Epistemology of Mathematics

Edited by Alan Baker (Swarthmore College)
Assistant editor: Sam Roberts (University of Sheffield)
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  1. Wigner's Puzzle on Applicability of Mathematics: On What Table to Assemble It?Catalin Barboianu - forthcoming - European Journal for Philosophy of Science.
    Attempts at solving what has been labeled as Eugene Wigner’s puzzle of applicability of mathematics are still far from arriving at an acceptable solution. The accounts developed to explain the “miracle” of applied mathematics vary in nature, foundation, and solution, from denying the existence of a genuine problem to designing structural theories based on mathematical formalism. Despite this variation, all investigations treated the problem in a unitary way with respect to the target, pointing to one or two ‘why’ or ‘how’ (...)
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  2. Mathematician's Call for Interdisciplinary Research Effort.Catalin Barboianu - 2013 - International Gambling Studies 13 (3):430-433.
    The article addresses the necessity of increasing the role of mathematics in the psychological intervention in problem gambling, including cognitive therapies. It also calls for interdisciplinary research with the direct contribution of mathematics. The current contributions and limitations of the role of mathematics are analysed with an eye toward the professional profiles of the researchers. An enhanced collaboration between these two disciplines is suggested and predicted.
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  3. Circularities In The Contemporary Philosophical Accounts Of The Applicability Of Mathematics In The Physical Universe.Catalin Barboianu - 2015 - Revista de Filosofie 61 (5):517-542.
    Contemporary philosophical accounts of the applicability of mathematics in physical sciences and the empirical world are based on formalized relations between the mathematical structures and the physical systems they are supposed to represent within the models. Such relations were constructed both to ensure an adequate representation and to allow a justification of the validity of the mathematical models as means of scientific inference. This article puts in evidence the various circularities (logical, epistemic, and of definition) that are present in these (...)
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  4. The "Unreasonable" Effectiveness of Mathematics: The Foundational Approach of the Theoretic Alternatives.Catalin Barboianu - 2015 - Revista de Filosofie 62 (1):58-71.
    The attempts of theoretically solving the famous puzzle-dictum of physicist Eugene Wigner regarding the “unreasonable” effectiveness of mathematics as a problem of analytical philosophy, started at the end of the 19th century, are yet far from coming out with an acceptable theoretical solution. The theories developed for explaining the empirical “miracle” of applied mathematics vary in nature, foundation and solution, from denying the existence of a genuine problem to structural theories with an advanced level of mathematical formalism. Despite this variation, (...)
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  5. Learning the Natural Numbers as a Child.Stefan Buijsman - 2019 - Noûs 53 (1):3-22.
    How do we get out knowledge of the natural numbers? Various philosophical accounts exist, but there has been comparatively little attention to psychological data on how the learning process actually takes place. I work through the psychological literature on number acquisition with the aim of characterising the acquisition stages in formal terms. In doing so, I argue that we need a combination of current neologicist accounts and accounts such as that of Parsons. In particular, I argue that we learn the (...)
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  6. Set-Theoretic Pluralism and the Benacerraf Problem.Justin Clarke-Doane - forthcoming - Philosophical Studies.
    Set-theoretic pluralism is an increasingly influential position in the philosophy of set theory (Balaguer [1998], Linksy and Zalta [1995], Hamkins [2012]). There is considerable room for debate about how best to formulate set-theoretic pluralism, and even about whether the view is coherent. But there is widespread agreement as to what there is to recommend the view (given that it can be formulated coherently). Unlike set-theoretic universalism, set-theoretic pluralism affords an answer to Benacerraf’s epistemological challenge. The purpose of this paper is (...)
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  7. The "Artificial Mathematician" Objection: Exploring the (Im)Possibility of Automating Mathematical Understanding.Sven Delarivière & Bart Van Kerkhove - 2017 - In B. Sriraman (ed.), Humanizing Mathematics and its Philosophy. Cham: Birkhäuser. pp. 173-198.
    Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer.
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  8. The Reality of Field’s Epistemological Challenge to Platonism.David Liggins - 2018 - Erkenntnis 83 (5):1027-1031.
    In the introduction to his Realism, mathematics and modality, and in earlier papers included in that collection, Hartry Field offered an epistemological challenge to platonism in the philosophy of mathematics. Justin Clarke-Doane Truth, objects, infinity: New perspectives on the philosophy of Paul Benacerraf, 2016) argues that Field’s challenge is an illusion: it does not pose a genuine problem for platonism. My aim is to show that Clarke-Doane’s argument relies on a misunderstanding of Field’s challenge.
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  9. Mathematical Knowledge, the Analytic Method, and Naturalism.Fabio Sterpetti - 2018 - In Sorin Bangu (ed.), Naturalizing Logico-Mathematical Knowledge. Approaches from Philosophy, Psychology and Cognitive Science. New York, Stati Uniti: pp. 268-293.
    This chapter tries to answer the following question: How should we conceive of the method of mathematics, if we take a naturalist stance? The problem arises since mathematical knowledge is regarded as the paradigm of certain knowledge, because mathematics is based on the axiomatic method. Moreover, natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some authors tried to naturalize (...)
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  10. Mathematical Knowledge and Naturalism.Fabio Sterpetti - forthcoming - Philosophia:1-23.
    How should one conceive of the method of mathematics, if one takes a naturalist stance? Mathematical knowledge is regarded as the paradigm of certain knowledge, since mathematics is based on the axiomatic method. Natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some naturalists try to naturalize mathematics relying on Darwinism. But several difficulties arise when one tries to naturalize (...)
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  11. The Argument From Agreement and Mathematical Realism.Pieranna Garavaso - 1992 - Journal of Philosophical Research 17:173-187.
    Traditionally, in the philosophy of mathematics realists claim that mathematical objects exist independently of the human mind, whereas idealists regard them as mental constructions dependent upon human thought.It is tempting for realists to support their view by appeal to our widespread agreement on mathematical results. Roughly speaking, our agreement is explained by the fact that these results are about the same mathematical objects. It is alleged that the idealist’s appeal to mental constructions precludes any such explanation. I argue that realism (...)
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  12. Proof and the Virtues of Shared Enquiry.Don Berry - forthcoming - Philosophia Mathematica:nkw022.
    This paper investigates an important aspect of mathematical practice: that proof is required for a finished piece of mathematics. If follows that non-deductive arguments — however convincing — are never sufficient. I explore four aspects of mathematical research that have facilitated the impressive success of the discipline. These I call the Practical Virtues: Permanence, Reliability, Autonomy, and Consensus. I then argue that permitting results to become established on the basis of non-deductive evidence alone would lead to their deterioration. This furnishes (...)
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  13. Why the Naïve Derivation Recipe Model Cannot Explain How Mathematicians’ Proofs Secure Mathematical Knowledge.Brendan Larvor - 2016 - Philosophia Mathematica 24 (3):401-404.
    The view that a mathematical proof is a sketch of or recipe for a formal derivation requires the proof to function as an argument that there is a suitable derivation. This is a mathematical conclusion, and to avoid a regress we require some other account of how the proof can establish it.
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  14. The Nature of Mathematical Knowledge.Charles Parsons & Philip Kitcher - 1986 - Philosophical Review 95 (1):129.
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  15. Mathematical Knowledge. [REVIEW]David D. Auerbach - 1977 - Philosophical Review 86 (2):247.
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  16. José Ferreirós. Mathematical Knowledge and the Interplay of Practices. [REVIEW]Dirk Schlimm - 2017 - Philosophia Mathematica 25 (1):139-143.
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  17. The Applicability of Mathematics to Physical Modality.Nora Berenstain - 2017 - Synthese 194 (9):3361-3377.
    This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The viewpresented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal structure of (...)
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  18. The Varieties of Indispensability Arguments.Marco Panza & Andrea Sereni - 2016 - Synthese 193 (2):469-516.
    The indispensability argument comes in many different versions that all reduce to a general valid schema. Providing a sound IA amounts to providing a full interpretation of the schema according to which all its premises are true. Hence, arguing whether IA is sound results in wondering whether the schema admits such an interpretation. We discuss in full details all the parameters on which the specification of the general schema may depend. In doing this, we consider how different versions of IA (...)
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  19. The Good, the Bad and the Ugly.Philip A. Ebert & Stewart Shapiro - 2009 - Synthese 170 (3):415-441.
    This paper discusses the neo-logicist approach to the foundations of mathematics by highlighting an issue that arises from looking at the Bad Company objection from an epistemological perspective. For the most part, our issue is independent of the details of any resolution of the Bad Company objection and, as we will show, it concerns other foundational approaches in the philosophy of mathematics. In the first two sections, we give a brief overview of the "Scottish" neo-logicist school, present a generic form (...)
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  20. The Role of Axioms in Mathematics.Kenny Easwaran - 2008 - Erkenntnis 68 (3):381-391.
    To answer the question of whether mathematics needs new axioms, it seems necessary to say what role axioms actually play in mathematics. A first guess is that they are inherently obvious statements that are used to guarantee the truth of theorems proved from them. However, this may neither be possible nor necessary, and it doesn’t seem to fit the historical facts. Instead, I argue that the role of axioms is to systematize uncontroversial facts that mathematicians can accept from a wide (...)
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  21. Godel's Disjunction: The Scope and Limits of Mathematical Knowledge.Leon Horsten & Philip Welch (eds.) - 2016 - Oxford University Press UK.
    To what extent can we hope to find answers to all mathematical questions? A famous theorem from Gödel entails that if our thinking capacities do not go beyond what an electronic computer is capable of, then there are indeed absolutely unsolvable mathematical problems. Thus it is of capital importance to find out whether human mathematicians can outstrip computers. Within this context, the contributions to this book critically examine positions about the scope and limits of human mathematical knowledge.
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  22. Logicism and its Philosophical Legacy.William Demopoulos - 2015 - Cambridge University Press.
    The idea that mathematics is reducible to logic has a long history, but it was Frege who gave logicism an articulation and defense that transformed it into a distinctive philosophical thesis with a profound influence on the development of philosophy in the twentieth century. This volume of classic, revised and newly written essays by William Demopoulos examines logicism's principal legacy for philosophy: its elaboration of notions of analysis and reconstruction. The essays reflect on the deployment of these ideas by the (...)
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  23. Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences.Jody Azzouni - 2008 - Cambridge University Press.
    Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things as well. Jody Azzouni argues that mathematical knowledge really is a special kind of knowledge with its own special means of gathering evidence. He analyses the linguistic (...)
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  24. Towards a Philosophy of Real Mathematics.David Corfield - 2005 - Cambridge University Press.
    In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of approaches to new thinking about the philosophy of mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing (...)
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  25. Mathematical Reasoning and Heuristics.Carlo Cellucci & Donald Gillies (eds.) - 2005 - College Publications.
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  26. On Principles In Sadi Carnot’s Theory (1824). Epistemological Reflections.Raffaele Pisano - 2010 - Almagest 2/1:128–179 2 (1):128-179.
    In 1824 Sadi Carnot published Réflexions sur la Puissance Motrice du Feu in which he founded almost the entire thermodynamics theory. Two years after his death, his friend Clapeyron introduced the famous diagram PV for analytically representing the famous Carnot’s cycle: one of the main and crucial ideas presented by Carnot in his booklet. Twenty-five years later, in order to achieve the modern version of the theory, Kelvin and Clausius had to reject the caloric hypothesis, which had influenced a few (...)
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  27. Proof and Knowledge in Mathematics.Michael Detlefsen (ed.) - 1992 - Routledge.
    These questions arise from any attempt to discover an epistemology for mathematics. This collection of essays considers various questions concerning the nature of justification in mathematics and possible sources of that justification. Among these are the question of whether mathematical justification is _a priori_ or _a posteriori_ in character, whether logical and mathematical differ, and if formalization plays a significant role in mathematical justification.
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  28. Of Proofs, Mathematicians, and Computers.Astrik Yepremyan - unknown
    As computers become a more prevalent commodity in mathematical research and mathematical proof, the question of whether or not a computer assisted proof can be considered a mathematical proof has become an ongoing topic of discussion in the mathematics community. The use of the computer in mathematical research leads to several implications about mathematics in the present day including the notion that mathematical proof can be based on empirical evidence, and that some mathematical conclusions can be achieved a posteriori instead (...)
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  29. Mathematical Reasoning with Diagrams From Intuition to Automation.Mateja Jamnik - 2001
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  30. Truth, Proof and Infinity a Theory of Constructions and Contructive Reasoning.Peter Fletcher - 1998
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  31. A Role for Quasi-Empiricism in Mathematics Education.Eduard Glas - 2014 - In Michael R. Matthews (ed.), International Handbook of Research in History, Philosophy and Science Teaching. Springer. pp. 731-753.
    Although there are quite a few directions in modern philosophy of mathematics that invoke some essential role for (quasi-)empirical material, this chapter will be devoted exclusively to what may be considered the seminal tradition. This enabled me to present the subject as one coherent whole and to forestall the discussion getting scattered in a diversity of directions without doing justice to any one of them. -/- Quasi-empiricism in this tradition is the view that the logic of mathematical inquiry is based, (...)
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  32. Benacerraf's Dilemma and Natural Realism for Arithmetic.Anoop K. Gupta - 2002 - Dissertation, University of Ottawa (Canada)
    A natural realist approach to the philosophy of arithmetic is defended by way of considering and arguing against contemporary attempts to solve Paul Benacerraf's dilemma . The first horn of the dilemma concerns the existence of abstract mathematical objects, which seems necessitated by a desire for a unified semantics. Benacerraf adopts an extensional semantics whereby the reference of terms for natural numbers must be abstract objects. The second horn concerns a desirable causal constraint on knowledge, according to which "for X (...)
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  33. What Is the Goal of Proof? A Reaction to "What Do Mathematicians Want?" by Don Fallis.Aaron Lercher - 2002 - Logique Et Analyse 45.
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  34. Leibniz's and Kant's Philosophical Ideas and the Development of Hilbert's Programme.Roman Murawski - 2002 - Logique Et Analyse 45.
  35. 2. Remarks On The Structuralistic Epistemology Of Mathematics.Izabella Bondecka-Krzykowska & Roman Murawski - 2006 - Logique Et Analyse 49:85-93.
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  36. Looking for New Mathematical Concepts for the Material World - Whitehead's Investigations Into Formal Epistemology.Bruno Leclercq - 2011 - Logique Et Analyse 54.
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  37. Michael Detlefsen, Ed., Proof and Knowledge in Mathematics. [REVIEW]Alasdair Urquhart - 1992 - Philosophy in Review 12:237-238.
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  38. Kinetic Proof of the Existence of God in the Face of Present Mathematics.Ewa Rydzynska - 1993 - Roczniki Filozoficzne 41 (3):134.
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  39. Relying on Reason: A Reliabilist Account of a Priori Mathematical Knowledge.Mark Valentine Mcevoy - 2003 - Dissertation, City University of New York
    Because mathematical Platonism construes mathematical objects as existing outside of space and time, it precludes their having any causal interactions. This has led some to object that mathematical Platonism cannot explain how we know anything about such objects. ;Process reliabilism sometimes evokes the converse objection. Since process reliabilism takes knowledge to be reliably produced true belief, it is sometimes said that the theory cannot explain the reliability of our mathematical beliefs, as we cannot interact causally with mathematical objects. If this (...)
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  40. Philip Kitcher: The Nature of Mathematical Knowledge. [REVIEW]John Fauvel - 1984 - Radical Philosophy 37:43.
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  41. A Defense of a Probabilistic Method of Establishing Mathematical Truths.Don Thomas Fallis - 1995 - Dissertation, University of California, Irvine
    One of the primary goals of mathematicians is to establish new mathematical truths. Toward this end, mathematicians are almost invariably theorem provers. However, there are several methods other than writing down a proof which seem to achieve this epistemic goal of establishing mathematical truths. For instance, Michael Rabin describes a probabilistic test for primality which establishes to an arbitrarily high degree of certainty that a number is prime. Nevertheless, the vast majority of mathematicians are unwilling to employ such probabilistic methods (...)
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  42. The Crucial Role of Proof: A Classical Defense Against Mathematical Empiricism.Catherine Allen Womack - 1993 - Dissertation, Massachusetts Institute of Technology
    Mathematical knowledge seems to enjoy special status not accorded to scientific knowledge: it is considered a priori and necessary. We attribute this status to mathematics largely because of the way we come to know it--through following proofs. Mathematics has come under attack from sceptics who reject the idea that mathematical knowledge is a priori. Many sceptics consider it to be a posteriori knowledge, subject to possible empirical refutation. In a series of three papers I defend the a priori status of (...)
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  43. On Mathematical Knowledge.Mark Jay Steiner - 1972 - Dissertation, Princeton University
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  44. Creativity, Thought and Mathematical Proof.G. Hanna, Ian Winchester & Ontario Institute for Studies in Education - 1990 - Ontario Institute for Studies in Education.
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  45. Michael Detlefsen, Ed., "Proof and Knowledge in Mathematics". [REVIEW]Brendan Larvor - 1994 - International Journal of Philosophical Studies 2 (1):149.
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  46. Computer Method Of Demonstration?Mieczyslaw Lubanski - 1983 - Bulletin of the Section of Logic 12 (4):165-169.
    It is generally accepted that mathematics is a science with the help of which theorems are proved, i.e. indubitable statements are reached pro- vided we accept the initial assumptions and the relevant rules of argumen- tation. The concept of proof which functions here appears to be a univocal well dened concept. And yet, as it seems, computers have caused irrevert- ible changes in this sphere making it essential to reconsider the problems of essence of mathematical proof. In other words, we (...)
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  47. Philip Kitcher, "The Nature of Mathematical Knowledge". [REVIEW]Donald Gillies - 1985 - Philosophical Quarterly 35:104-107.
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  48. Idealization in Mathematics: Husserl and Beyond.Guillermo Rosado Haddock - 2004 - Poznan Studies in the Philosophy of the Sciences and the Humanities 82:245-252.
    Husserl's contributions to the nature of mathematical knowledge are opposed to the naturalist, empiricist and pragmatist tendences that are nowadays dominant. It is claimed that mainstream tendences fail to distinguish the historical problem of the origin and evolution of mathematical knowledge from the epistemological problem of how is it that we have access to mathematical knowledge.
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  49. Thinking Analysis to the Process of Mathematical Creativity of Mathematicians.Zhang Xiao Gui - 2013 - Philosophy of Mathematics Education Journal 27.
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  50. Crisis and Return of Intuition in Hans Hahn’s Philosophy of Mathematics.Erhard Oeser - 1995 - Vienna Circle Institute Yearbook 3:247-258.
    In the long history of 2000 years of interaction between philosophy and mathematics three major problem areas have been dealt with, following the three classic disciplines logic, metaphysics and epistemology: the problem of truth of mathematical statements the problem of existence of mathematical objects and the problem of how to recognize mathematical objects.
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