In his book Modal Logic as Metaphysics, Timothy Williamson defends first-order necessitism using simplicity as a powerful argument. However, simplicity is decomposed into two different, even antagonistic, sides: elegance and parsimony. On the one hand, elegance is the property of theories possessing few and simple principles that allow them to deploy all their theoretical power; on the other hand, parsimony is the property of theories having the fair and necessary number of ontological entities that allow such theories give an account (...) of themselves. Since necessitism endorses Barcan Formulae for the sake of elegance, it is committed to a vast number of contingently non-concrete objects, so one may think that it is not qualitatively parsimonious. I argue that necessitism could be viewed as an additive case in the sense that Alan Baker characterizes the adjective so quantitative parsimony should be considered when it comes to necessitism instead of qualitative parsimony. (shrink)

In the introduction to his famous book, La Science et l’hypothèse, Poincaré remarks on the necessary role and legitimacy of hypotheses. He establishes a triple classification of hypotheses, dividing them into verifiable, useful, and apparent. However, in chapter 9, entitled ‘Les hypothèses en physique’, he gives a slightly different triadic classification: natural hypotheses, indifferent hypotheses, and real generalizations. The origin of this second classification is a lecture given at the International Congress of Physics, Paris, 1900. What are the similarities and (...) differences between these two classifications? My main purpose is to provide a possible equivalence between the two classifications in order to clarify the role of hypotheses in Poincaré’s epistemology of natural science. The discussion will be based on the two fundamental texts mentioned above and on the contrast between Poincaré’s use of the notion of hypothesis and that of some of his contemporaries. (shrink)

Ethologists and others studying animal behavior in a spirit are in need of a descriptive language and method that are neither anachronistically bound by behaviorist scruples nor prematurely committed to particular Just such an interim descriptive method can be found in intentional system theory. The use of intentional system theory is illustrated with the case of the apparently communicative behavior of vervet monkeys. A way of using the theory to generate data - including usable, testable data - is sketched. The (...) underlying assumptions of this approach can be seen to ally it directly with theorizing in evolutionary biology, which has recently come under attack from Stephen Gould and Richard Lewontin, who castigate it as the Their arguments, which are strongly analogous to B. F, Skinner's arguments against point to certain pitfalls that attend the careless exercise of such thinking (and rival varieties of thinking as well), but do not constitute a fundamental objection to either adaptationist theorizing or its cousin, intentional system theory. (shrink)

Retention of structure across theory change has been invoked in support of a ‘structural’ alternative to more traditional entity-based scientific realism. In that context the transition from Newtonian mechanics to the Special Theory of Relativity is often regarded as a very significant instance of structural preservation, or retention, associated with correspondence-based recovery. The joint derivation, from a small set of elementary and ontologically neutral assumptions, of both the Galilei and the Lorentz transformation exemplifies the virtues of structural approaches to the (...) foundations of physical theories. The common origination of the resulting two relativistic frameworks sheds light on both the successes and the limitations of correspondence claims. However, the cognitive-operational character of the basic assumptions lends no support to the structural realist’s ‘inference to the best explanation’. (shrink)

Euclidean geometry, statics, and classical mechanics, being in some sense the simplest physical theories based on a full-fledged mathematical apparatus, are well suited to a historico-philosophical analysis of the way in which a physical theory differs from a purely mathematical theory. Through a series of examples including Newton’s Principia and later forms of mechanics, we will identify the interpretive substructure that connects the mathematical apparatus of the theory to the world of experience. This substructure includes models of experiments, models of (...) measurement, and modular connections with partial theories. It evolves during the life of a theory as physicists learn how to apply it in various contexts. It should nevertheless be regarded as an integral part of a genuine physical theory since the theory would otherwise degenerate into pure mathematics. (shrink)

Science and religion have been described as the “two dominant forces in our culture”. As such, the relation between them has been a matter of intense debate, having profound implications for deeper understanding of our place in the universe. One position naturally associated with scientists of a materialistic outlook is that science and religion are contradictory, incompatible worldviews; however, a great deal of recent literature criticises this “conflict thesis” as simple-minded, essentially ignorant of the nature of religion and its philosophical (...) and theological underpinnings. In this first part of a two-part article, I set out the wide-ranging background required for a proper understanding of the debate as a preliminary for the second part, in which Ian Barbour ’s influential four-fold typology of science-religion relations is critically assessed, leading to the conclusion that the conflict model is not to be so easily dismissed. (shrink)

In a recent paper Samir Okasha has suggested an application of Arrow’s impossibility theorem to theory choice. When epistemic virtues are interpreted as ‘voters’ in charge of ranking competing theories, and there are more than two theories at stake, the final ordering is bound to coincide with the one proposed by one of the voters, provided a number of seemingly reasonable conditions are in place. In a similar spirit, Jacob Stegenga has shown that Arrow’s theorem applies to the amalgamation of (...) evidence; the ‘voters’ here are the different sources of evidence. As with Okasha’s proposal, it is not clear how to avoid Arrow’s pessimistic conclusion.In this paper we develop a novel argument that purports to show that, in typical examples, Arrow’s result does not obtain when dealing with evidence amalgamation. The reason is that we cannot escape the well-known Duhem problem: the evidence actually confirms complex conjunctions that include various auxiliary hypotheses. We argue that confirmational holism induces us to restrict the domain of a reasonable amalgamation function, thus violating one of Arrow’s conditions. The upshot is that we are now able to see the Duhem problem under a different, positive light – namely, as a phenomenon that makes the aggregation of the evidence possible in the first place, when there are at least three options on the table. (shrink)

To explain their role in discovery and contrast them with theory-driven research, philosophers of science have characterized exploratory experiments in terms of what they lack: namely, that they lack direction from what have been called “local theories” of the target system or object under investigation. I argue that this is incorrect: it’s not whether or not there is direction from a local theory that matters, but instead how such a theory is used to direct an experiment that matters. Appealing to (...) contemporary exploratory experiments that involve the use of experimental techniques—specifically, examples where scientists explore the interaction of neural activity and human behavior by magnetically stimulating brains—I argue that local theories of a target system can inform auxiliary hypotheses in exploratory experiments, which direct these experiments. These examples illustrate how local theories can direct the exploration of target systems where researchers do not aim to evaluate these theories. (shrink)

On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system, that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for (...) number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind of number being represented. In response, we propose that the ANS represents not only natural numbers, but also non-natural rational numbers. It does not represent irrational numbers, however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research. (shrink)

In this article, I analyze Quine’s conception of science, which is a radical defence of extensionalism on the grounds that first‐order logic is the most adequate logic for science. I examine some criticisms addressed to it, which show the role of modalities and probabilities in science and argue that Quine’s treatment of probability minimizes the intensional character of scientific language and methods by considering that probability is extensionalizable. But this extensionalizing leads to untenable results in some cases and is not (...) consistent with the fact that Quine himself admits confirmation which includes probability. Quine’s extensionalism does not account for this fact and then seems unrealistic, even if science ought to be extensional in so far as it is descriptive and mathematically expressible. (shrink)

in this paper i assess Pierre Duhem’s criticism of crucial experiments, and argue that we have reasons to vindicate their existence. i then analize the experiments carried out by Fizeau and Foucault in 1850, and criticize Duhem’s interpretation of their results. i contend that the results of crucial experiences can be described in terms that are neutral with respect to the competing theories. For that reason an experimental result is relatively stable and open to different interpretations in the light of (...) new theories. i conclude by distinguishing successful crucial experiments from decisive experiments. The former are meant to provide evidence that confirms one theory and disconfirms its rivals, but not necessarily to provide sufficient reasons for accepting or rejecting one of those theories. (shrink)

Le développement et la complexité, qui ont caractérisé les débats sur la structure conceptuelle des sciences au xxe siècle, ont déterminé la naissance d'une nouvelle discipline, l'histoire de l'épistémologie, avec l'objectif de tracer une histoire critique de notre « patrimoine épistémologique » avec des méthodologies plus appropriées de nature historique et théorique. Une histoire critique de la philosophie des sciences nous montre beaucoup de « Wenden » au-delà de celles plus bien connues, néopositiviste et post-néopositiviste ; en Italie et en (...) France, dans les dix premières années du xxe siècle, a pris naissance une tradition de recherche épistémologique de nature néorationaliste, avec des noyaux théoriques spécifiques pour l'importance accordée à l'historicité de la science. L'épistémologie néorationaliste italo-française a privilégié l'analyse des rapports entre la pensée mathématique et la pensée physique, telle que l'on peut la caractériser comme une véritable épistémologie de la physique mathématique; de plus, cette approche différente, dessinée d'abord par Federigo Enriques et ensuite par Gaston Bachelard, Albert Lautman et Ferdinand Gonseth, a permis de comprendre déjà dès les années trente la « philosophie implicite » dans les travaux de Kurt Godel et d'Hermann Weyl. (shrink)

En los Discorsi Galileo afirma haber realizado un experimento, cuyo resultado fue negativo, para determinar si la luz tarda tiempo en propagarse. En este trabajo analizo ese experimento como crucial entre las hipótesis rivales según las cuales la luz se propaga con velocidad infinita o con velocidad finita. Procuro determinar las hipótesis auxiliares y la carga teórica que presupone el diseño experimental. Argumento que las hipótesis presupuestas son razonables y que la carga teórica es muy baja. Sostengo que en principio (...) es posible un experimento crucial exitoso, esto es, con resultado positivo, como el concebido por Galileo. Concluyo que la hipótesis de que la luz se propaga con velocidad finita no es refutable mediante un experimento de esta clase, por lo cual cualquier resultado negativo es compatible con ambas hipótesis rivales. Como consecuencia de ello, la hipótesis de que la luz, o cualquier otra interacción física, se propagan con velocidad infinita resulta inverificable. (shrink)

The conventional interpretation that Pierre Duhem condemned outright any type of thought experiment in Ernst Mach’s sense should be, at least in large part, rejected. Although Duhem placed particular emphasis on the perils of thought experiments that Mach had overlooked or at least underestimated, he retained the core idea of Mach’s theory, according to which thought experiments cannot break free from the ultimate authority of real-world experiments. This similarity between Duhem’s and Mach’s views about thought experiments is not the only (...) one. Just as there was in Duhem’s criticism of “expériences fictives” a tendency leading him to give voice to one of Mach’s basic empiricist claims, so also there was in Mach’s interpretation of thought experiments a tendency in the direction of Duhem’s conventionalism. If Mach’s and Duhem’s conceptions of thought experiments are compared, there results an importantly similar, although not identical, tension that should properly be taken into account in the more general comparison between Mach and Duhem. (shrink)

Although to live is to face problems, the general concept of a problem has been significantly understudied. So much so, that the publication of Polya’s delightful How to Solve It caused quite a stir. And, although the concept of a conceptual problem is philosophical because it is deep and occurs across fields, from mathematics to politics, no philosophers have produced any memorable studies of it. Moreover, the word ‘problem’ is absent from most philosophical reference works. There are plenty of texts (...) on particular problems such as those of evil and induction, but none on the general concept of a conceptual problem—as if it were not a problematic concept. Worse, we are seldom told that doing original mathematics, science or technology is to wrestle with original problems. Nor have we been told that the hardest problems are likely to be inverse, in that they go from effect to cause, or from sketch to implementation, or from conclusion to premises—as in diagnosing disease from symptoms, looking for a westward passage from Europe to Asia, and designing public policies to attack social issues. The existence of this significant gap in the philosophical literature justifies the publication of the present study. (shrink)

In this paper, I consider a variety of objections against the covering-law model of scientific explanation, show that Aristotle was already aware of them and had solutions for them, and argue that these solutions are correct. These solutions involve the notions of nonHumean causality and of essential properties. There are a great many familiar objections, both methodological and epistemological, to introducing these concepts into the methodology of science, but I show that these objections are based upon misunderstandings of these concepts.

New perspectives on Pierre Duhem’s The aim and structure of physical theory Content Type Journal Article DOI 10.1007/s11016-010-9467-3 Authors Anastasios Brenner, Department of Philosophy, Paul Valéry University-Montpellier III, Route De Mende, 34199 Montpellier cedex 5, France Paul Needham, Department of Philosophy, University of Stockholm, 10691 Stockholm, Sweden David J. Stump, Department of Philosophy, University of San Francisco, 2130 Fulton Street, San Francisco, CA 94117, USA Robert Deltete, Department of Philosophy, Seattle University, 901 12th Avenue, Seattle, WA 98122-1090, USA Journal Metascience (...) Online ISSN 1467-9981 Print ISSN 0815-0796 Journal Volume Volume 20 Journal Issue Volume 20, Number 1. (shrink)

One of the difficulties facing the philosopher of science today is the divide between historical epistemology and analytic philosophy of science. For over half a century these two traditions have followed independent and divergent paths. Historical epistemology, which originated in France in the early twentieth century, has recently been reformulated by a number of scholars such as Lorraine Daston, Ian Hacking, and Hans-Jörg Rheinberger. Elaborating novel historical methods, they seek to provide answers to major questions in the field. In the (...) light of this situation, my article examines the cultural barrier that explains the uneasy relationship between the two traditions. This barrier hinges on a number of factors—institutional, political and social—that are bound up with the philosophical issues in question. By resorting both to historical study and logical analysis, the new historical epistemology incites us to move beyond a rather sterile antagonism. (shrink)

This paper summarises the contributions to our Topical Collection on indeterminacy and underdetermination. The collection includes papers in ethics, metaethics, logic, metaphysics, epistemology, philosophy of science, philosophy of language and philosophy of computation.

The formalism used to describe evolutionary change in a multilevel setting can be used equally to re-describe the situation as one where all the selection occurs at the individual level. Thus, whether multilevel or individual-level selection occurs seems to be a matter of convention rather than fact. Yet, group selection is regarded by some as an important concept with factual rather than conventional elements. I flesh out an alternative position that regards groups as a target of selection in a way (...) that is not merely definitional fiat and provide a theoretical basis for this position. (shrink)

In this paper a unified theory of models and thought experiments is proposed by considering them as fictions, la Vaihinger. In order to reach this aim, the Hertzian and Botzmannian interpretation of theories as Bilder is reconsidered.

In this paper, I introduce a new challenge to moral realism: the skeptical argument from moral underdetermination. The challenge arises as a consequence of two recent projects in normative ethics. Both Parfit and a group called consequentializers have independently claimed that the main traditions of normative theories can agree on the set of correct particular deontic verdicts. Nonetheless, as Dietrich and List :421–479, 2017) and myself :191–221, 2018; Australas J Philos 97:511–527, 2019; Ethical Theory Moral Pract 24:999–1018, 2021a) have argued, (...) the traditions still disagree about why these are the correct verdicts. This means that we can understand the situation in terms of an idea from the philosophy of science, the underdetermination of theory by the evidence. Yet underdetermination figures in one of the most important skeptical challenges to scientific realism. I show how an analogous skeptical argument can be construed for the moral realm. I propose a standard form for that argument. I then defend it against three possible objections, arguing that it is at least as plausible as, if not more plausible than, its counterpart in the philosophy of science. (shrink)

I outline a dilemma for Derek Parfit’s project to vindicate moral realism. In On What Matters, Parfit argues that the best versions of three of the main moral traditions agree on a set of moral principles, which should make us more confident about the prospects of truth in ethics. I show that the result of this Convergence Argument can be interpreted in two ways. Either there remain three separate and deontically equivalent theories or there remains just one theory, the Triple (...) Theory. Both interpretations fail to deliver what Parfit is looking for. The first interpretation leads to a situation of underdetermination of theory choice that gives rise to a skeptical challenge. The second interpretation jettisons Parfit’s Conciliatory Project, that is, the reconciliation of the three moral traditions. The dilemma, I contend, is the result of Parfit failing to resolve two antithetical lines of thought. His search for the Trinity of moral theorizing must thus fail. (shrink)

ArgumentThe use of diagrams is pervasive in theoretical physics. Together with mathematical formulae and natural language, diagrams play a major role in theoretical modeling. They enrich the expressive power of physicists and help them to explore new theoretical ideas. Diagrams are not only heuristic or pedagogical tools, but they are also tools that enable developing the content of models into novel implications.

Dans ce travail, je vais m'efforcer de critiquer la thèse poppérienne de la dissymétrie entre les procédures de vérification et de réfutation des théories empiriques. Pour caractériser les théories empiriques, j'utiliserai le critère suivant: l'analyse des procédures discursives employées ne suffit pas comme base de discussion de leur valeur de vérité. On doit faire appel à quelque chose dont on considère que la donnée est en quelque façon, et du point de vue de la connaissance, extérieure à la théorie. Dans (...) le titre, j'ai employé «falsification» parce que c'était le terme habituel des traducteurs français de Popper jusqu'à une date récente. Je trouve cette terminologie peu claire et j'utiliserai les couples vérifier/réfuter, confirmer/infirmer. (shrink)

This article addresses the question of the mechanisms of the emergence of structure and meaning in the biological and physical sciences. It proceeds from an examination of the concept of intentionality and proposes a model of intentional behavior on the basis of results of computer simulations of structural and functional self-organization. Current attempts to endow intuitive aspects of meaningful complexity with operational content are analyzed and the metaphor of DNA as a computer program (the `genetic program') is critically examined in (...) relation to an alternative metaphor of DNA as data. It is argued that relatively simple networks of boolean automata can classify and recognize patterns of binary strings on the basis of non-programmed, self-generated criteria, but lack a capacity for self-observation and interpretation. To overcome this problem it is necessary to clarify the relationships between the goals and underlying mechanisms of a process and between a system and its environment. It will be shown that memory devices that record the histories of interactions are essential for models of conscious and unconscious intentional behavior and that the possibility of infinitely sophisticated - and therefore unprogrammable - machines cannot be avoided. It will be argued that the notion of infinite sophistication allows the ideas of self-organization and physical determinism to be reconciled. These models will be used to suggest how the voluntary aspect of decision-making in general can emerge out of functional self-organizing processes. The conclusion will introduce the notion of `underdetermination' of theories, which imposes an intrinsic limitation on models of complex natural systems - a limitation that, at the same time, may be precisely what makes possible mutual understanding and intersubjectivity. (shrink)

Since the revival of historicist philosophy of science in the 1960s many philosophers have acknowledged a debt to Duhem. But Duhem’s opinions are imperfectly understood and, as McMullin has shown in his (1970) and (1979), there are many strands in the current revival of historicism. We consider here Duhem’s views on the role of history in the appraisal of scientific theories. However, there is no single text offering Duhem’s views on the subject; rather, they are revealed during their application to (...) various historical and contemporary cases. Duhem’s most sustained examination of a contemporary case is his critique of Maxwell’s science and scientific methodology.Duhem’s critique of Maxwell is not a set of isolated or incidental dicta, but the expression of Duhem’s mature thought over the last quarter-century of his life, ranging from (at least) 1893 to his death in 1916. (shrink)

The research on which the present paper makes a point in aimed at designing a cognitive model of Albert Einstein's discovery that is based on fundamental Einstein's publications and placed, ideally, at a meso-level, between macro-historical and micro-cognitive reconstructions (e.g. protocol analysis). As in a cognitive-historical analysis, we will trace some discovery heuristics in the construction of representations, that are on a continuum with those we employ in ordinary problem solving. Firstly, some theory-specific, reflexive heuristics—named orientative heuristics—are traced: inner perfection, (...) explain-or-assume, explanatory correspondence, and covariance/invariance. Then, other well-known abstractive heuristics as analogical and imagistic reasoning, thought experiment, limiting case analysis (e.g. Nersessian 1992) are shown occurring in Einstein's key-publications. A sketch of a socio-cognitive model for his discovery is then presented following two suggestions: (a) an idea of Van Fraassen about discovery phases, and (b) the Humean distinction between beliefs and ideas. (shrink)

Designing models of complex phenomena is a difficult task in engineering that can be tackled by composing a number of partial models to produce a global model of the phenomena. We propose to embed the partial models in software agents and to implement their composition as a cooperative negotiation between the agents. The resulting multiagent system provides a global model of a phenomenon. We applied this approach in modelling two complex physiological processes: the heart rate regulation and the glucose-insulin metabolism. (...) Beyond the effectiveness demonstrated in these two applications, the idea of using models associated to software agents to give reason of complex phenomena is in accordance with current tendencies in epistemology, where it is evident an increasing use of computational models for scientific explanation and analysis. Therefore, our approach has not only a practical, but also a theoretical significance: agents embedding models are a technology suitable both to representing and to investigating reality. (shrink)

The empirical underdetermination of theories is a philosophical problem which until the last century has not seriously troubled actual science. The reason is that confirmation does not depend only on empirical consequences, and theoretical virtues allow to choose among empirically equivalent theories. Moreover, I argue that the theories selected in this way are not just pragmatically or aesthetically better, but more probably true. At present in quantum mechanics not even theoretical virtues allow to choose among many competing theories and interpretations, (...) but this is because none of them possess those virtues to a sufficient degree. However, first, we can hope for some future advancement. Second, even if no further progress came forth, all the most credited competitors agree on a substantial core of theoretical assumptions. Therefore underdetermination does not show that we cannot be realist on unobservable entities in general, but at most that in particular fields our inquiry may encounter some de facto limits. (shrink)

On the basis of Levin’s claim that truth is not a scientific explanatory factor, Michel Ghins argues that the “no miracle” argument (NMA) is not scientific, therefore scientific realism is not a scientific hypothesis, and naturalism is wrong. I argue that there are genuine senses of ‘scientific’ and ‘explanation’ in which truth can yield scientific explanations. Hence, the NMA can be considered scientific in the sense that it hinges on a scientific explanation, it follows a typically scientific inferential pattern (IBE), (...) and it is based on an empirical fact (the success of science). Scientific realism, in turn, is scientific in the sense that it is supported both by a meta-level scientific argument (the NMA), and by first level scientific arguments through semantic ascent and generalization. However, both the NMA and scientific realism are not purely scientific, since they go beyond properly scientific concerns, and require additional philosophical reasoning. In turn, naturalism is correct in the sense that philosophy is continuous with science, partly based on it, and potentially equally well warranted. Beside denying the scientific nature of the NMA, Ghins raises some objections to its cogency , to which I reply in the final section. (shrink)

I. Science, myth, magic: three components of knowledge, in other words three types of activity in man who, in interaction with his surrounding environment seeks to accomodate himself to the constraints which this environment imposes on him while at the same time seeing to his own immediate or far-reaching needs.

In the late 19th century great changes in theories of light and electricity were in direct conflict with certitude, the view that scientific knowledge is infallible. What is, then, the epistemic status of scientific theory? To resolve this issue Duhem and Poincaré proposed images of fallible knowledge, Instrumentalism and Conventionalism, respectively. Only in 1919–1922, after Einstein's relativity was published, he offered arguments to support Fallibilism, the view that certainty cannot be achieved in science. Though Einstein did not consider Duhem's Instrumentalism, (...) he argued against Poincaré's Conventionalism. Hitherto, Einstein's Fallibilism, as presented at first in a rarely known essay of 1919, was left in the dark. Recently, Howard obscured its meaning. Einstein's essay was never translated into English. In my paper I provide its translation and attempt to shed light on Einstein's view and its context; I also direct attention to Einstein's images of philosophical opportunism in scientific practice. (shrink)

Structural realism has been suggested as the best compromise in the debate on scientific realism. It proposes that we should be realist about the relational structure of the world, not its nature. However, it faces an important objection, first raised by Newman against Russell: if relations are not qualified, then the position is either trivial or collapses into empiricism, but if relations are too strongly qualified, then it is no longer SR. A way to overcome this difficulty is to talk (...) about modal, or nomological relations instead of purely extensional relations. I argue that this is insufficient, for then, SR collapses into modal empiricism. I suggest, however, that ME could be the best position in the debate on scientific realism. 1Introduction2Objections to Structural Realism3How to Escape Newman’s Objection4Which Modal Relations Are Retained in Theory Change?5Are Modal Relations Real?6Relativity and Fundamentality7Is Modal Empiricism Really Empiricism?8Could Modal Empiricism Be the Best of Both Worlds? (shrink)

What is mathematics about? And if it is about some sort of mathematical reality, how can we have access to it? This is the problem raised by Plato, which still today is the subject of lively philosophical disputes. This book traces the history of the problem, from its origins to its contemporary treatment. It discusses the answers given by Aristotle, Proclus and Kant, through Frege's and Russell's versions of logicism, Hilbert's formalism, Gödel's platonism, up to the the current debate on (...) Benacerraf's dilemma and the indispensability argument. Through the considerations of themes in the philosophy of language, ontology, and the philosophy of science, the book aims at offering an historically-informed introduction to the philosophy of mathematics, approached through the lenses of its most fundamental problem. (shrink)