The issue of what constitutes a valid logical inference is a difficult question. At a minimum, we believe a permissible step in a proof must provide the reader with rational grounds to believe that the new step is a logically necessary consequence of previous assertions. However, this begs the question of what constitutes these rational grounds. Formalist accounts typically describe valid rules of inferences as those that can be found by applying one of the explicit rules of inference in (...) the formal system in which the proof is couched. However, philosophers of mathematics find such a description unhelpful because many inferences in the proofs that mathematicians actually produce cannot be expressed in a formal language, at least not without seriously distorting the semantic content of the inference (e.g., Larvor, 2012). In this chapter, we investigate mathematicians’ perceptions of a particular kind of inference in a particular setting. Specifically, we shed light on what types of graphical inferences mathematicians find permissible in a real analysis proof. Following Larvor (in press), we examine mathematicians’ reactions to metrical graphical inferences (i.e., inferences whose validity depends on the accuracy of the graph that was drawn and whose validity can be changed by minor deformations) and nonmetrical graphical inferences (i.e., inferences whose validity does not depend on a the accuracy of the graph and whose validity is not vulnerable to local deformations). The goal of this chapter is threefold. First, we demonstrate that most mathematicians reject metrical graphical inferences as impermissible in a real analysis proof. Second, we show that many mathematicians regard nonmetrical graphical inferences as permissible in a real analysis proofs. Third, we illustrate how mathematicians collectively disagree on the permissibility of nonmetrical inferences. (shrink)

We continue the investigation of the first paper where we studied logics with various negations including empirical negation and co-negation. We established how such logics can be treated uniformly with R. Sylvan's CCω as the basis. In this paper we use this result to obtain cut-free labelled sequent calculi for the logics.

I analyze a number of the quantum no-signalling proofs (Ghirardi et al. 1980, Bussey 1982, Jordan 1983, Shimony 1985, Redhead 1987, Eberhard and Ross 1989, Sherer and Busch 1993). These purport to show that the EPR correlations cannot be exploited for transmitting signals, i.e., are not causal. First, I show that these proofs can be mathematically unified; they are disguised versions of a single theorem. Second, I argue that these proofs are circular. The essential theorem relies upon the tensor product (...) representation for combined systems, which has no physical basis in the von Neumann axioms. Historically, the construction of this representation scheme by von Neumann and Weyl built no-signalling assumptions into the quantum theory. Signalling between the wings of the EPR-Bell experiments is unlikely but is not ruled out empirically by the class of proofs considered. (shrink)

Deductive proof checking programs are the most popular form of logic CAI. Whatever the reason for their widespread use, the proliferation and continuous development of these programs is evident. Contemporary proof checkers cover a wider variety of texts and rule sets, and offer more helpful editing, diagnostic, and remedial features than were once provided. These programs appear to be prime candidates for developing in the direction of "intelligent" CAI (ICAI). The primary thrust of ICAI is to build programs (...) that make use of information about learner strengths and weaknesses, the content of the subject matter being taught, and techniques for teaching various kinds of subject matter. This is a tall order by any standard, but there are signs that some initial progress is being made in the area of logic CAI. In particular, the expert system approach for offering strategic advice during proof construction is being explored by some projects. (shrink)

I focus on three preoccupations of recent writings on proof. -/- I. The role and possible effects of empirical reasoning in mathematics. Do recent developments (specifically, the computer-assisted proof of the 4CT) point to something essentially new as regards the need for and/or effects of using broadly empirical and inductive reasoning in mathematics? In particular, should we see such things as the computer-assisted proof of the 4CT as pointing to the existence of mathematical truths of (...) which we cannot have a priori knowledge? -/- 2. The role of formalization in proof. What are the patterns ofinference according to which mathematical reasoning naturally proceeds? Are they of 'local' character (i.e. sensitive to the subject-matter of the reasoning concerned) or 'global' character (i.e. invariant across all subject-matters)? Finally, what if any relationship is there (a) between the patterns of inference manifest in a proof and its explanatory capacity and (b) between explanatory capacity and rigor? -/- 3. Diagrams and their role in mathematical reasoning. What essentially is diagrammatic reasoning, and what is the nature and basis of its usefulness? Can it play a justificative role in the development of mathematical knowledge and, more particularly, in genuine proof? Finally, does the use of diagrammatic reasoning force an adjustment either in our conception of rigor or in our view of its importance? (shrink)

Mathematical proof is the primary form of justification for mathematical knowledge, but in order to count as a proper justification for a piece of mathematical knowl- edge, a mathematical proof must be rigorous. What does it mean then for a mathematical proof to be rigorous? According to what I shall call the standard view, a mathematical proof is rigorous if and only if it can be routinely translated into a formal proof. The standard view is (...) almost an orthodoxy among contemporary mathematicians, and is endorsed by many logicians and philosophers, but it has also been heavily criticized in the philosophy of mathematics literature. Progress on the debate between the proponents and opponents of the standard view is, however, currently blocked by a major obstacle, namely the absence of a precise formulation of it. To remedy this deficiency, I undertake in this paper to provide a precise formulation and a thorough evaluation of the standard view of mathematical rigor. The upshot of this study is that the standard view is more robust to criticisms than it transpires from the various arguments advanced against it, but that it also requires a certain conception of how mathematical proofs are judged to be rigorous in mathematical practice, a conception that can be challenged on empirical grounds by exhibiting rigor judgments of mathematical proofs in mathematical practice conflicting with it. (shrink)

This Element looks at the contemporary debate on the nature of mathematical rigour and informal proofs as found in mathematical practice. The central argument is for rigour pluralism: that multiple different models of informal proof are good at accounting for different features and functions of the concept of rigour. To illustrate this pluralism, the Element surveys some of the main options in the literature: the 'standard view' that rigour is just formal, logical rigour; the models of proofs as arguments (...) and dialogues; the recipe model of proofs as guiding actions and activities; and the idea of mathematical rigour as an intellectual virtue. The strengths and weaknesses of each are assessed, thereby providing an accessible and empirically-informed introduction to the key issues and ideas found in the current discussion. (shrink)

In this paper, I propose that applying the methods of data science to “the problem of whether mathematical explanations occur within mathematics itself” (Mancosu 2018) might be a fruitful way to shed new light on the problem. By carefully selecting indicator words for explanation and justification, and then systematically searching for these indicators in databases of scholarly works in mathematics, we can get an idea of how mathematicians use these terms in mathematical practice and with what frequency. The results of (...) this empirical study suggest that mathematical explanations do occur in research articles published in mathematics journals, as indicated by the occurrence of explanation indicators. When compared with the use of justification indicators, however, the data suggest that justifications occur much more frequently than explanations in scholarly mathematical practice. The results also suggest that justificatory proofs occur much more frequently than explanatory proofs, thus suggesting that proof may be playing a larger justificatory role than an explanatory role in scholarly mathematical practice. (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)

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)

This Article explores settlement incentives under three different burden of proof rules. The conventional burden of proof is a discontinuous step-function, jumping from no damages to full damages at the 0.5 jury confidence level. Continuous burdens of proof, by contrast, would permit sanctions to steadily increase as juror confidence rises from 0 to 1, with no discontinuity. Linear burdens, which have received extensive attention in prior literature, escalate sanctions steadily across the whole range of confidence levels, while (...) the logistic burden takes a nonlinear form. Using a data simulation approach guided by the empirical realities of American civil litigation, I consider the incentives that each of these rules creates for parties contemplating settlement, using a model in which parties make divergent forecasts of their expected outcomes at trial due to optimism bias. Based on this analysis, I conclude that a linear burden would likely raise our settlement rate by a modest amount, except in very large cases and in “easy” cases, in which an unbiased person would predict that a trial factfinder would have a level of confidence in liability quite close to either zero or one. I also compare the expected error rate of the settlements that each rule produces, and find that the linear rule modestly lowers the expected error rate of settlement overall, although this benefit does not hold for easy cases or those with very high damages. Lastly, I conduct a similar analysis for the logistic burden, finding that it induces a similar quality and quantity of settlements as we currently achieve using conventional burdens. (shrink)

Reviewed Works:Reuben Hersh, Proving is Convincing and Explaining.Philip J. Davis, Visual Theorems.Gila Hanna, H. Niels Jahnke, Proof and Application.Daniel Chazan, High School Geometry Students' Justification for Their Views of Empirical Evidence and Mathematical Proof.

It has been claimed that computer-assisted proof utilizes empirical evidence in a manner unheard of in traditional mathematics and therefore its employment forces us to modify our conception of proof. This paper provides a critical survey of some arguments for this claim. It starts by revisiting a well known paper by Thomas Tymoczko on the computer proof of the Four-Color Theorem. Drawing on some ideas from the works of Tyler Burge and others, it then considers a (...) way to see the philosophical significance of computer proof that casts doubts on the claim. (shrink)

An increasing number of learning goals refer to the acquisition of cognitive skills that can be described as ‘resource-based,’ as they require the availability, coordination, and integration of multiple underlying resources such as skills and knowledge facets. However, research on the support of cognitive skills rarely takes this resource-based nature explicitly into account. This is mirrored in prior research on mathematical argumentation and proof skills: Although repeatedly highlighted as resource-based, for example relying on mathematical topic knowledge, methodological knowledge, mathematical (...) strategic knowledge, and problem-solving skills, little evidence exists on how to support mathematical argumentation and proof skills based on its resources. To address this gap, a quasi-experimental intervention study with undergraduate mathematics students examined the effectiveness of different approaches to support both mathematical argumentation and proof skills and four of its resources. Based on the part-/whole-task debate from instructional design, two approaches were implemented during students’ work on proof construction tasks: (i) asequential approachfocusing and supporting each resource of mathematical argumentation and proof skills sequentially after each other and (ii) aconcurrent approachfocusing and supporting multiple resources concurrently. Empirical analyses show pronounced effects of both approaches regarding the resources underlying mathematical argumentation and proof skills. However, the effects of both approaches are mostly comparable, and only mathematical strategic knowledge benefits significantly more from the concurrent approach. Regarding mathematical argumentation and proof skills, short-term effects of both approaches are at best mixed and show differing effects based on prior attainment, possibly indicating an expertise reversal effect of the relatively short intervention. Data suggests that students with low prior attainment benefited most from the intervention, specifically from the concurrent approach. A supplementary qualitative analysis showcases how supporting multiple resources concurrently alongside mathematical argumentation and proof skills can lead to a synergistic integration of these during proof construction and can be beneficial yet demanding for students. Although results require further empirical underpinning, both approaches appear promising to support the resources underlying mathematical argumentation and proof skills and likely also show positive long-term effects on mathematical argumentation and proof skills, especially for initially weaker students. (shrink)

We claim that physics has been constructed because three “philosophical” principles have been respected, namely, realism, locality, and consistency. These principles lead to an interpretation of quantum mechanics (QM) in terms of local hidden-variables theories (LHV). In order to prove that LHV have not been refuted, we analyze the empirical proofs of Bell's inequalities and we argue that none is loophole-free. Then we propose a restricted QM that does not contain measurement postulates and that does not claim that all (...) state vectors (self-adjoint operators) are states (observables). The contradiction of such restricted QM with Bell's inequality cannot be shown as a theorem, but only by the design of a loophole-free experiment. Finally, we argue that noise has been underestimated in quantum theory. It does not appear in QM, but it is essential in quantum field theory. We conjecture that noise will prevent the violation of Bell's inequality. (shrink)

Part I presents a model of interactive computation and a metric for expressiveness, Part II relates interactive models of computation to physics, and Part III considers empirical models from a philosophical perspective. Interaction machines, which extend Turing Machines to interaction, are shown in Part I to be more expressive than Turing Machines by a direct proof, by adapting Gödel's incompleteness result, and by observability metrics. Observation equivalence provides a tool for measuring expressiveness according to which interactive systems are (...) more expressive than algorithms. Refinement of function equivalence by observation of outer interactive behavior and inner computation steps is examined. The change of focus from algorithms specified by computable functions to interaction specified by observation equivalence captures the essence of empirical computer science. Part II relates interaction in models of computation to observation in the natural sciences. Explanatory power in physics is specified by the same observability metric as expressiveness in interactive systems. Realist models of inner structure are characterized by induction, abduction, and Occam's Razor. Interactive realism extends the hidden-variable model of Einstein to hidden interfaces that provide extra degrees of freedom to formulate hypotheses with testable predictions conforming with quantum theory. Greater expressiveness of collaborative computational observers (writers) than single observers implies that hidden-interface models are more expressive than hidden-variable models. By providing a common foundation for empirical computational and physical models we can use precise results about computational models to establish properties of physical models. Part III shows that the evolution in computing from algorithms to interaction parallels that in physics from rationalism to empiricism. Plato's cave metaphor is interactively extended from Platonic rationalism to empiricism. The Turing test is extended to TMs with hidden interfaces that express interactive thinking richer than the traditional Turing test. Interactive (nonmonotonic) extensions of logic such as the closed-world assumption suggest that interactiveness is incompatible with monotonic logical inference. Procedure call, atomicity of transactions, and taking a fixed point are techniques for closing open systems similar to "preparation" followed by "observation" of a physical system. Pragmatics is introduced as a framework for extending logical models with a fixed syntax and semantics to multiple-interface models that support collaboration among clients sharing common resources. (shrink)

Who carries the burden of proof in analytic philosophical debates, and how can this burden be satisfied? As it turns out, the answer to this joint question yields a fundamental challenge to the very conduct of metaphysics in analytic philosophy. Empirical research presented in this book indicates that the vastly predominant goal pursued in analytic philosophical dialogues lies not in discovering truths or generating knowledge, but merely in prevailing over one’s opponents. Given this goal, the book examines how (...) most effectively to allocate and discharge the burden of proof. It focuses on premises that must prudently be avoided because a burden of proof on them could never be satisfied, and in particular discusses unsupportable bridge premises across inference barriers, like Hume’s barrier between ‘is’ and ‘ought’, or the barrier between the content of our talk or thought, and the world beyond such content. Employing this content/world barrier for a critical assessment of mainstream analytic philosophical methods, this book argues that we must prudently avoid invoking intuitions or other content of thought or talk in support of claims about the world beyond content, that is, metaphysically significant claims. Yet as content-located evidence is practically indispensable to metaphysical debates throughout analytic philosophy, from ethics to the philosophy of mathematics, this book reaches the startling conclusion that all such metaphysical debates must, prudently, be terminated. (shrink)

중세 스콜라 학자들은 ‘신의 존재’에 대해 많은 증명과 그 증명에 대한 비판을 남겨 놓았다. 그렇지만 이런 증명들에 대한 논의가자연과학이 발달한 현대에도 과연 의미를 지닐 수 있을까? 본 연구에서는 가장 유명한 두 증명을 비교해 보았다.????신학대전????에 나오는 ‘다섯가지 길’을 통해 ‘경험론적 증명’의 대표자 토마스 아퀴나스는 가장 유명한 신존재 증명 중에 하나인 안셀무스의????프로슬로기온????에 기원을 둔 존재론적 증명을 비판했다. 본 논문에서는 이를 소개한 후, 토마스의 안셀무스 비판이 과연 정당한가를 살펴보았다. 이를 토대로 두 사상가들이 지녔던 상이한 전제들과 의도들이 무엇인가를 기술했다. 그런데 안셀무스의 증명은 그가 전제했던 (...) 플라톤적인 체계와 실재 관념을 받아들이지 않는 경험론자들에게는 아무런 설득력도 지니지 못할 것이다. 토마스 아퀴나스의 “다섯 가지 길”도 그곳에 포함되어 있는 아리스토텔레스적인 전제가의문시되는 곳에서는 그 설득력을 잃어버린다. 이처럼 신존재 증명에 대한 평가는 모두 평가자 자신의 형이상학적 입장에 따라 다르게 판단된다. 우리는 각자 자신이 가지고 있는 형이상학적이거나 신앙적인 전제에 따라 어떠한 종류의 신존재증명을 인정하거나일부 또는 일체의 신존재증명을 부정할 수도 있다. 그러나 다양한신존재증명은 다른 입장의 문제점을 드러나게 해 주고, 인간이 직접 전체적으로 포괄할 수 없는 신적인 실재에 대한 이해가능성을한 측면에서라도 높여 줄 수 있다. 신존재증명들간의 차이에 대한성찰은 우리 현대인의 사고틀도 경직되어 있지 않은가를 돌아보게해 준다. 개별적인 신존재 증명은 이를 접하는 각자가 지니고 있는전제에 기반해서 실존적인 결단을 도와줄 이성적인 성찰로서의 기능할 수 있을 것이다. (shrink)

We investigated whether mathematicians typically agree about the qualities of mathematical proofs. Between-mathematician consensus in proof appraisals is an implicit assumption of many arguments made by philosophers of mathematics, but to our knowledge the issue has not previously been empirically investigated. We asked a group of mathematicians to assess a specific proof on four dimensions, using the framework identified by Inglis and Aberdein (2015). We found widespread disagreement between our participants about the aesthetics, intricacy, precision and utility of (...) the proof, suggesting that a priori assumptions about the consistency of mathematical proof appraisals are unreasonable. (shrink)

This book's activities highlight the important cycle of exploration, conjecture, and justification in all five mathematical strands. Students recognize patterns and make conjectures, learn the value of a counterexample, explore the strengths and weaknesses of visual proofs, discover the power of algebraic representations, and learn that theoretical approaches can substantiate empirical results. The supplemental CD-ROM features interactive electronic activities, master copies of activity pages for students, and additional readings for teachers. --publisher description.

During the course of about ten years, Wittgenstein revised some of his most basic views in philosophy of mathematics, for example that a mathematical theorem can have only one proof. This essay argues that these changes are rooted in his growing belief that mathematical theorems are ‘internally’ connected to their canonical applications, i.e. , that mathematical theorems are ‘hardened’ empirical regularities, upon which the former are supervenient. The central role Wittgenstein increasingly assigns to empirical regularities had profound (...) implications for all of his later philosophy; some of these implications (particularly to rule following) are addressed in the essay. (shrink)

Mathematical investigation, when done well, can confer understanding. This bare observation shouldn’t be controversial; where obstacles appear is rather in the effort to engage this observation with epistemology. The complexity of the issue of course precludes addressing it tout court in one paper, and I’ll just be laying some early foundations here. To this end I’ll narrow the field in two ways. First, I’ll address a specific account of explanation and understanding that applies naturally to mathematical reasoning: the view proposed (...) by Philip Kitcher and Michael Friedman of explanation or understanding as involving the unification of theories that had antecedently appeared heterogeneous. For the second narrowing, I’ll take up one specific feature (among many) of theories and their basic concepts that is sometimes taken to make the theories and concepts preferred: in some fields, for some problems, what is counted as understanding a problem may involve finding a way to represent the problem so that it (or some aspect of it) can be visualized. The final section develops a case study which exemplifies the way that this consideration – the potential for visualizability – can rationally inform decisions as to what the proper framework and axioms should be. The discussion of unification (in sections 3 and 4) leads to a mathematical analogue of Goodman’s problem of identifying a principled basis for distinguishing grue and green. Just as there is a philosophical issue about how we arrive at the predicates we should use when making empirical predictions, so too there is an issue about what properties best support many kinds of mathematical reasoning that are especially valuable to us. The issue becomes pressing via an examination of some physical and mathematical cases that make it seem unlikely that treatments of unification can be as straightforward as the philosophical literature has hoped. Though unification accounts have a grain of truth (since a phenomenon (or cluster of phenomena) called “unification” is in fact important in many cases) we are far from an analysis of what “unification” is.. (shrink)

This chapter contains sections titled: Hegel's Discussion of the Proofs On “Proof” and “Existence” The Proofs, Religious Elevation, and the Communion of Spirit The Multiplicity of Proofs and the One God The Cosmological Proof The Teleological Proof The Ontological Proof The Dialectic of the Proofs and the Speculative Reversal Hegel's Proofs Today.

Modern thinkers regard logic as a purely formal discipline like number theory, and not to be confused with any empirical discipline such as cognitive psychology, which may seek to characterize how people actually reason. Opposed to this is the traditional view that even a formal logic can be cognitively veridical – descriptive of procedures people actually follow in arriving at their deductive judgments (logic as Laws of Thought). In a cognitively veridical logic, any formal proof that a deductive (...) judgment, intuitively arrived at, is valid should ideally conform to the method the reasoning subject has used to arrive at that judgment. More specifically, it should reveal the actual reckoning process that the reasoning subject more or less consciously carries out when they make a deductive inference. That the common logical words used in everyday reasoning – words such as 'and', 'if,''some', 'is''not,' and 'all'– have fixed positive and negative charges has escaped the notice of modern logic. The present paper shows how, by unconsciously recognizing 'not' and 'all' as 'minus-words', while recognizing 'and', 'some', and 'is' as 'plus words', a child can intuitively reckon, for example, 'not (−) all (−) dogs are (+) friendly' as equivalent to 'some (+) dogs aren't (−) friendly': −(−D+F) = +D−F. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Empirical Questions Deserve Empirical AnswersColin MartindaleWhat is wrong with the current state of humanistic literary studies? On the theoretical level, we find various types of postmodernism, none of which makes much sense. On the other hand, there are approaches such as Marxism, Feminism, and the New Historicism. One can at least understand the contentions of such theorists, but these contentions are generally quite implausible. If poetry were (...) an effective means of maintaining the status quo or of oppressing anyone, one wonders why this was never noticed by politicians or generals. Now that the fact has been “discovered,” one wonders why it is not put to use by the power elite. What has led literary theory to stray onto such strange grounds? I believe that the faulty methods used by many literary theorists offer us at least part of an answer.Humanistic critics have no reasonable way of deciding among alternative answers if theorists happen to ask a question that is answerable—i.e., if they propose an empirical investigation. Their only method is one of producing examples and trying to persuade the reader. 1 Can a scientific approach provide something better? Certainly it can. First, the scientist does not pose unanswerable questions. Second, science provides methods for resolving disputes, for answering questions that are answerable. These methods are the scientific apparatus of counting, measuring, experimenting, and so on. Humanistic inquiry is simply not as effective as scientific inquiry. If a question is meaningful, it is by definition an empirical question. Science is the only method to us available for answering such questions. Humanism has no decision-making methods, no objective methods of choosing amongst competing hypotheses. [End Page 347]To paraphrase Plato, literary critics hypothesize many fine things but know nothing of how to test them. Science already has the methods. We merely need to apply them. To give an example from my own research, Raymond and a number of other literary scholars tried to say that across the course of the nineteenth century, the content of French poetry became more primary process (Freud), dedifferentiated (Werner), associative (Wundt), or mythic (Cassirer) in its content. 2 Unless one can translate Raymond’s comments into these theoretical terms, it is difficult to understand what in the world he is talking about, let alone to test his contention. Because literary critics usually make up their psychology as they go along, it is difficult to understand what they say. Because they have only a proof-by-example method, it is difficult to believe what they say. If we do make the translation and develop a metric to measure the degree to which poetic content is primary process, we can test Raymond’s hypothesis and find out if it is correct. 3 We can not only find out if it is correct. We can fit curves to the data, compute rates of change, relate rates of change to other variables, and so on. One cannot fit curves to what Lotman aptly called “literary chit-chat.” 4 Without such an objective measure of primary process content, one would no doubt assert that twentieth-century poetry is even more mythic in its content, because the poets themselves have said so. 5 In fact, the metric confirms Plato’s warning that poets lie.Some literary theorists, of course, do things that are at least close to science. Numbers would be of great help to such theorists if only as mnemonic devices. Given that people cannot hold more than about seven things in mind at once, the task of sorting out and comparing hundreds of texts is very difficult unless some counting is done. 6 Humanists can count, but they seem not to realize the value of this ability. Lévi-Strauss and the structuralists were doing what might be called zero-degree science. 7 It is reasonable to segment texts and look for regularities. It is not reasonable to look for regularities before one has determined the reliability of the segmentations: that is, before one has determined whether people agree as to how to segment texts. To compute a reliability coefficient, one has to use numbers, and one has to know the formula. Humanists don’t... (shrink)

In this paper I propose a novel interpretation of Kant’s proof of the existence of the outer world in the Refutation of Idealism. According to this interpretation, Kant’s proof does not provide a regressive explanation of our capacity to determine the temporal order of our experiences. Rather, it expresses a counterfactual reflection on what it takes for something to be actual in contrast to being merely imagined. On the ground of this reflection, Kant argues against the Cartesian sceptic (...) that, even if all our representations of empirical objects other than ourselves failed to be veridical, we would still know a priori that in every situation in which we, as thinking things, actually exist, something outside us in space must necessarily exist. (shrink)

What do mathematicians mean when they use terms such as ‘deep’, ‘elegant’, and ‘beautiful’? By applying empirical methods developed by social psychologists, we demonstrate that mathematicians' appraisals of proofs vary on four dimensions: aesthetics, intricacy, utility, and precision. We pay particular attention to mathematical beauty and show that, contrary to the classical view, beauty and simplicity are almost entirely unrelated in mathematics.

The most fundamental subject and aim of the Islamic belief system is the subject of maʿrifatullah (knowing Allah). Studies on this subject are mostly called ithbāt al-wājib (the demonstration of God) in the literature. They are considered the most valuable work for kalām, philosophy and mysticism schools. Kalām schools started to use this conceptualization intensively after Fakhr al-Dīn al-Rāzī, mainly under the influence of Ibn Sīnā. Sūfis, on the other hand, most participated in these studies based on the theory of (...) the unity of existence. In kalām, all the principles of belief are ultimately grounded to explain and strengthen faith in Allah. For this reason, belief in Allah is referred to as aṣl al-uṣūl (the origin of all principles). Throughout history, scholars of kalām discussed the creedal issues of Islam in the Muslim society while it tried to answer the creedal issues abroad by the expanding Islamic geography with studies on ithbāt al-wājib. In this framework, the problems of prophecy and the hereafter are included in kalām. Since the beginning of the history of kalām, it has used the theory of atomism to a large extent, especially in defending the existence of Allah against materialists. The systematisation of Islamic philosophy and mysticism in time, the kalām cosmological argument, the argument from contingency of philosophers and the argument from kashf and ilhām of Sūfis strengthened this area from three different branches. On the other hand, the interaction of these schools within themselves and the diversification of the interlocutors of studies on ithbāt al-wājib brought these arguments closer together. The convergence in terms of terminology, subject and methodology, especially after Shahrestānī and Rāzī, revealed the tradition of philosophical theology. In this process, theologians began to use both the concepts of temporality (hudūth) and contingency (imkān) to use ontological, cosmological and teleological proofs. Dawānī, one of the eminent figures of these studies in the last periods of Ottomans, is the subject of our research, due to his skills in utilising these proofs of ithbāt al-wājib and his influence in the studies after him. His review of the matter can be observed in his effort in his works where he deals with the so-called deficiencies and errors of his predecessors regarding ithbāt al-wājib. For this reason, many annotations and commentaries were made for his treatises and ideas until the 19th century. Especially, the advance of materialism in the West in this century showed great influence on many scholars and intellectuals in the Ottoman Empire seems to be a parallel historical experience. Therefore, it has become necessary to utilise the arguments for the existence of Allah by supporting each individual argument together. In this regard, in our study, to see the process, we also discussed the views of Aḥmad Nūrī, one of the scholars in the last period Ottoman, who dealt with the common arguments about ithbāt al-wājib produced by the three primary schools of Islamic thought: kalām, Islamic philosophy and mysticism. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Nature of Proof in PsychiatryPaul Lieberman (bio)Keywordspsychotherapy process, knowledge and psychiatry, externalism, WittgensteinThis vivid clinical report illustrates recognizably, and provocatively, a number of routine, but often unexamined, clinical questions. In its few paragraphs, it depicts challenges that each practitioner confronts, and, in the flux of clinical work, addresses, however implicitly and imperfectly, every day: From what data, and by what processes, does a clinical formulation, or way (...) of understanding, come into being? How do we appraise its accuracy and adequacy? Of what value is even a well-substantiated understanding, in the mind of the therapist, to the patient? What is the power of empathy as a treatment tool? How do we assess someone's status as a “good candidate for therapy?” When, typically, only the outlines of a problem are visible, and its origins and implications are largely hidden, by what right do we recommend psychotherapy, and with what conception, at what level of detail, of what its outcome will be? These are among the questions that Dr. Bailey's story challenges us to address.They are clinical questions, of course, because they deal with empirical matters of fact—psychological processes, outcomes studies, and evidence. But they are also philosophical because, as Freud insisted and Dr. Bailey shows, information obtained in the clinical situation often has power and authority that lead us to believe we understand what is going on, understand the other person, even in the absence of scientifically tested hypotheses, or the clear application of research-based evidence. Interactions between people—including clinical interactions—have a life and ability to engage us. Their meanings can seem so transparent and convincing that no doubt or questioning may seem cogent. At their most powerful, such experiences can reawaken us to important realms of experience and feeling which we might, otherwise, forget, ignore, suppress, deny, or in other ways keep out of awareness; they have the potential to arouse us and reconnect us with truths that, once revived, can seem both valid and important—perhaps the kind of experience that we may derive from works of art or literature (in fact, Dr. Bailey's account could be seen as a clinical parable—meant to be neither true nor false, but instructive). But even less dramatic examples are commonplace: we do routinely claim to know and be able to interpret and understand the actions of other people from our everyday interactions with and observations of them.What is less clear is how to account for these convictions and how to appraise their value, because they do not derive from established scientific procedures. Clinical intuition has a role throughout medicine, but we usually expect it to be subjected later to more rigorous tests to establish the correct diagnosis. Yet, as Dr. Bailey's case illustrates, clinical intuition in the psychiatric encounter seems somehow different. When Colin [End Page 225] leads Dr. Bailey to the conclusion that a “painful lack of wounds” has made him depressed, we find this credible. It seems similarly credible that Dr. Bailey might be anxious about his own capacity to defend his young family, because he says he is, that he perceives Colin as testing his manhood or sincerity, because that seems to fit Colin's temperament, that being accurately understood is helpful, because most of us have had that experience and valued it, and that, most generally, an absence of challenge or of activity could be depressing. But suppose a patient believed that a painful lack of wounds produced diabetes instead of depression, or suppose he had developed epilepsy and attributed that to an absence of combat experience. We might see these as ‘pre-scientific’ or ‘hypothesis-generating’ (or worse). By contrast, it seems timid to consider psychiatric formulations to be always and only provisional or hypothetical.In what ways and to what extent should we expect psychiatry to be different, and why? Several considerations suggest that psychiatry's reasons are not, always and everywhere, the same as those of general medicine, nor are its objectives.In general medicine, we distinguish between signs and symptoms, on the one hand, and “underlying” disease processes, on the other. But in psychiatry, this distinction can be... (shrink)

The objective of the paper is to provide a critical exposition of Henry Sidgwick's theory of "proof" in ethics, by means of a restatement and a critique of relevant sections of Book IV of The Methods of Ethics and an article in the 1879 volume of Mind. It is concluded that Sidgwick's thought contains two fundamental unresolved tensions. One of these relates to whether "proof" is to be treated as a normative or an empirical matter. On the (...) one hand, Sidgwick clearly wants to offer a ground for ethics whose epistemic force would be universal; on the other, he accepts Mill's "considerations determining the mind to accept" as a definition. The second unresolved tension relates to the question whether abstract transcendent axioms or the familiar rules of common sense morality constitute the ultimate court of appeal in ethical decisionmaking. (shrink)

K. R. Popper distinguished between two main uses of logic, the demonstrational one, in mathematical proofs, and the derivational one, in the empirical sciences. These two uses are governed by the following methodological constraints: in mathematical proofs one ought to use minimal logical means (logical minimalism), while in the empirical sciences one ought to use the strongest available logic (logical maximalism). In this paper I discuss whether Popper’s critical rationalism is compatible with a revision of logic in the (...) empirical sciences, given the condition of logical maximalism. Apparently, if one ought to use the strongest logic in the empirical sciences, logic would remain immune to criticism and, thus, non-revisable. I will show that critical rationalism is theoretically compatible with a revision of logic in the empirical sciences. However, a question that remains to be clarified by the critical rationalists is what kind of evidence would lead them to revise the system of logic that underlies a physical theory, such as quantum mechanics? Popper’s falsificationist methodology will be compared with the recently advocated extension of the abductive methodology from the empirical sciences to logic by T. Williamson, since both of them arrive at the same conclusion concerning the status of classical logic. (shrink)

Moore's proof of an external world is a piece of reasoning whose premises, in context, are true and warranted and whose conclusion is perfectly acceptable, and yet immediately seems flawed. I argue that neither Wright's nor Pryor's readings of the proof can explain this paradox. Rather, one must take the proof as responding to a sceptical challenge to our right to claim to have warrant for our ordinary empirical beliefs, either for any particular empirical belief (...) we might have, or for belief in the existence of an external world itself. I show how Wright's and Pryor's positions are of interest when taken in connection with Humean scepticism, but that it is only linking it with Cartesian scepticism which can explain why the proof strikes us as an obvious failure. (shrink)

I criticize a recent paper by Thomas Tymoczko in which he attributes fundamental philosophical significance and novelty to the lately-published computer-assisted proof of the four color theorem (4CT). Using reasoning precisely analogous to that employed by Tymoczko, I argue that much of traditional mathematical proof must be seen as resting on what Tymoczko must take as being "empirical" evidence. The new proof of the 4CT, with its use of what Tymoczko calls "empirical" evidence is therefore (...) not so novel as he maintains. Finally, without attempting to give a full account of the notion of empirical mathematical evidence, I sketch a view showing how the use of calculation injects an empirical ingredient into proof. (shrink)

This article aims to evaluate the purported empirical character of computer-assisted proof, as suggested by Thomas Tymoczko and others. Tymoczko famously argued that the proof of the Four-Color Theorem introduced a new, empirical method of proof, forcing us to modify the traditional conception of mathematical argument as a priori reasoning. Detlefsen and Luker contended that Tymoczko’s suggestion entailed that typically mathematical proofs were empirical. My chief interest is to raise some objections to a line (...) of thought common to both of these arguments, with a view to outlining an account of the a priori which allows thepossibility of a priori knowledge obtained by appeal to computers or through testimony. Drawing on some recent discussions by Tyler Burge, this account gives a broad construal of the non-justificatory, ‘enabling’ role that experience is held to play in knowledge and cognition, allowing us to argue that the purported empirical character of the appeal to computers pertains only to the role experience plays in enabling our access to the a priori warrant provided by computer proof. (shrink)

In the last 5 years, the controversy about whether or not gauge transformations can be empirically significant has intensified. On the one hand, Greaves and Wallace developed a framework according to which, under some circumstances, gauge transformations can be empirically significant—and Teh further supported this result by using the constrained Hamiltonian formalism. On the other hand, Friederich claims to have proved that gauge transformation can never be empirically significant. In this article, I accomplish two tasks. First, I argue that there (...) are strong reasons to resist Friederich’s proof because one of its assumptions is, at the very least, highly controversial. Second, I argue that despite criticism by Brading and Brown and Friederich, ‘t Hooft’s beam-splitter experiment is indeed a concrete example of a case where a local gauge symmetry has empirical significance. By shedding light on these two points, this article shows that recent arguments that claim gauge transformations cannot be empirically significant are not satisfactory. (shrink)

Nudges have gained popularity as a behavioral change tool that aims to facilitate the selection of the sensible choice option by altering the way choice options are presented. Although nudges are designed to facilitate these choices without interfering with people’s prior preferences, both the relation between individuals’ prior preferences and nudge effectiveness, as well as the notion that nudges ‘facilitate’ decision-making have received little empirical scrutiny. Two studies examine the hypothesis that a social proof nudge is particularly effective (...) when people have no clear prior preference, either because people are indifferent (in a colour-categorization task; Study 1, N = 255) or because people experience a choice conflict (making shopping decisions about meat products; Study 2, N = 97). Both studies employed a social proof nudge to steer participants’ choices. The potential facilitating effect of the nudge was tested using a mouse-tracker paradigm that implicitly assessed experienced uncertainty during decision-making. Results showed that the nudge was effective in steering participants’ decisions; the facilitation effect (i.e., reduced uncertainty regarding the decision) was only observed for conflicting preferences, but not under indifference. A better understanding of when and how nudges can influence individuals’ behavior may help in deciding whether nudges are an appropriate policy tool for changing particular undesirable behavior. (shrink)

The possibility of spectrum inversion has been debated since it was raised by Locke and is still discussed because of its implications for functionalist theories of conscious experience . This paper provides a mathematical formulation of the question of spectrum inversion and proves that such inversions, and indeed bijective scramblings of color in general, are logically possible. Symmetries in the structure of color space are, for purposes of the proof, irrelevant. The proof entails that conscious experiences are not (...) identical with functional relations. It leaves open the empirical possibility that functional relations might, at least in part, be causally responsible for generating conscious experiences. Functionalists can propose causal accounts that meet the normal standards for scientific theories, including numerical precision and novel prediction; they cannot, however, claim that, because functional relationships and conscious experiences are identical, any attempt to construct such causal theories entails a category error. (shrink)

We have seen that despite Feferman's results Gödel's second theorem vitiates the use of Hilbert-type epistemological programs and consistency proofs as a response to mathematical skepticism. Thus consistency proofs fail to have the philosophical significance often attributed to them.This does not mean that consistency proofs are of no interest to philosophers. We know that a ‘non-pathological’ consistency proof for a system S will use methods which are not available in S. When S is as strong a system as we (...) are willing to entertain seriously then a consistency proof for it will yield no epistemological gain. But in other cases philosophers might argue that the proof uses methods which are merely different rather than stronger than those available in the system in question. This claim has been made, for example, in the case of the constructive consistency proofs for elementary number theory. Similar philosophical investigations can be made on relative consistency proofs, since these differ from each other in the principles they employ. For example, most relative consistency proofs can be carried out within elementary number theory, but without using the theory of real numbers, no one has been able to prove the consistency of Quine's ML relative to that of his NF.What about the consistency of all mathematics or of some strong system for set theory? How do we answer the skeptic? Since here a convincing proof is not possible, we have established that the skeptic demands too much. We cannot be certain that our axioms are free from contradiction and must treat them as hypotheses which may be abandoned or modified in the face of further mathematical experience. This attitude is taken by many foundational workers who also go on to voice opinions about thelikelihood that various systems are consistent. Since these opinions are variously supported by appeals to the clarity of the mathematical concept formalized, the existence or non-existence of ‘weird’ models for the system and actual empirical experience with the system, this is surely a fruitful area for philosophical research. (shrink)

The literature on mathematical explanation contains numerous examples of explanatory, and not so explanatory proofs. In this paper we report results of an empirical study aimed at investigating mathematicians’ notion of explanatoriness, and its relationship to accounts of mathematical explanation. Using a Comparative Judgement approach, we asked 38 mathematicians to assess the explanatory value of several proofs of the same proposition. We found an extremely high level of agreement among mathematicians, and some inconsistencies between their assessments and claims in (...) the literature regarding the explanatoriness of certain types of proofs. (shrink)

This paper argues that we should reject G. E. Moore’s anti-skeptical argument as it is presented in “Proof of an External World.” However, the reason I offer is different from traditional objections. A proper understanding of Moore’s “proof” requires paying attention to an important distinction between two forms of skepticism. I call these Ontological Skepticism and Epistemic Skepticism. The former is skepticism about the ontological status of fundamental reality, while the latter is skepticism about our empirical knowledge. (...) Philosophers often assume that Moore’s response to “external world skepticism” deals exclusively with the former, not the latter. But this is a mistake. I shall argue that Moore’s anti-skeptical argument targets an ontological form of skepticism. Thus, the conclusion is an ontological claim about fundamental reality, while the premises are epistemic claims. If this is correct, then the conclusion outstrips the scope of its premises and proves too much. (shrink)

Moore’s proof of an external world is a piece of reasoning whose premises, in context, are true and warranted and whose conclusion is perfectly acceptable, and yet immediately seems flawed. I argue that neither Wright’s nor Pryor’s readings of the proof can explain this paradox. Rather, one must take the proof as responding to a sceptical challenge to our right to claim to have warrant for our ordinary empirical beliefs, either for any particular empirical belief (...) we might have, or for belief in the existence of an external world itself. I show how Wright’s and Pryor’s positions are of interest when taken in connection with Humean scepticism, but that it is only linking it with Cartesian scepticism which can explain why the proof strikes us as an obvious failure. (shrink)

A common reaction among idealist philosophers to the classical syntactic characterization of proof so crisply articulated by Tarski is an urgent but inchoate Angst that something momentous is missing, an awesome intimation of bereftness. The simple fact is that in many pursuits proof involves an empirical appeal, an operation that Tarski excludes from the domain of proof and assigns to the company of confirmation. In Tarski’s terms, empirical statements never even admit of the predicate true, (...) let alone proved, unless perhaps they hanker after truth or proof in the limit in some dimly understood and always unsuccessful way. Some sources of queasiness about the syntactic account are not hard to pinpoint. On the syntactic account, the injunction “Prove it!” can never be fulfilled, or even well-formedly uttered, in regard to an empirical statement, ┏∅, therefore ∅┓ is valid for any well-formed ∅ in a given first-order language, for any theorem, t, ┏∅, therefore t┓ is an impeccable proof for any well-formed ∅ in the classical languages. It is easy to think, with Bradley, that there is an important and legitimate, if not formally well-articulated sense of proof, characteristic of much of philosophical reasoning and argumentation, which is badly misrepresented by the syntactic conception. The kind of proof often sought after in theology, and sought after and found in the law, and the experimental and social sciences, is not purely syntactic. Idealists, of course, have frequently expressed dissatisfaction with the Tarski conception of truth. We will here extend this skepticism to the Tarski notion of proof. (shrink)

This paper discusses critically what simulation models of the evolution of cooperation can possibly prove by examining Axelrod’s “Evolution of Cooperation” (1984) and the modeling tradition it has inspired. Hardly any of the many simulation models in this tradition have been applicable empirically. Axelrod’s role model suggested a research design that seemingly allowed to draw general conclusions from simulation models even if the mechanisms that drive the simulation could not be identified empirically. But this research design was fundamentally flawed. At (...) best such simulations can claim to prove logical possibilities, i.e. they prove that certain phenomena are possible as the consequence of the modeling assumptions built into the simulation, but not that they are possible or can be expected to occur in reality. I suggest several requirements under which proofs of logical possibilities can nevertheless be considered useful. Sadly, most Axelrod-style simulations do not meet these requirements. It would be better not to use this kind of simulations at all. (shrink)

With the discovery of consistent non-Euclidean geometries, the a priori status of Euclidean proof was radically undermined. In response, philosophers proposed two revisionary interpretations of the practice: some argued that Euclidean proof is a purely formal system of deductive logic; others suggested that Euclidean reasoning is empirical, employing concepts derived from experience. I argue that both interpretations fail to capture the true nature of our geometrical thought. Euclidean proof is not a system of pure logic, but (...) one in which our grasp of the content of geometrical concepts plays a central role; moreover, our grasp of this content is a priori. (shrink)

In recent years, new forms of transnational regulation have emerged, filling the void created by the failure of governments and international institutions to effectively regulate transnational corporations. Among the variety of initiatives addressing social and environmental problems, a growing number of certification systems have appeared in various sectors, particularly agrifood. Most initiatives rely on independent third-party certification to verify compliance with a standard, as it is seen as the most credible route for certification. The effects of third-party audits, however, still (...) need to be empirically investigated. This article provides a critical assessment of the notion of ‘evidence’ which is at the heart of auditing practices. It focuses on the case of the Roundtable on Sustainable Palm Oil (RSPO) and fieldwork carried out in Indonesia, the world’s largest producer of palm oil. In this country, some non-governmental organizations decided to participate in the RSPO in order to use this platform to tackle the issue of land conflicts. They managed to include important clauses regarding indigenous and land rights in the RSPO standard. In practice, however, auditors rarely recognize as valid evidence the forms of proof put forward by local communities. As a result, the whole process risks compounding local power imbalances. (shrink)

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, and so there is no reason to believe that computer-assisted proofs are not a priori. Thanks are due to Michael Levin, David Corfield, and an anonymous referee for Philosophia Mathematica for their helpful comments. Earlier versions of this paper were presented at the Hofstra University Department of Mathematics colloquium series, and at the 2005 New Jersey Regional Philosophical Association; I am grateful to both audiences for their comments. CiteULike Connotea Del.icio.us What's this? (shrink)