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Andrew Arana [33]Andrew Peter Arana [1]
  1. On the relationship between plane and solid geometry.Andrew Arana & Paolo Mancosu - 2012 - Review of Symbolic Logic 5 (2):294-353.
    Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.
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  2. Purity of Methods.Michael Detlefsen & Andrew Arana - 2011 - Philosophers' Imprint 11.
    Throughout history, mathematicians have expressed preference for solutions to problems that avoid introducing concepts that are in one sense or another “foreign” or “alien” to the problem under investigation. This preference for “purity” (which German writers commonly referred to as “methoden Reinheit”) has taken various forms. It has also been persistent. This notwithstanding, it has not been analyzed at even a basic philosophical level. In this paper we give a basic analysis of one conception of purity—what we call topical purity—and (...)
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  3. On the alleged simplicity of impure proof.Andrew Arana - 2017 - In Roman Kossak & Philip Ording (eds.), Simplicity: Ideals of Practice in Mathematics and the Arts. Springer. pp. 207-226.
    Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and (...)
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  4. On the Depth of Szemeredi's Theorem.Andrew Arana - 2015 - Philosophia Mathematica 23 (2):163-176.
    Many mathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemerédi's theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep by many mathematicians, and so makes for a good case on which (...)
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  5.  46
    Purity and Explanation: Essentially Linked?Andrew Arana - 2023 - In Carl Posy & Yemima Ben-Menahem (eds.), Mathematical Knowledge, Objects and Applications: Essays in Memory of Mark Steiner. Springer. pp. 25-39.
    In his 1978 paper “Mathematical Explanation”, Mark Steiner attempts to modernize the Aristotelian idea that to explain a mathematical statement is to deduce it from the essence of entities figuring in the statement, by replacing talk of essences with talk of “characterizing properties”. The language Steiner uses is reminiscent of language used for proofs deemed “pure”, such as Selberg and Erdős’ elementary proofs of the prime number theorem avoiding the complex analysis of earlier proofs. Hilbert characterized pure proofs as those (...)
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  6.  94
    (1 other version)Logical and semantic purity.Andrew Arana - 2008 - ProtoSociology 25:36-48.
    Many mathematicians have sought ‘pure’ proofs of theorems. There are different takes on what a ‘pure’ proof is, though, and it’s important to be clear on their differences, because they can easily be conflated. In this paper I want to distinguish between two of them.
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  7. Imagination in mathematics.Andrew Arana - 2016 - In Amy Kind (ed.), The Routledge Handbook of the Philosophy of Imagination. New York: Routledge. pp. 463-477.
    This article will consider imagination in mathematics from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras.
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  8.  75
    On Formally Measuring and Eliminating Extraneous Notions in Proofs.Andrew Arana - 2009 - Philosophia Mathematica 17 (2):189-207.
    Many mathematicians and philosophers of mathematics believe some proofs contain elements extraneous to what is being proved. In this paper I discuss extraneousness generally, and then consider a specific proposal for measuring extraneousness syntactically. This specific proposal uses Gentzen's cut-elimination theorem. I argue that the proposal fails, and that we should be skeptical about the usefulness of syntactic extraneousness measures.
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  9. Purity in Arithmetic: some Formal and Informal Issues.Andrew Arana - 2014 - In Godehard Link (ed.), Formalism and Beyond: On the Nature of Mathematical Discourse. Boston: De Gruyter. pp. 315-336.
    Over the years many mathematicians have voiced a preference for proofs that stay “close” to the statements being proved, avoiding “foreign”, “extraneous”, or “remote” considerations. Such proofs have come to be known as “pure”. Purity issues have arisen repeatedly in the practice of arithmetic; a famous instance is the question of complex-analytic considerations in the proof of the prime number theorem. This article surveys several such issues, and discusses ways in which logical considerations shed light on these issues.
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  10. Takeuti's proof theory in the context of the Kyoto School.Andrew Arana - 2019 - Jahrbuch Für Philosophie Das Tetsugaku-Ronso 46:1-17.
    Gaisi Takeuti (1926–2017) is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. He extensively extended Hilbert's program in the sense that he formulated Gentzen's sequent calculus, conjectured that cut-elimination holds for it (Takeuti's conjecture), and obtained several stunning results in the 1950–60s towards the solution of his conjecture. Though he has been known chiefly as a great mathematician, he wrote many papers in English and Japanese where he expressed his philosophical thoughts. In particular, he used (...)
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  11.  71
    Mathematical Hygiene.Andrew Arana & Heather Burnett - 2023 - Synthese 202 (4):1-28.
    This paper aims to bring together the study of normative judgments in mathematics as studied by the philosophy of mathematics and verbal hygiene as studied by sociolinguistics. Verbal hygiene (Cameron 1995) refers to the set of normative ideas that language users have about which linguistic practices should be preferred, and the ways in which they go about encouraging or forcing others to adopt their preference. We introduce the notion of mathematical hygiene, which we define in a parallel way as the (...)
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  12. Idéaux de preuve : explication et pureté.Andrew Arana - 2022 - In Andrew Arana & Marco Panza (eds.), Précis de philosophie de la logique et des mathématiques, Volume 2, philosophie des mathématiques. Paris: Editions de la Sorbonne. pp. 387-425.
    Why do mathematics often give several proofs of the same theorem? This is the question raised in this article, introducing the notion of an epistemic ideal and discussing two such ideals, the explanatoriness and purity of proof.
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  13.  47
    On the difficulty of discovering mathematical proofs.Andrew Arana & Will Stafford - 2023 - Synthese 202 (2):1-29.
    An account of mathematical understanding should account for the differences between theorems whose proofs are “easy” to discover, and those whose proofs are difficult to discover. Though Hilbert seems to have created proof theory with the idea that it would address this kind of “discovermental complexity”, much more attention has been paid to the lengths of proofs, a measure of the difficulty of _verifying_ of a _given_ formal object that it is a proof of a given formula in a given (...)
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  14. 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 (...)
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  15.  22
    Solovay's theorem cannot be simplified.Andrew Arana - 2001 - Annals of Pure and Applied Logic 112 (1):27-41.
    In this paper we consider three potential simplifications to a result of Solovay’s concerning the Turing degrees of nonstandard models of arbitrary completions of first-order Peano Arithmetic (PA). Solovay characterized the degrees of nonstandard models of completions T of PA, showing that they are the degrees of sets X such that there is an enumeration R ≤T X of an “appropriate” Scott set and there is a family of functions (tn)n∈ω, ∆0 n(X) uniformly in n, such that lim tn(s) s→∞.
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  16. L'infinité des nombres premiers : une étude de cas de la pureté des méthodes.Andrew Arana - 2011 - Les Etudes Philosophiques 97 (2):193.
    Une preuve est pure si, en gros, elle ne réfère dans son développement qu’à ce qui est « proche » de, ou « intrinsèque » à l’énoncé à prouver. L’infinité des nombres premiers, un théorème classique de l’arithmétique, est un cas d’étude particulièrement riche pour les recherches philosophiques sur la pureté. Deux preuves différentes de ce résultat sont ici considérées, à savoir la preuve euclidienne classique et une preuve « topologique » plus récente proposée par Furstenberg. D’un point de vue (...)
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  17. Possible m-diagrams of models of arithmetic.Andrew Arana - 2005 - In Stephen Simpson (ed.), Reverse Mathematics 2001. Association for Symbolic Logic.
    In this paper I begin by extending two results of Solovay; the first characterizes the possible Turing degrees of models of True Arithmetic (TA), the complete first-order theory of the standard model of PA, while the second characterizes the possible Turing degrees of arbitrary completions of P. I extend these two results to characterize the possible Turing degrees of m-diagrams of models of TA and of arbitrary complete extensions of PA. I next give a construction showing that the conditions Solovay (...)
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  18. Proof theory in philosophy of mathematics.Andrew Arana - 2010 - Philosophy Compass 5 (4):336-347.
    A variety of projects in proof theory of relevance to the philosophy of mathematics are surveyed, including Gödel's incompleteness theorems, conservation results, independence results, ordinal analysis, predicativity, reverse mathematics, speed-up results, and provability logics.
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  19.  87
    Arithmetical independence results using higher recursion theory.Andrew Arana - 2004 - Journal of Symbolic Logic 69 (1):1-8.
    We extend an independence result proved in our earlier paper "Solovay's Theorem Cannot Be Simplified" (Annals of Pure and Applied Logic 112 (2001)). Our method uses the Barwise.
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  20. Plane and Solid Geometry: A Note on Purity of Methods.Andrew Arana & Paolo Mancosu - 2014 - In Giorgio Venturi, Marco Panza & Gabriele Lolli (eds.), From Logic to Practice: Italian Studies in the Philosophy of Mathematics. Cham: Springer International Publishing. pp. 23--31.
     
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  21.  70
    Descartes and the cylindrical helix.Paolo Mancosu & Andrew Arana - 2010 - Historia Mathematica 37 (3):403-427.
    In correspondence with Mersenne in 1629, Descartes discusses a construction involving a cylinder and what Descartes calls a “helice.” Mancosu has argued that by “helice” Descartes was referring to a cylindrical helix. The editors of Mersenne’s correspondence (Vol. II), and Henk Bos, have independently argued that, on the con- trary, by “helice” Descartes was referring to the Archimedean spiral. We argue that identifying the helice with the cylindrical helix makes better sense of the text. In the process we take a (...)
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  22. Précis de philosophie de la logique et des mathématiques, Volume 2, philosophie des mathématiques.Andrew Arana & Marco Panza (eds.) - 2022 - Paris: Editions de la Sorbonne.
    The project of this Précis de philosophie de la logique et des mathématiques (vol. 1 under the direction of F. Poggiolesi and P. Wagner, vol. 2 under the direction of A. Arana and M. Panza) aims to offer a rich, systematic and clear introduction to the main contemporary debates in the philosophy of mathematics and logic. The two volumes bring together the contributions of thirty researchers (twelve for the philosophy of logic and eighteen for the philosophy of mathematics), specialists in (...)
     
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  23. The changing practices of proof in mathematics: Gilles Dowek: Computation, proof, machine. Cambridge: Cambridge University Press, 2015. Translation of Les Métamorphoses du calcul, Paris: Le Pommier, 2007. Translation from the French by Pierre Guillot and Marion Roman, $124.00HB, $40.99PB. [REVIEW]Andrew Arana - 2017 - Metascience 26 (1):131-135.
    Review of Dowek, Gilles, Computation, Proof, Machine, Cambridge University Press, Cambridge, 2015. Translation of Les Métamorphoses du calcul, Le Pommier, Paris, 2007. Translation from the French by Pierre Guillot and Marion Roman.
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  24. Review of D. Corfield's Toward A Philosophy Of Real Mathematics. [REVIEW]Andrew Arana - 2007 - Mathematical Intelligencer 29 (2).
    When mathematicians think of the philosophy of mathematics, they probably think of endless debates about what numbers are and whether they exist. Since plenty of mathematical progress continues to be made without taking a stance on either of these questions, mathematicians feel confident they can work without much regard for philosophical reflections. In his sharp–toned, sprawling book, David Corfield acknowledges the irrelevance of much contemporary philosophy of mathematics to current mathematical practice, and proposes reforming the subject accordingly.
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  25. (1 other version)Review of M. Giaquinto's Visual thinking in mathematics. [REVIEW]Andrew Arana - 2009 - Analysis 69 (2):401-403.
    Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late nineteenth century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis (in the sense of the infinitesimal calculus) received much attention in the nineteenth century. They helped instigate what Hans Hahn called a “crisis of intuition”, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this “crisis” as (...)
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  26. Review of Ferreiros and Gray's The Architecture of Modern Mathematics. [REVIEW]Andrew Arana - 2008 - Mathematical Intelligencer 30 (4).
    This collection of essays explores what makes modern mathematics ‘modern’, where ‘modern mathematics’ is understood as the mathematics done in the West from roughly 1800 to 1970. This is not the trivial matter of exploring what makes recent mathematics recent. The term ‘modern’ (or ‘modernism’) is used widely in the humanities to describe the era since about 1900, exemplified by Picasso or Kandinsky in the visual arts, Rilke or Pound in poetry, or Le Corbusier or Loos in architecture (a building (...)
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  27. Review of S. Feferman's in the light of logic. [REVIEW]Andrew Arana - 2005 - Mathematical Intelligencer 27 (4).
    We review Solomon Feferman's 1998 essay collection In The Light of Logic (Oxford University Press).
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  28.  64
    T. Button and S. Walsh. Philosophy and Model Theory. Oxford University Press, Oxford, 2018, xvi + 517 pp. [REVIEW]Andrew Arana - 2020 - Bulletin of Symbolic Logic 26 (3-4):287-292.
  29.  96
    Review of Computability: Turing, Gödel, Church, and Beyond. [REVIEW]Andrew Arana - 2015 - Notre Dame Philosophical Reviews 3 (20).