The small, the tiny, and the infinitesimal have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that written by Errett Bishop about Keisler’s book Elementary Calculus: an Infinitesimal Approach. In this skit we investigate both the argument itself, and some of its roots in Bishop George Berkeley’s criticism of Leibnizian and Newtonian Calculus. We also explore some of the consequences to students for whom the infinitesimal approach is congenial. The (...) casual mathematical reader may be satisfied to read the text of the five act play, whereas the others may wish to delve into the 130 footnotes, some of which contain elucidation of the mathematics or comments on the history. (shrink)

Events between which we have no epistemic reason to discriminate have equal epistemic probabilities. Bertrand’s chord paradox, however, appears to show this to be false, and thereby poses a general threat to probabilities for continuum sized state spaces. Articulating the nature of such spaces involves some deep mathematics and that is perhaps why the recent literature on Bertrand’s Paradox has been almost entirely from mathematicians and physicists, who have often deployed elegant mathematics of considerable sophistication. At the same time, the (...) philosophy of probability has been left out. In particular, left out entirely are the philosophical ground of the principle of indifference, the nature of the principle itself, the stringent constraint this places on the mathematical representation of the principle needed for its application to continuum sized event spaces, and what these entail for rigour in developing the paradox itself. This book puts the philosophy and its entailments back in and in so doing casts a new light on the paradox, giving original analyses of the paradox, its possible solutions, the source of the paradox, the philosophical errors we make in attempting to solve it and what the paradox proves for the philosophy of probability. The book finishes with the author’s proposed solution—a solution in the spirit of Bertrand’s, indeed—in which an epistemic principle more general than the principle of indifference offers a principled restriction of the domain of the principle of indifference. (shrink)

The philosophy of mathematics of the last few decades is commonly distinguished into mainstream and maverick, to which a ‘third way’ has been recently added, the philosophy of mathematical practice. In this paper the limitations of these trends in the philosophy of mathematics are pointed out, and it is argued that they are due to the fact that all of them are based on a top-down approach, that is, an approach which explains the nature of mathematics in terms of some (...) general unproven assumption. As an alternative, a bottom-up approach is proposed, which explains the nature of mathematics in terms of the activity of real individuals and interactions between them. This involves distinguishing between mathematics as a discipline and the mathematics embodied in organisms as a result of biological evolution, which however, while being distinguished, are not opposed. Moreover, it requires a view of mathematical proof, mathematical definition and mathematical objects which is alternative to the top-down approach. (shrink)

Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways between these two monstrous habitats. I start from Jeffrey Jerome Cohen’s influential account of monster culture and explore how well mathematical monsters fit each of his seven theses. The mathematical monsters I discuss are drawn primarily from three distinct but overlapping domains. Firstly, late nineteenth-century mathematicians made numerous unsettling discoveries that threatened their understanding of their own discipline and challenged their intuitions. The great French mathematician (...) Henri Poincaré characterised these anomalies as ‘monsters’, a name that stuck. Secondly, the twentieth-century philosopher Imre Lakatos composed a seminal work on the nature of mathematical proof, in which monsters play a conspicuous role. Lakatos coined such terms as ‘monster-barring’ and ‘monster-adjusting’ to describe strategies for dealing with entities whose properties seem to falsify a conjecture. Thirdly, and most recently, mathematicians dubbed the largest of the sporadic groups ‘the Monster’, because of its vast size and uncanny properties, and because its existence was suspected long before it could be confirmed. (shrink)

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre (...) Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)

To explore the relation between mathematical models and reality, four different domains of reality are distinguished: observer-independent reality, personal reality, social reality and mathematical/formal reality. The concepts of personal and social reality are strongly inspired by constructivist ideas. Mathematical reality is social as well, but constructed as an autonomous system in order to make absolute agreement possible. The essential problem of mathematical modelling is that within mathematics there is agreement about ‘truth’, but the assignment of mathematics to informal reality is (...) not itself formally analysable, and it is dependent on social and personal construction processes. On these levels, absolute agreement cannot be expected. Starting from this point of view, repercussion of mathematical on social and personal reality, the historical development of mathematical modelling, and the role, use and interpretation of mathematical models in scientific practice are discussed. (shrink)

What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...) argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (shrink)

Philosophers of logic are particularly interested in understanding the aims, epistemology, and methodology of logic. This raises the question of how the philosophy of logic should go about these enquires. According to the practice-based approach, the most reliable method we have to investigate the methodology and epistemology of a research field is by considering in detail the activities of its practitioners. This holds just as true for logic as it does for the recognised empirical and abstract sciences. If we wish (...) to systematically understand the aims and epistemology of logic, we are best placed achieving this by looking at what logicians do, rather than reflecting upon the nature of logic itself. In this entry, we outline the motivations for a practice-based approach towards the philosophy of logic and highlight its advantages over more “top-down” approaches, which attempt to infer conclusions about the nature of logic and its methodology on the basis of traditional assumptions about knowledge, metaphysics, or logic itself. We end by addressing several prominent concerns raised against the practice-based approach. (shrink)

Ethical discourse and fallacy theory come together in a natural way over concepts such as bias, prejudice, preconceived opinion, prototypical and stereotypical thinking, dogmatism and loyalty. By and large, these are concepts that have not been sufficiently worked up to bear the theoretical weight either of ethics or of logic. The present paper seeks to ameliorate this situation. It proposes that situations describable by any such concepts partition into (a) the rationally and morally regrettable and (b) the rationally and morally (...) impeccable, and in any event, unavoidable. Finding (b) will come as a surprise to some theorists. (shrink)

Es wird die Frage untersucht, ob die Annahme der Existenz von universellen Normen für die Annäherung der Wissenschaft an die Wahrheit nicht in praxi lediglich Gegenwartszentrismus und Ethnozentrismus heißt. Die griechisch-römische Zivilisation förderte die Wissenschaft durch ihre Demokratie, aber andere Zivilisationen haben sehr wertvolle Datensammlungen geliefert. Die Universalität der Wissenschaft impliziert u. a. daß, wo verschiedene Zivilisationen mit ihren Wissenschaften einander begegnen, sie von einander lernen können.

This title offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.

During the first half of the twentieth century, mainstream answers to the foundational crisis, mainly triggered by Russell and Gödel, remained largely perfectibilist in nature. Along with a general naturalist wave in the philosophy of science, during the second half of that century, this idealist picture was finally challenged and traded in for more realist ones. Next to the necessary preliminaries, the present paper proposes a structured view of various philosophical accounts of mathematics indebted to this general idea, laying the (...) ground for a desirable integration of their strenghts. (shrink)

No one will dispute, looking at the history of mathematics, that there are plenty of moments where mathematics is “in trouble”, when paradoxes and inconsistencies crop up and anomalies multiply. This need not lead, however, to the view that mathematics is intrinsically inconsistent, as it is compatible with the view that these are just transient moments. Once the problems are resolved, consistency (in some sense or other) is restored. Even when one accepts this view, what remains is the question what (...) mathematicians do during such a transient moment? This requires some method or other to reason with inconsistencies. But there is more: what if one accepts the view that mathematics is always in a phase of transience? In short, that mathematics is basically inconsistent? Do we then not need a mathematics of inconsistency? This paper wants to explore these issues, using classic examples such as infinitesimals, complex numbers, and infinity. (shrink)

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

This paper describes an attempt to develop a program for teaching history and philosophy of mathematics to inservice mathematics teachers. I argue briefly for the view that philosophical positions and epistemological accounts related to mathematics have a significant influence and a powerful impact on the way mathematics is taught. But since philosophy of mathematics without history of mathematics does not exist, both philosophy and history of mathematics are necessary components of programs for the training of preservice as well as inservice (...) mathematics teachers. (shrink)

Abstract The first fractal constructions appeared in mathematics in the second half of the 19th century. Their history is divided into two periods. The first period lasted 100 years and is a good example of the method of proofs and refutations discovered by Lakatos. The modern history of these objects started 20 years ago, when Mandelbrot decided to create fractal geometry, a general theory concentrated on specific properties of fractals. His approach has been surprisingly effective. The aim of this paper (...) is to examine the reasons for Mandelbrot's success and for the present popularity of fractals. They are now known not only in mathematics but also in many fields of natural, social and applied sciences, and even in the arts. (shrink)

Since the birth of computing as an academic discipline, the disciplinary identity of computing has been debated fiercely. The most heated question has concerned the scientific status of computing. Some consider computing to be a natural science and some consider it to be an experimental science. Others argue that computing is bad science, whereas some say that computing is not a science at all. This survey article presents viewpoints for and against computing as a science. Those viewpoints are analyzed against (...) basic positions in the philosophy of science. The article aims at giving the reader an overview, background, and a historical and theoretical frame of reference for understanding and interpreting some central questions in the debates about the disciplinary identity of computer science. The article argues that much of the discussion about the scientific nature of computing is misguided due to a deep conceptual uncertainty about science in general as well as computing in particular. (shrink)

Derivationists, those wishing to explain the correctness and rigour of informal proofs in terms of associated formal proofs, are generally held to be supported by the success of the project of translating informal proofs into computer-checkable formal counterparts. I argue, however, that this project is a false friend for the derivationists because there are too many different associated formal proofs for each informal proof, leading to a serious worry of overgeneration. I press this worry primarily against Azzouni's derivation-indicator account, but (...) conclude that overgeneration is a major obstacle to a successful account of informal proofs in this direction. (shrink)

Scientific anomalies are observations and facts that contradict current scientific theories and they are instrumental in scientific theory change. Philosophers of science have approached scientific theory change from different perspectives as Darden (Theory change in science: Strategies from Mendelian genetics, 1991) observes: Lakatos (In: Lakatos, Musgrave (eds) Criticism and the growth of knowledge, 1970) approaches it as a progressive “research programmes” consisting of incremental improvements (“monster barring” in Lakatos, Proofs and refutations: The logic of mathematical discovery, 1976), Kuhn (The structure (...) of scientific revolutions, 1996) observes that changes in “paradigms” are instigated by a crisis from some anomaly, and Hanson (In: Feigl, Maxwell (eds) Current issues in the philosophy of science, 1961) proposes that discovery does not begin with hypothesis but with some “problematic phenomena requiring explanation”. Even though anomalies are important in all of these approaches to scientific theory change, there have been only few investigations into the specific role anomalies play in scientific theory change. Furthermore, much of these approaches focus on the theories themselves and not on how the scientists and their experiments bring about scientific change (Gooding, Experiment and the making of meaning: Human agency in scientific observation and experiment, 1990). To address these issues, this paper approaches scientific anomaly resolution from a meaning construction point of view. Conceptual integration theory (Fauconnier and Turner, Cogn Sci 22:133–187, 1996; The way we think: Conceptual blending and mind’s hidden complexities, 2002) from cognitive linguistics describes how one constructs meaning from various stimuli, such as text and diagrams, through conceptual integration or blending. The conceptual integration networks that describe the conceptual integration process characterize cognition that occurs unconsciously during meaning construction. These same networks are used to describe some of the cognition while resolving an anomaly in molecular genetics called RNA interference (RNAi) in a case study. The RNAi case study is a cognitive-historical reconstruction (Nersessian, In: Giere (ed) Cognitive models of science, 1992) that reconstructs how the RNAi anomaly was resolved. This reconstruction traces four relevant molecular genetics publications in describing the cognition necessary in accounting for how RNAi was resolved through strategies (Darden 1991), abductive reasoning (Peirce, In: Hartshorne, Weiss (eds) Collected papers, 1958), and experimental reasoning (Gooding 1990). The results of the case study show that experiments play a crucial role in formulating an explanation of the RNAi anomaly and the integration networks describe the experiments’ role. Furthermore, these results suggest that RNAi anomaly resolution is embodied. It is embodied in a sense that cognition described in the cognitive-historical reconstruction is experientially based. (shrink)

Mathematics educators suggest that students of all ages need to participate in productive forms of mathematical argument (NCTM, 2000). Accordingly, we developed two complementary frameworks for analyzing the emergence of mathematical argumentation in one second‐grade classroom. Children attempted to resolve contesting claims about the “space covered” by three different‐looking rectangles of equal area measure. Our first analysis renders the topology of the semantic structure of the classroom conversation as a directed graph. The graph affords clear “at a glance” visualization of (...) how various senses of mathematics—as imagined, as performed, and as historically rooted—were interrelated. The graph represents the emergence and intercoordination between conceptual and procedural knowledge in the ebb and flow of classroom conversation. The graph has not just descriptive, but also predictive power: Interconnectedness among the nodes of the graph representing the first 40 min of conversation predicted the structure of recall in the final ten minutes by children who had played little overt role in the conversation up to that point. The second, complementary framework draws upon Goffman's expanded repertoire of roles in speech to demonstrate how the teacher orchestrated this classroom conversation to establish coherent argument. In this second analysis, we establish how the teacher mediated between the everyday talk of her students and the discourse of mathematics. Her revoicing of student talk created interstices for identity and participation in the formation of the argument. Together, the two forms of analysis illuminate the emergence of mathematical argument at two levels: as collective structure and concurrently, as individual activity. (shrink)

The consensus among legal philosophers is probably that rule-based legal expert systems leave much to be desired as aids in legal decision-making. Why? What can we do about it? A bureaucrat administering some set of complex rules will ascertain the facts and apply the rules to them in order to discover their consequences for the case in hand. This process of deductive reasoning is characteristically bureaucratic.

Engineering deals with different problem situations than science, and theories in engineering are different to theories in science. So, the growth of knowledge in engineering is also different to that in science. Nonetheless, methodological issues in engineering epistemology can be explored by adapting frameworks already established in the philosophy of science. In this paper I use critical rationalism and Popper’s three worlds framework to investigate error elimination and the growth of knowledge in engineering. I discuss engineering failure arising from the (...) falsification of engineering theories, and present taxonomies of the sources of falsification and responses to falsification in engineering. From this I discuss contexts of research and design in engineering, ad hoc rescue of engineering theories, and engineering assurance. (shrink)

We discuss several aspects of legal arguments, primarily arguments about the meaning of statutes. First, we discuss how the requirements of argument guide the specification and selection of supporting cases and how an existing case base influences argument formation. Second, we present,our evolving taxonomy of patterns of actual legal argument. This taxonomy builds upon our much earlier work on argument moves and also on our more recent analysis of how cases are used to support arguments for the interpretation of legal (...) statutes. Third, we show how the theory of argument used by CABARET, a hybrid case-based/rule-based reasoner, can support many of the argument patterns in our taxonomy. (shrink)

Quine’s thesis of underdetermination is significantly weaker than it has been taken to be in the recent literature, for the following reasons: (i) it does not hold for all theories, but only for some global theories, (ii) it does not require the existence of empirically equivalent yet logically incompatible theories, (iii) it does not rule out the possibility that all perceived rivalry between empirically equivalent theories might be merely apparent and eliminable through translation, (iv) it is not a fundamental thesis (...) within Quine’s philosophy, and (v) it does not carry with it the anti-realistic consequences often associated with the thesis in recent debates. The paper analyzes Quine’s views on the matter and the changes they underwent over the years. A conjecture is put forth about why Quine’s thesis has been so widely misrepresented: Quine’s writings up to 1975 tackled primarily the formulation and justification of the thesis, but afterwards were concerned mostly with the question whether empirically equivalent rivals to the theory we hold are to be considered true also. When this latter discussion is read without bearing in mind Quine’s earlier formulation and justification of the thesis, his thesis seems to have stronger epistemic consequences than it actually does. A careful reading of his later writings shows, however, that the formulation of the thesis remained unchanged after 1975, and that his mature and considered views supported only a very mitigated version of the thesis. (shrink)

This article looks at recent work in cognitive science on mathematical cognition from the perspective of history and philosophy of mathematical practice. The discussion is focused on the work of Lakoff and Núñez, because this is the first comprehensive account of mathematical cognition that also addresses advanced mathematics and its history. Building on a distinction between mathematics as it is presented in textbooks and as it presents itself to the researcher, it is argued that the focus of cognitive analyses of (...) historical developments of mathematics has been primarily on the former, even if they claim to be about the latter. (shrink)

The ill?named debate between postmodernists and postlibertarians should be transcended; this requires the abandonment of both foundationalism and its converse, without abandoning common sense as well (which is no mean trick). Similarly, the debate over ?minimal statism? versus the planned economy is outdated. Instead of claiming to be in possession of foundations of our scientific?cum?political knowledge in broad terms, and instead of severely limiting our knowledge to given proofs, we offer the putative heuristics of critique in general and the critical (...) evaluations of contexts in particular. (shrink)

In this paper it is argued that the fundamental difference of the formal and the informal position in the philosophy of mathematics results from the collision of an object and a process centric perspective towards mathematics. This collision can be overcome by means of dialectical analysis, which shows that both perspectives essentially depend on each other. This is illustrated by the example of mathematical proof and its formal and informal nature. A short overview of the employed materialist dialectical approach is (...) given that rationalises mathematical development as a process of model production. It aims at placing more emphasis on the application aspects of mathematical results. Moreover, it is shown how such production realises subjective capacities as well as objective conditions, where the latter are mediated by mathematical formalism. The approach is further sustained by Polanyi’s theory of problem solving and Stegmaier’s philosophy of orientation. In particular, the tool and application perspective illuminates which role computer-based proofs can play in mathematics. (shrink)

Traditional theories of how children learn the positive integers start from infants' abilities in detecting the quantity of physical objects. Our target article examined this view and found no plausible accounts of such development. Most of our commentators appear to agree that no adequate developmental theory is presently available, but they attempt to hold onto a role for early enumeration. Although some defend the traditional theories, others introduce new basic quantitative abilities, new methods of transformation, or new types of end (...) states. A survey of these proposals, however, shows that they do not succeed in bridging the gap to knowledge of the integers. We suggest that a better theory depends on starting with primitives that are inherently structural and mathematical. (shrink)

ABSTRACT Historical structuralist views have been ontological. They either deny that there are any mathematical objects or they maintain that mathematical objects are structures or positions in them. Non-ontological structuralism offers no account of the nature of mathematical objects. My own structuralism has evolved from an early sui generis version to a non-ontological version that embraces Quine’s doctrine of ontological relativity. In this paper I further develop and explain this view.

My first paper on the Is/Ought issue. The young Arthur Prior endorsed the Autonomy of Ethics, in the form of Hume’s No-Ought-From-Is (NOFI) but the later Prior developed a seemingly devastating counter-argument. I defend Prior's earlier logical thesis (albeit in a modified form) against his later self. However it is important to distinguish between three versions of the Autonomy of Ethics: Ontological, Semantic and Ontological. Ontological Autonomy is the thesis that moral judgments, to be true, must answer to a realm (...) of sui generis non-natural PROPERTIES. Semantic autonomy insists on a realm of sui generis non-natural PREDICATES which do not mean the same as any natural counterparts. Logical Autonomy maintains that moral conclusions cannot be derived from non-moral premises.-moral premises with the aid of logic alone. Logical Autonomy does not entail Semantic Autonomy and Semantic Autonomy does not entail Ontological Autonomy. But, given some plausible assumptions Ontological Autonomy entails Semantic Autonomy and given the conservativeness of logic – the idea that in a valid argument you don’t get out what you haven’t put in – Semantic Autonomy entails Logical Autonomy. So if Logical Autonomy is false – as Prior appears to prove – then Semantic and Ontological Autonomy would appear to be false too! I develop a version of Logical Autonomy (or NOFI) and vindicate it against Prior’s counterexamples, which are also counterexamples to the conservativeness of logic as traditionally conceived. The key concept here is an idea derived in part from Quine - that of INFERENCE-RELATIVE VACUITY. I prove that you cannot derive conclusions in which the moral terms appear non-vacuously from premises from which they are absent. But this is because you cannot derive conclusions in which ANY (non-logical) terms appear non-vacuously from premises from which they are absent Thus NOFI or Logical Autonomy comes out as an instance of the conservativeness of logic. This means that the reverse entailment that I have suggested turns out to be a mistake. The falsehood of Logical Autonomy would not entail either the falsehood Semantic Autonomy or the falsehood of Ontological Autonomy, since Semantic Autonomy only entails Logical Autonomy with the aid of the conservativeness of logic of which Logical Autonomy is simply an instance. Thus NOFI or Logical Autonomy is vindicated, but it turns out to be a less world-shattering thesis than some have supposed. It provides no support for either non-cognitivism or non-naturalism. (shrink)