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  1. On The Rabbinical Exegesis of an Enhanced Biblical Value of Pi.Edward G. Belaga - 1991 - In Hardi Grant, Israel Kleiner & Abe Shenitzer (eds.), Proc. of the 17th Congress of the Canadian Society of History and Philosophy of Mathematics. Kingston.
    We present here a biblical exegesis of the value of Pi, PI_{Hebrew} = 3.141509 ..., from the well known verse 1 Kings 7:23. This verse is then compared to 2 Chronicles 4:2; the comparison provides independent supporting evidence for the exegesis.
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  2. Nominalism and Mathematical Intuition.Otávio Bueno - 2008 - ProtoSociology 25:89-107.
    As part of the development of an epistemology for mathematics, some Platonists have defended the view that we have intuition that certain mathematical principles hold, and intuition of the properties of some mathematical objects. In this paper, I discuss some difficulties that this view faces to accommodate some salient features of mathematical practice. I then offer an alternative, agnostic nominalist proposal in which, despite the role played by mathematical intuition, these difficulties do not emerge.
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  3. Intuition in Mathematics.Elijah Chudnoff - 2014 - In Barbara Held & Lisa Osbeck (eds.), Rational Intuition. Cambridge University Press.
    The literature on mathematics suggests that intuition plays a role in it as a ground of belief. This article explores the nature of intuition as it occurs in mathematical thinking. Section 1 suggests that intuitions should be understood by analogy with perceptions. Section 2 explains what fleshing out such an analogy requires. Section 3 discusses Kantian ways of fleshing it out. Section 4 discusses Platonist ways of fleshing it out. Section 5 sketches a proposal for resolving the main problem facing (...)
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  4. Awareness of Abstract Objects.Elijah Chudnoff - 2013 - Noûs 47 (4):706-726.
    Awareness is a two-place determinable relation some determinates of which are seeing, hearing, etc. Abstract objects are items such as universals and functions, which contrast with concrete objects such as solids and liquids. It is uncontroversial that we are sometimes aware of concrete objects. In this paper I explore the more controversial topic of awareness of abstract objects. I distinguish two questions. First, the Existence Question: are there any experiences that make their subjects aware of abstract objects? Second, the Grounding (...)
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  5. Intuitive Knowledge.Elijah Chudnoff - 2013 - Philosophical Studies 162 (2):359-378.
    In this paper I assume that we have some intuitive knowledge—i.e. beliefs that amount to knowledge because they are based on intuitions. The question I take up is this: given that some intuition makes a belief based on it amount to knowledge, in virtue of what does it do so? We can ask a similar question about perception. That is: given that some perception makes a belief based on it amount to knowledge, in virtue of what does it do so? (...)
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  6. Intuição e Conceito: A Transformação do Pensamento Matemático de Kant a Bolzano.Humberto de Assis Clímaco - 2014 - Dissertation, Universidade Federal de Goiás, Brazil
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  7. Numerical Cognition and Mathematical Realism.Helen De Cruz - 2016 - Philosophers' Imprint 16.
    Humans and other animals have an evolved ability to detect discrete magnitudes in their environment. Does this observation support evolutionary debunking arguments against mathematical realism, as has been recently argued by Clarke-Doane, or does it bolster mathematical realism, as authors such as Joyce and Sinnott-Armstrong have assumed? To find out, we need to pay closer attention to the features of evolved numerical cognition. I provide a detailed examination of the functional properties of evolved numerical cognition, and propose that they prima (...)
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  8. Forms and Roles of Diagrams in Knot Theory.Silvia De Toffoli & Valeria Giardino - 2014 - Erkenntnis 79 (4):829-842.
    The aim of this article is to explain why knot diagrams are an effective notation in topology. Their cognitive features and epistemic roles will be assessed. First, it will be argued that different interpretations of a figure give rise to different diagrams and as a consequence various levels of representation for knots will be identified. Second, it will be shown that knot diagrams are dynamic by pointing at the moves which are commonly applied to them. For this reason, experts must (...)
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  9. Mathematics and Conceptual Analysis.Antony Eagle - 2008 - Synthese 161 (1):67–88.
    Gödel argued that intuition has an important role to play in mathematical epistemology, and despite the infamy of his own position, this opinion still has much to recommend it. Intuitions and folk platitudes play a central role in philosophical enquiry too, and have recently been elevated to a central position in one project for understanding philosophical methodology: the so-called ‘Canberra Plan’. This philosophical role for intuitions suggests an analogous epistemology for some fundamental parts of mathematics, which casts a number of (...)
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  10. Kurt Gödels mathematische Anschauung und John P. Burgess’ mathematische Intuition.Eva-Maria Engelen - 2014 - XXIII Deutscher Kongress Für Philosophie Münster 2014, Konferenzveröffentlichung.
    John P. Burgess kritisiert Kurt Gödels Begriff der mathematischen oder rationalen Anschauung und erläutert, warum heuristische Intuition dasselbe leistet wie rationale Anschauung, aber ganz ohne ontologisch überflüssige Vorannahmen auskommt. Laut Burgess müsste Gödel einen Unterschied zwischen rationaler Anschauung und so etwas wie mathematischer Ahnung, aufzeigen können, die auf unbewusster Induktion oder Analogie beruht und eine heuristische Funktion bei der Rechtfertigung mathematischer Aussagen einnimmt. Nur, wozu benötigen wir eine solche Annahme? Reicht es nicht, wenn die mathematische Intuition als Heuristik funktioniert? Für (...)
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  11. Perceiving Necessity.Catherine Legg & James Franklin - 2017 - Pacific Philosophical Quarterly 98 (3).
    In many diagrams one seems to perceive necessity – one sees not only that something is so, but that it must be so. That conflicts with a certain empiricism largely taken for granted in contemporary philosophy, which believes perception is not capable of such feats. The reason for this belief is often thought well-summarized in Hume's maxim: ‘there are no necessary connections between distinct existences’. It is also thought that even if there were such necessities, perception is too passive or (...)
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  12. The Epistemology of Geometry I: The Problem of Exactness.Anne Newstead & Franklin James - 2010 - Proceedings of the Australasian Society for Cognitive Science 2009.
    We show how an epistemology informed by cognitive science promises to shed light on an ancient problem in the philosophy of mathematics: the problem of exactness. The problem of exactness arises because geometrical knowledge is thought to concern perfect geometrical forms, whereas the embodiment of such forms in the natural world may be imperfect. There thus arises an apparent mismatch between mathematical concepts and physical reality. We propose that the problem can be solved by emphasizing the ways in which the (...)
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  13. The Eternal Unprovability Filter – Part I.Kiran Pai - 2016 - Dissertation, Thinkstrike
    I prove both the mathematical conjectures P ≠ NP and the Continuum Hypothesis are eternally unprovable using the same fundamental idea. Starting with the Saunders Maclane idea that a proof is eternal or it is not a proof, I use the indeterminacy of human biological capabilities in the eternal future to show that since both conjectures are independent of Axioms and have definitions connected with human biological capabilities, it would be impossible to prove them eternally without the creation and widespread (...)
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  14. 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 (...)
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  15. Ortega y Gasset on Georg Cantor's Theory of Transfinite Numbers.Lior Rabi - 2016 - Kairos. Journal of Philosophy and Science (15):46-70.
    Ortega y Gasset is known for his philosophy of life and his effort to propose an alternative to both realism and idealism. The goal of this article is to focus on an unfamiliar aspect of his thought. The focus will be given to Ortega’s interpretation of the advancements in modern mathematics in general and Cantor’s theory of transfinite numbers in particular. The main argument is that Ortega acknowledged the historical importance of the Cantor’s Set Theory, analyzed it and articulated a (...)
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  16. Review of The Art of the Infinite by R. Kaplan, E. Kaplan 324p(2003).Michael Starks - 2016 - In Philosophy, Human Nature and the Collapse of Civilization Articles and Reviews 2006-2016 by Michael Starks 662p (2016). Michael Starks. pp. 619.
    This book tries to present math to the millions and does a pretty good job. It is simple and sometimes witty but often the literary allusions intrude and the text bogs down in pages of relentless math--lovely if you like it and horrid if you don´t. If you already know alot of math you will still probably find the discussions of general math, geometry, projective geometry, and infinite series to be a nice refresher. If you don´t know any and don´t (...)
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  17. Toward a Theoretical Account of Strategy Use and Sense-Making in Mathematics Problem Solving.H. J. M. Tabachneck, K. R. Koedinger & M. J. Nathan - 1994 - In Ashwin Ram & Kurt Eiselt (eds.), Proceedings of the Sixteenth Annual Conference of the Cognitive Science Society. Erlbaum.
    Much problem solving and learning research in math and science has focused on formal representations. Recently researchers have documented the use of unschooled strategies for solving daily problems -- informal strategies which can be as effective, and sometimes as sophisticated, as school-taught formalisms. Our research focuses on how formal and informal strategies interact in the process of doing and learning mathematics. We found that combining informal and formal strategies is more effective than single strategies. We provide a theoretical account of (...)
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  18. Hilbert's Finitism: Historical, Philosophical, and Metamathematical Perspectives.Richard Zach - 2001 - Dissertation, University of California, Berkeley
    In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented (...)
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