Numbers

Edited by Rafal Urbaniak (University of Ghent, University of Gdansk)
Assistant editors: Sam Roberts, Pawel Pawlowski
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  1. added 2020-01-28
    Number and Reality: Sources of Scientific Knowledge.Alex V. Halapsis - 2016 - ScienceRise 23 (6):59-64.
    Pythagoras’s number doctrine had a great effect on the development of science. Number – the key to the highest reality, and such approach allowed Pythagoras to transform mathematics from craft into science, which continues implementation of its project of “digitization of being”. Pythagoras's project underwent considerable transformation, but it only means that the plan in knowledge is often far from result.
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  2. added 2020-01-28
    Visa to Heaven: Orpheus, Pythagoras, and Immortality.Alex V. Halapsis - 2016 - ScienceRise 25 (8):60-65.
    The article deals with the doctrines of Orpheus and Pythagoras about the immortality of the soul in the context of the birth of philosophy in ancient Greece. Orpheus demonstrated the closeness of heavenly (divine) and earthly (human) worlds, and Pythagoras mathematically proved their fundamental identity. Greek philosophy was “an investment in the afterlife future”, being the product of the mystical (Orpheus) and rationalist (Pythagoras) theology.
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  3. added 2019-12-31
    Numbers and Manifolds.Peter Simons - 1982 - In Barry Smith (ed.), Parts and Moments. Studies in Logic and Formal Ontology. Munich: Philosophia. pp. 160-197.
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  4. added 2019-06-05
    Number Words as Number Names.Friederike Moltmann - 2017 - Linguistics and Philosophy 40 (4):331-345.
    This paper criticizes the view that number words in argument position retain the meaning they have on an adjectival or determiner use, as argued by Hofweber :179–225, 2005) and Moltmann :499–534, 2013a, 2013b). In particular the paper re-evaluates syntactic evidence from German given in Moltmann to that effect.
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  5. added 2019-03-01
    On the Reality of the Continuum Discussion Note: A Reply to Ormell, ‘Russell's Moment of Candour’, Philosophy: Anne Newstead and James Franklin.Anne Newstead - 2008 - Philosophy 83 (1):117-127.
    In a recent article, Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably infinite set. He rejects the conclusion of Cantor’s diagonal argument for the higher, non-denumerable infinity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only ‘well-defined’ real numbers as proper objects of study. In practice, this means excluding as (...)
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  6. added 2018-12-22
    Plato’s Philosophy of Cognition by Mathematical Modelling.Roman S. Kljujkov & Sergey Kljujkov - 2014 - Dialogue and Universalism 24 (3):110-115.
    By the end of his life Plato had rearranged the theory of ideas into his teaching about ideal numbers, but no written records have been left. The Ideal mathematics of Plato is present in all his dialogues. It can be clearly grasped in relation to the effective use of mathematical modelling. Many problems of mathematical modelling were laid in the foundation of the method by cutting the three-level idealism of Plato to the single-level “ideism” of Aristotle. For a long time, (...)
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  7. added 2018-12-08
    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 (...)
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  8. added 2018-11-22
    Arbitrary Reference, Numbers, and Propositions.Michele Palmira - 2018 - European Journal of Philosophy 26 (3):1069-1085.
    Reductionist realist accounts of certain entities, such as the natural numbers and propositions, have been taken to be fatally undermined by what we may call the problem of arbitrary identification. The problem is that there are multiple and equally adequate reductions of the natural numbers to sets (see Benacerraf, 1965), as well as of propositions to unstructured or structured entities (see, e.g., Bealer, 1998; King, Soames, & Speaks, 2014; Melia, 1992). This paper sets out to solve the problem by canvassing (...)
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  9. added 2018-10-01
    Deflationary Nominalism and Puzzle Avoidance.David Mark Kovacs - 2019 - Philosophia Mathematica 27 (1):88-104.
    In a series of works, Jody Azzouni has defended deflationary nominalism, the view that certain sentences quantifying over mathematical objects are literally true, although such objects do not exist. One alleged attraction of this view is that it avoids various philosophical puzzles about mathematical objects. I argue that this thought is misguided. I first develop an ontologically neutral counterpart of Field’s reliability challenge and argue that deflationary nominalism offers no distinctive answer to it. I then show how this reasoning generalizes (...)
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  10. added 2018-08-17
    Numbers and Propositions: Reply to Melia.Tim Crane - 1992 - Analysis 52 (4):253-256.
    Is the way we use propositions to individuate beliefs and other intentional states analogous to the way we use numbers to measure weights and other physical magnitudes? In an earlier paper [2], I argued that there is an important disanalogy. One and the same weight can be 'related to' different numbers under different units of measurement. Moreover, the choice of a unit of measurement is arbitrary,in the sense that which way we choose doesn't affect the weight attributed to the object. (...)
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  11. added 2018-07-16
    Maths, Logic and Language.Tetsuaki Iwamoto - 2018 - Geneva: Logic Forum.
    A work on the philosophy of mathematics (2017) -/- ‘Number’, such a simple idea, and yet it fascinated and absorbed the greatest proportion of human geniuses over centuries, not to mention the likes of Pythagoras, Euclid, Newton, Leibniz, Descartes and countless maths giants like Euler, Gauss and Hilbert, etc.. Einstein thought of pure maths as the poetry of logical ideas, the exactitude of which, although independent of experience, strangely seems to benefit the study of the objects of reality. And, interestingly (...)
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  12. added 2018-03-31
    Our Incorrigible Ontological Relations and Categories of Being.Julian M. Galvez Bunge (ed.) - 2017 - USA: Amazon.
    The object of this book is to present a radical novel conception of the ontological categories, their nature and epistemic importance. A conception that constitutes a challenge to the prevailing tenets, if not paradigms, of ontology today. The arguments and observations are given without addressing nor directly contesting the current theories on the subject. However, its author emphasises some of the main conclusions that entail from the new perspective, in particular regarding the role of philosophy among the sciences. Departing from (...)
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  13. added 2018-02-16
    Why Numbers Are Sets.Eric Steinhart - 2002 - Synthese 133 (3):343-361.
    I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's. I give a detailed mathematical demonstration that 0 is {} and for every natural number n, n is the set of all natural numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals.
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  14. added 2017-12-31
    To Reduce Nothingness Into a Reference by Falsity.Hazhir Roshangar - manuscript
    Assuming the absolute nothingness as the most basic object of thought, I present a way to refer to this object, by reducing it onto a primitive object that supersedes and comes right after the absolute nothingness. The new primitive object that is constructed can be regarded as a formal system that can generate some infinite variety of symbols. [The PDF here is outdated, for a recent draft please contact me.].
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  15. added 2017-11-28
    What We Talk About When We Talk About Numbers.Richard Pettigrew - manuscript
    In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism.
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  16. added 2017-11-09
    Fourteen Arguments in Favour of a Formalist Philosophy of Real Mathematics.Karlis Podnieks - 2015 - Baltic Journal of Modern Computing 3 (1):1-15.
    The formalist philosophy of mathematics (in its purest, most extreme version) is widely regarded as a “discredited position”. This pure and extreme version of formalism is called by some authors “game formalism”, because it is alleged to represent mathematics as a meaningless game with strings of symbols. Nevertheless, I would like to draw attention to some arguments in favour of game formalism as an appropriate philosophy of real mathematics. For the most part, these arguments have not yet been used or (...)
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  17. added 2017-11-06
    Chapter 6: Reifying Terms.Friederike Moltmann - 2013 - In Abstract Objects and the Semantics of Natural Language. Oxford: Oxford University Press.
    This chapter develops a semantics for 'reifying terms' of the sort 'the proposition that S', 'the fact that S', 'the property of being P', 'the number eight', 'the concept horse', 'the truth value true', 'the kind humane being'. This semantics is developed within the broader perspective of the ontology of natural language involving abstract objects only at its periphery, not its core.
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  18. added 2017-09-06
    Does Ontology Rest on a Mistake?Stephen Yablo & Andre Gallois - 1998 - Proceedings of the Aristotelian Society, Supplementary Volumes( 72:229-283.
    [Stephen Yablo] The usual charge against Carnap's internal/external distinction is one of 'guilt by association with analytic/synthetic'. But it can be freed of this association, to become the distinction between statements made within make-believe games and those made outside them-or, rather, a special case of it with some claim to be called the metaphorical/literal distinction. Not even Quine considers figurative speech committal, so this turns the tables somewhat. To determine our ontological commitments, we have to ferret out all traces of (...)
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  19. added 2017-02-10
    Non-Symbolic Halving in an Amazonian Indigene Group.Koleen McCrink, Elizabeth Spelke, Stanislas Dehaene & Pierre Pica - 2013 - Developmental Science 16 (3):451-462.
    Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non-human animals to generate coarse, non-symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an intuition of a more complex calculation, division, exists in an indigene group in the Amazon, the Mundurucu, whose language includes no words for large numbers. Mundurucu children were presented with a video event (...)
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  20. added 2016-09-21
    Speaks's Reduction of Propositions to Properties: A Benacerraf Problem.T. Scott Dixon & Cody Gilmore - 2016 - Thought: A Journal of Philosophy 5 (3):275-284.
    Speaks defends the view that propositions are properties: for example, the proposition that grass is green is the property being such that grass is green. We argue that there is no reason to prefer Speaks's theory to analogous but competing theories that identify propositions with, say, 2-adic relations. This style of argument has recently been deployed by many, including Moore and King, against the view that propositions are n-tuples, and by Caplan and Tillman against King's view that propositions are facts (...)
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  21. added 2016-08-30
    Figure, Ratio, Form: Plato's Five Mathematical Studies.Mitchell Miller - 1999 - Apeiron 32 (4):73-88.
    A close reading of the five mathematical studies Socrates proposes for the philosopher-to-be in Republic VII, arguing that (1) each study proposes an object the thought of which turns the soul towards pure intelligibility and that (2) the sequence of studies involves both a departure from the sensible and a return to it in its intelligible structure.
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  22. added 2016-04-17
    Frege, the Complex Numbers, and the Identity of Indiscernibles.Wenzel Christian Helmut - 2010 - Logique Et Analyse 53 (209):51-60.
    There are mathematical structures with elements that cannot be distinguished by the properties they have within that structure. For instance within the field of complex numbers the two square roots of −1, i and −i, have the same algebraic properties in that field. So how do we distinguish between them? Imbedding the complex numbers in a bigger structure, the quaternions, allows us to algebraically tell them apart. But a similar problem appears for this larger structure. There seems to be always (...)
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  23. added 2015-04-04
    Structure and the Concept of Number.Mark Eli Kalderon - 1995 - Dissertation, Princeton University
    The present essay examines and critically discusses Paul Benacerraf's antiplatonist argument of "What Numbers Could Not Be." In the course of defending platonism against Benacerraf's semantic skepticism, I develop a novel platonist analysis of the content of arithmetic on the basis of which the necessary existence of the natural numbers and the nature of numerical reference are explained.
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  24. added 2015-03-20
    Education Enhances the Acuity of the Nonverbal Approximate Number System.Manuela Piazza, Pierre Pica, Véronique Izard, Elizabeth Spelke & Stanislas Dehaene - 2013 - Psychological Science 24 (4):p.
    All humans share a universal, evolutionarily ancient approximate number system (ANS) that estimates and combines the numbers of objects in sets with ratio-limited precision. Interindividual variability in the acuity of the ANS correlates with mathematical achievement, but the causes of this correlation have never been established. We acquired psychophysical measures of ANS acuity in child and adult members of an indigene group in the Amazon, the Mundurucú, who have a very restricted numerical lexicon and highly variable access to mathematics education. (...)
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  25. added 2015-03-17
    Why the Numbers Should Sometimes Count.John T. Sanders - 1988 - Philosophy and Public Affairs 17 (1):3-14.
    John Taurek has argued that, where choices must be made between alternatives that affect different numbers of people, the numbers are not, by themselves, morally relevant. This is because we "must" take "losses-to" the persons into account (and these don't sum), but "must not" consider "losses-of" persons (because we must not treat persons like objects). I argue that the numbers are always ethically relevant, and that they may sometimes be the decisive consideration.
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  26. added 2014-10-09
    Musil's Imaginary Bridge.Achille C. Varzi - 2014 - The Monist 97 (1):30-46.
    In a calculation involving imaginary numbers, we begin with real numbers that represent concrete measures and we end up with numbers that are equally real, but in the course of the operation we find ourselves walking “as if on a bridge that stands on no piles”. How is that possible? How does that work? And what is involved in the as-if stance that this metaphor introduces so beautifully? These are questions that bother Törless deeply. And that Törless is bothered by (...)
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  27. added 2014-03-10
    On 'Average'.Christopher Kennedy & Jason Stanley - 2009 - Mind 118 (471):583 - 646.
    This article investigates the semantics of sentences that express numerical averages, focusing initially on cases such as 'The average American has 2.3 children'. Such sentences have been used both by linguists and philosophers to argue for a disjuncture between semantics and ontology. For example, Noam Chomsky and Norbert Hornstein have used them to provide evidence against the hypothesis that natural language semantics includes a reference relation holding between words and objects in the world, whereas metaphysicians such as Joseph Melia and (...)
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  28. added 2014-03-06
    Theoretical Implications of the Study of Numbers and Numerals in Mundurucu.Pierre Pica & Alain Lecomte - 2008 - Philosophical Psychology 21 (4):507 – 522.
    Developing earlier studies of the system of numbers in Mundurucu, this paper argues that the Mundurucu numeral system is far more complex than usually assumed. The Mundurucu numeral system provides indirect but insightful arguments for a modular approach to numbers and numerals. It is argued that distinct components must be distinguished, such as a system of representation of numbers in the format of internal magnitudes, a system of representation for individuals and sets, and one-to-one correspondences between the numerosity expressed by (...)
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  29. added 2014-03-06
    Exact Equality and Successor Function: Two Key Concepts on the Path Towards Understanding Exact Numbers.Véronique Izard, Pierre Pica, Elizabeth S. Spelke & Stanislas Dehaene - 2008 - Philosophical Psychology 21 (4):491 – 505.
    Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers can be generated by (...)
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  30. added 2014-01-16
    Pythagorean Powers or a Challenge to Platonism.Colin Cheyne & Charles R. Pigden - 1996 - Australasian Journal of Philosophy 74 (4):639 – 645.
    The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their (...)
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  31. added 2013-09-02
    Frege, Carnap, and Explication: ‘Our Concern Here Is to Arrive at a Concept of Number Usable for the Purpose of Science’.Gregory Lavers - 2013 - History and Philosophy of Logic 34 (3):225-41.
    This paper argues that Carnap both did not view and should not have viewed Frege's project in the foundations of mathematics as misguided metaphysics. The reason for this is that Frege's project was to give an explication of number in a very Carnapian sense — something that was not lost on Carnap. Furthermore, Frege gives pragmatic justification for the basic features of his system, especially where there are ontological considerations. It will be argued that even on the question of the (...)
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  32. added 2013-01-25
    Abstract Objects and the Semantics of Natural Language.Friederike Moltmann - 2013 - Oxford University Press.
    This book pursues the question of how and whether natural language allows for reference to abstract objects in a fully systematic way. By making full use of contemporary linguistic semantics, it presents a much greater range of linguistic generalizations than has previously been taken into consideration in philosophical discussions, and it argues for an ontological picture is very different from that generally taken for granted by philosophers and semanticists alike. Reference to abstract objects such as properties, numbers, propositions, and degrees (...)
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  33. added 2013-01-18
    Wide Sets, ZFCU, and the Iterative Conception.Christopher Menzel - 2014 - Journal of Philosophy 111 (2):57-83.
    The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence of “wide” sets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as to allow for the (...)
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  34. added 2013-01-05
    The Mapping of Numbers on Space : Evidence for a Logarithmic Intuition.Véronique Izard, Pierre Pica, Elizabeth Spelke & Stanislas Dehaene - 2008 - Médecine/Science 24 (12):1014-1016.
    Des branches entières des mathématiques sont fondées sur des liens posés entre les nombres et l’espace : mesure de longueurs, définition de repères et de coordonnées, projection des nombres complexes sur le plan… Si les nombres complexes, comme l’utilisation de repères, sont apparus relativement récemment (vers le XVIIe siècle), la mesure des longueurs est en revanche un procédé très ancien, qui remonte au moins au 3e ou 4e millénaire av. J-C. Loin d’être fortuits, ces liens entre les nombres et l’espace (...)
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  35. added 2013-01-05
    Exact and Approximate Arithmetic in an Amazonian Indigene Group.Pierre Pica, Cathy Lemer, Véronique Izard & Stanislas Dehaene - 2004 - Science 306 (5695):499-503.
    Is calculation possible without language? Or is the human ability for arithmetic dependent on the language faculty? To clarify the relation between language and arithmetic, we studied numerical cognition in speakers of Mundurukú, an Amazonian language with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their naming range. However, they fail in exact arithmetic with numbers larger than 4 (...)
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  36. added 2011-12-06
    Reference to Numbers in Natural Language.Friederike Moltmann - 2013 - Philosophical Studies 162 (3):499 - 536.
    A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are not primarily treated abstract objects, (...)
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  37. added 2011-11-21
    The Number of Planets, a Number-Referring Term?Friederike Moltmann - 2016 - In Philip A. Ebert & Marcus Rossberg (eds.), Abstractionism: Essays in Philosophy of Mathematics. Oxford University Press. pp. 113-129.
    The question whether numbers are objects is a central question in the philosophy of mathematics. Frege made use of a syntactic criterion for objethood: numbers are objects because there are singular terms that stand for them, and not just singular terms in some formal language, but in natural language in particular. In particular, Frege (1884) thought that both noun phrases like the number of planets and simple numerals like eight as in (1) are singular terms referring to numbers as abstract (...)
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  38. added 2011-11-21
    Infinite Numbers Are Large Finite Numbers.Jeremy Gwiazda - unknown
    In this paper, I suggest that infinite numbers are large finite numbers, and that infinite numbers, properly understood, are 1) of the structure omega + (omega* + omega)Ө + omega*, and 2) the part is smaller than the whole. I present an explanation of these claims in terms of epistemic limitations. I then consider the importance, part of which is demonstrating the contradiction that lies at the heart of Cantorian set theory: the natural numbers are too large to be counted (...)
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  39. added 2011-06-23
    The Cultural Phenomenology of Qualitative Quantity - Work in Progress - Introduction Autobiographical.Borislav Dimitrov - manuscript
    This study is about the Quality. Here I have dealt with the quality that differs significantly from the common understanding of quality /as determined quality/ that arise from the law of dialectics. This new quality is the quality of the quantity /quality of the quantitative changes/, noticed in philosophy by Plato as “quality of numbers”, and later developed by Hegel as “qualitative quantity. The difference between the known determined quality and qualitative quantity is evident in the exhibit form of these (...)
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  40. added 2011-03-03
    Aristotle and Modern Mathematical Theories of the Continuum.Anne Newstead - 2001 - In Demetra Sfendoni-Mentzou & James Brown (eds.), Aristotle and Contemporary Philosophy of Science. Peter Lang.
    This paper is on Aristotle's conception of the continuum. It is argued that although Aristotle did not have the modern conception of real numbers, his account of the continuum does mirror the topology of the real number continuum in modern mathematics especially as seen in the work of Georg Cantor. Some differences are noted, particularly as regards Aristotle's conception of number and the modern conception of real numbers. The issue of whether Aristotle had the notion of open versus closed intervals (...)
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