A Dedekind cut is a partition of the rational numbers into two non-empty sets A and B , such that all elements of A are less than all elements of B , and A contains no greatest element. The set B may or may not have a smallest element among the rationals. If B has a smallest element among the rationals, the cut corresponds to that rational. An irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is equated to one and only one cut of rationals [ citation needed ].
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I begin with a historical instruction with which I intend to place Dedekind's works in the history of mathematics. To introduce the concept To introduce the concept of cutting, which is the central concept of my research, I begin by stating how the cuts are incomplete for rational numbers and once these concepts are settled I try to move towards the completeness of the cuts in real numbers and how they are intimately linked to the concept of continuity. I also explain as an introduction how we can operate with the cuts.
Comienzo con una intruduc-cion historica con la que pretendo situar los trabajos de Dedekind en la historia de las matematicas. Save to Library. The main question in this research is how the idea of However, this idea is neither a purely philosophical nor a mathematical one. This is an interdisciplinary concept. Eldar Amirov. What are implicit definitions? The paper surveys different notions of implicit definition.
In particular , we offer an examination of a kind of definition commonly used in formal axiomatics, which in general terms is understood as providing a definition of the In particular , we offer an examination of a kind of definition commonly used in formal axiomatics, which in general terms is understood as providing a definition of the primitive terminology of an axiomatic theory.
We argue that such "structural definitions" can be semantically understood in two different ways, namely i as specifications of the meaning of the primitive terms of a theory and ii as definitions of higher-order mathematical concepts or structures.
We analyze these two conceptions of structural definition both in the history of modern axiomatics and in contemporary philosophical debates. Based on that, we give a systematic assessment of the underlying semantics of these two ways of understanding the definiens of such definitions, by considering alternative model-theoretic and inferential accounts of meaning. Eudoxos' Proportionenlehre und Dedekinds Reelle Zahlen.
Aus heutiger Perspektive ergibt sich aber noch ein weiterer interessanter Aspekt: Die 5. Riemann, par R. Dedekind et H. A road map of Dedekind's Theorem Dedekind's Theorem 66 states that there exists an infinite set.
Its proof invokes such apparently non-mathematical notions as the thought-world and the self. This paper discusses the content and context of Dedekind's proof. It is It is suggested that Dedekind took the notion of thought-world from Lotze. The influence of Kant and Bolzano on the proof is also discussed, and the reception of the proof in the mathematical and philosophical literature is covered in detail. With several examples, I suggest that this editorial work is to be understood as a mathematical With several examples, I suggest that this editorial work is to be understood as a mathematical activity in and of itself and provide evidence for it.
Brentanian Continua. The core idea of the theory is that boundaries and coincidences thereof belong to the essence of continua. Brentano is confident that he developed a full-fledged, To be clear, the theory of boundaries on which it relies, as well as the account of ontological dependence that Brentano develops alongside his theory of boundaries, constitute splendid achievements.
However, the passage from the theory of boundaries to the account of continuity is rather sketchy. I show that their paper provides an I show that their paper provides an arithmetical rewriting of Riemannian function theory, i. Then, through a detailed analysis of the paper and using elements of their correspondence, I suggest that Dedekind and Weber deploy a strategy of rewriting parts of mathematics using arithmetic, and that this strategy is essentially related to Dedekind's specific conception of numbers and arithmetic as intrinsically linked to the human mind.
Frede, Dedekind, and the Modern Epistemology of Arithmetic. In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components.
Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well as mathematical. The approach here is two-fold. Then I will consider those views from the perspective of modern philosophy of mathematics and in particular the empirical study of arithmetical cognition.
I aim to show that there is nothing to suggest that the axiomatic Dedekind approach could not provide a perfectly adequate basis for philosophy of arithmetic. In the XIX century in mathematics passes reforms of rigor and ground, begun by Cauchy and extended by Weierstrass.
Set theory was created as generalization of arithmetic, but it became the foundation of mathematics. The main problems of mathematical analysis: understanding the real number and continuity, stimulated the creation of four new concepts that appeared almost simultaneously, around The author of one of concepts, Richard Dedekind — , claimed the freedom to create math mathematical objects with the condition of their consistency.
In "Was sind und was sollen die Zahlen? The book is a re-edition of Russian translation of Richard Dedekind's book "What are numbers and what should they be? The preface by G. Concepts of a number of C. Heine, G. Cantor, R. Dedekind, and K. The Ch. Related Topics. Georg Cantor. Follow Following. Philosophy and history of mathematics.
History of Mathematics. History of Continuity and Infinitesimals. Logical and semantical notions. Set Theory. Ads help cover our server costs. Remember me on this computer. Enter the email address you signed up with and we'll email you a reset link.
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Idealization in mathematics/La idealizacion en la matematica