Related categories

39 found
Order:
  1. 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.
    Remove from this list   Direct download  
     
    Export citation  
     
    Bookmark  
  2. 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 (...)
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  3. 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 (...)
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    Bookmark   12 citations  
  4. ‘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. pp. 139-160.
    Remove from this list   Direct download  
     
    Export citation  
     
    Bookmark  
  5. 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 (...)
    Remove from this list   Direct download (6 more)  
     
    Export citation  
     
    Bookmark   11 citations  
  6. 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.
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  7. An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Stucture.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 (...)
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    Bookmark   30 citations  
  8. 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 (...)
    Remove from this list   Direct download  
     
    Export citation  
     
    Bookmark   3 citations  
  9. 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 (...)
    Remove from this list   Direct download (8 more)  
     
    Export citation  
     
    Bookmark   22 citations  
  10. 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 (...)
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  11. 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 (...)
    Remove from this list   Direct download (5 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  12. 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 (...)
    Remove from this list   Direct download  
     
    Export citation  
     
    Bookmark   4 citations  
  13. 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 (...)
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  14. Non-Deductive Methods in Mathematics.Alan Baker - 2010 - Stanford Encyclopedia of Philosophy.
  15. 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 Aspects and Mathematical Practice. 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 (...)
    Remove from this list   Direct download  
    Translate
     
     
    Export citation  
     
    Bookmark   4 citations  
  16. 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.
    Remove from this list   Direct download (2 more)  
    Translate
     
     
    Export citation  
     
    Bookmark  
  17. 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 (...)
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  18. 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 (...)
    Remove from this list   Direct download (10 more)  
     
    Export citation  
     
    Bookmark   21 citations  
  19. 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 (...)
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    Bookmark   14 citations  
  20. (Implications of Experimental Mathematics Jor the Philosoj) Hij of Mathematics1.Jonathan Borwein - 2008 - In Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America. pp. 33.
  21. 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, (...)
    Remove from this list   Direct download (8 more)  
     
    Export citation  
     
    Bookmark   11 citations  
  22. The Psychology of Mathematics Education and the Conjectural Nature of Experimental Mathematics.Aurel Pera - 2008 - Linguistic and Philosophical Investigations 7.
  23. Is There a Problem of Induction for Mathematics?Alan Baker - 2007 - In M. Potter (ed.), Mathematical Knowledge. Oxford University Press. pp. 57-71.
    Remove from this list  
    Translate
     
     
    Export citation  
     
    Bookmark   15 citations  
  24. 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 (...)
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  25. 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.
  26. 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 (...)
    Remove from this list  
     
    Export citation  
     
    Bookmark   12 citations  
  27. 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 (...)
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  28. 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 (...)
    Remove from this list   Direct download (7 more)  
     
    Export citation  
     
    Bookmark   7 citations  
  29. Thought-Experimentation and Mathematical Innovation.Eduard Glas - 1999 - Studies in History and Philosophy of Science Part A 30 (1):1-19.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    Bookmark   6 citations  
  30. The Epistemic Status of Probabilistic Proof.Don Fallis - 1997 - Journal of Philosophy 94 (4):165-186.
  31. Mathematical Proof and the Reliability of DNA Evidence.Don Fallis - 1996 - The American Mathematical Monthly 103 (6):491-497.
  32. What is Empirical in Mathematics?Philip L. Peterson - 1991 - Philosophia Mathematica (1):91-110.
    Remove from this list   Direct download (5 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  33. Non-Deductive Logic in Mathematics.James Franklin - 1987 - British Journal for the Philosophy of Science 38 (1):1-18.
    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 Fermat's Last Theorem and the Riemann Hypothesis, have had to be considered in terms of the evidence for and against them. It is argued here that it is not adequate to describe the relation of evidence to hypothesis as `subjective', `heuristic' or (...)
    Remove from this list   Direct download (10 more)  
     
    Export citation  
     
    Bookmark   30 citations  
  34. Toward Computer-Aided Induction: A Brief Review of Currently Implemented Aqval Programs.Ryszard Stanisław Michalski - 1977 - Dept. Of Computer Science, University of Illinois at Urbana-Champaign.
    Remove from this list  
     
    Export citation  
     
    Bookmark  
  35. Mathematics as an Experimental Science.Sidney Axinn - 1968 - Philosophia Mathematica (1-2):1-10.
    Remove from this list   Direct download (5 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  36. Mathematical Proof and Experimental Proof.Sr Arthur H. Copeland - 1966 - Philosophy of Science 33 (4):303-316.
    In studies of scientific methodology, surprisingly little attention has been given to tests of hypotheses. Such testing constitutes a methodology common to various scientific disciplines and is an essential factor in the development of science since it determines which theories are retained. The classical theory of tests is a major accomplishment but requires modification in order to produce a theory that accounts for the success of science. The revised theory is an analysis of the nondeductive aspect of scientific reasoning. It (...)
    Remove from this list   Direct download (5 more)  
     
    Export citation  
     
    Bookmark  
  37. Mathematics and Plausible Reasoning.George Polya - 1954 - Princeton: Princeton University Press.
    2014 Reprint of 1954 American Edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. This two volume classic comprises two titles: "Patterns of Plausible Inference" and "Induction and Analogy in Mathematics." This is a guide to the practical art of plausible reasoning, particularly in mathematics, but also in every field of human activity. Using mathematics as the example par excellence, Polya shows how even the most rigorous deductive discipline is heavily dependent on techniques of guessing, inductive (...)
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    Bookmark   70 citations  
  38. Mathematics and Plausible Reasoning: Induction and Analogy in Mathematics.George Polya - 1954 - Princeton, NJ, USA: Princeton University Press.
    Here the author of How to Solve It explains how to become a "good guesser." Marked by G. Polya's simple, energetic prose and use of clever examples from a wide range of human activities, this two-volume work explores techniques of guessing, inductive reasoning, and reasoning by analogy, and the role they play in the most rigorous of deductive disciplines.
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    Bookmark   76 citations  
  39. Is Mathematics a 'Deductive' Science?Harold R. Smart - 1929 - Philosophical Review 38 (3):232-245.