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.

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 (...) in solving mathematical tasks have shown early promise that deep artificial neural networks could learn symbolic mathematical processing. In this paper, I analyze the theoretical prospects of such neural networks in proving mathematical theorems. In particular, I focus on the question how such AI systems could be incorporated in practice to theorem proving and what consequences that could have. In the most optimistic scenario, this includes the possibility of autonomous automated theorem provers (AATP). Here I discuss whether such AI systems could, or should, become accepted as active agents in mathematical communities. (shrink)

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 (...) recognize whether a putative proof is correct is not infallible. In most cases, disagreement over the correctness of a putative proof is, however, evanescent. Once an error is spotted and communicated, the disagreement disappears. But this is not always the case. Sometimes it is recalcitrant; that is, it persists over time. In order to zoom in on this type of disagreement and explain its very possibility, we focus on a single case study: a decades-long (1921-1949) controversy between Federigo Enriques and Francesco Severi, two prominent exponents of the Italian school of algebraic geometry. We suggest that the instability of the mathematical community to which they belonged can be explained by the gap between an abstract criterion of rigor and local criteria of acceptability. It is this instability that made the existence of recalcitrant disagreement over putative proofs possible. We do not condemn speculative mathematics but rather its pretense of being rigorous mathematics. In this respect, we show that the overly self-confident Severi and the more intuitive, visionary Enriques had a completely different attitude. (shrink)

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 (...) their 1997, also computer-dependent, proof of 4CT. However, treating any specific number of ‘reducible’ configurations in an ‘unavoidable’ set as sufficient entails a minimum number as necessary and sufficient. We now show that the minimum number of such configurations can only be the one corresponding to the ‘unavoidable’ set of a ‘reducible’, 4-sided configuration identified by Alfred Kempe in his, seemingly fatally flawed, 1879 ‘proof’ of 4CT. Although Kempe appealed to putative properties of ‘Kempe’ chains in a graphical representation to fallaciously argue that a 5-sided configuration was also in the ‘unavoidable’ set, and ‘reducible’, we shall show why the flaw in his ‘proof’ is not fatal when the argument is expressed geometrically; and that, essentially, Kempe correctly argued that any planar map which admits a chromatic differentiation with a five-sided area C that shares non-zero boundaries with four, all differently coloured, neighbours can be 4-coloured. (shrink)

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 (...) the philosophical assumptions are justified as they are, or one needs to take a specific philosophical position to accept them. (shrink)

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 (...) beliefs about what is possible or necessary. My aim in this paper is to argue that intellectual traits of character, like thoroughness, autonomy, epistemic courage and open-mindedness, are relevant to modal epistemology. (shrink)

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 (...) is possible iff an ideal conceiver could conceive that P. I will draw attention to three major objections that can be identified in the modal metaphysics and epistemology literature: Chalmers’ Incoherency Objection, Sherratt’s Transparency Objection and Geirsson’s Irrelevancy Objection. Each of these objections raises a different worry regarding Menzies’ account: that the notion of an ideal conceiver is incoherent, that the account implies a controversial thesis, and that it fails to offer an explanation of our knowledge of modality. The aim of this paper is to defend the response-dependent account of modality against these three objections. (shrink)

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 (...) a process may be inaccessible. In this paper I argue that there are certain instances of epistemic opacity— particularly in computational methods such a computer simulations and machine learning processes—that do not arise from, are not responsive to, and are therefore not explained by the epistemic limitations of an agent. I call these instances agent-neutral and agent-independent instances of epistemic opacity respectively. As a result, I also argue that conventional accounts of epistemic opacity offer a limited understanding of the full spectrum of kinds and sources of epistemic opacity, particularly of the kind found in computational methods. In particular, as I will show below, the limitations of these accounts are reflected in the way they fail to provide satisfactory explanations when faced with certain instances of opacity. (shrink)

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 (...) attention. In this paper, we will consider the philosophical tensions that Steingart uncovers, and use them to argue that the best account of the epistemic status of the Classification Theorem will be essentially and ineliminably social. This forms part of the broader argument that in order to understand mathematical proofs, we must appreciate their social aspects. (shrink)

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.

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 (...) propose a fallibilist account of mathematical justification. I show that the main function of mathematical justification is to guarantee that the mathematical community can correct the errors that inevitably arise from our fallible practices. (shrink)

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 (...) new approach questions Hilbert’s Thesis, according to which a correct mathematical proof is in principle reducible to a formal proof, based on explicit axioms and logic. (shrink)

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 (...) depends on its targeted audience. History is useful here, because from it, we are able to follow the transition from a first-generation proof, which is usually non-explanatory, into its explanatory version. By tracking the similarities and differences between these proofs, we are able to gain a better understanding of what makes a proof explanatory according to mathematicians who have the relevant background to evaluate it as so. My first chapter discusses why history is important for understanding mathematical practices. I describe two kinds of history: one that presents a narrative of events, which influenced developments in mathematics both directly and indirectly, and another, typically used in mathematical research, which concentrates only on technical developments. I contend that both versions of the past benefit the philosopher. History used in research gives us an idea of what mathematicians desire or find to be important, while history written by historians shows us what effects these have on mathematical practices. The next two chapters are about explanatory proofs. My second chapter examines the main theories of mathematical explanation. I argue that these theories are short-sighted as they only consider what appears in a proof without considering the proof’s purported audience or background knowledge necessary to understand the proof. In the third chapter, I propose an alternative way of analyzing explanatory proofs. Here, I suggest looking at a theorem’s history, which includes its successive proofs, as well as the mathematicians who wrote them. From this, we can better understand how and why mathematicians prove theorems in multiple ways, which depends on the purposes of these theorems. The last chapter is a case study on the computer proof of the Four Color Theorem by Appel and Haken. Here, I compare and contrast what philosophers and mathematicians have had to say about the proof. I argue that the main philosophical worry regarding the theorem—its unsurveyability—did not make a strong impact on the mathematical community and would have hindered mathematical development in computer-assisted proofs. By studying the history of the theorem, we learn that Appel and Haken relied on the strategy of Kempe’s flawed proof from the 1800s (which, obviously, did not involve a computer). Two later proofs, also aided by computer, were developed using similar methods. None of these proofs are explanatory, but not because of their massive lengths. Rather, the methods used in these proofs are a series of calculations that exhaust all possible configurations of maps. (shrink)

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 $\mathbf{P} \neq \mathbf{NP}$ (...) problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof. (shrink)

ABSTRACT This paper examines consequences of the computer revolution in mathematics. By comparing its repercussions with those of conceptual developments that unfolded in the nineteenth century, I argue that the key epistemological lesson to draw from the two transformative periods is that effective and successful mathematical practices in science result from integrating the computational and conceptual styles of mathematics, and not that one of the two styles of mathematical reasoning is superior. Finally, I show that the methodology deployed by applied (...) mathematicians in modern scientific computing is a paradigmatic instance of this key lesson. (shrink)

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 (...) Tymoczko and some Wittgensteinian topics in the philosophy of mathematics such as the importance of the surveyability as a criterion for distinguishing mathematical proofs from empirical experiments. Our aim is to show that the “characteristic Wittgensteinian invention” (Mühlhölzer 2006) – the strong distinction between proofs and experiments – can shed some light in the conceptual confusions surrounding the Four-Colour Theorem. (shrink)

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 (...) aesthetic tradition, offering examples in support, and rejecting related alternative suggestions. Second, I argue that the proposal boasts considerable merits. Not only does it capture much of what we ordinarily think of as ugly, but it also comprises an objective criterion for ugliness, offers unity and comprehensiveness, and is informative and explanatorily potent. Third, I discuss a number of objections, thereby demonstrating that the proposal withstands reflective scrutiny. (shrink)

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; (...) and 3) mathematical propositions are necessary. Epistemologists who believe that some nonmathematical propositions, such as logical or ethical propositions, cannot be known a posteriori also typically appeal to the three factors cited above in defending their position. The primary purpose of this paper is to examine whether any of these alleged features of mathematical propositions establishes that knowledge of such propositions cannot be a posteriori. (shrink)

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.

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 (...) considered according to which experimental mathematics involves calculating instances of some general hypothesis. The paper concludes with the examination of some philosophical implications of this characterization. (shrink)

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 (...) from considering rigour of proof as a fixture, I discuss fixed models, invariant notions and fixed information about objects across theories. There are other fixtures, but it is enough to start with these. (shrink)

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 (...) categories cannot apply in any simple way. Our strategy consists in acknowledging the complexity of scientific practice, and trying to assess its rationality. Hence, while we are careful not to abstract away from the details of scientific practice, our approach is not strictly descriptive: our goal is to state in what conditions empirical science can rely on computer simulations. In order to do so, we need to adopt a renewed epistemological framework, whose categories would enable us to give a finer-grained, and better-fitted analysis of the rationality of scientific practice. (shrink)

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 (...) or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (shrink)

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 (...) assumption that there is no role for informal logic in mathematics. Next, it details the usefulness of argumentation theory in the understanding of mathematical practice, offering an impressively diverse set of examples, covering the history of mathematics, mathematics education and, perhaps surprisingly, formal proof verification. From there, the book demonstrates that mathematics also offers a valuable testbed for argumentation theory. Coverage concludes by defending attention to mathematical argumentation as the basis for new perspectives on the philosophy of mathematics. ​. (shrink)

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 (...) objections and reject the view that computer simulations produce knowledge because they are experiments. I conclude by comparing thought experiments and computer simulations, assuming that both are arguments. (shrink)

Informal logic is a method of argument analysis which is complementary to that of formal logic, providing for the pragmatic treatment of features of argumentation which cannot be reduced to logical form. The central claim of this paper is that a more nuanced understanding of mathematical proof and discovery may be achieved by paying attention to the aspects of mathematical argumentation which can be captured by informal, rather than formal, logic. Two accounts of argumentation are considered: the pioneering work of (...) Stephen Toulmin [The uses of argument, Cambridge University Press, 1958] and the more recent studies of Douglas Walton, [e.g. The new dialectic: Conversational contexts of argument, University of Toronto Press, 1998]. The focus of both of these approaches has largely been restricted to natural language argumentation. However, Walton’s method in particular provides a fruitful analysis of mathematical proof. He offers a contextual account of argumentational strategies, distinguishing a variety of different types of dialogue in which arguments may occur. This analysis represents many different fallacious or otherwise illicit arguments as the deployment of strategies which are sometimes admissible in contexts in which they are inadmissible. I argue that mathematical proofs are deployed in a greater variety of types of dialogue than has commonly been assumed. I proceed to show that many of the important philosophical and pedagogical problems of mathematical proof arise from a failure to make explicit the type of dialogue in which the proof is introduced. (shrink)

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.

This article is a brief formulation of a radical thesis. We start with the formalist doctrine that mathematical objects have no meanings; we have marks and rules governing how these marks can be combined. That's all. Then I go further by arguing that the signs of a formal system of mathematics should be considered as physical objects, and the formal operations as physical processes. The rules of the formal operations are or can be expressed in terms of the laws of (...) physics governing these processes. In accordance with the physicalist understanding of mind, this is true even if the operations in question are executed in the head. A truth obtained through (mathematical) reasoning is, therefore, an observed outcome of a neuro-physiological (or other physical) experiment. Consequently, deduction is nothing but a particular case of induction. (shrink)

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 (...) with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument. (shrink)

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 (...) with respect to the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way, namely, by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories. For these reasons mathematics poses problems of a quite distinctive kind for philosophy. Therefore philosophers have accorded special attention to ontological and epistemological questions concerning mathematics. (shrink)

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 (...) proofs, and can be sensibly regarded as a priori. We also show that the aforementioned impression is mistaken because it fails to distinguish between proof search (the context of discovery) and proof checking (the context of justification). By using mechanized proof assistants capable of producing certificates that can be independently checked, it is possible to carry out complex proofs without the need to trust arbitrary custom-written code. We only need to trust one fixed, small, and simple piece of software: the proof checker. This is not only possible in principle, but is in fact becoming a viable methodology for performing complicated mathematical reasoning. This is evinced by a new proof of the four-color theorem that appeared in 2005, and which was developed and checked in its entirety by a mechanical proof system. (shrink)

Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of mathematical reasoning closely resemble patterns of reasoning in nonmathematical domains. Hence the tools developed to understand informal reasoning, collectively known as argumentation theory, are also applicable to much mathematical argumentation. This chapter investigates some of the details of that application. Consideration is given to the many contrasting meanings (...) of the word “argument”; to some of the specific argumentation-theoretic tools that have been applied to mathematics, notably Toulmin layouts and argumentation schemes; to some of the different ways that argumentation is implicated in mathematical practices; and to the social aspects of mathematical argumentation. (shrink)

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 (...) intuitions and evaluating it with Deutsch’s arguments. Specifically, I argue that one would find these arguments persuasive if and only if one is already disposed to exhibit the relevant intuition. (shrink)

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 (...) in need of fixing, in the sense that they contain nontrivial mistakes. The claim that acceptable proofs must be transferable seems quite plausible. The claim that some acceptable proofs need fixing seems plausible too. Unfortunately, these attractive suggestions stand in tension with one another. I argue that the transferability requirement is the problem. Acceptable proofs need only satisfy a weaker requirement I call “corrigibility”. I explain why, despite appearances, the corrigibility standard is preferable to stricter alternatives. (shrink)

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 (...) empirical world in that it calls for exploration and offers surprises. This presents a major reason why thought experiments occur both in empirical sciences and in mathematics. The second claim is about a looming aporia that signals the limitation of thought experiments. This aporia arises when mathematical arguments cease to be fully accessible, thus violating a precondition for experimenting in thought. The contribution focuses on the work of Vladimir Voevodsky (1966–2017, Fields medalist in 2002) who argued that even very pure branches of mathematics cannot avoid inaccessibility of proof. Furthermore, he suggested that computer verification is a feasible path forward, but only if proof is not modeled in terms of formal logic. (shrink)

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.

According to quasi-empiricism, mathematics is very like a branch of natural science. But if mathematics is like a branch of science, and science studies real objects, then mathematics should study real objects. Thus a quasi-empirical account of mathematics must answer the old epistemological question: How is knowledge of abstract objects possible? This paper attempts to show how it is possible.The second section examines the problem as it was posed by Benacerraf in Mathematical Truth and the next section presents a way (...) of looking at abstract objects that purports to demythologize them. In particular, it shows how we can have empirical knowledge of various abstract objects and even how we might causally interact with them. (shrink)

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 (...) in their 1997, also computer-dependent, proof of 4CT. However, treating any specific number of ‘reducible’ configurations in an ‘unavoidable’ set as sufficient entails a minimum number as necessary and sufficient. We now show that the minimum number of such configurations can only be the one corresponding to the ‘unavoidable’ set of the single, ‘reducible’, 4-sided configuration identified by Alfred Kempe in his, seemingly fatally flawed, 1879 ‘proof’ of 4CT. We shall further show that although Kempe fallaciously concluded that a 5-sided configuration was also in the ‘unavoidable’ set, and appealed to unproven properties of ‘Kempe’ chains in a graphical representation to then argue for its ‘reducibility’, neither flaw in his ‘proof’ is fatal when the argument is expressed geometrically; and that, essentially, Kempe correctly argued that any planar map which admits a chromatic differentiation with a five-sided area C that shares non-zero boundaries with four, all differently coloured, neighbours can be 4-coloured. (shrink)

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 (...) şi care acordă astfel o atenţie deosebită practicii matematice. (shrink)

Monte Carlo simulations arrive at their results by introducing randomness, sometimes derived from a physical randomizing device. Nonetheless, we argue, they open no new epistemic channels beyond that already employed by traditional simulations: the inference by ordinary argumentation of conclusions from assumptions built into the simulations. We show that Monte Carlo simulations cannot produce knowledge other than by inference, and that they resemble other computer simulations in the manner in which they derive their conclusions. Simple examples of Monte Carlo simulations (...) are analysed to identify the underlying inferences. (shrink)