the adoption of different instances of$AC$and$FC$(§3). We turn to the possible interpretations of$FC$(§4), and advance a Semantic$FC$(§4.1), arguing that while it suits Frege’s definition of natural numbers (4.1.1), it cannot reasonably be imposed on definitions of real numbers (§4.1.2), for reasons only partly similar to those offered by Crispin Wright (§4.1.3). We then rehearse a recent exchange between Bob Hale and Vadim Batitzky to shed light on Frege’s conception of real numbers and magnitudes (§4.2). We argue that an Architectonic version of$FC$is indeed faithful to Frege’s definition of real numbers, and compatible with his views on natural ones. Finally, we consider how attributing different instances of$FC$to Frege and appreciating the role of the Architectonic$FC$can provide a more perspicuous understanding of his foundational program, by questioning common pictures of his logicism (§5).

The Epistemological Question of the Applicability of Mathematics

The question of the applicability of mathematics is an epistemological issue that was explicitly raised by Kant, and which has played different roles in the works of neo-Kantian philosophers, before becoming an essential issue in early analytic philosophy. This paper will first distinguish three main issues that are related to the application of mathematics: (1) indispensability arguments that are aimed at justifying mathematics itself; (2) philosophical justifications of the successful application of mathematics to scientific theories; and (3) discussions on the application of real numbers to the measurement of physical magnitudes. A refinement of this tripartition is suggested and supported by a historical investigation of the differences between Kant’s position on the problem, several neo-Kantian perspectives (Helmholtz and Cassirer in particular, but also Otto Hölder), early analytic philosophy (Frege), and late 19th century mathematicians (Grassmann, Dedekind, Hankel, and Bettazzi). Finally, the debate on the cogency of an application constraint in the definition of real numbers is discussed in relation to a contemporary debate in neo-logicism (Hale, Wright and some criticism by Batitksy), in order to suggest a comparison not only with Frege’s original positions, but also with the ideas of several neo-Kantian scholars, including Hölder, Cassirer, and Helmholtz.

Rescuing Poincaré from Richard’s Paradox

Poincaré in a 1909 lecture in Göttingen proposed a solution to the apparent incompatibility of two results as viewed from a definitionist perspective: on the one hand, Richard’s proof that the definitions of real numbers form a countable set and, on the other, Cantor’s proof that the real numbers make up an uncountable class. Poincaré argues that, Richard’s result notwithstanding, there is no enumeration of all definable real numbers. We apply previous research by Luna and Taylor on Richard’s paradox, indefinite extensibility and unrestricted quantification to evaluate Poincaré’s proposal. We emphasize that Poincaré’s solution involves an early recourse to indefinite extensibility and argue that his proposal, if it is to completely avoid Richard’s paradox, requires rejecting absolutely unrestricted quantification: Richard’s paradox provides a context in which paradox seems inescapable if unrestricted quantification is possible. In proposing his solution to the apparent conflict between Richard’s and Cantor’s results, Poincaré employs temporal expressions whose exact meaning he does not clarify. We suggest an interpretation of these expressions in terms of order of availability and briefly discuss its explanatory power in topics like paradoxes, limitation theorems and indefinite extensibility.

Axioms of real numbers, and the beginning teaching of the analysis

This paper provides an overview of the definitions of real numbers. We pose the axiomatic definition of correctness question. Provided examples of sets that are consistent with some definitions of real numbers, but not in with us usual set of real numbers. Attention is drawn to the mathematical analysis of teaching beginning and claims proof of clearance.

A tale of two algorithms

Real numbers are often a missing link in mathematical education. The standard working assumption in calculus courses is that there exists a system of ‘numbers’, extending the rational number system, adequate for measuring continuous quantities. Moreover, that such ‘numbers’ are in one-to-one correspondence with points on a ‘number line’. But typically real ‘numbers’ are not systematically presented via any constructive method. While taken for granted, they are one of the most commonly used mathematical objects. This paper proposes a geometric algorithm, extending the long division algorithm, which leads to a constructive definition of real numbers. It proceeds to describe a direct algorithm for adding ‘real numbers’. Combined use of the two algorithms enables a smooth and meaningful presentation, offering a double image (geometric and numerical) of real numbers in decimal notation. An early such presentation is of both conceptual and practical importance.

Russell’s Logicism

_Russell_ journal (home office): E:CPBRRUSSJOURTYPE2402\REVIEWS.242 : 2005-05-19 13:37 eviews RUSSELL’S LOGICISM I. G-G Stefano Donati. I fondamenti della matematica nel logicismo di Bertrand Russell [The Foundations of Mathematics in the Logicism of Bertrand Russell]. Florence: Atheneum, . Pp. . .. he measure of Italian writings on Russell’s logic and philosophy has been Tmodest; but now we have the longest essay on his logic in any language. Many of the mathematical topics are treated, as well as the much better known philosophical ones; the standard of referencing of texts is high. Yet the book is also among the most perplexing histories of my acquaintance. Rather than following a synchronic treatment that tracks the many interacting ways in which Russell’s theories changed over time, the author treats the subject diachronically , taking a topic or theme from the s or s onwards and sometimes up later writings, and then not always in chronological order. In addition, several significant topics and aspects are largely or wholly passed over. After the introduction, Chapter  (pp. –) describes Cantorian set theory , especially the treatment of finite and transfinite arithmetic. This is an important influence upon Russell that is often poorly treated by Russellians. Curiously, only one historical work in and around Cantor is cited. After establishing this major figure, one would expect to read next about an even greater influence on Russell: Peano, and the growth of mathematical logic. Instead Chapter  (pp. –) treats “The Russellian Foundation of [Finite] Arithmetic”, where Peano features only for his axioms; there is little historical account of the chief originator of mathematical logic, and nothing on his important followers Pieri and Padoa. Russell’s own definition of integers in terms of equivalent (well-ordered) classes is done in some detail, including comparisons with Frege and Peano’s theories. Chapter  (pp. –) treats Russell’s “extension” to real and complex numbers (including the relation numbers), and some of their relationships to geometry. The second definition of real numbers (positive and negative) in Principia Mathematica, usually ignored, is duly noted. Chapter  (pp. –) deals with Russell’s “crisis”, and covers not only the paradoxes and his attempted solutions but also the axiom of choice. It is nice to see the latter topic given the space that it deserves; the author might have stressed more on page  that Russell seems to have slightly anticipated russell: the Journal of Bertrand Russell Studies n.s.  (winter –): – The Bertrand Russell Research Centre, McMaster U.  - _Russell_ journal (home office): E:CPBRRUSSJOURTYPE2402\REVIEWS.242 : 2005-05-19 13:37  Reviews Zermelo in its discovery. Now come two jumbo chapters:  “Towards the Ramified Theory of Types” (pp. –), and  on the definitive versions (pp. –). The account begins with the first detailed discussion of Russell’s Principles of Mathematics (), with Donati’s book nearly half over. Chapter  then proceeds through the theory of denoting and goes up to the three possible solutions of the paradoxes that Russell was entertaining by . One solution was the substitutional theory, which Russell pursued for some time; the author reviews both it and the differing opinions of its merits that have been aired by historians in recent times. (However, the paradox that Russell found within it is not considered.) The next chapter starts with the vicious-circle principle and examines the type theories in the  paper and that (those?) given in Principia. An important limitation, indeed refutation, of logicism is missed on page ; since only a finite number of types can be defined, the upper end of the sequence of Cantor ’s transfinite numbers is not definable, so that the territory of the paradoxes of the greatest cardinal and ordinal cannot even be approached. The chapter concludes with various “objections”, such as queries over the dubious axioms. Zermelo’s axiomatization of set theory comes here, though it is hardly an objection to Russell but an alternative approach to the foundations of set theory. Finally comes Chapter , on “The Logicism of Russell” (pp. –). Inevitably this matter has turned up earlier: the concern here is with its most general features, and with the book manuscript of  on epistemology that Russell came to abandon. Also considered are some positions taken later by Gödel and Quine; the...