Results for 'Infinitesimals'

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  1. Infinitesimals as an Issue of Neo-Kantian Philosophy of Science.Thomas Mormann & Mikhail Katz - 2013 - Hopos: The Journal of the International Society for the History of Philosophy of Science (2):236-280.
    We seek to elucidate the philosophical context in which one of the most important conceptual transformations of modern mathematics took place, namely the so-called revolution in rigor in infinitesimal calculus and mathematical analysis. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind,and Weierstrass. The dominant current of philosophy in Germany at the time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our (...)
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  2. What Makes a Theory of Infinitesimals Useful? A View by Klein and Fraenkel.Vladimir Kanovei, K. Katz, M. Katz & Thomas Mormann - 2018 - Journal of Humanistic Mathematics 8 (1):108 - 119.
    Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein.
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  3. Numerical Infinities and Infinitesimals: Methodology, Applications, and Repercussions on Two Hilbert Problems.Yaroslav Sergeyev - 2017 - EMS Surveys in Mathematical Sciences 4 (2):219–320.
    In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than (...)
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  4. Solving Ordinary Differential Equations by Working with Infinitesimals Numerically on the Infinity Computer.Yaroslav Sergeyev - 2013 - Applied Mathematics and Computation 219 (22):10668–10681.
    There exists a huge number of numerical methods that iteratively construct approximations to the solution y(x) of an ordinary differential equation (ODE) y′(x) = f(x,y) starting from an initial value y_0=y(x_0) and using a finite approximation step h that influences the accuracy of the obtained approximation. In this paper, a new framework for solving ODEs is presented for a new kind of a computer – the Infinity Computer (it has been patented and its working prototype exists). The new computer is (...)
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  5. Berkeley: El origen de la crítica a los infinitesimales / Berkeley: The Origin of his Critics to Infinitesimals.Alberto Luis López - 2014 - Cuadernos Salmantinos de Filosofía 41 (1):195-217.
    BERKELEY: THE ORIGIN OF CRITICISM OF THE INFINITESIMALS Abstract: In this paper I propose a new reading of a little known George Berkeley´s work Of Infinites. Hitherto, the work has been studied partially, or emphasizing only the mathematical contributions, downplaying the philosophical aspects, or minimizing mathematical issues taking into account only the incipient immaterialism. Both readings have been pernicious for the correct comprehension of the work and that has brought as a result that will follow underestimated its importance, and (...)
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  6. The Exact (Up to Infinitesimals) Infinite Perimeter of the Koch Snowflake and its Finite Area.Yaroslav Sergeyev - 2016 - Communications in Nonlinear Science and Numerical Simulation 31 (1-3):21–29.
    The Koch snowflake is one of the first fractals that were mathematically described. It is interesting because it has an infinite perimeter in the limit but its limit area is finite. In this paper, a recently proposed computational methodology allowing one to execute numerical computations with infinities and infinitesimals is applied to study the Koch snowflake at infinity. Numerical computations with actual infinite and infinitesimal numbers can be executed on the Infinity Computer being a new supercomputer patented in USA (...)
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  7. THE LOGIC OF TIME AND THE CONTINUUM IN KANT's CRITICAL PHILOSOPHY.Riccardo Pinosio & Michiel van Lambalgen - manuscript
    We aim to show that Kant’s theory of time is consistent by providing axioms whose models validate all synthetic a priori principles for time proposed in the Critique of Pure Reason. In this paper we focus on the distinction between time as form of intuition and time as formal intuition, for which Kant’s own explanations are all too brief. We provide axioms that allow us to construct ‘time as formal intuition’ as a pair of continua, corresponding to time as ‘inner (...)
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  8. The Logic and Topology of Kant's Temporal Continuum.Riccardo Pinosio & Michiel van Lambalgen - manuscript
    In this article we provide a mathematical model of Kant?s temporal continuum that satisfies the (not obviously consistent) synthetic a priori principles for time that Kant lists in the Critique of pure Reason (CPR), the Metaphysical Foundations of Natural Science (MFNS), the Opus Postumum and the notes and frag- ments published after his death. The continuum so obtained has some affinities with the Brouwerian continuum, but it also has ‘infinitesimal intervals’ consisting of nilpotent infinitesimals, which capture Kant’s theory of (...)
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  9.  69
    Independence of the Grossone-Based Infinity Methodology From Non-Standard Analysis and Comments Upon Logical Fallacies in Some Texts Asserting the Opposite.Yaroslav Sergeyev - 2019 - Foundations of Science 24 (1):153-170.
    This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ①. The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of (...)
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  10. Fair Infinite Lotteries.Sylvia Wenmackers & Leon Horsten - 2013 - Synthese 190 (1):37-61.
    This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from non-standard analysis are brought to bear on the problem.
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  11. Discrete and Continuous: A Fundamental Dichotomy in Mathematics.James Franklin - 2017 - Journal of Humanistic Mathematics 7 (2):355-378.
    The distinction between the discrete and the continuous lies at the heart of mathematics. Discrete mathematics (arithmetic, algebra, combinatorics, graph theory, cryptography, logic) has a set of concepts, techniques, and application areas largely distinct from continuous mathematics (traditional geometry, calculus, most of functional analysis, differential equations, topology). The interaction between the two – for example in computer models of continuous systems such as fluid flow – is a central issue in the applicable mathematics of the last hundred years. This article (...)
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  12. À Maneira de Um Colar de Pérolas?André Porto - 2017 - Revista Portuguesa de Filosofia 73 (3-4):1381-1404.
    This paper offers an overview of various alternative formulations for Analysis, the theory of Integral and Differential Calculus, and its diverging conceptions of the topological structure of the continuum. We pay particularly attention to Smooth Analysis, a proposal created by William Lawvere and Anders Kock based on Grothendieck’s work on a categorical algebraic geometry. The role of Heyting’s logic, common to all these alternatives is emphasized.
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  13. Surreal Time and Ultratasks.Haidar Al-Dhalimy & Charles J. Geyer - 2016 - Review of Symbolic Logic 9 (4):836-847.
    This paper suggests that time could have a much richer mathematical structure than that of the real numbers. Clark & Read (1984) argue that a hypertask (uncountably many tasks done in a finite length of time) cannot be performed. Assuming that time takes values in the real numbers, we give a trivial proof of this. If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an ultratask (a sequence which (...)
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  14. Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics.Vladimir Kanovei, Mikhail G. Katz & Thomas Mormann - 2013 - Foundations of Science 18 (2):259-296.
    We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals (...)
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  15. Hermann Cohen’s Principle of the Infinitesimal Method: A Defense.Scott Edgar - forthcoming - Hopos: The Journal of the International Society for the History of Philosophy of Science.
    In Bertrand Russell's 1903 Principles of Mathematics, he offers an apparently devastating criticism of the neo-Kantian Hermann Cohen's Principle of the Infinitesimal Method and its History (PIM). Russell's criticism is motivated by his concern that Cohen's account of the foundations of calculus saddles mathematics with the paradoxes of the infinitesimal and continuum, and thus threatens the very idea of mathematical truth. This paper defends Cohen against that objection of Russell's, and argues that properly understood, Cohen's views of limits and (...) do not entail the paradoxes of the infinitesimal and continuum. Essential to that defense is an interpretation, developed in the paper, of Cohen's positions in the PIM as deeply rationalist. The interest in developing this interpretation is not just that it reveals how Cohen's views in the PIM avoid the paradoxes of the infinitesimal and continuum. It also reveals some of what is at stake, both historically and philosophically, in Russell's criticism of Cohen. (shrink)
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  16.  77
    Picturing the Infinite.Jeremy Gwiazda - manuscript
    The purpose of this note is to contrast a Cantorian outlook with a non-Cantorian one and to present a picture that provides support for the latter. In particular, I suggest that: i) infinite hyperreal numbers are the (actual, determined) infinite numbers, ii) ω is merely potentially infinite, and iii) infinitesimals should not be used in the di Finetti lottery. Though most Cantorians will likely maintain a Cantorian outlook, the picture is meant to motivate the obvious nature of the non-Cantorian (...)
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  17. More Trouble for Regular Probabilitites.Matthew W. Parker - manuscript
    In standard probability theory, probability zero is not the same as impossibility. But many have suggested that only impossible events should have probability zero. This can be arranged if we allow infinitesimal probabilities, but infinitesimals do not solve all of the problems. We will see that regular probabilities are not invariant over rigid transformations, even for simple, bounded, countable, constructive, and disjoint sets. Hence, regular chances cannot be determined by space-time invariant physical laws, and regular credences cannot satisfy seemingly (...)
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