I argue on the basis of an example, Fourier theory applied to the problem of vibration, that Field's program for nominalizing science is unlikely to succeed generally, since no nominalistic variant will provide us with the kind of physical insight into the phenomena that the standard theory supplies. Consideration of the same example also shows, I argue, that some of the motivation for mathematical fictionalism, particularly the alleged problem of cognitive access, is more apparent than real.

This is a charming and insightful contribution to an understanding of the "Science Wars" between postmodernist humanism and science, driving toward a resolution of the mutual misunderstanding that has driven the controversy. It traces the root of postmodern theory to a debate on the foundations of mathematics early in the 20th century, then compares developments in mathematics to what took place in the arts and humanities, discussing issues as diverse as literary theory, arts, and artificial intelligence. This is (...) a straightforward, easily understood presentation of what can be difficult theoretical concepts It demonstrates that a pattern of misreading mathematics can be seen both on the part of science and on the part of postmodern thinking. This is a humorous, playful yet deeply serious look at the intellectual foundations of mathematics for those in the humanities and the perfect critical introduction to the bases of modernism and postmodernism for those in the sciences. (shrink)

Alan Baker argues that mathematical objects play an indispensable explanatory role in science. There are several examples cited in the literature as solid candidates for such a role. We discuss two such examples and show that they are very different in their strength and (im)perfection, although both are recognized by the scientific community as examples of the best scientific explanations of particular phenomena. More specifically, it will be shown that the explanation of the cicada case has serious shortcomings compared (...) with the explanation of the case of Königsberg’s bridges. We will argue that the latter is a perfectly reliable scientific explanation that employs mathematical reasoning whereas the former is not. (shrink)

Reading the Timaeus as an early attempt at mathematizing natural science runs into serious difficulties. The so-called Platonic Solids are five in number, more by one than the traditional 'elements'. Plato provides a proportional ratio for these elements but this ratio fails to tie in with their geometrical features. Appealing to the authority of mathematics appears to be a rhetorical move with no further consequences.

a Mathematicians, like Proust and everyone else, are at their best when writing about their first lovea (TM) a ] They are among the very best we have; and their best is very good indeed. a ] One approaches this book with high hopes. Happily, one is not disappointed. a ]In paperback it might well have become a best seller. a ]read it. From The Mathematical Intelligencer Mathematics is shaped by the consistent concerns and styles of powerful minds a (...) three of which are represented here. Kaca (TM)s work is marked by deep commitment and breadth of inquiry. Rota is the easiest of these authors to reada ]a delight: witty and urbane, with a clear and interesting agenda and an astonishing intellectual range. To read him is to be a part of a pleasant and rewarding conversation. Jacob T. Schwartz attacks problems ina ]computer science, mathematical economics, computer and instructional a ] No matter how slippery the problem, Schwartz manages to capture a substantial chunk of it in his mathematical net. a Mathematics Magazine This is a volume of essays and reviews that delightfully explore mathematics in all its moods a from the light and the witty, and humorous to serious, rational, and cerebral. Topics include: logic, combinatorics, statistics, economics, artificial intelligence, computer science, and applications of mathematics broadly. You will also find history and philosophy covered, including discussion of the work of Ulam, Kant, Heidegger among others. (shrink)

In Part III of his Remarks on the Foundations of Mathematics Wittgenstein deals with what he calls the surveyability of proofs. By this he means that mathematical proofs can be reproduced with certainty and in the manner in which we reproduce pictures. There are remarkable similarities between Wittgenstein's view of proofs and Hilbert's, but Wittgenstein, unlike Hilbert, uses his view mainly in critical intent. He tries to undermine foundational systems in mathematics, like logicist or set theoretic ones, by (...) stressing the unsurveyability of the proof-patterns occurring in them. Wittgenstein presents two main arguments against foundational endeavours of this sort. First, he shows that there are problems with the criteria of identity for the unsurveyable proof-patterns, and second, he points out that by making these patterns surveyable, we rely on concepts and procedures which go beyond the foundational frameworks. When we take these concepts and procedures seriously, mathematics does not appear as a uniform system, but as a mixture of different techniques. (shrink)

The connection between understanding and explanation has recently been of interest to philosophers. Inglis and Mejía-Ramos (Synthese, 2019) propose that within mathematics, we should accept a functional account of explanation that characterizes explanations as those things that produce understanding. In this paper, I start with the assumption that this view of mathematical explanation is correct and consider what we can consequently learn about mathematical explanation. I argue that this view of explanation suggests that we should shift the question of (...) explanation away from why-questions and towards a “what’s going on here” question. Additionally, I argue that when we recognize the connection between understanding and explanation we naturally see how more than just proof can be explanatory. I expand this point by detailing how definitions and diagrams can be explanatory. In all, we see that when we take seriously the connection between understanding and explanation, we get a better sense of how explanation arises within mathematics. (shrink)

This contribution, by a mathematician, to the Common Knowledge symposium “Fuzzy Studies” examines some mechanisms that seem essential for the “ratchet effect” that, in Michael Tomasello's use of the term, refers to the ability of human cultures to preserve their achievements even through serious crises and even where preservation entails substantial loss. By taking the word culture to refer to any group of individuals who closely cooperate over an extended period, this article evaluates mathematicians and mathematics as its (...) main example. The assessment of the enterprise of mathematics as a culture corresponds to the author's personal experience — and mathematics, he argues, is both historically old and simply structured, despite the formidable complexity of mathematical knowledge as it has been compiled over the centuries. The preserving, restructuring, and enlarging of this knowledge may be regarded as the main cultural goals of mathematics and are the objects of study here. This article concentrates first on the techniques used in pursuing the tasks of preservation, enlargement, and restructuring — techniques such as counting methods and geometric diagrams — and it emphasizes their use in communication, when they function as media. The cultural techniques of mathematics, known to and used by all members of that culture, are media that, through intensity of communication, add symbolic functions to mere procedures. (The prototypical example of a cultural technique, in this sense, is language.) Knowledge, as the additional symbolic value, is distributed and at the same time preserved in the process of permanent communication. (shrink)

It is argued that Hellman's arguments purporting to demonstrate that constructive mathematics cannot cope with unbounded operators on a Hilbert space are seriously flawed, and that there is no evidence that his thesis is correct.

This book argues that we can only understand transformations of nature studies in the Scientific Revolution if we take seriously the interaction between practitioners and scholars. These are not in opposition, however. Theory and practice are end points on a continuum, with some participants interested only in the practical, others only in the theoretical, and most in the murky intellectual and material world in between. It is this borderland where influence, appropriation, and collaboration have the potential to lead to new (...) methods, new subjects of enquiry, and new social structures of natural philosophy and science. The case for connection between theory and practice can be most persuasively drawn in the area of mathematics, which is the focus of this book. Practical mathematics was a growing field in early modern Europe and these essays are organised into three parts which contribute to the debate about the role of mathematical practice in the Scientific Revolution. First, they demonstrate the variability of the identity of practical mathematicians, and of the practices involved in their activities in early modern Europe. Second, readers are invited to consider what practical mathematics looked like and that although practical mathematical knowledge was transmitted and circulated in a wide variety of ways, participants were able to recognize them all as practical mathematics. Third, the authors show how differences and nuances in practical mathematics typically depended on the different contexts in which it was practiced: social, cultural, political, and economic particularities matter. Historians of science, especially those interested in the Scientific Revolution period and the history of mathematics will find this book and its ground-breaking approach of particular interest. (shrink)

This work, in continuity with the article published by Ferdinando Casolaro and Giovanna Della Vecchia in Vol 5, 2017 of this series, in which we noted that in the centuries since the eight century B.C. at the 13th century A.D. the evolution of Astronomy and historical events have influenced the development of Mathematics, intends to demonstrate how the Architecture and Literature of the following centuries have further conditioned the development of the sciences in Italy and, in particular, of (...) class='Hi'>Mathematics, identifying the affinities and, in some cases, the coincidences between the different forms of thought that invite us to make a serious reflection on the uniqueness of culture. (shrink)

Glaucon in Plato's _Republic_ fails to grasp intermediates. He confuses pursuing a goal with achieving it, and so he adopts ‘mathematical platonism’. He says mathematical objects are eternal. Socrates urges a seriously debatable, and seriously defensible, alternative centered on the destruction of hypotheses. He offers his version of geometry and astronomy as refuting the charge that he impiously ‘ponders things up in the sky and investigates things under the earth and makes the weaker argument the stronger’. We relate his account (...) briefly to mathematical developments by Plato's associates Theaetetus and Eudoxus, and then to the past 200 years' developments in geometry. (shrink)

Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel (...) version of mathematical realism. She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics. (shrink)

W. V. Quine’s systematic development of mathematical logic has been widely praised for the new material presented and for the clarity of its exposition. This revised edition, in which the minor inconsistencies observed since its first publication have been eliminated, will be welcomed by all students and teachers in mathematics and philosophy who are seriously concerned with modern logic. Max Black, in Mind, has said of this book, “It will serve the purpose of inculcating, by precept and example, standards (...) of clarity and precision which are, even in formal logic, more often pursued than achieved.”. (shrink)

This Element introduces a young field, the 'philosophy of mathematical practice'. We first offer a general characterisation of the approach to the philosophy of mathematics that takes mathematical practice seriously and contrast it with 'mathematical philosophy'. The latter is traced back to Bertrand Russell and the orientation referred to as 'scientific philosophy' that was active between 1850 and 1930. To give a better sense of the field, the Element further contains two examples of topics studied, that of mathematical structuralism (...) and visual thinking in mathematics. These are in part presented from a methodological point of view, focussing on mathematics as an activity and questions related to how mathematics develops. In addition, the Element contains several examples from mathematics, both historical and contemporary, to illustrate and support the philosophical points. (shrink)

This chapter discusses the significance of mathematics in natural theology. It suggests that the existence of an independent noetic realm of mathematics should encourage an openness to the possibility of further metaphysical riches to be explored. Engagement with mathematics is only a part of our mental experience. In itself it can give just a hint of what might be meant by the spiritual. The realm of the divine is yet more distant still, but just as arithmetic may (...) have led our ancestors to begin an exploration that would eventually lead them into the riches of higher mathematics, so taking mathematical reality seriously might be the start of a journey which will lead to greater discoveries about the scope and nature of reality. In this way, mathematics has a modest preparatory part to play in the approach to natural theology, and may also contribute to natural theology through its role as a component in a more complex form of encounter with reality. (shrink)

Schistosomiasis, a vector-borne chronically debilitating infectious disease, is a serious public health concern for humans and animals in the affected tropical and sub-tropical regions. We formulate and theoretically analyze a deterministic mathematical model with snail and bovine hosts. The basic reproduction number R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document} is computed and used to investigate the local stability of the model’s steady states. Global stability of the endemic equilibrium is carried out by constructing a suitable Lyapunov (...) function. Sensitivity analysis shows that the basic reproduction number is most sensitive to the model parameters related to the contaminated environment, namely: shedding rate of cercariae by snails, cercariae to miracidia survival probability, snails-miracidia effective contact rate and natural death rate of miracidia and cercariae. Numerical results show that when no intervention measures are implemented, there is an increase of the infected classes, and a rapid decline of the number of susceptible and exposed bovines and snails. Effects of the variation of some of the key sensitive model parameters on the schistosomiasis dynamics as well as on the initial disease transmission threshold parameter R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document} are graphically depicted. (shrink)

This paper investigates the determinacy of mathematics. We begin by clarifying how we are understanding the notion of determinacy before turning to the questions of whether and how famous independence results bear on issues of determinacy in mathematics. From there, we pose a metasemantic challenge for those who believe that mathematical language is determinate, motivate two important constraints on attempts to meet our challenge, and then use these constraints to develop an argument against determinacy and discuss a particularly (...) popular approach to resolving indeterminacy, before offering some brief closing reflections. We believe our discussion poses a serious challenge for most philosophical theories of mathematics, since it puts considerable pressure on all views that accept a non-trivial amount of determinacy for even basic arithmetic. (shrink)

Charles Chihara's new book develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries. The view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true. Chihara builds upon his previous work, in which (...) he presented a new system of mathematics, the constructibility theory, which did not make reference to, or presuppose, mathematical objects. Now he develops the project further by analysing mathematical systems currently used by scientists to show how such systems are compatible with this nominalistic outlook. He advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam. And Chihara presents a rationale for the nominalistic outlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings.A Structural Account of Mathematics will be required reading for anyone working in this field. (shrink)

Approaches to the interpretation of physical theories provide accounts of how physical meaning accrues to the mathematical structure of a theory. According to many standard approaches to interpretation, meaning relations are captured by maps from the mathematical structure of the theory to statements expressing its empirical content. In this article I argue that while such accounts adequately address meaning relations when exact models are available or perturbation theory converges, they do not fare as well for models that give rise to (...) divergent perturbative expansions. Since truncations of divergent perturbative expansions often play a critical role in establishing the empirical adequacy of a theory, this is a serious deficiency. I show how to augment state-space semantics, a view developed by Beth and van Fraassen, to capture perturbatively evaluated observables even in cases where perturbation theory is divergent. This new semantics establishes a sense in which the calculations that underwrite the empirical adequacy of a theory are both meaningful and true, but requires departure from the assumption that physical meaning is captured entirely by the exact models of a theory. (shrink)

Legal scholarship exploring the nature of evidence and the process of juridical proof has had a complex relationship with formal modeling. As evident in so many fields of knowledge, algorithmic approaches to evidence have the theoretical potential to increase the accuracy of fact finding, a tremendously important goal of the legal system. The hope that knowledge could be formalized within the evidentiary realm generated a spate of articles attempting to put probability theory to this purpose. This literature was both insightful (...) and frustrating. Much light was shed on the legal system, but it also quickly became evident that the tools of probability theory were in many ways ill-constructed for the task. Fundamental incompatibilities between the structure of legal decision making and the extant formal tools were identified, and it became evident that many of the purported explanations of legal phenomena were internally inconsistent. As a consequence, interest in this type of formal modeling declined, and attention was directed toward different kinds of explanations of the phenomena. Perhaps under the influence of a recent trend toward various types of formal modeling in legal scholarship, a recent burst of articles, rather than attempting to explain the macro structure of trials, which was the previous object of interest, attempts to quantify the probative value of various items of evidence in ways consistent with the formal features of various probability theories, and then to study decision making from that perspective. For example, the value of evidence is often purported to be its likelihood ratio, that is, the probability of discovering or receiving the evidence given a hypothesis (e.g., the defendant did it) divided by the probability of discovering or receiving the evidence given the negation of the hypothesis (the defendant didn't do it). Alternatively, the value of evidence is purported (more contextually) to be the information gain it provides, defined as the increase in probability it provides for a hypothesis above the probability of the hypothesis based on the other available evidence. Both conceptions then assume that all of the various probability assessments conform or ought to conform to the dictates of Bayes' theorem (that maintains consistency among such assessments); empirical studies are then done testing the extent to which this is so and proposing how the law can increase the probability that it is so. The general criticisms of using Bayes' theorem as a formal model of juridical proof are well known and were integral to the last wave of interest in formal modeling of the evidentiary process. This paper thus for the most part puts aside that more general issue, and focuses specifically on mathematical modeling of the value of particular items of evidence. The paper demonstrates that formal modeling has only limited value in explaining the value of legal evidence, much more limited than those constructing and discussing the models assume, and thus that the conclusions they draw about the value of evidence are unwarranted. This is done through a discussion of four recent examples that attempt to quantify evidence relating to, respectively, carpet fibers, infidelity, DNA random-match evidence, and character evidence used to impeach a witness. This article thus makes two contributions. First, and most importantly, it is another demonstration of the complex relationship between algorithmic tools and legal decision making. Second, at a minimum it points out serious pitfalls for analytical or empirical studies of juridical proof. (shrink)

Mathematical statements arising from program verification are believed to be much easier to deal with than statements coming from serious mathematics. At least this is true for “normal programming”.

This eighth book of the author on gambling math presents in accessible terms the cold mathematics behind the sparkling slot machines, either physical or virtual. It contains all the mathematical facts grounding the configuration, functionality, outcome, and profits of the slot games. Therefore, it is not a so-called how-to-win book, but a complete, rigorous mathematical guide for the slot player and also for game producers, being unique in this respect. As it is primarily addressed to the slot player, its (...) goal is to present practical applications of the mathematical models of slot games, in order to provide numerical results that a player can use as criteria for gaming decisions or just as information for any slot game and any predicted winning event. These results are focused on probability and expected value, these being the most important parameters for decisional criteria in slots. The book is packed with plenty of figures, tables, and formulas. The content is organized so that readers can skip the theoretical parts and go picking the practical results (numerical, in tables of values where possible, or ready-to-compute formulas) for the desired situation. The practical results are gathered in the last chapter, titled "Practical Applications and Numerical Results," the largest part of the book, for the most popular categories of slot machines, namely with 3, 5, 9, and 16 reels. Any other category of slot games is covered in the theoretical part of the book, where the general formulas apply not only to existing slot games, but also to possible future slot games of any design and configuration. The author does not just throw the slot mathematics to the audience and run away, but offers an ultimate practical contribution with the chapter "How to estimate the number of stops and the symbol distribution on a reel", a surprise for both players and producers, where one can see that mathematics provides players with some statistical methods as well as methods based on physical measurements for retrieving these missing data. Having these data along with the mathematical results of this book, anyone can generate the PAR sheet of any slot machine. In the last decade, mathematics has been taken more and more seriously into account in gaming, as being the essence that governs the games of chance and the only rigorous tool providing information on optimal play, where possible. For the popular game of slots, mathematics already fulfilled its duty by providing all the data that it can provide and that cannot be found on the display of the slot machines - it is all here in this book. (shrink)

Geometrical and physical intuition, both untutored andcultivated, is ubiquitous in the research, teaching,and development of mathematics. A number ofmathematical ``monsters'', or pathological objects, havebeen produced which – according to somemathematicians – seriously challenge the reliability ofintuition. We examine several famous geometrical,topological and set-theoretical examples of suchmonsters in order to see to what extent, if at all,intuition is undermined in its everyday roles.

The existence of structures with non-trivial authomorphisms (such as the automorphism of the field of complex numbers onto itself that swaps the two roots of – 1) has been held by Burgess and others to pose a serious difficulty for mathematical structuralism. This paper proposes a model-theoretic solution to the problem. It suggests that mathematical structuralists identify the “position” of an n-tuple in a mathematical structure with the type of that n-tuple in the expansion of the structure that has (...) a name for every element in it. (shrink)

The main topic of this essay is symbolic mathematics or the method of symbolic construction, which I trace to the end of the sixteenth century when Franciscus Vieta invented the algebraic symbolism and started to use the word ‘symbolic’ in the relevant, non-ontological sense. This approach has played an important role for many of the great inventions in modern mathematics such as the introduction of the decimal place-value system of numeration, Descartes’ analytic geometry, and Leibniz’s infinitesimal calculus. It (...) was also central for the rigorization movement in mathematics in the late nineteenth century, as well as for the mathematics of modern physics in the 20th century.However, the nature of symbolic mathematics has been concealed and confused due to the strong influence of the heritage from the Euclidean and Aristotelian traditions. This essay sheds some light on what has been concealed by approaching some of the crucial issues from a historical perspective. Furthermore, I argue that the conception of modern mathematics as symbolic mathematics was essential to Wittgenstein’s approach to the foundations and nature of mathematics. This connection between Wittgenstein’s thought and symbolic mathematics provides the resources for countering the still prevalent view that he defended an uttrely idiosyncratic conception, disconnected from the progress of serious science. Instead, his project can be seen as clarifying ideas that have been crucial to the development of mathematics since early modernity. (shrink)

In the ten years following the publication of Galileo Galilei's Discorsi e dimostrazioni matematiche intorno a due nuove scienze , the new science of motion was intensely debated in Italy, France and northern Europe. Although Galileo's theories were interpreted and reworked in a variety of ways, it is possible to identify some crucial issues on which the attention of natural philosophers converged, namely the possibility of complementing Galileo's theory of natural acceleration with a physical explanation of gravity; the legitimacy of (...) Galileo's methods of proof and of his theory of the composition of continuous magnitudes; and the adequacy of the experimental evidence in favor of his theory.Through his published works and his correspondence, Marin Mersenne contributed in a significant way to each of these discussions. In the following, I shall provide a sketch of Mersenne's multiple engagement with the new science of motion, while trying to account for his growing skepticism concerning Galileo's law of free fall. Whereas in the 1630s, Mersenne still firmly believed in the validity of this law and tried to devise experimental and mathematical proofs in its favor, in the 1640s he developed serious doubts regarding the possibility of constructing an exact science of motion. As we shall see, Mersenne's evolving attitude sheds an interesting light on his views concerning the relation between physics and mathematics.--------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- -----------------------------------------. (shrink)

The take-over of the philosophy of mathematics by mathematical logic is not complete. The central problems examined in this book lie in the fringe area between the two, and by their very nature will no doubt continue to fall partly within the philosophical re mainder. In seeking to treat these problems with a properly sober mixture of rhyme and reason, I have tried to keep philosophical jargon to a minimum and to avoid excessive mathematical compli cation. The reader with (...) a philosophical background should be familiar with the formal syntactico-semantical explications of proof and truth, especially if he wishes to linger on Chapter 1, after which it is easier philosophical sailing; while the mathematician need only know that to "explicate" a concept consists in clarifying a heretofore vague notion by proposing a clearer (sometimes formal) definition or formulation for it. More seriously, the interested mathematician will find occasional recourse to EDWARD'S Encyclopedia of Philos ophy (cf. bibliography) highly rewarding. Sections 2. 5 and 2. 7 are of interest mainly to philosophers. The bibliography only contains works referred to in the text. References are made by giving the author's surname followed by the year of publication, the latter enclosed in parentheses. When the author referred to is obvious from the context, the surname is dropped, and even the year of publication or "ibid. " may be dropped when the same publication is referred to exclusively over the course of several paragraphs. (shrink)

[First paragraph of article] Many modern readers may be inclined to agree with ROBERTRECORDE, when in his Whetstone of Witte (1557) he gives as his reason for taking (( a paire of paralleles, or Gemowe lines of one lengthe, thus: == )) to be his sign of equality - (( bicause noe. 2. thynges, can be moare equalle )). To some readers this choice of the equality symbol may seem so evidently a foreordination, that they cannot readily understand how any (...) other symbol could have been proposed as a serious rival. They experience a very perceptible shock when informed that RECORDE'S symbol had several competitors and that it nearly perished in its fierce struggle for existence. (shrink)

We observe that Putnam’s model-theoretic argument against determinacy of the concept of second-order quantification or that of the set is harmless to the nominalist. It serves as a good motivation for the nominalist philosophy of mathematics. But in the end it can lead to a serious challenge to the nominalist account of mathematical objectivity if some minimal assumptions about the relation between mathematical objectivity and logical objectivity are made. We consider three strategies the nominalist might take to meet (...) this challenge, and we argue that all these strategies are untenable. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:American Journal of Philology 125.1 (2004) 140-144 [Access article in PDF] C. J. Tuplin and T. E. Rihll, eds. Science and Mathematics in Ancient Greek Culture. Foreword by Lewis Wolpert. Oxford: Oxford University Press, 2002. xvi + 379 pp. 21 black-and white ills. 3 tables. Cloth, $80. It has become something of a truism to say that, whatever their ambitions for abstraction, scientists remain profoundly caught up in (...) the life of their culture. Yet, that is a truism only at a certain level. No one is surprised to hear that Euclid had to eat and drink just like everyone else and relied on royal patronage for the leisure to pursue his studies, but it may occasion surprise to learn that the shape of his number theory should owe something to the particular way in which his eye fell upon a multiplication table. That Galen was influenced by his teachers is not news; but that his commitment to a particular school of philosophy could cause him to see and not see certain things when he was conducting dissections—here is something of interest. The papers in this volume, which were first delivered at a conference in Liverpool in 1996, have it as their common task to show how ancient forms of life influenced the pursuit and acquisition of knowledge on the part of scientists like Euclid and Galen.Heading the collection is a study by Andrew Barker of some idiosyncratic features of Greek harmonic theory. Archytas, Aristotle, and Ptolemy each attempted to characterize the physical basis of the two opposite sound qualities, oxusand barus. As Aristotle himself observed, the terms are metaphors drawn from the tactile realm, and trouble arose because in that realm they are heterogeneous opposites, i.e., the proper opposites of "sharp" and "heavy" are "dull" and "light." Barker uncovers a corresponding uncertainty in the texts about whether these sound qualities should be analyzed as something more like weight or something more like sharpness. Thus, an excessive trust in the descriptive adequacy of ordinary language terms led these thinkers to frustration.Harry Hine's paper on earthquakes and vulcanology deals with ancient knowledge of two phenomena that elicited the attention of scientist and non-scientist alike. Like Barker, Hine draws our attention to a terminological deficiency: although there was a fairly elaborate vocabulary for seismoi, neither Greek nor Latin possessed a generic noun that could properly be translated as [End Page 140] "volcano." This deficiency seems to be correlated with the fact that earthquakes form a regular part of the meteorological curriculum, while volcanoes usually received independent treatment in works such as the Aetna. Particularly intriguing is Hine's suggestion that the opinion of locals, especially sailors, may have been the source of certain scientific theses about volcanoes, e.g., that Aetna's behavior was closely correlated with the weather.The difference between ancient astronomy as practiced and as it was popularly believed to operate is the subject of Alan Bowen's paper. His central question is, when did astronomers first set out to make predictions of eclipses? The surprising answer is that, despite popular beliefs to the contrary, such prediction was never practiced by astronomers in a serious way before Ptolemy. Along with scrutiny of the misunderstanding in Herodotus, Diodorus, and Pliny, the paper offers thorough analysis of the most well-documented "prediction," that of a lunar eclipse that took place the night before the battle of Pydna in 167 B.C.E., which was supposedly foreseen by the Roman (!) general C. Sulpicius Gallus. Bowen casts doubts upon the entire tradition, showing how adherence to an ideology of the well-rounded commander led the sources to inflate his achievement: while some reports have Gallus simply assuaging his troops' fears at the event, others have him explaining its causes or even foretelling it. Bowen's skepticism is well founded, and one must regard these supposed feats of eclipse-prediction as a kind of science fiction, prescient yet false all the same.R. Hanna's subject is the format and function of the parapegma calendar first devised by... (shrink)

A lot of philosophical energy has been devoted recently in trying to determine if mathematics can contribute to our understanding of physical phenomena. Not many philosophers are interested, though, if the converse makes sense, i.e., if our cognitive interaction (scientific or otherwise) with the physical world can be helpful (in an explanatory or non-explanatory way) in our efforts to make sense of mathematical facts. My aim in this paper is to try to fill this important lacuna in the recent (...) literature. My answer to the question of this paper is negative. As I will argue, there are serious problems with the main reasons for believing in the first place that it is possible to have physical understanding of mathematical facts. (shrink)

There has been considerable discussion in the literature of one kind of identity problem that mathematical structuralism faces: the automorphism problem, in which the structure is unable to individuate the mathematical entities in its domain. Shapiro (Philos Math 16(3):285–309, 2008) has partly responded to these concerns. But I argue here that the theory faces an even more serious kind of identity problem, which the theory can’t overcome staying within its remit. I give two examples to make the point.

How mathematicians conceive of the nature of mathematics is reflected in the metaphors they use to talk about it. In this paper I investigate a change in the use of metaphors in the late nineteenth and early twentieth centuries. In particular, I argue that the metaphor of mathematics as a tree was used systematically by Pasch and some of his contemporaries, while that of mathematics as a building was deliberately chosen by Hilbert to reflect a different view (...) of mathematics. By taking these metaphors seriously we attain a new vantage point for understanding historical changes in conceptions of mathematics. (shrink)

The paper is devoted to the analysis and identification of new philosophical aspects of the problem of justification of modern mathematics according to which to the end of the 20th century the most exact of sciences had experienced new shocks associated with the crisis of excessive complexity of the mathematical theories. In the context of justification of mathematics philosophical conclusion consists in the fact that from a methodological point of view for general assessment of whether mathematics is (...) developed or not just the lack of crises is more negative phenomenon than its too long trouble-free state. Accomplished fact is that a computer can be regarded as one of the serious revolutionary technological inventions of the last century, which has had a huge influence on mathematics. In this development of progress of contemporary mathematics, more precisely, in its instrumental technologies the new philosophical problems are revealed - this is the role of computers and the new standards of mathematical thinking. In addition, with the development of contemporary mathematics and the emergence of increasingly complicated and long proofs, they lose their methodological advantage - the property of visibility and convincingness. Within the philosophy of mathematics this problem has not been discussed yet, possibly because the leading mathematicians haven’t expressed their opinions on the subject. After that, we can wait for a methodologically important period clarifying details, which can make the process itself foundations of mathematics philosophical consummation as the crisis complexity; in mathematics it is not a ‘systemic crisis‘ carrying the potential danger of the destruction of all mathematical knowledge. (shrink)

Kitcher's philosophical approach has moved from the reflection on the nature of mathematical knowledge to an explicit social concern about science, because he considers seriously the relevance of democratic values to scientific activity. Focal issues in this trajectory - from the internal perspective to the external - have been naturalism and scientific progress, which includes studies of the uses of scientific findings in the social milieu. Within this intellectual context, the chapter pays particular attention to his epistemological and methodological evolution. (...) The analysis of Philip Kitcher's contents on progress begins with mathematics, a conception that follows a naturalist perspective. Thereafter, the growth of science comes to the front line, an advancement that he views according to realism and cognitive naturalism. Later, the social concern about science receives a visible consideration, when his vision of scientific undertaking is characterized following modest realism and social naturalism. After these four steps (philosophical context, progress in mathematics, the growth in science, and the social concern about science), there is an analysis of his philosophical-methodological framework in retrospective. This is continued by the presentation of the origins of this book and the bibliography related to this thinker. (shrink)

The paper aims to capture a form of naturalism that can be found “built-in” in phenomenology, namely the idea to take science or mathematics on its own, without postulating extraneous normative “molds” on it. The paper offers a detailed comparison of Penelope Maddy’s naturalism about mathematics and Husserl’s approach to mathematics in Formal and Transcendental Logic. It argues that Maddy’s naturalized methodology is similar to the approach in the first part of the book. However, in the second (...) part Husserl enters into a transcendental clarification of the evidences and presuppositions of the mathematicians’ work, thus “transcendentalizing” his otherwise naturalist approach to mathematics. The result is a moderately revisionist view that takes the existing mathematical practices seriously, calls for reflection on them, and eventually gives suggestions for revisions if needed. (shrink)

This paper examines contemporary attempts to explicate the explanatory role of mathematics in the physical sciences. Most such approaches involve developing so-called mapping accounts of the relationships between the physical world and mathematical structures. The paper argues that the use of idealizations in physical theorizing poses serious difficulties for such mapping accounts. A new approach to the applicability of mathematics is proposed.

The late Imre Lakatos once hoped to found a school of dialectical philosophy of mathematics. The aim of this paper is to ask what that might possibly mean. But Lakatos's philosophy has serious shortcomings. The paper elaborates a conception of dialectical philosophy of mathematics that repairs these defects and considers the work of three philosophers who in some measure fit the description: Yehuda Rav, Mary Leng and David Corfield.

his paper explores the relationship between Kant’s views on the metaphysical foundations of Newtonian mathematical physics and his more general transcendental philosophy articulated in the Critique of pure reason. I argue that the relationship between the two positions is very close indeed and, in particular, that taking this relationship seriously can shed new light on the structure of the transcendental deduction of the categories as expounded in the second edition of the Critique.Author Keywords: Kant; Mathematical physics; Transcendental deduction.

The explanatory filter is a proposed method to detect design in nature with the aim of refuting Darwinian evolution. The explanatory filter borrows its logical structure from the theory of statistical hypothesis testing but we argue that, when viewed within this context, the filter runs into serious trouble in any interesting biological application. Although the explanatory filter has been extensively criticized from many angles, we present the first rigorous criticism based on the theory of mathematical statistics.

It is reported that in reply to John Wisdom’s request in 1944 to provide a dictionary entry describing his philosophy, Wittgenstein wrote only one sentence: “He has concerned himself principally with questions about the foundations of mathematics”. However, an understanding of his philosophy of mathematics has long been a desideratum. This was the case, in particular, for the period stretching from the Tractatus Logico-Philosophicus to the so-called transitional phase. Marion’s book represents a giant leap forward in this direction. (...) In the preface, Marion provides a diagnosis for why it has taken such a long time to obtain an accurate picture of Wittgenstein’s ideas in the foundations of mathematics. When the Remarks on the Foundations of Mathematics came out in 1956 the reception by specialists, such as Kreisel, was negative. This led many Wittgenstein scholars to set aside the issues in the foundations of mathematics and go on with the business of the day, focusing on Wittgenstein’s philosophy of language and psychology. However, these early negative appraisals were severely limited by two facts. First of all, the nature of the Remarks was a hindrance to an understanding of what Wittgenstein was up to. The book was edited by Wittgenstein’s literary executors by collating together passages from several manuscripts dating from 1937 to 1944, cutting, however, many passages in between remarks. Second, many of the original manuscripts and typescripts of the transitional phase were not available and thus it was virtually impossible to obtain a balanced picture of Wittgenstein’s development. Finally, a stark contraposition between the early and late Wittgenstein did not encourage people to work seriously on the development of Wittgenstein’s position from the Tractatus through the transitional period to the later writings. (shrink)

In Metaphysics B.2 and M.2, Aristotle gives a series of arguments against Platonic mathematical objects. On the view he targets, mathematicals are substances somehow intermediate between Platonic forms and sensible substances. I consider two closely related passages in B2 and M.2 in which he argues that Platonists will need intermediates not only for geometry and arithmetic, but also for the so-called mixed mathematical sciences, and ultimately for all sciences of sensibles. While this has been dismissed as mere polemics, I show (...) that the argument is given in earnest, as Aristotle is committed to its key premises. Further, the argument reveals that Annas’ uniqueness problem is not the only reason a Platonic ontology needs intermediates. Finally, since Aristotle’s objection to intermediates for the mixed mathematical sciences is one he takes seriously, so that it is unlikely that his own account of mathematical objects would fall prey to it, the argument casts doubt on a common interpretation of his philosophy of mathematics. (shrink)

To explore the impact of academic burden on the physical and mental health of primary school students, combined with the results of the Programme for International Student Assessment report in 2018, the relationship among the development of mathematical literacy, mathematics academic burden, and the physical and mental health of primary school students is studied. First, the relationship between mathematical literacy and mathematics anxiety is analyzed, and related influencing factors and measurement methods of mathematics anxiety are introduced. A (...) questionnaire is then designed for primary school students’ mathematical stress, and the reliability and validity of the designed questionnaire are tested. Finally, a questionnaire survey is conducted on students, parents, and teachers in the third, fourth, and fifth grades of three standardized public primary schools. The results of the questionnaire survey show that students, teachers, and parents have a general understanding of the mathematics academic burden of primary school students at this stage. A total of 70% of teachers believe that primary school students have a heavy mathematics burden; 50% of parents think that primary school students are under heavy academic stress; 70% of primary school students believe that the heavy mathematics burden leads to reduced sleep time and extracurricular activities, which has a serious impact on the physical and mental health of primary school students. This research provides a reference for improving the current balance between education and students’ physical and mental health in China. (shrink)

Since the beginning of the twentieth century, prominent authors including Jean Piaget have drawn attention to Edmond Goblot’s account of mathematical thought experiments. But his contribution to today’s debate has been neglected so far. The main goal of this article is to reconstruct and discuss Goblot’s account of logical operations (the term he used for thought experiments in mathematics) and its interpretation by Piaget against the theoretical background of two open questions in today’s debate: (1) the relationship between empirical (...) and mathematical thought experiments and (2) the question of whether mathematical thought experiments can play a justificatory function in proofs. The main corollary of this analysis is that Piaget’s interpretation is seriously flawed and insufficiently appreciative of important theses of Goblot’s account. First, Goblot can be easily defended against Piaget’s main criticism, and second, Goblot developed ideas about mathematical thought experiments that still deserve attention. (shrink)

This long-awaited volume is a must-read for anyone with a serious interest in philosophy of mathematics. The book falls into two parts, with the primary focus of the first on ontology and structuralism, and the second on intuition and epistemology, though with many links between them. The style throughout involves unhurried examination from several points of view of each issue addressed, before reaching a guarded conclusion. A wealth of material is set before the reader along the way, but (...) a reviewer wishing to summarize the author's views crisply will be frustrated. The chapter-by-chapter survey below conveys at best a very incomplete and imperfect impression of the work's virtues, and even of its contents, falling short even of supplying a full menu for the banquet of food for thought that Parsons serves up to his readers. (shrink)

Linnebo and Pettigrew's critique in this journal of categorical foundations well emphasizes that the particulars of various categorical foundations matter, and that mathematical practice must be a major consideration. But several categorists named by the authors as proposing categorical foundations do not propose foundations, notably Awodey, and the article's description of current textbook practice seems inaccurate. They say that categorical foundations have justificatory autonomy if and only if mathematics can be justified simply by its practice. Do they seriously believe (...) philosophers may be able to justify some currently unpractised mathematics better than current practice justifies itself? (shrink)

It turns out, time and time again, in order to make serious progress in f.o.m., we need to take actual reasoning and actual development into account at precisely the proper level. If we take these into account too much, then we are faced with information that is just too difficult to create an exact science around - at least at a given state of development of f.o.m. And if we take these into account too little, our findings will not (...) have the relevance to mathematical practice that could be achieved. (shrink)