Recent years have seen an explosion of empirical data concerning arithmetical cognition. In this paper that data is taken to be philosophically important and an outline for an empirically feasible epistemological theory of arithmetic is presented. The epistemological theory is based on the empirically well-supported hypothesis that our arithmetical ability is built on a protoarithmetical ability to categorize observations in terms of quantities that we have already as infants and share with many nonhuman animals. It is argued here that (...) arithmetical knowledge developed in such a way cannot be totally conceptual in the sense relevant to the philosophy of arithmetic, but neither can arithmetic understood to be empirical. Rather, we need to develop a contextual a priori notion of arithmetical knowledge that preserves the special mathematical characteristics without ignoring the roots of arithmetical cognition. Such a contextual a priori theory is shown not to require any ontologically problematic assumptions, in addition to fitting well within a standard framework of general epistemology. (shrink)

In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components. Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well as (...) mathematical. In this paper, I will argue that Dedekind’s approach can be seen as a precursor to modern structuralism and as such, it enjoys many advantages over Frege’s logicism. I also show that from a modern perspective, Frege’s criticism of abstraction and psychologism is one-sided and fails against the psychological processes that modern research suggests to be at the heart of numerical cognition. The approach here is twofold. First, through historical analysis, I will try to build a clear image of what Frege’s and Dedekind’s views on arithmetic were. Then, I will consider those views from the perspective of modern philosophy of mathematics, and in particular, the empirical study of arithmetical cognition. I aim to show that there is nothing to suggest that the axiomatic Dedekind approach could not provide a perfectly adequate basis for philosophy of arithmetic. (shrink)

In recent years philosophers have used results from cognitive science to formulate epistemologies of arithmetic :5–18, 2001). Such epistemologies have, however, been criticised, e.g. by Azzouni, for interpreting the capacities found by cognitive science in an overly numerical way. I offer an alternative framework for the way these psychological processes can be combined, forming the basis for an epistemology for arithmetic. The resulting framework avoids assigning numerical content to the Approximate Number System and Object Tracking System, two (...) systems that have so far been the basis of epistemologies of arithmetic informed by cognitive science. The resulting account is, however, only a framework for an epistemology: in the final part of the paper I argue that it is compatible with both platonist and nominalist views of numbers by fitting it into an epistemology for ante rem structuralism and one for fictionalism. Unsurprisingly, cognitive science does not settle the debate between these positions in the philosophy of mathematics, but I it can be used to refine existing epistemologies and restrict our focus to the capacities that cognitive science has found to underly our mathematical knowledge. (shrink)

In recent years philosophers have used results from cognitive science to formulate epistemologies of arithmetic :5–18, 2001). Such epistemologies have, however, been criticised, e.g. by Azzouni, for interpreting the capacities found by cognitive science in an overly numerical way. I offer an alternative framework for the way these psychological processes can be combined, forming the basis for an epistemology for arithmetic. The resulting framework avoids assigning numerical content to the Approximate Number System and Object Tracking System, two (...) systems that have so far been the basis of epistemologies of arithmetic informed by cognitive science. The resulting account is, however, only a framework for an epistemology: in the final part of the paper I argue that it is compatible with both platonist and nominalist views of numbers by fitting it into an epistemology for ante rem structuralism and one for fictionalism. Unsurprisingly, cognitive science does not settle the debate between these positions in the philosophy of mathematics, but I it can be used to refine existing epistemologies and restrict our focus to the capacities that cognitive science has found to underly our mathematical knowledge. (shrink)

This is the first complete English translation of Gottlob Frege's Grundgesetze der Arithmetik, with introduction and annotation. The importance of Frege's ideas within contemporary philosophy would be hard to exaggerate. He was, to all intents and purposes, the inventor of mathematical logic, and the influence exerted on modern philosophy of language and logic, and indeed on general epistemology, by the philosophical framework.

The goal of the research programme I describe in this article is a realist epistemology for arithmetic which respects arithmetic's special epistemic status (the status usually described as a prioricity) yet accommodates naturalistic concerns by remaining fundamentally empiricist. I argue that the central claims which would allow us to develop such an epistemology are (i) that arithmetical truths are known through an examination of our arithmetical concepts; (ii) that (at least our basic) arithmetical concepts are accurate (...) mental representations of elements of the arithmetical structure of the independent world; (iii) that (ii) obtains in virtue of the normal functioning of our sensory apparatus. The first of these claims protects arithmetic's special epistemic status relative, for example, to the laws of physics, the second preserves the independence of arithmetical truth, and the third ensures that we remain empiricists. Preliminaries Justifying and grounding concepts Cameras and filters An epistemology for arithmetic Concluding remarks. (shrink)

This paper ties together three threads of discussion about the following question: in accepting a system of axioms S, what else are we thereby warranted in accepting, on the basis of accepting S? First, certain foundational positions in the philosophy of mathematics are said to be epistemically stable, in that there exists a coherent rationale for accepting a corresponding system of axioms of arithmetic, which does not entail or otherwise rationally oblige the foundationalist to accept statements beyond the logical (...) consequences of those axioms. Second, epistemic stability is said to be incompatible with the implicit commitment thesis, according to which accepting a system of axioms implicitly commits the foundationalist to accept additional statements not immediately available in that theory. Third, epistemic stability stands in tension with the idea that in accepting a system of axioms S, one thereby also accepts soundness principles for S. We offer a framework for analysis of sets of implicit commitment which reconciles epistemic stability with the latter two notions, and argue that all three ideas are in fact compatible. (shrink)

The philosophy of arithmetic of Wittgenstein's Tractatus is outlined and the central role played in it by the general notion of operation is pointed out. Following which, the language, the axioms and the rules of a formal theory of operations, extracted from the Tractatus, are presented and a theorem of interpretability of the equational fragment of Peano's Arithmetic into such a formal theory is proven.

In 1913, in a draft for a new Preface for the second edition of the Logical Investigations, Edmund Husserl reveals to his readers that "The source of all my studies and the first source of my epistemological difficul­ties lies in my first works on the philosophy of arithmetic and mathematics in general", i.e. his Habilitationsschrift and the Philosophy of Arithmetic: "I carefully studied the consciousness constituting the amount, first the collec­tive consciousness (consciousness of quantity, of multiplicity) in its (...) simplest and higher levels (consciousness of sums, sums of sums etc.). I immediately separated proper (intuitive) and symbolic consciousness, in the characteriza­tion of the former I hit the radical difference of categorial consciousness [...] and sensuous consciousness of unity." Later on, in the Third Investigation, Husserl makes some very specific claims, that are of considerable importance to assess the development of his early works and their relation to his later phenomenology: "This first work of mine (an elaboration of my Habilitationsschrift, [...], 1887) should be compared with all assertions of the present work on compounds, moments of unity, complexes, wholes and objects of higher order. I am sorry that in many recent treatments of the doctrine of "Gestalt-qualities", this work has mostly been ignored, though quite a lot of the thought-content of later treatments by Cornelius, Meinong etc., of questions of analysis, apprehension of plurality and complexion is already to be found, differently expressed, in my Philosophy of Arithmetic. I think it would still be of use today to consult this work on the phenomenological and ontological issues in question, especially since it is the first work that attached importance to acts and objects of higher order and investigated them thoroughly." Hence, at the time of the Ideas, Husserl retrospectively considers his first works4 as being still relevant for phenomenological issues. Not only does Husserl advance a very interesting priority claim with respect to Von Ehrenfels’ development of the notion of Gestalt and Meinong’s development of Gegenstandstheorie, but also a strong affirmation of continuity and coherence of his position from 1887 all the way up to 1913, encompassing the alleged “revolution” in his position from psychologism to anti-psycho­logism in the 1890s. Indeed, according to much of the recent secondary literature, in 1894, right in the middle of the ten “incubation” years between the Philosophy of Arithmetic and the Logical Investigations, Frege’s destruc­tive review would have converted Husserl to antipsychologism practically overnight. This gives us two conflicting interpretations: on the one hand, Husserl himself in 1913 still seems to approve of the Philosophy of Arithmetic and even considers it to contain valuable phenomenological material, on the other, it is routinely dismissed by much of the secondary literature as hope­lessly psychologistic. So which one is it: do we have a phenomenological arithmetic or a psychologistic arithmetic in Husserl’s first book? On balance, I think that Husserl in his Philosophy of Arithmetic developed a position that does not fall prey to the exaggerated and poorly aimed critiques of Frege, while at the same time, as a descriptive psychology of the genesis and constitution of number, it can certainly be considered as providing phenome­nologically meaningful analyses, though of course not made from within an explicitly transcendental phenomenological framework. (shrink)

The topic of this paper is our knowledge of the natural numbers, and in particular, our knowledge of the basic axioms for the natural numbers, namely the Peano axioms. The thesis defended in this paper is that knowledge of these axioms may be gained by recourse to judgements of probability. While considerations of probability have come to the forefront in recent epistemology, it seems safe to say that the thesis defended here is heterodox from the vantage point of traditional (...) philosophy of mathematics. So this paper focuses on providing a preliminary defense of this thesis, in that it focuses on responding to several objections. Some of these objections are from the classical literature, such as Frege's concern about indiscernibility and circularity, while other are more recent, such as Baker's concern about the unreliability of small samplings in the setting of arithmetic. Another family of objections suggests that we simply do not have access to probability assignments in the setting of arithmetic, either due to issues related to the~$\omega$-rule or to the non-computability and non-continuity of probability assignments. Articulating these objections and the responses to them involves developing some non-trivial results on probability assignments, such as a forcing argument to establish the existence of continuous probability assignments that may be computably approximated. In the concluding section, two problems for future work are discussed: developing the source of arithmetical confirmation and responding to the probabilistic liar. (shrink)

Part 1 of this article exposed a tension between Poincaré’s views of arithmetic and geometry and argued that it could not be resolved by taking geometry to depend on arithmetic. Part 2 aims to resolve the tension by supposing not merely that intuition’s role is to justify induction on the natural numbers but rather that it also functions to acquaint us with the unity of orders and structures and show practices to fit or harmonize with experience. I argue (...) that in this manner, intuition serves the epistemological function of warranting generalizations and justifying practices. In particular, it justifies the application of group-theoretic notions in geometry but not the use of set-theoretic notions in arithmetic. (shrink)

Can visual thinking be a means of discovery in elementary analysis, as well as a means of illustration and a stimulus to discovery? The answer to the corresponding question for geometry and arithmetic seems to be ‘yes’ (Giaquinto [1992], [1993]), and so a positive answer might be expected for elementary analysis too. But I argue here that only in a severely restricted range of cases can visual thinking be a means of discovery in analysis. Examination of persuasive visual routes (...) to two simple theorems (Rolle, Bolzano) shows that they are not ways of discovering the theorems; the type of visual thinking involved can never be used to discover analytic theorems of a certain generality. The hypothesis that visual thinking is never a means of discovering the existence or nature of the limit of some infinite process is considered, and a likely counter-example is set out. It is still possible that restricted theorems can be discovered visually: an example from Littlewood is examined in detail and not found wanting. Even when visual thinking is not a means of discovery it can provide the idea for a proof in a direct way; an example is presented (Intermediate Value Theorem). In conclusion: it may be possible to discover theorems in elementary real analysis by visual means, but only theorems of a restricted kind; however, visual thinking in analysis can be very useful in other ways. (shrink)

As the editors of the series, New Literary Theory, proclaim in the preface of the book, the purpose of the series is to make more room in literary theory for playful and accessible approaches to li...

This paper is an attempt to bring together two separated areas of research: classical mathematics and metamathematics on the one side, non-monotonic reasoning on the other. This is done by simulating nonmonotonic logic through antitonic theory extensions. In the first half, the specific extension procedure proposed here is motivated informally, partly in comparison with some well-known non-monotonic formalisms. Operators V and, more generally, U are obtained which have some plausibility when viewed as giving nonmonotonic theory extensions. In the second half, (...) these operators are treated from a mathematical and metamathematical point of view. Here an important role is played by U -closed theories and U -fixed points. The last section contains results on V-closed theories which are specific for V. (shrink)

In Pantsar (2014), an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain (...) the ... (shrink)

One of the most important questions in the young field of numerical cognition studies is how humans bridge the gap between the quantity-related content produced by our evolutionarily ancient brains and the precise numerical content associated with numeration systems like Indo-Arabic numerals. This gap problem is the main focus of this paper. The aim here is to evaluate the extent to which cultural factors can help explain how we come to think about numbers beyond the subitizing range. To do this, (...) I summarize Clark’s notion of a difference maker in explaining complex causation that criss-crosses between mind and world and apply it to Menary’s Open MIND. MIND Group, Frankfurt, 2015a) discussion of mathematical cognition as a case of enculturation. I argue that while Menary’s views on enculturation can help explain what makes the difference between numerate and anumerate cultures, it cannot help specify what makes the difference between numerate and anumerate individuals. I argue that features of enculturation do not provide an account of innovation capable of explaining how individuals manage to improve and modify the practices of their cultural niche. This is because Menary’s construal of the role of enculturation in the development of mathematical cognition focuses mostly on the inheritance and transmission of practices, not on their origins, which involve individual-level understanding, rather than population-level practices and pressures. The upshot is that culture provides the necessary background conditions against which individuals can innovate. This role is crucial in the development of numerical abilities—crucial, but explanatorily limited. (shrink)

I try to reconstruct how Frege thought to reconcile the cognitive value of arithmetic with its analytical nature. There is evidence in Frege's texts that the epistemological formulation of the context principle plays a decisive role; it provides a way of obtaining concepts which are truly fruitful and whose contents cannot be grasped beforehand. Taking the definitions presented in the Begriffsschrift,I shall illustrate how this schema is intended to work.

Alberto Casullo ("Necessity, Certainty, and the A Priori", Canadian Journal of Philosophy 18, 1988) argues that arithmetical propositions could be disconfirmed by appeal to an invented scenario, wherein our standard counting procedures indicate that 2 + 2 != 4. Our best response to such a scenario would be, Casullo suggests, to accept the results of the counting procedures, and give up standard arithmetic. While Casullo's scenario avoids arguments against previous "disconfirming" scenarios, it founders on the assumption, common to scenario (...) and response, that arithmetic might be independent of standard counting procedures. Here I show, by attention to tallying as the simplest form of counting, that this assumption is incoherent: given standard counting procedures, then (on pain of irrationality) arithmetical theory follows. (shrink)

The paper traces the development and the role of syntactic reduction in Edmund Husserl’s early writings on mathematics and logic, especially on arithmetic. The notion has its origin in Hermann Hankel’s principle of permanence that Husserl set out to clarify. In Husserl’s early texts the emphasis of the reductions was meant to guarantee the consistency of the extended algorithm. Around the turn of the century Husserl uses the same idea in his conception of definiteness of what he calls “mathematical (...) manifolds.” The paper argues that the notion anticipates the notion of reduction in term rewrite theory in computer science. The role of the reduction for Husserl is, however, primarily epistemological: its purpose is to impart clarity to formal mathematics. (shrink)

I make an attempt at the description of the delicate role of the standard model of arithmetic for the syntax of formal systems. I try to assess whether the possible instability in the notion of finiteness deriving from the nonstandard interpretability of arithmetic affects the very notions of syntactic metatheory and of formal system. I maintain that the crucial point of the whole question lies in the evaluation of the phenomenon of formalization. The ideas of Skolem, Zermelo, Beth (...) and Carnap (among others) on the problem are discussed. ‘A tries to explain to B the meaning of negation. Finally A gives up, saying: “You don’t understand what I mean, and I am not going to explain any longer,” to which B replies: “Yes, I see what you mean, and I am glad you are willing to continue your explanations”’. G. Mannoury, reported by E. W. Beth (Beth, 1963, 489). (shrink)

In Die Grundlagen der Arithmetik, Frege attempted to introduce cardinalnumbers as logical objects by means of a second-order abstraction principlewhich is now widely known as ``Hume's Principle'' (HP): The number of Fsis identical with the number of Gs if and only if F and G are equinumerous.The attempt miscarried, because in its role as a contextual definition HP fails tofix uniquely the reference of the cardinality operator ``the number of Fs''. Thisproblem of referential indeterminacy is usually called ``the Julius Caesar (...) problem''.In this paper, Frege's treatment of the problem in Grundlagen is critically assessed. In particular, I try to shed new light on it by paying special attention to the framework of his logicism in which it appears embedded. I argue, among other things, that the Caesar problem, which is supposed to stem from Frege's tentative inductive definition of the natural numbers, is only spurious, not genuine; that the genuine Caesar problem deriving from HP is a purely semantic one and that the prospects of removing it by explicitly defining cardinal numbers as objects which are not classes are presumably poor for Frege. I conclude by rejecting two closely connected theses concerning Caesar put forward by Richard Heck: (i) that Frege could not abandon Axiom V because he could not solve the Julius Caesar problem without it; (ii) that (by his own lights) his logicist programme in Grundgesetze der Arithmetik failed because he could not overcome that problem. (shrink)

Frege's concern in GGI §10 is neither with the epistemological issue of how we come to know about value-ranges, nor with the semantic-metaphysical issue of whether we have said enough about such objects in order to ensure that any kind of reference to them is possible. The problem which occupies Frege in GGI §10 is the general problem according to which we ‘cannot yet decide’, for any arbitrary function, what value ‘’ has if ‘ℵ’ is a canonical value-range name. This (...) is a problem with the ‘reference’ of value-range names, but only in the weak sense that, if we do not exercise care, value-range terms might become ‘bedeutungslos’ for purely formal reasons. Frege addresses the general problem only for the primitive function- and object-names he has already introduced into his concept-script. I argue that this methodology was perfectly intentional: his intention for GG in general, on display in GGI §10, is to check, for each primitive function- and object-name, as it is introduced into concept-script, whether it interacts with the other primitive names which have already been introduced in such a way that these atomic combinations of primitive names do not become bedeutungslos. If there is a risk of producing a bedeutungslos combination, Frege will make an arbitrary stipulation to ensure that logical hygiene is maintained. I argue that this interpretation does not violate some of the other principal commitments of GG. (shrink)

This book is written so as to be ‘accessible to philosophers without a mathematical background’. The reviewer can assure the reader that this aim is achieved, even if only by focusing throughout on just one example of an arithmetical truth, namely ‘7+5=12’. This example’s familiarity will be reassuring; but its loneliness in this regard will not. Quantified propositions — even propositions of Goldbach type — are below the author’s radar.The author offers ‘a new kind of arithmetical epistemology’, one which (...) ‘respects certain important intuitions’ 1 : apriorism, realism, and empiricism. The book contains some clarification of these ‘isms’, and some thoughtful critiques of major positions regarding them, as espoused by such representative figures as Boghossian, Bealer, Peacocke, Field, Bostock, Maddy, Locke, Kant, C.I. Lewis, Ayer, Quine, Fodor, and McDowell. The philosophical reader will find some interest and value in these wider-ranging discussions. Our concern in this review, however, is to examine closely the original positive proposal on offer.Arithmetical truths, the author maintains, are conceptual truths. Knowing truths like 7+5=12 involves no ‘epistemic reliance on any empirical evidence’; but that, she says, is not to claim ‘epistemic independence of the senses altogether’. She wants to show that "experience grounds our concepts … and then mere conceptual examination enables us to learn arithmetical truths ." Concepts that are ‘appropriately sensitive’ to ‘the nature of [an independent] reality’ she calls grounded. Because of the role of grounded concepts, ‘arithmetical truths explain our arithmetical beliefs in the right sort of way for those beliefs to count as knowledge’ .In the context of her concentration on the special nature of arithmetical knowledge, the author offers what could strike some bystanders as an unnecessarily over-ambitious account of knowledge tout court. Knowledge, for the author, is "true belief which … ". (shrink)

The proportional weight view in epistemology of disagreement generalizes the equal weight view and proposes that we assign to judgments of different people weights that are proportional to their epistemic qualifications. It is shown that if the resulting degrees of confidence are to constitute a probability function, they must be the weighted arithmetic means of individual degrees of confidence, while if the resulting degrees of confidence are to obey the Bayesian rule of conditionalization, they must be the weighted (...) geometric means of individual degrees of confidence. The double bind entails that the proportional weight view (and its moderate adjustment in favor of one’s own judgment) is inconsistent with Bayesianism. (shrink)

The prospects for realistic interpretation of the nature of initial mathematical truths and objects are considered in the article. The arguments of realism, reasons impeding its recognition among philosophers of mathematics as well as the ways to eliminate these reasons are discussed. It is proven that the absence of acceptable ontological interpretation of mathematical realism is the main obstacle to its recognition. This paper explicates the introductory positions of this interpretation and presents a realistic interpretation of the arithmetical component of (...) mathematics. In summary, we should like to note that such constructions, as it is shown to us, ought to bring the direct use not only for the philosophical foundation of mathematics but for mathematics itself. In the justification of the author’s conclusions based on the works of famous mathematicians of the twentieth century, interpreting their findings in a broad historical and philosophical context. To illustrate his point, the author gives examples of arithmetic and geometry - both Euclidean and non-Euclidean. (shrink)

This article aims first at showing that Russell's general doctrine according to which all mathematics is deducible ‘by logical principles from logical principles’ does not require a preliminary reduction of all mathematics to arithmetic. In the Principles, mechanics (part VII), geometry (part VI), analysis (part IV–V) and magnitude theory (part III) are to be all directly derived from the theory of relations, without being first reduced to arithmetic (part II). The epistemological importance of this point cannot be overestimated: (...) Russell's logicism does not only contain the claim that mathematics is no more than logic, it also contains the claim that the differences between the various mathematical sciences can be logically justified—and thus, that, contrary to the arithmetization stance, analysis, geometry and mechanics are not merely outgrowths of arithmetic. The second aim of this article is to set out the neglected Russellian theory of quantity. The topic is obviously linked with the first, since the mere existence of a doctrine of magnitude, in a work dated from 1903, is a sign of a distrust vis-à-vis the arithmetization programme. After having shown that, despite the works of Cantor, Dedekind and Weierstrass, many mathematicians at the end of the 19th Century elaborated various axiomatic theories of the magnitude, I will try to define the peculiarity of the Russellian approach. I will lay stress on the continuity of the logicist's thought on this point: Whitehead, in the Principia, deepens and generalizes the first Russellian 1903 theory. (shrink)

In Pantsar, an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain the (...) third one: why arithmetical knowledge appears to be necessary. A Kripkean analysis of necessity is used as an example to show that a proper analysis of the relevant possible worlds can explain arithmetical necessity in a sufficiently strong form. (shrink)

Objective Apeiron was founded in 1966 and has developed into one of the oldest and most distinguished journals dedicated to the study of ancient philosophy, ancient science, and, in particular, of problems that concern both fields. Apeiron is committed to publishing high-quality research papers in these areas of ancient Greco-Roman intellectual history; it also welcomes submission of articles dealing with the reception of ancient philosophical and scientific ideas in the later western tradition. The journal appears quarterly. Articles are peer-reviewed on (...) a double-blind basis. Topics ancient philosophy ancient science reception of ancient philosophical and scientific ideas in the later western tradition Article formats Original Articles. (shrink)

This paper presents a defense of Epistemic Arithmetic as used for a formalization of intuitionistic arithmetic and of certain informal mathematical principles. First, objections by Allen Hazen and Craig Smorynski against Epistemic Arithmetic are discussed and found wanting. Second, positive support is given for the research program by showing that Epistemic Arithmetic can give interesting formulations of Church's Thesis.

Theorems of arithmetic are used, perhaps essentially, to reach conclusions about the natural world. This applicability can be explained in a natural way by analogy with the applicability of statements of law to the world. ;In order to carry out an ontological argument for my thesis, I assume the existence of universals as a working hypothesis. I motivate a theory of laws according to which statements of law are singular statements about scientific properties. Such statements entail generalizations about instances (...) of those properties. My task, then, is to show that theorems of arithmetic, in application, are similar to statements of law in this respect. On the basis of my working hypothesis, I argue that there are inclining reasons to think that cardinal numbers are scientific properties whose instances occur in the natural world. The works of Mill, Frege and Benacerraf play central roles in these arguments. ;In general, I urge that there are no conclusive epistemological or ontological factors which preclude arithmetical truths from being laws of nature. Consequently, I think we can extend any semantic theory which is adequate for statements of law to theorems of arithmetic. This would apply even to a theory of predication which does not involve commitment to universals. So, to the degree that my argument succeeds with the working hypothesis, it also supports the unconditioned thesis that arithmetical truths are laws of nature. (shrink)

The late nineteenth century gradually witnessed a liberalization of the kinds of mathematical object and forms of mathematical reasoning permissible in physical argumentation. The construction of theories of units illustrates the slow and difficult spread of new “algebraic” modes of mathematical intelligibility, developed by leading mathematicians from the 1830s onwards, into elementary arithmetical pedagogy, experimental physics, and fields of physical practice like telegraphic engineering. A watershed event in this process was a clash that took place during 1878 between J. D. (...) Everett and James Thomson over the meaning and algebraic manipulation of dimensional formulae. This precipitated the emergence of rival “Maxwellian” and “Thomsonian” approaches towards interpreting and applying “dimensional” equations, which expressed the relationship between derived and fundamental units in an absolute system of measurement. What at first looks like a dispute over a seemingly esoteric mathematical tool for unit conversion turns out to concern Everett’s break with arithmetical algebra in the representation and manipulation of physical quantities. This move prompted a vigorous rebuttal from Thomsonian defenders of an orthodox “arithmetical empiricism” on epistemological, semantic, or pedagogical grounds. Their resistance in Victorian Britain to a shift in mathematical intelligibility is suggestive of the difficult birth of theoretical physics, in which the intermediate steps of a mathematical argument need have no direct physical meaning. (shrink)

philosophical practice has taken a new turn since it survived the large scale problems and debates which characterized its early beginnings in an African environment and intellectual community. The metaphilosophical issues then concerned about its status, relevance and methodology appropriate or usable for doing it. Although the issues that troubled African philosophers then may have subsided, yet some of them have and are still expressing reservations on the possibility of having Africa‟s indigenous ways of knowing, just as they deny the (...) possibility of „African physics‟ or „African arithmetic‟. Paulin Hountondji, a leading African philosopher, is reputed for denying African traditional thought as philosophy, which he prefers to type as ethnophilosophy, simply because it thrives on orality and other ethnographical materials like proverbs, parables, folklores, fables, songs etc. For him, the piece, at best can qualify as ethnographical or anthropological monographs as opposed to philosophical work which relies on written texts and documentation on the basis of which “theoretical knowledge and significant intellectual exchange and innovation can” be achieved in Africa. Hountondji‟s position is, to say the least, exclusionist, since it denies and debars African modes of thought and heritage a position in the on-going philosophical conversation or discourse. The paper shares Hountondji‟s vision of adoption of an attitude of critical, scientific and skeptical orientation in African societies. However, it rejects the views of Hountondji and other scholars who deny African intellectual and cognitive systems and argues that their position rests on one sided conception or dimension of epistemology. The other intention of the paper is to show that philosophical practice is as old as the history of mankind in Africa, though Hountondj has expressed the view that philosophy as an academic discipline started in African Universities only in the 1960’s and 1970’s. (shrink)

Why would we want to develop artificial human-like arithmetical intelligence, when computers already outperform humans in arithmetical calculations? Aside from arithmetic consisting of much more than mere calculations, one suggested reason is that AI research can help us explain the development of human arithmetical cognition. Here I argue that this question needs to be studied already in the context of basic, non-symbolic, numerical cognition. Analyzing recent machine learning research on artificial neural networks, I show how AI studies could potentially (...) shed light on the development of human numerical abilities, from the proto-arithmetical abilities of subitizing and estimating to counting procedures. Although the current results are far from conclusive and much more work is needed, I argue that AI research should be included in the interdisciplinary toolbox when we try to explain the development and character of numerical cognition and arithmetical intelligence. This makes it relevant also for the epistemology of mathematics. (shrink)

Sobre la base que aportan las notas manuscritas de David Hilbert para cursos sobre geometría, el artículo procura contextualizar y analizar una de las contribuciones más importantes y novedosas de su célebre monografía Fundamentos de la geometría, a saber: el cálculo de segmentos lineales. Se argumenta que, además de ser un resultado matemático importante, Hilbert depositó en su aritmética de segmentos un destacado significado epistemológico y metodológico. En particular, se afirma que para Hilbert este resultado representaba un claro ejemplo de (...) uno de los rasgos más fructíferos y atractivos de su nuevo método axiomático formal, o sea, la capacidad de descubrir y exhibir conexiones estructurales o internas entre diferentes teorías matemáticas. On the basis of a set of unpublished notes for lecture courses on geometry, the paper seeks to contextualize and analyze one of the most important and original contributions of David Hilbert's celebrated monograph Foundations of Geometry, namely its arithmetic of line segments. It is argued that Hilbert attributed to his arithmetic of segments an important epistemological and methodological meaning, in addition to its relevance as an original mathematical result. In particular, it is claimed that for Hilbert his arithmetic of segments represented a clear example of one of the most fruitful and attractive traits of his new formal axiomatic method, i.e., the power to discover and exhibit inner or structural connections among different mathematical theories. (shrink)

In the late XVII century in England has establishes the school of “political arithmetic”, which goal consisted in the analysis of social phenomena on the basis of quantitative indicators. Its main representatives became William Petty, John Graunt and Charles Davenant (1656-1714). The latter left a mark in the history of England as a philosopher, politician and publicist, who made a significant contribution to the development and implementation of the methods of “political arithmetic”. The object of this research is (...) the views of the English thinker, reflected in his pamphlets and treatises of the 1690’s. The subject is the Davenant’s views on the principles and tasks of “political arithmetic” in the context of his political theory. Special attention is given to correlation between Davenant’s views on the development of “political arithmetic’ and his concept of public administration. The scientific novelty lies in the comprehensive examination of the political and economic views of Charles Davenant within the framework of his attitude on the method of “political arithmetic”. The author delivers a thesis on the importance of the political and epistemological context, which formed Davenant’s views on the quantitative indicators. Grasp of quantitative data Davenant correlated directly with the practice of administration. According to the philosopher, a public official having the capacity for “evaluation” and proper interpretation of data could avoid many mistakes in decision-making and reliance on “bad” advices. The thinker suggested that possession of quantitative data and their competent interpretation could increase the efficiency of administration. (shrink)

Wright claims that his and Hale’s abstractionist neologicist project is primarily epistemological in aim. Its epistemological aims include establishing the possibility of a priori mathematical knowledge, and establishing the possibility of reference to abstract mathematical objects. But, as Wright acknowledges, there is a question of how neologicist epistemology applies to actual, ordinary mathematical beliefs. I take up this question, focusing on arithmetic. Following a suggestion of Hale and Wright, I consider the possibility that the neologicist account provides an (...) idealised reconstruction of how people acquire cognition of numbers or arithmetical knowledge. I conclude that it is unlikely that this account can be used to reconstruct even idealised answers to pressing questions about our actual epistemology. (shrink)

A natural realist approach to the philosophy of arithmetic is defended by way of considering and arguing against contemporary attempts to solve Paul Benacerraf's dilemma . The first horn of the dilemma concerns the existence of abstract mathematical objects, which seems necessitated by a desire for a unified semantics. Benacerraf adopts an extensional semantics whereby the reference of terms for natural numbers must be abstract objects. The second horn concerns a desirable causal constraint on knowledge, according to which "for (...) X to know that S is true requires some causal relation to obtain between X and the referents of the names, predicates, and quantifiers of S". Within the philosophical tradition, Benacerraf's dilemma crystallizes the tension between realists and empiricists . It is shown that natural realism and naturalism meet. ;Both horns of the dilemma are amended.objects are conceived along conceptualist lines such that their existence is not mind-independent. The causal constraint is refined by drawing upon the work of Mark Steiner . Natural realism is the notion that the truth-values of arithmetical statements are recognition-transcendent. Natural realism explains why one and only one arithmetic is applicable. It is in some sense discovered, which is captured by the idea that arithmetical statements have truth-values even if one is not able to adduce what they are. ;It is argued that with proper emendation, like that of Philip Kitcher , empiricism as envisioned by J. S. Mill is defensible. Furthermore, the epistemology of arithmetical knowledge is divided into two tiers. The first principles of arithmetic are acquired by causal interaction with physical objects. As Kitcher has shown, empiricism functions at the first-tier. The a priori is revised such that arithmetical knowledge generated from the first-tier is considered non-empirical. ;It is argued that pragmatists' indispensability argument, once qualified, allows that arithmetic's applicability is one reason to consider natural realism for that domain . Furthermore, this thesis provides a case where aspects of Hilary Putnam's early and later views are rendered consistent. By way of utilizing both Putnam's writing in favour of realism , and those that break with them , it is suggested that he should not have abandoned his earlier view. In a nutshell, the realization of knowledge can depend upon agents' values, methods, and so on, without forcing the abandonment of realism . Putnam's early view sets the criteria for when one can be a realist about a given domain. (shrink)

This is a dialogue in the philosophy of mathematics that focuses on these issues: Are the Peano axioms for arithmetic epistemologically irrelevant? What is the source of our knowledge of these axioms? What is the epistemological relationship between arithmetical laws and the particularities of number?

We show that the P 2 0 sentence, and explore some of theconsequences of that fact. This paper summarizes recent workby the author with N. C. A. da Costa on the P

Truth in Philosophy of Language

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According to Hartry Field, the mathematical Platonist is hostage of a dilemma. Faced with the request of explaining the mathematicians’ reliability, one option could be to maintain that the mathematicians are reliably responsive to a realm populated with mathematical entities; alternatively, one might try to contend that the mathematical realm conceptually depends on, and for this reason is reliably reflected by, the mathematicians’ (best) opinions; however, both alternatives are actually unavailable to the Platonist: the first one because it is in (...) tension with the idea that mathematical entities are causally ineffective, the second one because it is in tension with the suggestion that mathematical entities are mind-independent. John Divers and Alexander Miller have tried to reject the conclusion of this argument—according to which Platonism is inconsistent with a satisfactory epistemology for arithmetic—by redescribing the second horn of the dilemma in light of Crispin Wright’s notion of judgment-dependent truth; in particular they have contended that once arithmetical truth is conceived in this way the Platonist can have a substantial epistemology which does not conflict with the idea that the mathematical entities exist mind-independently. In this paper I analyze Wright’s notion of judgment-dependent truth, and reject Divers and Miller’s argument for the conclusion that arithmetical truth can be so characterized. In the final part, I address the worry that my argument generalizes very quickly to the conclusion that no area of discourse could be characterized as judgment-dependent. As against this conclusion, I indicate under what conditions—notably not satisfied in Divers and Miller’s case, but possibly satisfied in others—a discourse’s judgment-dependency can be successfully vindicated. (shrink)