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  1. Observation and Intuition.Justin Clarke-Doane & Avner Ash - forthcoming - In Carolin Antos, Neil Barton & Venturi Giorgio (eds.), Palgrave Companion to the Philosophy of Set Theory.
    The motivating question of this paper is: ‘How are our beliefs in the theorems of mathematics justified?’ This is distinguished from the question ‘How are our mathematical beliefs reliably true?’ We examine an influential answer, outlined by Russell, championed by Gödel, and developed by those searching for new axioms to settle undecidables, that our mathematical beliefs are justified by ‘intuitions’, as our scientific beliefs are justified by observations. On this view, axioms are analogous to laws of nature. They are postulated (...)
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  2. Reasoning by Analogy in Mathematical Practice.Francesco Nappo & Nicolò Cangiotti - 2023 - Philosophia Mathematica 31 (2):176-215.
    In this paper, we offer a descriptive theory of analogical reasoning in mathematics, stating general conditions under which an analogy may provide genuine inductive support to a mathematical conjecture (over and above fulfilling the merely heuristic role of ‘suggesting’ a conjecture in the psychological sense). The proposed conditions generalize the criteria of Hesse in her influential work on analogical reasoning in the empirical sciences. By reference to several case studies, we argue that the account proposed in this paper does a (...)
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  3. Non-deductive justification in mathematics.A. C. Paseau - 2023 - Handbook of the History and Philosophy of Mathematical Practice.
    In mathematics, the deductive method reigns. Without proof, a claim remains unsolved, a mere conjecture, not something that can be simply assumed; when a proof is found, the problem is solved, it turns into a “result,” something that can be relied on. So mathematicians think. But is there more to mathematical justification than proof? -/- The answer is an emphatic yes, as I explain in this article. I argue that non-deductive justification is in fact pervasive in mathematics, and that it (...)
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  4. Unrealistic Models in Mathematics.William D'Alessandro - 2022 - Philosophers' Imprint.
    Models are indispensable tools of scientific inquiry, and one of their main uses is to improve our understanding of the phenomena they represent. How do models accomplish this? And what does this tell us about the nature of understanding? While much recent work has aimed at answering these questions, philosophers' focus has been squarely on models in empirical science. I aim to show that pure mathematics also deserves a seat at the table. I begin by presenting two cases: Cramér’s random (...)
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  5. Experimenting with Triangles.Valeria Giardino - 2022 - Axiomathes 32 (1):55-77.
    Is there anything like an experiment in mathematics? And if this is the case, what would distinguish a mathematical experiment from a mathematical thought experiment? In the present paper, a framework for the practice of mathematics will be put forward, which will consider mathematics as an experimenting activity and as a proving activity. The relationship between these two activities will be explored and more importantly a distinction between thought-experiments, real experiments, quasi experiments and proofs in pure mathematics will be provided. (...)
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  6. Inference to the best explanation as supporting the expansion of mathematicians’ ontological commitments.Marc Lange - 2022 - Synthese 200 (2):1-26.
    This paper argues that in mathematical practice, conjectures are sometimes confirmed by “Inference to the Best Explanation” as applied to some mathematical evidence. IBE operates in mathematics in the same way as IBE in science. When applied to empirical evidence, IBE sometimes helps to justify the expansion of scientists’ ontological commitments. Analogously, when applied to mathematical evidence, IBE sometimes helps to justify mathematicians' in expanding the range of their ontological commitments. IBE supplements other forms of non-deductive reasoning in mathematics, avoiding (...)
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  7. Applied versus situated mathematics in ancient Egypt: bridging the gap between theory and practice.Sandra Visokolskis & Héctor Horacio Gerván - 2022 - European Journal for Philosophy of Science 12 (1):1-30.
    This historiographical study aims at introducing the category of “situated mathematics” to the case of Ancient Egypt. However, unlike Situated Learning Theory, which is based on ethnographic relativity, in this paper, the goal is to analyze a mathematical craft knowledge based on concrete particulars and case studies, which is ubiquitous in all human activity, and which even covers, as a specific case, the Hellenistic style, where theoretical constructs do not stand apart from practice, but instead remain grounded in it.The historiographic (...)
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  8. Legitimate Mathematical Methods.James Robert Brown - 2020 - Croatian Journal of Philosophy 20 (1):1-6.
    A thought experiment involving an omniscient being and quantum mechanics is used to justify non-deductive methods in mathematics. The twin prime conjecture is used to illustrate what can be achieved.
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  9. Bayesian perspectives on mathematical practice.James Franklin - 2020 - Handbook of the History and Philosophy of Mathematical Practice.
    Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as the Riemann hypothesis, have had to be considered in terms of the evidence for and against them. In recent decades, massive increases in computer power have permitted the gathering of huge amounts of numerical evidence, both for conjectures in pure mathematics and (...)
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  10. Analogy and Invention Some Remarks on Poincaré’s Analysis Situs Papers.Claudio Bartocci - 2018 - In Hassan Tahiri (ed.), The Philosophers and Mathematics: Festschrift for Roshdi Rashed. Cham: Springer Verlag. pp. 1-23.
    The primary role played by analogy in Henri Poincaré’s work, and in particular in his “analysis situs” papers, is emphasized. Poincaré’s “sixth example” and his construction of the homology sphere are discussed in detail.
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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 End of Proof’? The integration of different mathematical cultures as experimental mathematics comes of age.Henrik Kragh Sørensen - 2016 - In Brendan Larvor (ed.), Mathematical Cultures: The London Meetings 2012-2014. Springer International Publishing. pp. 139-160.
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  13. Knowledge of Mathematics without Proof.Alexander Paseau - 2015 - British Journal for the Philosophy of Science 66 (4):775-799.
    Mathematicians do not claim to know a proposition unless they think they possess a proof of it. For all their confidence in the truth of a proposition with weighty non-deductive support, they maintain that, strictly speaking, the proposition remains unknown until such time as someone has proved it. This article challenges this conception of knowledge, which is quasi-universal within mathematics. We present four arguments to the effect that non-deductive evidence can yield knowledge of a mathematical proposition. We also show that (...)
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  14. Malament–Hogarth Machines and Tait’s Axiomatic Conception of Mathematics.Sharon Berry - 2014 - Erkenntnis 79 (4):893-907.
    In this paper I will argue that Tait’s axiomatic conception of mathematics implies that it is in principle impossible to be justified in believing a mathematical statement without being justified in believing that statement to be provable. I will then show that there are possible courses of experience which would justify acceptance of a mathematical statement without justifying belief that this statement is provable.
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  15. An Aristotelian Realist Philosophy of Mathematics: Mathematics as the science of quantity and structure.James Franklin - 2014 - London and New York: Palgrave MacMillan.
    An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...)
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  16. The Web as a Tool for Proving.Petros Stefaneas & Ioannis M. Vandoulakis - 2014 - In Harry Halpin & Alexandre Monnin (eds.), Philosophical Engineering: Toward a Philosophy of the Web. Wiley-Blackwell. pp. 149-167.
    This is the first interdisciplinary exploration of the philosophical foundations of the Web, a new area of inquiry that has important implications across a range of domains. - Contains twelve essays that bridge the fields of philosophy, cognitive science, and phenomenology. - Tackles questions such as the impact of Google on intelligence and epistemology, the philosophical status of digital objects, ethics on the Web, semantic and ontological changes caused by the Web, and the potential of the Web to serve as (...)
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  17. That We See That Some Diagrammatic Proofs Are Perfectly Rigorous.Jody Azzouni - 2013 - Philosophia Mathematica 21 (3):323-338.
    Mistaken reasons for thinking diagrammatic proofs aren't rigorous are explored. The main result is that a confusion between the contents of a proof procedure (what's expressed by the referential elements in a proof procedure) and the unarticulated mathematical aspects of a proof procedure (how that proof procedure is enabled) gives the impression that diagrammatic proofs are less rigorous than language proofs. An additional (and independent) factor is treating the impossibility of naturally generalizing a diagrammatic proof procedure as an indication of (...)
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  18. Analogical arguments in mathematics.Paul Bartha - 2013 - In Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics. Springer. pp. 199--237.
  19. Non-deductive Logic in Mathematics: The Probability of Conjectures.James Franklin - 2013 - In Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics. Springer. pp. 11--29.
    Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...)
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  20. Experimental mathematics, computers and the a priori.Mark McEvoy - 2013 - Synthese 190 (3):397-412.
    In recent decades, experimental mathematics has emerged as a new branch of mathematics. This new branch is defined less by its subject matter, and more by its use of computer assisted reasoning. Experimental mathematics uses a variety of computer assisted approaches to verify or prove mathematical hypotheses. For example, there is “number crunching” such as searching for very large Mersenne primes, and showing that the Goldbach conjecture holds for all even numbers less than 2 × 1018. There are “verifications” of (...)
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  21. Non-Language Thinking in Mathematics.Dieter Lohmar - 2012 - Axiomathes 22 (1):109-120.
    After a brief outline of the topic of non-language thinking in mathematics the central phenomenological tool in this concern is established, i.e. the eidetic method. The special form of eidetic method in mathematical proving is implicit variation and this procedure entails three rules that are established in a simple geometrical example. Then the difficulties and the merits of analogical thinking in mathematics are discussed in different aspects. On the background of a new phenomenological understanding of the performance of non-language thinking (...)
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  22. The Web as A Tool For Proving.Petros Stefaneas & Ioannis M. Vandoulakis - 2012 - Metaphilosophy 43 (4):480-498.
    The Web may critically transform the way we understand the activity of proving. The Web as a collaborative medium allows the active participation of people with different backgrounds, interests, viewpoints, and styles. Mathematical formal proofs are inadequate for capturing Web-based proofs. This article claims that Web provings can be studied as a particular type of Goguen's proof-events. Web-based proof-events have a social component, communication medium, prover-interpreter interaction, interpretation process, understanding and validation, historical component, and styles. To demonstrate its claim, the (...)
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  23. Mathematical instrumentalism, Gödel’s theorem, and inductive evidence.Alexander Paseau - 2011 - Studies in History and Philosophy of Science Part A 42 (1):140-149.
    Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though some versions of mathematical instrumentalism (...)
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  24. Non-deductive methods in mathematics.Alan Baker - 2010 - Stanford Encyclopedia of Philosophy.
  25. By parallel reasoning: the construction and evaluation of analogical arguments.Paul Bartha - 2010 - New York: Oxford University Press.
    In this work, Paul Bartha proposes a normative theory of analogical arguments and raises questions and proposes answers regarding the criteria for evaluating analogical arguments, the philosophical justification for analogical reasoning, and the place of scientific analogies in the context of theoretical confirmation.
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  26. Exploratory experimentation in experimental mathematics: A glimpse at the PSLQ algorithm.Henrik Kragh Sørensen - 2010 - In Benedikt Löwe & Thomas Müller (eds.), PhiMSAMP: philosophy of mathematics: sociological aspsects and mathematical practice. London: College Publications. pp. 341--360.
    In the present paper, I go beyond these examples by bringing into play an example that I nd more experimental in nature, namely that of the use of the so-called PSLQ algorithm in researching integer relations between numerical constants. It is the purpose of this paper to combine a historical presentation with a preliminary exploration of some philosophical aspects of the notion of experiment in experimental mathematics. This dual goal will be sought by analysing these aspects as they are presented (...)
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  27. Experimental mathematics in the 1990s: A second loss of certainty?Henrik Kragh Sørensen - 2010 - Oberwolfach Reports (12):601--604.
    In this paper, I describe some aspects of the phenomenon of "experimental mathematics" in order to discuss whether it constitutes a subdiscipline or a particular style of mathematics. My conclusion is that neither of these notions accurately capture the complex culture of experimental mathematics.
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  28. A Genetic Interpretation of Neo-Pythagorean Arithmetic.Ioannis M. Vandoulakis - 2010 - Oriens - Occidens 7:113-154.
    The style of arithmetic in the treatises the Neo-Pythagorean authors is strikingly different from that of the "Elements". Namely, it is characterised by the absence of proof in the Euclidean sense and a specific genetic approach to the construction of arithmetic that we are going to describe in our paper. Lack of mathematical sophistication has led certain historians to consider this type of mathematics as a feature of decadence of mathematics in this period [Tannery 1887; Heath 1921]. The alleged absence (...)
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  29. By Parallel Reasoning: The Construction and Evaluation of Analogical Arguments.Paul Bartha - 2009 - Oxford and New York: Oxford University Press USA.
    By Parallel Reasoning is the first comprehensive philosophical examination of analogical reasoning in more than forty years designed to formulate and justify standards for the critical evaluation of analogical arguments. It proposes a normative theory with special focus on the use of analogies in mathematics and science. In recent decades, research on analogy has been dominated by computational theories whose objective has been to model analogical reasoning as a psychological process. These theories have devoted little attention to normative questions. In (...)
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  30. Probabilistic proofs and transferability.Kenny Easwaran - 2009 - Philosophia Mathematica 17 (3):341-362.
    In a series of papers, Don Fallis points out that although mathematicians are generally unwilling to accept merely probabilistic proofs, they do accept proofs that are incomplete, long and complicated, or partly carried out by computers. He argues that there are no epistemic grounds on which probabilistic proofs can be rejected while these other proofs are accepted. I defend the practice by presenting a property I call ‘transferability’, which probabilistic proofs lack and acceptable proofs have. I also consider what this (...)
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  31. Experimental Mathematics.Alan Baker - 2008 - Erkenntnis 68 (3):331-344.
    The rise of the field of “ experimental mathematics” poses an apparent challenge to traditional philosophical accounts of mathematics as an a priori, non-empirical endeavor. This paper surveys different attempts to characterize experimental mathematics. One suggestion is that experimental mathematics makes essential use of electronic computers. A second suggestion is that experimental mathematics involves support being gathered for an hypothesis which is inductive rather than deductive. Each of these options turns out to be inadequate, and instead a third suggestion is (...)
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  32. Implications of Experimental Mathematics for the Philosophy of Mathematics.Jonathan Borwein - 2008 - In Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America. pp. 33.
  33. The epistemological status of computer-assisted proofs.Mark McEvoy - 2008 - Philosophia Mathematica 16 (3):374-387.
    Several high-profile mathematical problems have been solved in recent decades by computer-assisted proofs. Some philosophers have argued that such proofs are a posteriori on the grounds that some such proofs are unsurveyable; that our warrant for accepting these proofs involves empirical claims about the reliability of computers; that there might be errors in the computer or program executing the proof; and that appeal to computer introduces into a proof an experimental element. I argue that none of these arguments withstands scrutiny, (...)
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  34. The psychology of mathematics education and the conjectural nature of experimental mathematics.Aurel Pera - 2008 - Linguistic and Philosophical Investigations 7.
  35. Two Ways of Analogy: Extending the Study of Analogies to Mathematical Domains.Dirk Schlimm - 2008 - Philosophy of Science 75 (2):178-200.
    The structure-mapping theory has become the de-facto standard account of analogies in cognitive science and philosophy of science. In this paper I propose a distinction between two kinds of domains and I show how the account of analogies based on structure-preserving mappings fails in certain (object-rich) domains, which are very common in mathematics, and how the axiomatic approach to analogies, which is based on a common linguistic description of the analogs in terms of laws or axioms, can be used successfully (...)
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  36. Is there a problem of induction for mathematics?Alan Baker - 2007 - In M. Potter (ed.), Mathematical Knowledge. Oxford University Press. pp. 57-71.
  37. Thought Experiments in Science, Philosophy, and Mathematics.James Robert Brown - 2007 - Croatian Journal of Philosophy 7 (1):3-27.
    Most disciplines make use of thought experiments, but physics and philosophy lead the pack with heavy dependence upon them. Often this is for conceptual clarification, but occasionally they provide real theoretical advances. In spite of their importance, however, thought experirnents have received rather little attention as a topic in their own right until recently. The situation has improved in the past few years, but a mere generation ago the entire published literature on thought experiments could have been mastered in a (...)
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  38. Mathematical reasoning: induction, deduction and beyond.David Sherry - 2006 - Studies in History and Philosophy of Science Part A 37 (3):489-504.
    Mathematics used to be portrayed as a deductive science. Stemming from Polya, however, is a philosophical movement which broadens the concept of mathematical reasoning to include inductive or quasi-empirical methods. Interest in inductive methods is a welcome turn from foundationalism toward a philosophy grounded in mathematical practice. Regrettably, though, the conception of mathematical reasoning embraced by quasi-empiricists is still too narrow to include the sort of thought-experiment which Mueller describes as traditional mathematical proof and which Lakatos examines in Proofs and (...)
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  39. Analogy and diagonal argument.Zbigniew Tworak - 2006 - Logic and Logical Philosophy 15 (1):39-66.
    In this paper, I try to accomplish two goals. The first is to provide a general characterization of a method of proofs called — in mathematics — the diagonal argument. The second is to establish that analogical thinking plays an important role also in mathematical creativity. Namely, mathematical research make use of analogies regarding general strategies of proof. Some of mathematicians, for example George Polya, argued that deductions is impotent without analogy. What I want to show is that there exists (...)
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  40. Selmer Bringsjord and Michael Zenzen. Superminds: People Harness Hypercomputation, and More. Studies in Cognitive Systems, Volume 29. Dordrecht: Kluwer Academic Publishers, 2003. Pp. xxx + 339. ISBN 1-4020-1094-X. [REVIEW]E. Mendelson - 2005 - Philosophia Mathematica 13 (2):228-230.
  41. Essay on Analogy in Mathematics.André van Es - 2004 - Philosophy of Mathematics Education Journal 18.
  42. What Do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians.Don Fallis - 2002 - Logique Et Analyse 45.
    Several philosophers have used the framework of means/ends reasoning to explain the methodological choices made by scientists and mathematicians (see, e.g., Goldman 1999, Levi 1962, Maddy 1997). In particular, they have tried to identify the epistemic objectives of scientists and mathematicians that will explain these choices. In this paper, the framework of means/ends reasoning is used to study an important methodological choice made by mathematicians. Namely, mathematicians will only use deductive proofs to establish the truth of mathematical claims. In this (...)
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  43. Crunchy Methods in Practical Mathematics.Michael Wood - 2001 - Philosophy of Mathematics Education Journal 14.
    This paper focuses on the distinction between methods which are mathematically "clever", and those which are simply crude, typically repetitive and computer intensive, approaches for "crunching" out answers to problems. Examples of the latter include simulated probability distributions and resampling methods in statistics, and iterative methods for solving equations or optimisation problems. Most of these methods require software support, but this is easily provided by a PC. The paper argues that the crunchier methods often have substantial advantages from the perspectives (...)
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  44. The nature of progress in mathematics: the significance of analogy.Hourya Benis-Sinaceur - 2000 - In Emily Grosholz & Herbert Breger (eds.), The growth of mathematical knowledge. Boston: Kluwer Academic Publishers. pp. 281--293.
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  45. The Reliability of Randomized Algorithms.D. Fallis - 2000 - British Journal for the Philosophy of Science 51 (2):255-271.
    Recently, certain philosophers of mathematics (Fallis [1997]; Womack and Farach [(1997]) have argued that there are no epistemic considerations that should stop mathematicians from using probabilistic methods to establish that mathematical propositions are true. However, mathematicians clearly should not use methods that are unreliable. Unfortunately, due to the fact that randomized algorithms are not really random in practice, there is reason to doubt their reliability. In this paper, I analyze the prospects for establishing that randomized algorithms are reliable. I end (...)
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  46. Analogy and the growth of mathematical knowledge.Eberhard Knobloch - 2000 - In Emily Grosholz & Herbert Breger (eds.), The growth of mathematical knowledge. Boston: Kluwer Academic Publishers. pp. 295--314.
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  47. Thought-experimentation and mathematical innovation.Eduard Glas - 1999 - Studies in History and Philosophy of Science Part A 30 (1):1-19.
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  48. The Incidence of Analogical Procedures in the Emergence of Mathematical Concepts. Newton and Leibniz: a Case Study.Sandra Visokolskis - 1998 - Philosophica 62 (2).
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  49. Analogical reasoning and early mathematics learning.Patricia A. Alexander, C. Stephen White & Martha Daugherty - 1997 - In Lyn D. English (ed.), Mathematical reasoning: analogies, metaphors, and images. Mahwah, N.J.: L. Erlbaum Associates. pp. 117--147.
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  50. Mathematical reasoning: analogies, metaphors, and images.Lyn D. English (ed.) - 1997 - Mahwah, N.J.: L. Erlbaum Associates.
    Presents the latest research on how reasoning with analogies, metaphors, metonymies, and images can facilitate mathematical understanding. For math education, educational psychology, and cognitive science scholars.
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