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Profile: Andrew Arana (University of Illinois, Urbana-Champaign)
  1. Andrew Arana & Paolo Mancosu (2012). On the Relationship Between Plane and Solid Geometry. 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. Andrew Arana & Brice Halimi (2011). L'infinité des nombres premiers : une étude de cas de la pureté des méthodes. Les Etudes Philosophiques 2 (2):193-213.
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  3. Michael Detlefsen & Andrew Arana (2011). Purity of Methods. Philosophers' Imprint 11 (2).
    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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  4. Andrew Arana (2010). Proof Theory in Philosophy of Mathematics. 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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  5. Paolo Mancosu & Andrew Arana (2010). Descartes and the Cylindrical Helix. Historia Mathematica 37 (3):403-427.
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  6. Andrew Arana (2009). On Formally Measuring and Eliminating Extraneous Notions in Proofs. Philosophia Mathematica 17 (2):208–219.
    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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  7. Andrew Arana (2009). Review of M. Giaquinto's Visual Thinking in Mathematics. [REVIEW] Analysis 69:401-403.
    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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  8. Andrew Arana (2009). Visual Thinking in Mathematics • by Marcus Giaquinto. Analysis 69 (2):401-403.
  9. Andrew Arana (2008). Review of Ferreiros and Gray's The Architecture of Modern Mathematics. [REVIEW] 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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  10. Andrew Arana (2008). Logical and Semantic Purity. 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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  11. Andrew Arana (2007). Review of D. Corfield's Toward A Philosophy Of Real Mathematics. [REVIEW] 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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  12. Andrew Arana (2007). Review of D. Corfield, Toward a Philosophy of Real Mathematics. [REVIEW] Mathematical Intelligencer 29 (2).
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  13. Andrew Arana (2005). Possible M-Diagrams of Models of Arithmetic. In Stephen Simpson (ed.), Reverse Mathematics 2001.
    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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  14. Andrew Arana (2005). Review of S. Feferman's in the Light of Logic. [REVIEW] Mathematical Intelligencer 27 (4).
    We review Solomon Feferman's 1998 essay collection In The Light of Logic (Oxford University Press).
     
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  15. Andrew Arana (2004). Arithmetical Independence Results Using Higher Recursion Theory. 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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  16. Warren Goldfarb, Jeremy Avigad, Andrew Arana, Geoffrey Hellman, Dana Scott & Michael Kremer (2004). Of the Association for Symbolic Logic. Bulletin of Symbolic Logic 10 (3):438.
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  17. Warren Goldfarb, Erich Reck, Jeremy Avigad, Andrew Arana, Geoffrey Hellman, Colin McLarty, Dana Scott & Michael Kremer (2004). Palmer House Hilton Hotel, Chicago, Illinois April 23–24, 2004. Bulletin of Symbolic Logic 10 (3).
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  18. Andrew Arana (2001). Solovay's Theorem Cannot Be Simplified. 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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