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Computer assisted theorem proving is an increasingly important part of mathematical methodology, as well as a long-standing topic in artificial intelligence (AI) research. However, the current generation of theorem proving software have limited functioning in terms of providing new proofs. Importantly, they are not able to discriminate interesting theorems and proofs from trivial ones. In order for computers to develop further in theorem proving, there would need to be a radical change in how the software functions. Recently, machine learning results (...) |
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If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, are fallible. Their ability to (...) |
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All acknowledged proofs of the Four Colour Theorem (4CT) are computerdependent. They appeal to the existence, and manual identification, of an ‘unavoidable’ set containing a sufficient number of explicitly defined configurations—each evidenced only by a computer as ‘reducible’—such that at least one of the configurations must occur in any chromatically distinguished, putatively minimal, planar map. For instance, Appel and Haken ‘identified’ 1,482 such configurations in their 1977, computer-dependent, proof of 4CT; whilst Neil Robertson et al ‘identified’ 633 configurations as sufficient (...) |
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All accepted proofs of the Four Colour Theorem (4CT) are computer-dependent; and appeal to the existence, and manual identification, of an ‘unavoidable’ set containing a sufficient number of explicitly defined configurations—each evidenced only by a computer as ‘reducible’—such that at least one of the configurations must occur in any chromatically distinguished, minimal, planar map. For instance, Appel and Haken ‘identified’ 1,482 such configurations in their 1977, computer-dependent, proof of 4CT; whilst Neil Robertson et al ‘identified’ 633 configurations as sufficient in (...) |
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In 1977, the first computer-assisted proof of a mathematical theorem was presented by K. Appel and W. Haken. The proof was met with a lot of criticism from both mathematicians and philosophers. In this paper, I present some examples of computer-assisted proofs, including Appel and Haken’s work. Then, I analyze the most famous arguments against the equal acceptance of computer-based and human-based proofs in mathematics and examine the philosophical assumptions behind the presented criticism. In the conclusion, I talk about whether (...) |
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Deutsch 2010 (The Review of Philosophy and Psychology 1: 447–460) claims that hypothetical scenarios are evaluated using arguments, not intuitions, and therefore experiments on intuitions are philosophically inconsequential. Using the Gettier case as an example, he identifies three arguments that are supposed to point to the right response to the case. In the paper, I present the results of studies ran on Polish, Indian, Spanish, and American participants that suggest that there’s no deep difference between evaluating the Gettier case with (...) |
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What grounds the truth of modal statements? And how do we get to know about what is possible or necessary? One of the most prominent anti-realist perspectives on the nature of modality, due to Peter Menzies, is the response-dependent account of modal concepts. Typically, offering a response-dependent account of a concept means defining it in terms of dispositions to elicit certain mental states from suitable agents under suitable circumstances. Menzies grounded possibility and necessity in the conceivability-response of ideal conceivers: P (...) |
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A proof P of a theorem T is transferable when a typical expert can become convinced of T solely on the basis of their prior knowledge and the information contained in P. Easwaran has argued that transferability is a constraint on acceptable proof. Meanwhile, a proof P is fixable when it’s possible for other experts to correct any mistakes P contains without having to develop significant new mathematics. Habgood-Coote and Tanswell have observed that some acceptable proofs are both fixable and (...) |
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This book develops a naturalistic aesthetic theory that accounts for aesthetic phenomena in mathematics in the same terms as it accounts for more traditional aesthetic phenomena. Building upon a view advanced by James McAllister, the assertion is that beauty in science does not confine itself to anecdotes or personal idiosyncrasies, but rather that it had played a role in shaping the development of science. Mathematicians often evaluate certain pieces of mathematics using words like beautiful, elegant, or even ugly. Such evaluations (...) |
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Conventional accounts of epistemic opacity, particularly those that stem from the definitive work of Paul Humphreys, typically point to limitations on the part of epistemic agents to account for the distinct ways in which systems, such as computational methods and devices, are opaque. They point, for example, to the lack of technical skill on the part of an agent, the failure to meet standards of best practice, or even the nature of an agent as reasons why epistemically relevant elements of (...) |
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In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very little (...) |
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According to the received view, genuine mathematical justification derives from proofs. In this article, I challenge this view. First, I sketch a notion of proof that cannot be reduced to deduction from the axioms but rather is tailored to human agents. Secondly, I identify a tension between the received view and mathematical practice. In some cases, cognitively diligent, well-functioning mathematicians go wrong. In these cases, it is plausible to think that proof sets the bar for justification too high. I then (...) |
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Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ (...) |
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The Four-Colour Theorem (4CT) proof, presented to the mathematical community in a pair of papers by Appel and Haken in the late 1970's, provoked a series of philosophical debates. Many conceptual points of these disputes still require some elucidation. After a brief presentation of the main ideas of Appel and Haken’s procedure for the proof and a reconstruction of Thomas Tymoczko’s argument for the novelty of 4CT’s proof, we shall formulate some questions regarding the connections between the points raised by (...) |
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O tendinţă relativ nouă în filosofia contemporană a matematicii este reprezentată de nemulţumirea manifestată de un număr din ce în ce mai mare de filosofi faţă de viziunea tradiţională asupra matematicii ca având un statut special ce poate fi surprins doar cu ajutorul unei epistemologii speciale. Această nemulţumire i-a determinat pe mulţi să propună o nouă perspectivă asupra matematicii – una care ia în serios aspecte până acum neglijate de filosofia matematicii, precum latura sociologică, istorică şi empirică a cercetării matematice (...) |
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The pluralist sheds the more traditional ideas of truth and ontology. This is dangerous, because it threatens instability of the theory. To lend stability to his philosophy, the pluralist trades truth and ontology for rigour and other ‘fixtures’. Fixtures are the steady goal posts. They are the parts of a theory that stay fixed across a pair of theories, and allow us to make translations and comparisons. They can ultimately be moved, but we tend to keep them fixed temporarily. Apart (...) |
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Computer simulations are widely used in current scientific practice, as a tool to obtain information about various phenomena. Scientists accordingly rely on the outputs of computer simulations to make statements about the empirical world. In that sense, simulations seem to enable scientists to acquire empirical knowledge. The aim of this paper is to assess whether computer simulations actually allow for the production of empirical knowledge, and how. It provides an epistemological analysis of present-day empirical science, to which the traditional epistemological (...) |
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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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Written by experts in the field, this volume presents a comprehensive investigation into the relationship between argumentation theory and the philosophy of mathematical practice. Argumentation theory studies reasoning and argument, and especially those aspects not addressed, or not addressed well, by formal deduction. The philosophy of mathematical practice diverges from mainstream philosophy of mathematics in the emphasis it places on what the majority of working mathematicians actually do, rather than on mathematical foundations. -/- The book begins by first challenging the (...) |
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It is often claimed that scientists can obtain new knowledge about nature by running computer simulations. How is this possible? I answer this question by arguing that computer simulations are arguments. This view parallels Norton’s argument view about thought experiments. I show that computer simulations can be reconstructed as arguments that fully capture the epistemic power of the simulations. Assuming the extended mind hypothesis, I furthermore argue that running the computer simulation is to execute the reconstructing argument. I discuss some (...) |
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The philosophy of mathematics has largely abandoned foundational studies, but is still fixated on theorem proving, logic and number theory, and on whether mathematical knowledge is certain. That is not what mathematics looks like to, say, a knot theorist or an industrial mathematical modeller. The "computer revolution" shows that mathematics is a much more direct study of the world, especially its structural aspects. |
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In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent (...) |
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If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is also the case (...) |
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The original proof of the four-color theorem by Appel and Haken sparked a controversy when Tymoczko used it to argue that the justification provided by unsurveyable proofs carried out by computers cannot be a priori. It also created a lingering impression to the effect that such proofs depend heavily for their soundness on large amounts of computation-intensive custom-built software. Contra Tymoczko, we argue that the justification provided by certain computerized mathematical proofs is not fundamentally different from that provided by surveyable (...) |
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The domain of modal epistemology tackles questions regarding the sources of our knowledge of modalities (i.e., possibility and necessity), and what justifies our beliefs about modalities. Virtue epistemology, on the other hand, aims at explaining epistemological concepts like knowledge and justification in terms of properties of the epistemic subject, i.e., cognitive capacities and character traits. While there is extensive literature on both domains, almost all attempts to analyze modal knowledge elude the importance of the agent’s intellectual character traits in justifying (...) |
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Computer simulations are often claimed to be opaque and thus to lack transparency. But what exactly is the opacity of simulations? This paper aims to answer that question by proposing an explication of opacity. Such an explication is needed, I argue, because the pioneering definition of opacity by P. Humphreys and a recent elaboration by Durán and Formanek are too narrow. While it is true that simulations are opaque in that they include too many computations and thus cannot be checked (...) |
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Douglas Walton’s multitudinous contributions to the study of argumentation seldom, if ever, directly engage with argumentation in mathematics. Nonetheless, several of the innovations with which he is most closely associated lend themselves to improving our understanding of mathematical arguments. I concentrate on two such innovations: dialogue types (§1) and argumentation schemes (§2). I argue that both devices are much more applicable to mathematical reasoning than may be commonly supposed. |
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What does it mean to trust the results of a computer simulation? This paper argues that trust in simulations should be grounded in empirical evidence, good engineering practice, and established theoretical principles. Without these constraints, computer simulation risks becoming little more than speculation. We argue against two prominent positions in the epistemology of computer simulation and defend a conservative view that emphasizes the difference between the norms governing scientific investigation and those governing ordinary epistemic practices. |
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Empiricist theories of knowledge are attractive for they offer the prospect of a unitary theory of knowledge based on relatively well understood physiological and cognitive processes. Mathematical knowledge, however, has been a traditional stumbling block for such theories. There are three primary features of mathematical knowledge which have led epistemologists to the conclusion that it cannot be accommodated within an empiricist framework: 1) mathematical propositions appear to be immune from empirical disconfirmation; 2) mathematical propositions appear to be known with certainty; (...) |
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Computation and Simulation have always played a role in economics – whether it be pure economic theory or any variant of applied, especially policy-oriented, macro- and microeconomics or what has increasingly come to be called empirical or experimental economics. Computations and simulations are also intrinsically dynamic. This triptych – computation, simulation and dynamic – is given natural foundations, mainly as a result of developments in the mathematics underpinnings in the potentials of computing, using digital technology. A running theme in this (...) |
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The ArgumentThe belief in the existence of eternal mathematical truth has been part of this science throughout history. Bourbaki, however, introduced an interesting, and rather innovative twist to it, beginning in the mid-1930s. This group of mathematicians advanced the view that mathematics is a science dealing with structures, and that it attains its results through a systematic application of the modern axiomatic method. Like many other mathematicians, past and contemporary, Bourbaki understood the historical development of mathematics as a series of (...) |
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The aim of this paper is to provide epistemic reasons for investigating the notions of informal rigour and informal provability. I argue that the standard view of mathematical proof and rigour yields an implausible account of mathematical knowledge, and falls short of explaining the success of mathematical practice. I conclude that careful consideration of mathematical practice urges us to pursue a theory of informal provability. |
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The distinction between analytic and synthetic propositions, and with that the distinction between a priori and a posteriori truth, is being abandoned in much of analytic philosophy and the philosophy of most of the sciences. These distinctions should also be abandoned in the philosophy of mathematics. In particular, we must recognize the strong empirical component in our mathematical knowledge. The traditional distinction between logic and mathematics, on the one hand, and the natural sciences, on the other, should be dropped. Abstract (...) |
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We argue that (1) a purported example of an infinite inference we humans can actually perform admits a faithful, finitary description, and (2) infinite inference contravenes any view which does not grant our minds uncomputable powers. These arguments block the strategy, dating back to Carnap's Logical Syntax of Language, of using infinitary inference rules to secure the determinacy of arithmetical truth on conventionalist grounds. |
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The Euclidean ideal of mathematics as well as all the foundational schools in the philosophy of mathematics have been contested by the new approach, called the “maverick” trend in the philosophy of mathematics. Several points made by its main representatives are mentioned – from the revisability of actual proofs to the stress on real mathematical practice as opposed to its idealized reconstruction. Main features of real proofs are then mentioned; for example, whether they are convincing, understandable, and/or explanatory. Therefore, the (...) |
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My dissertation focuses on mathematical explanation found in proofs looked at from a historical point of view, while stressing the importance of mathematical practices. Current philosophical theories on explanatory proofs emphasize the structure and content of proofs without any regard to external factors that influence a proof’s explanatory power. As a result, the major philosophical views have been shown to be inadequate in capturing general aspects of explanation. I argue that, in addition to form and content, a proof’s explanatory power (...) |
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In spite of their practical importance, the connections between technology and mathematics have not received much scholarly attention. This article begins by outlining how the technology–mathematics relationship has developed, from the use of simple aide-mémoires for counting and arithmetic, via the use of mathematics in weaving, building and other trades, and the introduction of calculus to solve technological problems, to the modern use of computers to solve both technological and mathematical problems. Three important philosophical issues emerge from this historical résumé: (...) |
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Ugliness is a neglected topic in contemporary analytic aesthetics. This is regrettable given that this topic is not just genuinely fascinating, but could also illuminate other areas in the field, seeing as ugliness, albeit unexplored, does feature rather prominently in several debates in aesthetics. This paper articulates a ‘deformity-related’ conception of ugliness. Ultimately, I argue that deformity, understood in a certain way, and displeasure, jointly suffice for ugliness. First, I motivate my proposal, by locating a ‘deformity-related’ conception of ugliness in (...) |
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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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The paper comments on a rather uncommon approach to mathematics called physicalist formalism. According to this view, the formal systems mathematicians concern with are nothing more and nothing less than genuine physical systems. I give a brief review on the main theses, then I provide some arguments, concerning mostly with the practice of mathematics and the uniqueness of formal systems, aiming to show the implausibility of this radical view. |
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This paper attempts to show how the “big data” paradigm is changing science through offering access to millions of database elements in real time and the computational power to rapidly process those data in ways that are not initially obvious. In order to gain a proper understanding of these changes and their implications, we propose applying an extended cognition model to the novel scenario. |
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This contribution defends two claims. The first is about why thought experiments are so relevant and powerful in mathematics. Heuristics and proof are not strictly and, therefore, the relevance of thought experiments is not contained to heuristics. The main argument is based on a semiotic analysis of how mathematics works with signs. Seen in this way, formal symbols do not eliminate thought experiments (replacing them by something rigorous), but rather provide a new stage for them. The formal world resembles the (...) |
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Where should computer simulations be located on the ‘usual methodological map’ which distinguishes experiment from theory? Specifically, do simulations ultimately qualify as experiments or as thought experiments? Ever since Galison raised that question, a passionate debate has developed, pushing many issues to the forefront of discussions concerning the epistemology and methodology of computer simulation. This review article illuminates the positions in that debate, evaluates the discourse and gives an outlook on questions that have not yet been addressed. |
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