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Dedekind's Mathematical Structuralism: From Galois Theory to Numbers, Sets, and Functions [chapter]

José Ferreirós, Erich H. Reck
2020 The Prehistory of Mathematical Structuralism  
After that, the essayturns to Dedekind's more "foundational" contributions, i.e., his writings on the real numbers, the natural numbers, and set theory.  ...  This includes his resolute acceptance of actually infinite systems, understood within a "logical" framework, and studied not just axiomatically, but also in terms of isomorphisms and related notions (since  ...  Acknowledgment We would like to thank Georg Schiemer for helpful comments on an earlier version of this essay.  ... 
doi:10.1093/oso/9780190641221.003.0003 fatcat:mteiyd43s5dkbbiaconnykeaua

A new kind of numbers, the Non-Dedekindian Numbers, and the extension to them of the notion of algorithmic randomness [article]

Gavriel Segre
2007 arXiv   pre-print
A new number system, the set of the non-Dedekindian numbers, is introduced and characterized axiomatically.  ...  As a particular case, the notion of algorithmic randomness for the particular hyperreal number system of Non-Standard Analysis is explicitly analyzed.  ...  Such a similarity naturally induces to investigate which kind of different numbers' systems we obtain by replacing Dedekind's Axiom with different choices in the imposed cardinality of the intersection  ... 
arXiv:math/0612590v3 fatcat:xkpotj7fhnhl7pnkseyujtjf5a

"If Numbers Are to Be Anything At All, They Must Be Intrinsically Something": Bertrand Russell and Mathematical Structuralism [chapter]

Jeremy Heis
2020 The Prehistory of Mathematical Structuralism  
Bertrand Russell was one of the first philosophers to recognize clearly the philosophically innovative nature of Richard Dedekind's philosophy of arithmetic: a position we now describe as non-eliminative  ...  But Russell's response was deeply negative: "If [numbers] are to be anything at all, they must be intrinsically something" (Principles of Mathematics, §242).  ...  Thanks also to Bahram Assadian and to audiences at the University of California, Riverside, and the Logic Seminar at the University of California, Irvine, for very helpful feedback and criticisms.  ... 
doi:10.1093/oso/9780190641221.003.0012 fatcat:yztvpo7fajdnxkqwjtwd2fv4de

Irrational numbers in English language textbooks, 1890–1915: Constructions and postulates for the completeness of the real numbers

R.P Burn
1992 Historia Mathematica  
of this century; to Frank Smithies for directing me toward Peano's Formulaire and for reminiscences of G.  ...  Hardy; and to Ivor Grattan-Guinness for extensive and detailed advice which has enhanced almost every paragraph of this paper.  ...  Fine cited Stolz and Gmeiner [1902] but gave his own treatment of the number system both with axioms and by construction, the axioms bearing an obvious relation to those of Veblen's paper [ 19051.  ... 
doi:10.1016/0315-0860(92)90074-l fatcat:vq5yf7sdsjb2lj3gdqiobgmyom

Peano's concept of number

Hubert C Kennedy
1974 Historia Mathematica  
Dedekind's analysis in was sind und was sollen die Zahlen?  ...  "Sul concetto di numero" ["On the concept of number"] [Peano 1959 [Peano , 80-1091 [l] In it he simplified his system by eliminating the undefined term symbolized by =, and the axioms relating to it.  ...  In order to be fair to those who were already subscribers when the combination offer was first announced, we offer all subscribers a copy of May's Bibliography and Research Manual of the History of Mathematics  ... 
doi:10.1016/0315-0860(74)90031-7 fatcat:qnpzirkkzrf6tpwrqhfd6zhkdm

Numbers [article]

David Pierce
2011 arXiv   pre-print
The von Neumann construction of the ordinal numbers includes a construction of natural numbers as a special kind of ordinal.  ...  This paper grew out of the observation that the possibilities of proof by induction and definition by recursion are often confused. The paper reviews the distinctions.  ...  The consistency of Dedekind's definition of a simply infinite system is self-evident.  ... 
arXiv:1104.5311v1 fatcat:fvs6humpmjf43leivp2mxdzyma

Weierstrass's construction of the irrational numbers

J. Christopher Tweddle
2010 Mathematische Semesterberichte  
We present an overview of the development of the irrational numbers due to Karl Weierstrass. This construction was first presented during lectures in the 1860s in Berlin.  ...  Several of his students (Kossak, Horwitz, von Dantscher and Pincherle, to name a few) gave accounts in lecture notes from the courses.  ...  Smith for his feedback on earlier drafts of this work, as well as his insight into the work of Pincherle [19] .  ... 
doi:10.1007/s00591-010-0082-6 fatcat:dyrohgrasrhu5grxhmkjtdepwi

The Introduction of Irrational Numbers

Philip E. B. Jourdain
1908 Mathematical Gazette  
of rational numbers and the system of points on a straight line, to introduce real numbers as “Schnitte” (or, more exactly, as classes of all the rationals which satisfy certain conditions),§ and then  ...  From the Cantor and Dedekind point of view of this creation of numbers,* which appears at first (see below) to be logically irreproachable, the unprovable nature of the corresponding Cantor-Dedekind axiom  ... 
doi:10.1017/s0025557200242824 fatcat:tmmjjgeywndmnhwr5t6wqi4hdq

Stevin Numbers and Reality

Karin Usadi Katz, Mikhail G. Katz
2011 Foundations of Science  
We explore the potential of Simon Stevin's numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis.  ...  Acknowledgments The authors are grateful to Piotr B laszczyk, Paolo Giordano, Thomas Mormann, and David Tall for valuable comments that helped improve the text.  ...  of the traditional number system.  ... 
doi:10.1007/s10699-011-9228-9 fatcat:domhxhs3abfirilzctnx2wfmim

What we talk about when we talk about numbers

Richard Pettigrew
2018 Annals of Pure and Applied Logic  
Similarly, Dedekind's axioms for a complete ordered field characterise the subject matter of real analysis (Dedekind, 1872) .  ...  Extreme Structure Realism just says that those other systems do not belong to the subject matter of number theory, real analysis, or group theory, respectively.  ... 
doi:10.1016/j.apal.2018.08.009 fatcat:p4hkcayph5gztpntvz6ttro4qu

The Introduction of Irrational Numbers

Philip E. B. Jourdain
1908 Mathematical Gazette  
., the President of the Association for 1907. The Branch consists of Members and Associates.  ...  JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive.  ...  From the Cantor and Dedekind point of view of this creation of numbers,* which appears at first (see below) to be logically irreproachable, the unprovable nature of the corresponding Cantor-Dedekind axiom  ... 
doi:10.2307/3602961 fatcat:xmotu2du3rffdbryzewlhdqjqa

The Continuum as a Type of Order: An Exposition of the Modern Theory. With an Appendix on the Transfinite Numbers

Philip E. B. Jourdain, Edward V. Huntington
1906 Mathematical Gazette  
It was Dedekind and Cantor in 1872, and Cantor in 1882, who first gave us a clear and purely arithmetical account of the system of the real numbers and exactly what one means by saying: " the system of  ...  It was Dedekind and Cantor in 1872, and Cantor in 1882, who first gave us a clear and purely arithmetical account of the system of the real numbers and exactly what one means by saying: " the system of  ... 
doi:10.2307/3603619 fatcat:b4f4zvzli5fe7i5c2flabg25ki

Number theory and elementary arithmetic†

JEREMY AVIGAD
2003 Philosophia Mathematica  
I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context.  ...  I am grateful to Andrew Arana, Steve Awodey, Patrick Cegielski, Teddy Seidenfeld, Wilfried Sieg, Neil Tennant, and the referees for comments, suggestions, and corrections.  ...  are easily shown to be equivalent, or at least have equivalent properties, on the basis of minimal systems of axioms.  ... 
doi:10.1093/philmat/11.3.257 fatcat:pbrezlovffbjdprrez32ypc5vy

From Numbers to Rings: The Early History of Ring Theory

Israel Kleiner
1998 Elemente der Mathematik  
As a final comment, the recent paper of Richard Taylor and Andrew Wiles, filling a gap in Wiles' previously announced proof of Fermat's Last Theorem, is entitled "Ring-theoretic properties of certain Hecke  ...  Two axioms give the closure of the system under the operations, and there is the requirement of an identity in the definition of the ring.  ...  Under one of the operations (addition) the system forms a group -he gives its axioms. The second operation (multiplication) is associative and distributes over the first.  ... 
doi:10.1007/s000170050029 fatcat:uvqa4gsdpvbw3pxiiacdub47ne

The real numbers - a survey of constructions [article]

Ittay Weiss
2015 arXiv   pre-print
We present a comprehensive survey of constructions of the real numbers (from either the rationals or the integers) in a unified fashion, thus providing an overview of most (if not all) known constructions  ...  ranging from the earliest attempts to recent results, and allowing for a simple comparison-at-a-glance between different constructions.  ...  sequences of rational numbers and Dedekind's construction by means of cuts of rational numbers, named after him.  ... 
arXiv:1506.03467v1 fatcat:kylpjq6kkbfr3ayf3xqbchu3yy
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