While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental (...) mathematics on the one hand and computerized formal proofs on the other hand. Along the way, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed. (shrink)

Mathematical rigor is commonly formulated by mathematicians and philosophers using the notion of proof gap: a mathematical proof is rig­orous when there is no gaps in the mathematical reasoning of the proof. Any philosophical approach to mathematical rigor along this line requires then an account of what a proof gap is. However, the notion of proof gap makes sense only relatively to a given conception of valid mathematical reasoning, i.e., to a given conception of the validity of mathematical (...) inference. A proof gap can in particular be conceived as a failure in drawing a valid mathematical inference. The aim of this paper is to discuss two possible views of the validity of math­ematical inference with respect to their capacity to yield a plausible account of the intuitive notion(s) of proof gap present in mathematical practice. The first view is the one provided by the contemporary standards of mathematical rigor: a mathematical inference is valid if and only if its conclusion can be formally derived from its premises. We will argue that this conception does not lead to a plausible account of the intuitive notion(s) of proof gap. The second view is based on a new account of the validity of inference proposed by Prawitz: an inference is valid if and only if it consists in an operation that provides a ground for its conclusion given (previously obtained) grounds for its premises. We will first specify Prawitz's account to mathematical inference and we will then argue that the resulting ground-based account is able to capture various intuitive notions of proof gap as different types of failure in drawing valid mathematical inferences. We conclude that the ground-based account ap­pears of particular interest for the philosophy of mathematical practice, and we finally raise several challenges facing a full development of a ground-based account of the notions of mathematical rigor, proof gap and the validity of mathematical inference. (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)

A proof is given that there does not exist an event, that is not already in the past for some possible distant observer at the (our) moment that the latter is "now" for us. Such event is as "legally" past for that distant observer as is the moment five minutes ago on the sun for us (irrespective of the circumstance that the light of the sun cannot reach us in a period of five minutes). Only an extreme positivism: "that which (...) cannot yet be observed does not yet exist", can possibly withstand the conclusion concerned. Therefore, there is determinism, also in micro-physics. (shrink)

This paper argues that a successful philosophical analysis of models and simulations must accommodate an account of mathematically rigorous results. Such rigorous results may be thought of as genuinely model-specific contributions, which can neither be deduced from fundamental theory nor inferred from empirical data. Rigorous results provide new indirect ways of assessing the success of models and simulations and are crucial to understanding the connections between different models. This is most obvious in cases where rigorous results map different models on (...) to one another. Not only does this put constraints on the extent to which performance in specific empirical contexts may be regarded as the main touchstone of success in scientific modelling, it also allows for the transfer of warrant across different models. Mathematically rigorous results can thus come to be seen as not only strengthening the cohesion between scientific strategies of modelling and simulation, but also as offering new ways of indirect confirmation. (shrink)

Open peer commentary on the article “Design Research as a Variety of Second-Order Cybernetic Practice” by Ben Sweeting. Upshot: I reflect on the theme of honesty in research and discuss the adjoining requirements of rigor from an academic perspective. Central to my discussion is Glanville’s assertion that what researchers - from either science or design - presented was not what they actually thought and did.

This paper examines whether Bruno Bauer's critical assessment of Jewish emancipation in Prussia is consistent with his other republican writings in the 1840s. It argues that Bauer's political position is a form of republican rigorism, according to which human emancipation requires identification with universal interests, and not the defence of particular identities. Rigorism involves the elimination of internal as well as external heteronomous influences, and implies shifting the boundaries between the juridical and the moral realms as defined by Kant. Subjects' (...) incentives for adopting the maxims that govern their external action are taken to be relevant to the assessment of the emancipatory potential of their deeds and claims. The paper distinguishes degrees of rigorism, and shows that Bauer's specific conclusions on the Jewish question, while not entirely unfounded by his political logic, are not strictly implied by his premises. Contrasting positions in his work are illustrated. (shrink)

The present paper examines the role of exact results in the theory of many‐body physics, and specifically the example of the Mermin‐Wagner theorem, a rigorous result concerning the absence of phase transitions in low‐dimensional systems. While the theorem has been shown to hold for a wide range of many‐body models, it is frequently ‘violated’ by results derived from the same models using numerical techniques. This raises the question of how scientists regulate their theoretical commitments in such cases, given that the (...) models, too, are often described as approximations to the underlying ‘full’ many‐body problem. (shrink)

Analyses in Greek geometry are traditionally seen as heuristic devices. However, many occurrences of analysis in formal treatises are difficult to justify in such terms. I show that Greek analysies of geometrics can also serve formal mathematical purposes, which are arguably incomplete without which their associated syntheses are arguably incomplete. Firstly, when the solution of a problem is preceded by an analysis, the analysis latter proves rigorously that there are no other solutions to the problem than those offered in the (...) synthesis. Secondly, whenever some construction assumption beyond ruler and compass is made, the problem is not only solvable by that assumption but is in fact equivalent to that assumption in a rigorous sense. (shrink)

Against the modern tendency to considerate just the formal-empirical knowledge as Science, and this one mathematized as much as possible, here many declarations of Aristotle are raised in order to show that the scientific rigour is not univocal but analogic: it is determined according to the nature of the object to be known. This is a so elementary epistemological rule that not knowing it is understood by that philosopher as apaideusia, i.e., lack of basic education in the knowledge sphere in (...) general. To the Stagirite, a pepaideuménos, i.e., a well educated spirit requires to every subject just the kind of rigour which is adequate to it, attempting not to impose whether the demonstrative or the mathematical rigour to the subject. It is specially truth for the theological Science, whose subject, by being transcendent, asks for a sui generis rigour which has to be so faithful as possible to it. (shrink)

Criteria of acceptability for mathematical proofs are field-dependent. In topology, though not in most other domains, it is sometimes acceptable to appeal to visual intuition to support inferential steps. In previous work :829–842, 2014; Lolli, Panza, Venturi From logic to practice, Springer, Berlin, 2015; Larvor Mathematical cultures, Springer, Berlin, 2016) my co-author and I aimed at spelling out how topological proofs work on their own terms, without appealing to formal proofs which might be associated with them. In this article, I (...) address two criticisms that have been raised in Tatton-Brown against our approach: that it leads to a form of relativism according to which validity is equated with social agreement and that it implies an antiformalizability thesis according to which it is not the case that all rigorous mathematical proofs can be formalized. I reject both criticisms and suggest that our previous case studies provide insight into the plausibility of two related but quite different theses. (shrink)

Part I of the present work outlined the rigorous application of information theory to a quantum mechanical system in a thermodynamic equilibrium state. The general formula developed there for the best-guess density operator $\hat \rho$ was indeterminate because it involved in an essential way an unspecified prior probability distribution over the continuumD H of strong equilibrium density operators. In Part II mathematical evaluation of $\hat \rho$ is completed after an epistemological analysis which leads first to the discretization ofD H and (...) then to the adoption of a suitable indifference axiom to delimit the set of admissible prior distributions. Finally, quantal formulas for information-theoretic and thermodynamic entropies are contrasted, and the entire work is summarized. (shrink)

Rigorous proof is supposed to guarantee that the premises invoked imply the conclusion reached, and the problem of rigor may be described as that of bringing together the perspectives of formal logic and mathematical practice on how this is to be achieved. This problem has recently raised a lot of discussion among philosophers of mathematics. We survey some possible solutions and argue that failure to understand its terms properly has led to misunderstandings in the literature.

In "Moses the Egyptian"--the centerpiece of Rigorism of Truth, the German philosopher Hans Blumenberg addresses two defining figures in the intellectual history of the twentieth century: Sigmund Freud and Hannah Arendt. Unpublished during his lifetime, this essay analyzes Freud's Moses and Monotheism and Arendt's Eichmann in Jerusalem, and discovers in both a principled rigidity that turns into recklessness because it is blind to the politics of the unknown. Offering striking insights into the importance of myth in politics and the extent (...) to which truth can be tolerated in adversity, the essay also provides one of the few instances where Blumenberg reveals his thinking about Judaism and Zionism. Rigorism of Truth also includes commentaries by Ahlrich Meyer that give a fuller understanding of the philosopher's engagement with Freud, Arendt, and the Eichmann trial, as well as situating these reflections in the broader context of Blumenberg's life and thought. (shrink)

Although the target article provides strong evidence against the locationist view, evidence for the constructionist view is inconclusive, because co-activation of brain regions does not necessarily imply connectivity between them. We propose a rigorous approach wherein connectivity between co-activated regions is first modeled using exploratory Granger causality, and then confirmed using dynamic causal modeling or Bayesian modeling.

A task of any moral theory is to account for both the rigidity and the flexibility of moral rules. Utilitarianism faces the problem of building rigidity into a framework that tends towards objectionable flexibility. Kantianism faces the problem of building flexibility into a framework that tends towards objectionable rigidity. I offer an argument on this front on behalf of Kantians. I show how Kantians can maintain that actions are right and wrong "in themselves," while still maintaining that such actions can (...) be corrupted under certain "nonideal" circumstances. (shrink)

In this paper I discuss Hans Blumenberg’s The Rigorism of Truth, a short polemic that criticizes Freud and Hannah Arendt for placing a misplaced faith in the liberatory potential of rational truth in moments of historical disaster. The secondary literature suggests that this piece exhibits either all the signs of a late, Romantic capitulation to the ‘need’ for myth, or Blumenberg’s failure to recognize his own faith and debts to the ‘mythology’ of reason’s emancipatory hopes. My argument hinges on the (...) claim that these readings put undue emphasis on the philosophical anthropology component of Blumenberg’s work. Instead, I offer a new reading of the essay, in keeping with an alternative reading of his theory of myth. The essay transforms, then, from a polemic regarding the need for myth, into a nuanced description of the ways in which we can overestimate our capacity to overcome it. (shrink)

It is difficult to imagine that a Royal Institute of Physics would organize an annual lecture series on the theme ‘conceptions of physics’. Similarly, it is quite improbable that a Royal Institute of Astronomy would even contemplate inviting speakers for a lecture series called ‘conceptions of astronomy’. What, then, is so special about philosophy that the theme of this lecture series does not appear to be altogether outlandish? Is it, perhaps, that philosophy is the reflective discipline par excellence, so that (...) the very nature of philosophy is a topic that belongs to its domain of investigations? Or is there something more serious at issue? Let me first tell you how the question concerning the nature of philosophy arose for me as a young student, and how I tried to answer it, quite naively, by endorsing Edmund Husserl's programme of turning philosophy into a rigorous science. In a second section, I shall offer three diagnoses of the fact that all such attempts seem to have failed. Finally, I'd like to say a few words on my own research agenda in philosophy and on a public function of philosophers. (shrink)

The ArgumentIt has long been apparent that in the nineteenth century, mathematics in France and England developed along different lines. The differences, which might well be labelled stylistic, are most easy to see on the foundational level. At first this may seem surprising because it is such a fundamental area, but, upon reflection, it is to be expected. Ultimately discussions about the foundations of mathematics turn on views about what mathematics is, and this is a question which is answered by (...) a variety of different groups including mathematicians, students, curricular planners, parents, etc. Mathematical practice rests on some kind of mixture of the answers to this fundamental question which come from these diverse groups. Comparing the cultural matrices which supported mathematics in France and Britain in the first decades of the nineteenth century sheds light on the real though often subtle differences in the ways the subject was pursued in the two countries. (shrink)

The Clauser–Horne–Shimony–Holt (CHSH) inequality is a constraint that local hidden variable theories must obey. Quantum Mechanics predicts a violation of this inequality in certain experimental settings. Treatments of this subject frequently make simplifying assumptions about the probability spaces available to a local hidden variable theory, such as assuming the state of the system is a discrete or absolutely continuous random variable, or assuming that repeated experimental trials are independent and identically distributed. In this paper, we do two things: first, show (...) that the CHSH inequality holds even for completely general state variables in the measure-theoretic setting, and second, demonstrate how to drop the assumption of independence of subsequent trials while still being able to perform a hypothesis test that will distinguish Quantum Mechanics from local theories. The statistical strength of such a test is computed. (shrink)

En la epistemología actual, hay dos actitudes opuestas en relación con las ciencias empíricas.Por una parte, aparecen como herramienta esencial para el avance del conocimiento. Por otro lado, existe duda sobre las bases metafísicas y epistemológicas de esa confianza en el saber científico, lo cual ha llevado a la ciencia por caminos de escepticismo y pragmatismo. Este trabajo se propone aportar filosóficamente a la racionalidad y al estatuto ontológico de la física, teniendo como punto de partida algunas obras del filósofo (...) de la ciencia Evandro Agazzi. El artículo que aquí se presenta introduce al pensamiento de Agazzi y a asuntos nucleares de su epistemología. Posteriormente, define los conceptos de “rigor” y “objetividad” según los entiende Agazzi,finalmente, establece criterios de rigor y objetividad para la física, mostrando de qué manera se verifican en dos experimentos clásicos. Con base en estas ideas, se demuestra que la física, como ciencia que es, cuenta con criterios de rigor y objetividad que le permiten un alcance efectivo de lo real, respondiendo así al desafío formalista y pragmatista. Así pues, el artículo no se agota en una descripción del pensamiento de Agazzi, sino que aplica sus ideas al ámbito concreto de la física, explicitando ideas que no han sido lo suficientemente explicitadas por el filósofo italiano. El artículo comienza introduciendo al pensamiento de Agazzi y a asuntos nucleares de su epistemología. Posteriormente, define los conceptos de rigor y objetividad según los entiende Agazzi, y, finalmente, establece criterios de rigor y objetividad para la física, mostrando de qué manera se verifican en dos experimentos clásicos. Con base en estas ideas, se demuestra que la física, como ciencia que es, cuenta con criterios de rigor y objetividad que le permiten un alcance efectivo de lo real, respondiendo así al desafío formalista y pragmatista. Así pues, el artículo no se agota en una descripción del pensamiento de Agazzi, sino que aplicará sus ideas al ámbito concreto de la física, explicitando ideas que no han sido lo suficientemente explicitadas por el filósofo italiano. (shrink)

The Rigor of a Certain Inhumanity is a rich and passionate, playful and perceptive work of philosophical analysis.

Euler famously used diagrams to illustrate syllogisms in his Lettres à une princesse d’Allemagne [1]. His diagrams are usually seen as suffering from a fatal “ambiguity problem” [11]: as soon as they involve intersecting circles, which are required for the representation of existential statements, it becomes unclear what exactly may be read off from them, and as Hammer & Shin conclusively showed, any set of reading conventions can lead to erroneous conclusions. I claim that Euler diagrams can, however, be used (...) rigorously, if they are read in conjunction with the premises they are supposed to illustrate. More precisely, I give rigorous “heterogeneous” inference rules (in the sense of Barwise and Etchemendy) – rules whose premises are a sentence and a diagram and whose conclusion is a sentence – which allow to use them safely. I conclude that one should abandon the preconception that diagrams can only be used rigorously if they can be given a context-independent semantics. Finally, I suggest that context-dependence is a widespread feature of diagrams: for instance, Mumma [12] noticed that what may be read off from a Euclidean diagram depends not only on the diagram’s appearance, but also on the way it was constructed. (shrink)

American taxpayers spend $30 billion annually funding biomedical research. By some estimates, half of the results from these studies can't be replicated elsewhere-the science is simply wrong. Often, research institutes and academia emphasize publishing results over getting the right answers, incentivizing poor experimental design, improper methods, and sloppy statistics. Bad science doesn't just hold back medical progress, it can sign the equivalent of a death sentence. How are those with breast cancer helped when the cell on which 900 papers are (...) based turns out not to be a breast cancer cell at all? How effective could a new treatment for ALS be when it failed to cure even the mice it was initially tested on? In Rigor Mortis, award-winning science journalist Richard F. Harris reveals these urgent issues with vivid anecdotes, personal stories, and interviews with the nation's top biomedical researchers. We need to fix our dysfunctional biomedical system-now. (shrink)

In this second edition, Joseph Rychlak has retained his analysis of the philosophical antecedents of psychology and, at the same time, has considerably revised more complicated material illustration rigorous humanism to make the book more accessible for students. Rychlak here offers an analysis of the philosophical traditions underlying the social sciences and shows how functionalism came to dominate the modern science of psychology in America.

Many arguments found in the physics literature involve concepts that are not well-defined by the usual standards of mathematics. I argue that physicists are entitled to employ such concepts without rigorously defining them so long as they restrict the sorts of mathematical arguments in which these concepts are involved. Restrictions of this sort allow the physicist to ignore calculations involving these concepts that might lead to contradictory results. I argue that such restrictions need not be ad hoc, but can sometimes (...) be justified by considering some of the metaphysical issues surrounding the question of the applicability of mathematics to physical reality. 1 Introduction 2 Rejecting inferential permissiveness 3 The agreement problem 4 Independent objections to the liberal view. (shrink)

Following a suggestion of Feys, we use “rigorous implication” as a translation of Ackermann's strenge Implikation ([1]). Interest in Ackermann's system stems in part from the fact that it formalizes the properties of a strong, natural sort of implication which provably avoids standard implicational paradoxes, and which is consequently a good candidate for a formalization of entailment (considered as a narrower relation than that of strict implication). Our present purpose will not be to defend this suggestion, but rather to present (...) some information about rigorous implication. In particular, we show first that the structure of modalities (in the sense of Parry [4]) in Ackermann's system is identical with the structure of modalities in Lewis's S4, and secondly that (Ackermann's apparent conjecture to the contrary notwithstanding) it is possible to define modalities with the help of rigorous implication. (shrink)

This paper outlines a framework that may provide a mathematically rigorous quantum field theory. The framework relies upon the methods of nonstandard analysis. A theory of nonstandard inner product spaces and operators on these spaces is first developed. This theory is then applied to construct nonstandard Fock spaces which extend the standard Fock spaces. Then a rigorous framework for the field operators of quantum field theory is presented. The results are illustrated for the case of Klein-Gordon fields.

The present paper focuses on a particular class of models intended to describe and explain the physical behaviour of systems that consist of a large number of interacting particles. Such many-body models are characterized by a specific Hamiltonian (energy operator) and are frequently employed in condensed matter physics in order to account for such phenomena as magnetism, superconductivity, and other phase transitions. Because of the dual role of many-body models as models of physical sys-tems (with specific physical phenomena as their (...) explananda) as well as mathematical structures, they form an important sub-class of scientific models, from which one can expect to draw general conclusions about the function and functioning of models in science, as well as to gain specific insight into the challenge of modelling complex systems of correlated particles in condensed matter physics. In particular, it is argued that many-body models contribute novel elements to the process of inquiry and open up new avenues of cross-model confirmation and model-based understanding. In contradistinction to phenomenological models, which have received comparatively more philosophical attention, many-body models typically gain their strength not from ‘empirical fit’ per se, but from their being the result of a constructive application of mature formalisms, which frees them from the grip of both ‘fundamental theory’ and an overly narrow conception of ‘empirical success’. (shrink)

In his seminars on the death penalty, Derrida consistently describes Kant's arguments in favor of capital punishment as “rigorous” and explicitly relates that rigor to the mechanisms of execution and the subsequent rigor mortis of the corpse. ‘Rigor’ has also often been a contested term in descriptions of deconstruction: different commentators have either deplored or celebrated the presence or the absence of rigor in Derrida's work. Derrida himself uses the term a good deal throughout his career, (...) usually in a positive sense, although he also at least once, in passing, suggests the need to question the rigor of the concept of rigor itself. In this paper, I will outline the place of Kant in the Death Penalty Seminars and suggest that it is the very rigor attributed to Kant that makes him (rather than some other writers—whether supporters or opponents of the death penalty—whose arguments seem less rigorous to Derrida) an exemplary object for deconstructive attention, not for the first time in Derrida's work. Broadening the focus beyond the texts Derrida explicitly analyzes, I suggest that this kind of attention can also be fruitfully brought to bear on some more general arguments in Kant about right and justice. In conclusion, I suggest some implications of this situation for the still difficult issue of the more general relation between deconstruction and critique in the Kantian sense. (shrink)

Building on the works of Popkin, Gilson, Fackenheim, and Kaufmann, the author expands the scope of what can be treated by the hypothesis of skepticism or pursuit of rigor in the modern world.

Past research on people's moral judgments about moral dilemmas has revealed a connection between utilitarian judgment and reflective cognitive style. This has traditionally been interpreted as reflection is conducive to utilitarianism. However, recent research shows that the connection between reflective cognitive style and utilitarian judgments holds only when participants are asked whether the utilitarian option is permissible, and disappears when they are asked whether it is recommended. To explain this phenomenon, we propose that reflective cognitive style is associated with a (...) greater moral leniency—that is, a greater tendency to be tolerant of moral violations, and that moral leniency predicts utilitarian judgment when utilitarian judgment is measured through permissibility. In Study 1 (N = 192), we design a set of vignettes to assess moral leniency. In Studies 2 and 3 (N = 455, 428), we show that reflective cognitive style is indeed associated with greater moral leniency, and that moral leniency mediates the connection between cognitive style and utilitarian judgment. We discuss the implication of our results for the interpretation of the relationship between utilitarianism and reflective cognitive style. (shrink)

Duhem's portrayal of the history of mathematics as manifesting calm and regular development is traced to his conception of mathematical rigor as an essentially static concept. This account is undermined by citing controversies over rigorous demonstration from the eighteenth and twentieth centuries.

This paper critically examines and evaluates Yacin Hamami’s reconstruction of the standard view of mathematical rigor. We will argue that the reconstruction offered by Hamami is premised on a strong and controversial epistemological thesis and a strong and controversial thesis in the philosophy of mind. Secondly, we will argue that Hamami’s reconstruction of the standard view robs it of its original philosophical rationale, i.e. making sense of the notion of rigor in mathematical practice.

Edmund Husserl, the founder of modern phenomenology, always insisted that philosophy is not just a scholarly discipline, but can and must aspire to the status of a ‘strict’ or ‘rigorous science’ (strenge Wissenschaft). Heidegger, by contrast, began his winter lectures in 1929 by dismissing what he called the ‘delusion’ that philosophy was or could be either a discipline or a science as the most disastrous debasement of its innermost essence. To understand what Husserl had in mind, it is important to (...) begin by remembering that the word Wissenschaft has a wider extension than the word ‘science’. German distinguishes the Naturwissenschaften from the Geisteswissenschaften, or human sciences, which Husserl and Heidegger both believed could be perfectly ‘rigorous’ in their own way. Speakers of English, by contrast, tend to draw a threefold distinction among the natural sciences, the social sciences, and the humanities. (shrink)