Contents
73 found
Order:
1 — 50 / 73
  1. Strict Finitism's Unrequited Love for Computational Complexity.Noel Arteche - manuscript
    As a philosophy of mathematics, strict finitism has been traditionally concerned with the notion of feasibility, defended mostly by appealing to the physicality of mathematical practice. This has led the strict finitists to influence and be influenced by the field of computational complexity theory, under the widely held belief that this branch of mathematics is concerned with the study of what is “feasible in practice”. In this paper, I survey these ideas and contend that, contrary to popular belief, complexity theory (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  2. Frege's Basic Law V and Cantor's Theorem.Manuel Bremer - manuscript
    The following essay reconsiders the ontological and logical issues around Frege’s Basic Law (V). If focuses less on Russell’s Paradox, as most treatments of Frege’s Grundgesetze der Arithmetik (GGA)1 do, but rather on the relation between Frege’s Basic Law (V) and Cantor’s Theorem (CT). So for the most part the inconsistency of Naïve Comprehension (in the context of standard Second Order Logic) will not concern us, but rather the ontological issues central to the conflict between (BLV) and (CT). These ontological (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  3. A Puzzle about Sums.Andrew Y. Lee - forthcoming - Oxford Studies in Metaphysics.
    A famous mathematical theorem says that the sum of an infinite series of numbers can depend on the order in which those numbers occur. Suppose we interpret the numbers in such a series as representing instances of some physical quantity, such as the weights of a collection of items. The mathematics seems to lead to the result that the weight of a collection of items can depend on the order in which those items are weighed. But that is very hard (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  4. Cosmic Topology, Underdetermination, and Spatial Infinity.Patrick James Ryan - 2024 - European Journal for Philosophy of Science 14 (17):1-28.
    It is well-known that the global structure of every space-time model for relativistic cosmology is observationally underdetermined. In order to alleviate the severity of this underdetermination, it has been proposed that we adopt the Cosmological Principle because the Principle restricts our attention to a distinguished class of space-time models (spatially homogeneous and isotropic models). I argue that, even assuming the Cosmological Principle, the topology of space remains observationally underdetermined. Nonetheless, I argue that we can muster reasons to prefer various topological (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  5. Øystein vs Archimedes: A Note on Linnebo’s Infinite Balance.Daniel Hoek - 2023 - Erkenntnis 88 (4):1791-1796.
    Using Riemann’s Rearrangement Theorem, Øystein Linnebo (2020) argues that, if it were possible to apply an infinite positive weight and an infinite negative weight to a working scale, the resulting net weight could end up being any real number, depending on the procedure by which these weights are applied. Appealing to the First Postulate of Archimedes’ treatise on balance, I argue instead that the scale would always read 0 kg. Along the way, we stop to consider an infinitely jittery flea, (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   1 citation  
  6. Lower and Upper Estimates of the Quantity of Algebraic Numbers.Yaroslav Sergeyev - 2023 - Mediterranian Journal of Mathematics 20:12.
    It is well known that the set of algebraic numbers (let us call it A) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using ①-based infinite numbers is applied to measure the set A (where the number ① is called grossone). Our interest to this methodology is explained by the fact that in certain cases where cardinals allow one to say only whether a set is countable (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  7. Nothing Infinite: A Summary of Forever Finite.Kip Sewell - 2023 - Rond Media Library.
    In 'Forever Finite: The Case Against Infinity' (Rond Books, 2023), the author argues that, despite its cultural popularity, infinity is not a logical concept and consequently cannot be a property of anything that exists in the real world. This article summarizes the main points in 'Forever Finite', including its overview of what debunking infinity entails for conceptual thought in philosophy, mathematics, science, cosmology, and theology.
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  8. Forever Finite: The Case Against Infinity (Expanded Edition).Kip K. Sewell - 2023 - Alexandria, VA: Rond Books.
    EXPANDED EDITION (eBook): -/- Infinity Is Not What It Seems...Infinity is commonly assumed to be a logical concept, reliable for conducting mathematics, describing the Universe, and understanding the divine. Most of us are educated to take for granted that there exist infinite sets of numbers, that lines contain an infinite number of points, that space is infinite in expanse, that time has an infinite succession of events, that possibilities are infinite in quantity, and over half of the world’s population believes (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  9. The Potential in Frege’s Theorem.Will Stafford - 2023 - Review of Symbolic Logic 16 (2):553-577.
    Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. We re-explore this idea and discover that in the setting of the (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  10. How can a line segment with extension be composed of extensionless points?Brian Reese, Michael Vazquez & Scott Weinstein - 2022 - Synthese 200 (2):1-28.
    We provide a new interpretation of Zeno’s Paradox of Measure that begins by giving a substantive account, drawn from Aristotle’s text, of the fact that points lack magnitude. The main elements of this account are (1) the Axiom of Archimedes which states that there are no infinitesimal magnitudes, and (2) the principle that all assignments of magnitude, or lack thereof, must be grounded in the magnitude of line segments, the primary objects to which the notion of linear magnitude applies. Armed (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  11. Some paradoxes of infinity revisited.Yaroslav Sergeyev - 2022 - Mediterranian Journal of Mathematics 19:143.
    In this article, some classical paradoxes of infinity such as Galileo’s paradox, Hilbert’s paradox of the Grand Hotel, Thomson’s lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described. It is shown that the surprising counting system of an Amazonian tribe, Pirah ̃a, working with only three numerals (one, two, many) can help us to change our perception (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  12. Takeuti's well-ordering proofs revisited.Andrew Arana & Ryota Akiyoshi - 2021 - Mita Philosophy Society 3 (146):83-110.
    Gaisi Takeuti extended Gentzen's work to higher-order case in 1950's–1960's and proved the consistency of impredicative subsystems of analysis. He has been chiefly known as a successor of Hilbert's school, but we pointed out in the previous paper that Takeuti's aimed to investigate the relationships between "minds" by carrying out his proof-theoretic project rather than proving the "reliability" of such impredicative subsystems of analysis. Moreover, as briefly explained there, his philosophical ideas can be traced back to Nishida's philosophy in Kyoto's (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  13. Bolzano’s Mathematical Infinite.Anna Bellomo & Guillaume Massas - 2021 - Review of Symbolic Logic:1-55.
    Bernard Bolzano (1781–1848) is commonly thought to have attempted to develop a theory of size for infinite collections that follows the so-called part–whole principle, according to which the whole is always greater than any of its proper parts. In this paper, we develop a novel interpretation of Bolzano’s mature theory of the infinite and show that, contrary to mainstream interpretations, it is best understood as a theory of infinite sums. Our formal results show that Bolzano’s infinite sums can be equipped (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   2 citations  
  14. Are infinite explanations self-explanatory?Alexandre Billon - 2021 - Erkenntnis 88 (5):1935-1954.
    Consider an infinite series whose items are each explained by their immediate successor. Does such an infinite explanation explain the whole series or does it leave something to be explained? Hume arguably claimed that it does fully explain the whole series. Leibniz, however, designed a very telling objection against this claim, an objection involving an infinite series of book copies. In this paper, I argue that the Humean claim can, in certain cases, be saved from the Leibnizian “infinite book copies” (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   3 citations  
  15. Existence Is Evidence of Immortality.Michael Huemer - 2021 - Noûs 55 (1):128-151.
    Time may be infinite in both directions. If it is, then, if persons could live at most once in all of time, the probability that you would be alive now would be zero. But if persons can live more than once, the probability that you would be alive now would be nonzero. Since you are alive now, with certainty, either the past is finite, or persons can live more than once.
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   6 citations  
  16. Hermann Cohen’s Principle of the Infinitesimal Method: A Defense.Scott Edgar - 2020 - Hopos: The Journal of the International Society for the History of Philosophy of Science 10 (2):440-470.
    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 infinitesimals (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   2 citations  
  17. Apophatic Finitism and Infinitism.Jan Heylen - 2019 - Logique Et Analyse 62 (247):319-337.
    This article is about the ontological dispute between finitists, who claim that only finitely many numbers exist, and infinitists, who claim that infinitely many numbers exist. Van Bendegem set out to solve the 'general problem' for finitism: how can one recast finite fragments of classical mathematics in finitist terms? To solve this problem Van Bendegem comes up with a new brand of finitism, namely so-called 'apophatic finitism'. In this article it will be argued that apophatic finitism is unable to represent (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  18. On the Coherence of Strict Finitism.Auke Alesander Montesano Montessori - 2019 - Kriterion - Journal of Philosophy 33 (2):1-14.
    Strict finitism is the position that only those natural numbers exist that we can represent in practice. Michael Dummett, in a paper called Wang’s Paradox, famously tried to show that strict finitism is an incoherent position. By using the Sorites paradox, he claimed that certain predicates the strict finitist is committed to are incoherent. More recently, Ofra Magidor objected to Dummett’s claims, arguing that Dummett fails to show the incoherence of strict finitism. In this paper, I shall investigate whether Magidor (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  19. Infinite barbarians.Daniel Nolan - 2019 - Ratio 32 (3):173-181.
    This paper discusses an infinite regress that looms behind a certain kind of historical explanation. The movement of one barbarian group is often explained by the movement of others, but those movements in turn call for an explanation. While their explanation can again be the movement of yet another group of barbarians, if this sort of explanation does not stop somewhere we are left with an infinite regress of barbarians. While that regress would be vicious, it cannot be accommodated by (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   3 citations  
  20. Independence of the Grossone-Based Infinity Methodology from Non-standard Analysis and Comments upon Logical Fallacies in Some Texts Asserting the Opposite.Yaroslav D. 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 these (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   1 citation  
  21. Revisão de ' Os Limites Exteriores da Razão ' (The Outer Limits of Reason)por Noson Yanofsky 403p (2013) (revisão revisada 2019).Michael Richard Starks - 2019 - In Delírios Utópicos Suicidas no Século XXI Filosofia, Natureza Humana e o Colapso da Civilization- Artigos e Comentários 2006-2019 5ª edição. Las Vegas, NV USA: Reality Press. pp. 188-202.
    Eu dou uma revisão detalhada de "os limites exteriores da razão" por Noson Yanofsky de uma perspectiva unificada de Wittgenstein e psicologia evolutiva. Eu indico que a dificuldade com tais questões como paradoxo na linguagem e matemática, incompletude, undecidabilidade, computabilidade, o cérebro eo universo como computadores, etc., todos surgem a partir da falta de olhar atentamente para o nosso uso da linguagem no apropriado contexto e, consequentemente, a falta de separar questões de fato científico a partir de questões de como (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  22. A Categorical Characterization of Accessible Domains.Patrick Walsh - 2019 - Dissertation, Carnegie Mellon University
    Inductively defined structures are ubiquitous in mathematics; their specification is unambiguous and their properties are powerful. All fields of mathematical logic feature these structures prominently: the formula of a language, the set of theorems, the natural numbers, the primitive recursive functions, the constructive number classes and segments of the cumulative hierarchy of sets. -/- This dissertation gives a mathematical characterization of a species of inductively defined structures, called accessible domains, which include all of the above examples except the set of (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   1 citation  
  23. Enseignement et apprentissage de l’infini : aspects philosophiques, épistemologiques et didactiques.Pascale Boulais, R. Brouzet, Viviane Durand-Guerrier, Maha Majaj, David Marino, Francoise Monnoyeur & Martine Vergnac - 2018 - In Pascale Boulais, R. Brouzet, Viviane Durand-Guerrier, Maha Majaj, David Marino, Francoise Monnoyeur & Martine Vergnac (eds.), Mathématiques en scène des ponts entre les disciplines. Paris, France: pp. 246-255.
    Résumé – Nous nous intéressons à l’enseignement et l’apprentissage de l’infini en classe de mathématiques en considérant les différences et les relations entre infini potentiel et infini actuel. Nous présentons les principaux éléments de notre étude philosophique, épistémologique et didactique, ainsi que trois situations visant à conduire un travail explicite avec les élèves sur ces questions en début de lycée. ---------------------------------------------------------------------------------------------------- --------------------------------- Abstract – We are interested in the teaching and learning of infinite in mathematics class, taking into account the relations (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  24. Numerical infinities applied for studying Riemann series theorem and Ramanujan summation.Yaroslav Sergeyev - 2018 - In AIP Conference Proceedings 1978. AIP. pp. 020004.
    A computational methodology called Grossone Infinity Computing introduced with the intention to allow one to work with infinities and infinitesimals numerically has been applied recently to a number of problems in numerical mathematics (optimization, numerical differentiation, numerical algorithms for solving ODEs, etc.). The possibility to use a specially developed computational device called the Infinity Computer (patented in USA and EU) for working with infinite and infinitesimal numbers numerically gives an additional advantage to this approach in comparison with traditional methodologies studying (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   1 citation  
  25. 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 (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   3 citations  
  26. 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 the (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   4 citations  
  27. Eternity a History.Yitzhak Y. Melamed (ed.) - 2016 - New York, New York: Oxford University Press USA.
    Eternity is a unique kind of existence that is supposed to belong to the most real being or beings. It is an existence that is not shaken by the common wear and tear of time. Over the two and half millennia history of Western philosophy we find various conceptions of eternity, yet one sharp distinction between two notions of eternity seems to run throughout this long history: eternity as timeless existence, as opposed to eternity as existence in all times. Both (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  28. Finitism, Divisibilty, and the Beginning of the Universe: Replies to Loke and Dumsday.Stephen Puryear - 2016 - Australasian Journal of Philosophy 94 (4):808-813.
    Some philosophers contend that the past must be finite in duration, because otherwise reaching the present would have involved the sequential occurrence of an actual infinity of events, which they regard as impossible. I recently developed a new objection to this finitist argument, to which Andrew Ter Ern Loke and Travis Dumsday have replied. Here I respond to the three main points raised in their replies.
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   7 citations  
  29. 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 and (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   3 citations  
  30. Aristotelian finitism.Tamer Nawar - 2015 - Synthese 192 (8):2345-2360.
    It is widely known that Aristotle rules out the existence of actual infinities but allows for potential infinities. However, precisely why Aristotle should deny the existence of actual infinities remains somewhat obscure and has received relatively little attention in the secondary literature. In this paper I investigate the motivations of Aristotle’s finitism and offer a careful examination of some of the arguments considered by Aristotle both in favour of and against the existence of actual infinities. I argue that Aristotle has (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   6 citations  
  31. In search of $$\aleph _{0}$$ ℵ 0 : how infinity can be created.Markus Pantsar - 2015 - Synthese 192 (8):2489-2511.
    In this paper I develop a philosophical account of actual mathematical infinity that does not demand ontologically or epistemologically problematic assumptions. The account is based on a simple metaphor in which we think of indefinitely continuing processes as defining objects. It is shown that such a metaphor is valid in terms of mathematical practice, as well as in line with empirical data on arithmetical cognition.
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   16 citations  
  32. The Olympic medals ranks, lexicographic ordering and numerical infinities.Yaroslav Sergeyev - 2015 - The Mathematical Intelligencer 37 (2):4-8.
    Several ways used to rank countries with respect to medals won during Olympic Games are discussed. In particular, it is shown that the unofficial rank used by the Olympic Committee is the only rank that does not allow one to use a numerical counter for ranking – this rank uses the lexicographic ordering to rank countries: one gold medal is more precious than any number of silver medals and one silver medal is more precious than any number of bronze medals. (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   4 citations  
  33. On Multiverses and Infinite Numbers.Jeremy Gwiazda - 2014 - In Klaas Kraay (ed.), God and the Multiverse. Routledge. pp. 162-173.
    A multiverse is comprised of many universes, which quickly leads to the question: How many universes? There are either finitely many or infinitely many universes. The purpose of this paper is to discuss two conceptions of infinite number and their relationship to multiverses. The first conception is the standard Cantorian view. But recent work has suggested a second conception of infinite number, on which infinite numbers behave very much like finite numbers. I will argue that that this second conception of (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   1 citation  
  34. Finitism and the Beginning of the Universe.Stephen Puryear - 2014 - Australasian Journal of Philosophy 92 (4):619-629.
    Many philosophers have argued that the past must be finite in duration because otherwise reaching the present moment would have involved something impossible, namely, the sequential occurrence of an actual infinity of events. In reply, some philosophers have objected that there can be nothing amiss in such an occurrence, since actually infinite sequences are ‘traversed’ all the time in nature, for example, whenever an object moves from one location in space to another. This essay focuses on one of the two (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   12 citations  
  35. Boring Infinite Descent.Tuomas E. Tahko - 2014 - Metaphilosophy 45 (2):257-269.
    In formal ontology, infinite regresses are generally considered a bad sign. One debate where such regresses come into play is the debate about fundamentality. Arguments in favour of some type of fundamentalism are many, but they generally share the idea that infinite chains of ontological dependence must be ruled out. Some motivations for this view are assessed in this article, with the conclusion that such infinite chains may not always be vicious. Indeed, there may even be room for a type (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   46 citations  
  36. Mathematical Platonism and the Nature of Infinity.Gilbert B. Côté - 2013 - Open Journal of Philosophy 3 (3):372-375.
    An analysis of the counter-intuitive properties of infinity as understood differently in mathematics, classical physics and quantum physics allows the consideration of various paradoxes under a new light (e.g. Zeno’s dichotomy, Torricelli’s trumpet, and the weirdness of quantum physics). It provides strong support for the reality of abstractness and mathematical Platonism, and a plausible reason why there is something rather than nothing in the concrete universe. The conclusions are far reaching for science and philosophy.
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   8 citations  
  37. Two concepts of completing an infinite number of tasks.Jeremy Gwiazda - 2013 - The Reasoner 7 (6):69-70.
    In this paper, two concepts of completing an infinite number of tasks are considered. After discussing supertasks, equisupertasks are introduced. I suggest that equisupertasks are logically possible.
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   1 citation  
  38. Set Size and the Part–Whole Principle.Matthew W. Parker - 2013 - Review of Symbolic Logic (4):1-24.
    Recent work has defended “Euclidean” theories of set size, in which Cantor’s Principle (two sets have equally many elements if and only if there is a one-to-one correspondence between them) is abandoned in favor of the Part-Whole Principle (if A is a proper subset of B then A is smaller than B). It has also been suggested that Gödel’s argument for the unique correctness of Cantor’s Principle is inadequate. Here we see from simple examples, not that Euclidean theories of set (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   15 citations  
  39. Science, Religion, and Infinity.Graham Oppy - 2012 - In J. B. Stump & Alan G. Padgett (eds.), The Blackwell Companion to Science and Christianity. Chichester, UK: Wiley. pp. 430-440.
    This chapter contains sections titled: * Brief History * How We Talk * Science and Infinity * Religion and Infinity * Concluding Remarks * Notes * References * Further Reading.
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  40. Interpretation of percolation in terms of infinity computations.Yaroslav Sergeyev, Dmitri Iudin & Masaschi Hayakawa - 2012 - Applied Mathematics and Computation 218 (16):8099-8111.
    In this paper, a number of traditional models related to the percolation theory has been considered by means of new computational methodology that does not use Cantor’s ideas and describes infinite and infinitesimal numbers in accordance with the principle ‘The part is less than the whole’. It gives a possibility to work with finite, infinite, and infinitesimal quantities numerically by using a new kind of a compute - the Infinity Computer – introduced recently in [18]. The new approach does not (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   3 citations  
  41. Wittgenstein And Labyrinth Of ‘Actual Infinity’: The Critique Of Transfinite Set Theory.Valérie Lynn Therrien - 2012 - Ithaque 10:43-65.
    In order to explain Wittgenstein’s account of the reality of completed infinity in mathematics, a brief overview of Cantor’s initial injection of the idea into set- theory, its trajectory and the philosophic implications he attributed to it will be presented. Subsequently, we will first expound Wittgenstein’s grammatical critique of the use of the term ‘infinity’ in common parlance and its conversion into a notion of an actually existing infinite ‘set’. Secondly, we will delve into Wittgenstein’s technical critique of the concept (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   2 citations  
  42. On the Infinite in Mereology with Plural Quantification.Massimiliano Carrara & Enrico Martino - 2011 - Review of Symbolic Logic 4 (1):54-62.
    In Lewis reconstructs set theory using mereology and plural quantification (MPQ). In his recontruction he assumes from the beginning that there is an infinite plurality of atoms, whose size is equivalent to that of the set theoretical universe. Since this assumption is far beyond the basic axioms of mereology, it might seem that MPQ do not play any role in order to guarantee the existence of a large infinity of objects. However, we intend to demonstrate that mereology and plural quantification (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   2 citations  
  43. Higher order numerical differentiation on the Infinity Computer.Yaroslav Sergeyev - 2011 - Optimization Letters 5 (4):575-585.
    There exist many applications where it is necessary to approximate numerically derivatives of a function which is given by a computer procedure. In particular, all the fields of optimization have a special interest in such a kind of information. In this paper, a new way to do this is presented for a new kind of a computer - the Infinity Computer - able to work numerically with finite, infinite, and infinitesimal number. It is proved that the Infinity Computer is able (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   5 citations  
  44. Philosophy of Probability: Foundations, Epistemology, and Computation.Sylvia Wenmackers - 2011 - Dissertation, University of Groningen
    This dissertation is a contribution to formal and computational philosophy. -/- In the first part, we show that by exploiting the parallels between large, yet finite lotteries on the one hand and countably infinite lotteries on the other, we gain insights in the foundations of probability theory as well as in epistemology. Case 1: Infinite lotteries. We discuss how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. The solution boils down to the introduction (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   5 citations  
  45. Rez. „Adam Drozdek: In the Beginning Was the Apeiron: Infinity in Greek Philosophy, Stuttgart: Steiner, 2008“. [REVIEW]Sergiusz Kazmierski - 2010 - Bryn Mawr Classical Review 2010.
    Es ist das Verdienst der Arbeit von Adam Drozdek, in einem noch grösseren historischen Umfang sowie mit einer noch stärkeren thematischen Gewichtung und Stringenz als dies bereits Sinnige getan hat, nicht nur die entscheidendste Phase der griechischen Philosophie, sondern auch der Mathematik, ausgehend vom physikalischen und mathematischen Infinitätsgedanken dargestellt zu haben.
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
  46. Cantor’s Proof in the Full Definable Universe.Laureano Luna & William Taylor - 2010 - Australasian Journal of Logic 9:10-25.
    Cantor’s proof that the powerset of the set of all natural numbers is uncountable yields a version of Richard’s paradox when restricted to the full definable universe, that is, to the universe containing all objects that can be defined not just in one formal language but by means of the full expressive power of natural language: this universe seems to be countable on one account and uncountable on another. We argue that the claim that definitional contexts impose restrictions on the (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   4 citations  
  47. Lagrange Lecture: Methodology of numerical computations with infinities and infinitesimals.Yaroslav Sergeyev - 2010 - Rendiconti Del Seminario Matematico dell'Università E Del Politecnico di Torino 68 (2):95–113.
    A recently developed computational methodology for executing numerical calculations with infinities and infinitesimals is described in this paper. The approach developed has a pronounced applied character and is based on the principle “The part is less than the whole” introduced by the ancient Greeks. This principle is applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The point of view on infinities and infinitesimals (and in general, on Mathematics) presented in this paper (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   2 citations  
  48. Counting systems and the First Hilbert problem.Yaroslav Sergeyev - 2010 - Nonlinear Analysis Series A 72 (3-4):1701-1708.
    The First Hilbert problem is studied in this paper by applying two instruments: a new methodology distinguishing between mathematical objects and mathematical languages used to describe these objects; and a new numeral system allowing one to express different infinite numbers and to use these numbers for measuring infinite sets. Several counting systems are taken into consideration. It is emphasized in the paper that different mathematical languages can describe mathematical objects (in particular, sets and the number of their elements) with different (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   2 citations  
  49. Cantor on Infinity in Nature, Number, and the Divine Mind.Anne Newstead - 2009 - American Catholic Philosophical Quarterly 83 (4):533-553.
    The mathematician Georg Cantor strongly believed in the existence of actually infinite numbers and sets. Cantor’s “actualism” went against the Aristotelian tradition in metaphysics and mathematics. Under the pressures to defend his theory, his metaphysics changed from Spinozistic monism to Leibnizian voluntarist dualism. The factor motivating this change was two-fold: the desire to avoid antinomies associated with the notion of a universal collection and the desire to avoid the heresy of necessitarian pantheism. We document the changes in Cantor’s thought with (...)
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark   8 citations  
  50. Infinity and Metaphysics.Daniel Nolan - 2009 - In Robin Le Poidevin, Simons Peter, McGonigal Andrew & Ross P. Cameron (eds.), The Routledge Companion to Metaphysics. New York: Routledge. pp. 430-439.
    This introduction to the roles infinity plays in metaphysics includes discussion of the nature of infinity itself; infinite space and time, both in extent and in divisibility; infinite regresses; and a list of some other topics in metaphysics where infinity plays a significant role.
    Remove from this list   Download  
     
    Export citation  
     
    Bookmark  
1 — 50 / 73