James Gow's A Short History of Greek Mathematics provided the first full account of the subject available in English, and it today remains a clear and thorough guide to early arithmetic and geometry. Beginning with the origins of the numerical system and proceeding through the theorems of Pythagoras, Euclid, Archimedes and many others, the Short History offers in-depth analysis and useful translations of individual texts as well as a broad historical overview of the development of mathematics. (...) Parts I and II concern Greek arithmetic, including the origin of alphabetic numerals and the nomenclature for operations; Part III constitutes a complete history of Greek geometry, from its earliest precursors in Egypt and Babylon through to the innovations of the Ionic, Sophistic, and Academic schools and their followers. Particular attention is given to Pythagorus, Euclid, Archimedes, and Ptolemy, but a host of lesser-known thinkers receive deserved attention as well. (shrink)

We propose that, for the purpose of studying theoretical properties of the knowledge of an agent with Artificial General Intelligence (that is, the knowledge of an AGI), a pragmatic way to define such an agent’s knowledge (restricted to the language of Epistemic Arithmetic, or EA) is as follows. We declare an AGI to know an EA-statement φ if and only if that AGI would include φ in the resulting enumeration if that AGI were commanded: “Enumerate all the EA-sentences which you (...) know.” This definition is non-circular because an AGI, being capable of practical English communication, is capable of understanding the everyday English word “know” independently of how any philosopher formally defines knowledge; we elaborate further on the non-circularity of this circular-looking definition. This elegantly solves the problem that different AGIs may have different internal knowledge definitions and yet we want to study knowledge of AGIs in general, without having to study different AGIs separately just because they have separate internal knowledge definitions. Finally, we suggest how this definition of AGI knowledge can be used as a bridge which could allow the AGI research community to import certain abstract results about mechanical knowing agents from mathematical logic. (shrink)

Empiricism in philosophy is either a method or a theory. The two are separable: one might hold that all knowledge is empirical but that philosophy does something other than add to our knowledge, e.g., that it clarifies concepts; or one might hold that philosophy’s method is empirical and that one of the things known in that way is that not all knowledge is empirical, e.g., mathematics. And what is the empirical? If it is knowledge based on observation, then what (...) is it that can be observed? If philosophy is in method empirical, the range of the observable must be broad, perhaps including mental processes, human history, social institutions such as language, and even the difference between good and evil.In Peirce’s... (shrink)

Theory of Mind (ToM) is the term used for our ability to predict and explain the behaviour of ourselves and others. Accounts of this theory have so far fallen into two competing types: Simulation Theory and 'Theory Theory'. In contrast with Theory Theory, Simulation Theory argues that we predict behaviour not by employing a model of people, but by replicating others' thoughts and feelings. This book presents a novel defence of Simulation Theory, reviewing the major challenges against it and positing (...) the theory as the most effective method for exploring how we know each other and ourselves. Drawing on key research in the field, chapters reopen the debates surrounding Theory of Mind and cover a variety of topics including schizophrenia with implications for experimental social psychology. In the past, one of the greatest criticisms against Simulation Theory is that it cannot explain systematic error in Theory of Mind. This book explores the rapidly developing heuristics and biases programme, pioneered by Kahneman and Tversky, to suggest that a novel bias mismatch defence available to Simulation Theory explains these systematic errors. Simulation Theory: A psychological and philosophical consideration will appeal to a range of researchers and academics, including psychologists from the fields of cognitive, social and developmental psychology, as well as philosophers, psychotherapists and practitioners looking for further research on Theory of Mind. The book will also be of relevance to those interested in autism, since it offers a new approach to Theory of Mind which explains central symptoms in autistic subjects. (shrink)

Between December 1855 and March 1856, a public dispute raged, in British national newspapers and locally published pamphlets, between two teachers at the University of Oxford: the mathematical lecturer Francis Ashpitel and Bartholomew Price, the professor of natural philosophy. The starting point for these exchanges was the particularly poor results that had come out of the final mathematics examinations in Oxford that December. Ashpitel, as one of the examiners, stood accused of setting questions that were too difficult for the (...) ordinary student, with the consequence that, in Price’s view, further mathematical study in Oxford – never as robust as in Cambridge – would be discouraged. We examine this short-lived affair, and use it not only to gain insight into the status of mathematical study in Oxford in the mid-nineteenth century, but also to point towards the increasing importance of competitive examinations in British public life at that time. (shrink)

Mathematics is one of the most successful human endeavors—a paradigm of precision and objectivity. It is also one of our most puzzling endeavors, as it seems to deliver non-experiential knowledge of a non-physical reality consisting of numbers, sets, and functions. How can the success and objectivity of mathematics be reconciled with its puzzling features, which seem to set it apart from all the usual empirical sciences? This book offers a short but systematic introduction to the philosophy of (...) mathematics. Readers are introduced to all of the classical approaches to the field, including logicism, formalism, intuitionism, empiricism, and structuralism. The book also contains accessible introductions to some more specialized issues, such as mathematical intuition, potential infinity, the iterative conception of sets, and the search for new mathematical axioms. The exposition is always closely informed by ongoing research in the field and sometimes draws on the author’s own contributions to this research. This means that Gottlob Frege—a German mathematician and philosopher widely recognized as one of the founders of analytic philosophy—figures prominently in the book, both through his own views and his criticism of other thinkers. (shrink)

Originally published in 1949. This meticulously researched book presents a comprehensive outline and discussion of Aristotle’s mathematics with the author's translations of the greek. To Aristotle, mathematics was one of the three theoretical sciences, the others being theology and the philosophy of nature. Arranged thematically, this book considers his thinking in relation to the other sciences and looks into such specifics as squaring of the circle, syllogism, parallels, incommensurability of the diagonal, angles, universal proof, gnomons, infinity, agelessness of (...) the universe, surface of water, meteorology, metaphysics and mechanics such as levers, rudders, wedges, wheels and inertia. The last few short chapters address ‘problems’ that Aristotle posed but couldn’t answer, related ethics issues and a summary of some short treatises that only briefly touch on mathematics. (shrink)

In a very influential paper Rota stresses the relevance of mathematical beauty to mathematical research, and claims that a piece of mathematics is beautiful when it is enlightening. He stops short, however, of explaining what he means by ‘enlightening’. This paper proposes an alternative approach, according to which a mathematical demonstration or theorem is beautiful when it provides understanding. Mathematical beauty thus considered can have a role in mathematical discovery because it can guide the mathematician in selecting which (...) hypothesis to consider and which to disregard. Thus aesthetic factors can have an epistemic role qua aesthetic factors in mathematical research. (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)

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as the Riemann hypothesis, have had to be considered in terms of the evidence for and against them. In recent decades, massive increases in computer power have permitted the gathering of huge amounts of numerical evidence, both for conjectures in pure (...) mathematics and for the behavior of complex applied mathematical models and statistical algorithms. Mathematics has therefore become (among other things) an experimental science (though that has not diminished the importance of proof in the traditional style). We examine how the evaluation of evidence for conjectures works in mathematical practice. We explain the (objective) Bayesian view of probability, which gives a theoretical framework for unifying evidence evaluation in science and law as well as in mathematics. Numerical evidence in mathematics is related to the problem of induction; the occurrence of straightforward inductive reasoning in the purely logical material of pure mathematics casts light on the nature of induction as well as of mathematical reasoning. (shrink)

One of the chief concerns of the young Descartes was with what he, and others, termed “physico-mathematics”. This signalled a questioning of the Scholastic Aristotelian view of the mixed mathematical sciences as subordinate to natural philosophy, non explanatory, and merely instrumental. Somehow, the mixed mathematical disciplines were now to become intimately related to natural philosophical issues of matter and cause. That is, they were to become more ’physicalised’, more closely intertwined with natural philosophising, regardless of which species of natural (...) philosophy one advocated. A curious, short-lived yet portentous epistemological conceit lay at the core of Descartes’ physico-mathematics—the belief that solid geometrical results in the mixed mathematical sciences literally offered windows into the realm of natural philosophical causation—that in such cases one could literally “see the causes”. Optics took pride of place within Descartes’ physico-mathematics project, because he believed it offered unique possibilities for the successful vision of causes. This paper traces Descartes’ early physico-mathematical program in optics, its origins, pitfalls and its successes, which were crucial in providing Descartes resources for his later work in systematic natural philosophy. It explores how Descartes exploited his discovery of the law of refraction of light—an achievement well within the bounds of traditional mixed mathematical optics—in order to derive—in the manner of physico-mathematics—causal knowledge about light, and indeed insight about the principles of a “dynamics” that would provide the laws of corpuscular motion and tendency to motion in his natural philosophical system. (shrink)

The aim of this paper is to provide epistemic reasons for investigating the notions of informal rigour and informal provability. I argue that the standard view of mathematical proof and rigour yields an implausible account of mathematical knowledge, and falls short of explaining the success of mathematical practice. I conclude that careful consideration of mathematical practice urges us to pursue a theory of informal provability.

An increasing number of learning goals refer to the acquisition of cognitive skills that can be described as ‘resource-based,’ as they require the availability, coordination, and integration of multiple underlying resources such as skills and knowledge facets. However, research on the support of cognitive skills rarely takes this resource-based nature explicitly into account. This is mirrored in prior research on mathematical argumentation and proof skills: Although repeatedly highlighted as resource-based, for example relying on mathematical topic knowledge, methodological knowledge, mathematical strategic (...) knowledge, and problem-solving skills, little evidence exists on how to support mathematical argumentation and proof skills based on its resources. To address this gap, a quasi-experimental intervention study with undergraduate mathematics students examined the effectiveness of different approaches to support both mathematical argumentation and proof skills and four of its resources. Based on the part-/whole-task debate from instructional design, two approaches were implemented during students’ work on proof construction tasks: (i) asequential approachfocusing and supporting each resource of mathematical argumentation and proof skills sequentially after each other and (ii) aconcurrent approachfocusing and supporting multiple resources concurrently. Empirical analyses show pronounced effects of both approaches regarding the resources underlying mathematical argumentation and proof skills. However, the effects of both approaches are mostly comparable, and only mathematical strategic knowledge benefits significantly more from the concurrent approach. Regarding mathematical argumentation and proof skills, short-term effects of both approaches are at best mixed and show differing effects based on prior attainment, possibly indicating an expertise reversal effect of the relatively short intervention. Data suggests that students with low prior attainment benefited most from the intervention, specifically from the concurrent approach. A supplementary qualitative analysis showcases how supporting multiple resources concurrently alongside mathematical argumentation and proof skills can lead to a synergistic integration of these during proof construction and can be beneficial yet demanding for students. Although results require further empirical underpinning, both approaches appear promising to support the resources underlying mathematical argumentation and proof skills and likely also show positive long-term effects on mathematical argumentation and proof skills, especially for initially weaker students. (shrink)

In Plato’s eponymous dialogue, Timaeus, the main character presents the universe as an (almost) perfect sphere filled by tiny, invisible particles having the form of four regular polyhedrons. At first glance, such a construction may seem close to an atomistic theory. However, one does not find any text in Antiquity that links Timaeus’ cosmology to the atomists, while Aristotle opposes clearly Plato to the latter. Nevertheless, Plato is commonly presented in contemporary literature as some sort of atomist, sometimes as supporting (...) a form of so-called ‘mathematical atomism’. However, the term ‘atomism’ is rarely defined when applied to Plato. Since it covers many different theories, it seems that this term has almost as different meanings as different authors. The purpose of this article is to consider whether it is correct to connect Timaeus’ cosmology to some kind of ‘atomism’, however this term may be understood. Its purpose is double: to obtain a better understanding of the cosmology of the Timaeus, and to consider the different modern ‘atomistic’ interpretations of this cosmology. In short, we would like to show that such a claim, in any form whatsoever, is misleading, an impediment to the understanding of the dialogue, and more generally of Plato’s philosophy. (shrink)

This essay is an attempt to take stock of what has been done by those who have worked on the foundations of mathematics and to suggest, very inadequately in so short a space, what may be a satisfactory approach to this subject for one who is not an expert in it. The subject is one that neither mathematicians nor philosophers can any longer afford to ignore, but in the technicalities of which they may not be very interested.

Over the years, mathematics and statistics have become increasingly important in the social sciences1 . A look at history quickly confirms this claim. At the beginning of the 20th century most theories in the social sciences were formulated in qualitative terms while quantitative methods did not play a substantial role in their formulation and establishment. Moreover, many practitioners considered mathematical methods to be inappropriate and simply unsuited to foster our understanding of the social domain. Notably, the famous Methodenstreit also (...) concerned the role of mathematics in the social sciences. Here, mathematics was considered to be the method of the natural sciences from which the social sciences had to be separated during the period of maturation of these disciplines. All this changed by the end of the century. By then, mathematical, and especially statistical, methods were standardly used, and their value in the social sciences became relatively uncontested. The use of mathematical and statistical methods is now ubiquitous: Almost all social sciences rely on statistical methods to analyze data and form hypotheses, and almost all of them use (to a greater or lesser extent) a range of mathematical methods to help us understand the social world. Additional indication for the increasing importance of mathematical and statistical methods in the social sciences is the formation of new subdisciplines, and the establishment of specialized journals and societies. Indeed, subdisciplines such as Mathematical Psychology and Mathematical Sociology emerged, and corresponding journals such as The Journal of Mathematical Psychology (since 1964), The Journal of Mathematical Sociology (since 1976), Mathematical Social Sciences (since 1980) as well as the online journals Journal of Artificial Societies and Social Simulation (since 1998) and Mathematical Anthropology and Cultural Theory (since 2000) were established. What is more, societies such as the Society for Mathematical Psychology (since 1976) and the Mathematical Sociology Section of the American Sociological Association (since 1996) were founded. Similar developments can be observed in other countries. The mathematization of economics set in somewhat earlier (Vazquez 1995; Weintraub 2002). However, the use of mathematical methods in economics started booming only in the second half of the last century (Debreu 1991). Contemporary economics is dominated by the mathematical approach, although a certain style of doing economics became more and more under attack in the last decade or so. Recent developments in behavioral economics and experimental economics can also be understood as a reaction against the dominance (and limitations) of an overly mathematical approach to economics. There are similar debates in other social sciences. It is, however, important to stress that problems of one method (such as axiomatization or the use of set theory) can hardly be taken as a sign of bankruptcy of mathematical methods in the social sciences tout court. This chapter surveys mathematical and statistical methods used in the social sciences and discusses some of the philosophical questions they raise. It is divided into two parts. Sections 1 and 2 are devoted to mathematical methods, and Sections 3 to 7 to statistical methods. As several other chapters in this handbook provide detailed accounts of various mathematical methods, our remarks about the latter will be rather short and general. Statistical methods, on the other hand, will be discussed in-depth. (shrink)

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as Fermat's Last Theorem and the Riemann Hypothesis, have had to be considered in terms of the evidence for and against them. It is argued here that it is not adequate to describe the relation of evidence to hypothesis as `subjective', (...) `heuristic' or `pragmatic', but that there must be an element of what it is rational to believe on the evidence, that is, of non-deductive logic. (shrink)

This article develops a criticism of Alain Badiou’s assertion that “mathematics is ontology.” I argue that despite appearances to the contrary, Badiou’s case for bringing set theory and ontology together is problematic. To arrive at this judgment, I explore how a case for the identification of mathematics and ontology could work. In short, ontology would have to be characterised to make it evident that set theory can contribute to it fundamentally. This is indeed how Badiou proceeds in (...) Being and Event. I review his descriptions of the ontological problematic at some length here, only to argue that set theory is a poor fit. Although philosophers working on questions of being were certainly occupied with matters of oneness and the part-whole relationship, I argue that Badiou’s discussion of philosophical sources points towards a mereological treatment, not a set-theoretic one. Finally, I suggest that Badiou’s philosophical interpretations of key set-theoretic results are better understood as some sort of analogising between mathematics, ontology, and philosophical anthropology. (shrink)

To be asked to provide a short paper on Wittgenstein's views on mathematical proof is to be given a tall order . Close to one half of Wittgenstein's writings after 1929 concerned mathematics, and the roots of his discussions, which contain a bewildering variety of underdeveloped and sometimes conflicting suggestions, go deep to some of the most basic and difficult ideas in his later philosophy. So my aims in what follows are forced to be modest. I shall sketch (...) an intuitively attractive philosophy of mathematics and illustrate Wittgenstein's opposition to it. I shall explain why, contrary to what is often supposed, that opposition cannot be fully satisfactorily explained by tracing it back to the discussions of following a rule in the Philosophical Investigations and Remarks on the Foundations of Mathematics . Finally, I shall try to indicate very briefly something of the real motivation for Wittgenstein's more strikingly deflationary suggestions about mathematical proof, and canvass a reason why it may not in the end be possible to uphold them. (shrink)

In the Reality we know, we cannot say if something is infinite whether we are doing Physics, Biology, Sociology or Economics. This means we have to be careful using this concept. Infinite structures do not exist in the physical world as far as we know. So what do mathematicians mean when they assert the existence of ω? There is no universally accepted philosophy of mathematics but the most common belief is that mathematics touches on another worldly absolute truth. (...) Many mathematicians believe that mathematics involves a special perception of an idealized world of absolute truth. This comes in part from the recognition that our knowledge of the physical world is imperfect and falls short of what we can apprehend with mathematical thinking. The objective of this paper is to present an epistemological rather than an historical vision of the mathematical concept of infinity that examines the dialectic between the actual and potential infinity. (shrink)

In a short article, published in an earlier volume of Philosophy 1 under the title “Philosophy and Mathematics,” I tried to explain the current conception of pure mathematics as the study of abstract structure by construction and elaboration of appropriate axiomatic formalisms. In the present paper I propose to consider certain philosophical problems, of interest to philosophers and mathematicians alike, which have their origin in the relation between such formalisms and any applications to experience that they may (...) possess. Consideration of problems of this kind is no new undertaking, and in Reichenbach's Wahrscheinlichkeitslehre , for instance, a considerable amount of space is devoted to the Anwendungsproblem or problem of application of the formal calculus under consideration. Most such discussions, however, are at bottom an appendage to an account of the formalism itself, and the author's interest is primarily mathematical. The result is that the philosophical issues involved are not given due weight; and it is these philosophical issues that I wish to discuss in the present paper. (shrink)

What role does mathematics play in cognitive science today, what role should mathematics play in cognitive science tomorrow? The cautious short answers are: to the factual question, a rather modest role, except in peripheral areas; to the normative question, a far greater role, as the periphery’s place is reevaluated and as both cognitive science and mathematics grow. This paper aims at providing more detailed, perhaps more contentious answers.

In chapter 3, we reflected on the view that the fallacies on the traditional list are inherently dialectical. The answer proposed there was that, with the possible exception of, e.g., begging the question and many questions, they are not. The aim of the present chapter is to cancel theispossibility by showing that begging the question and many questions are not in fact dialectical fallacies. The reason for this is not that question-begging and many questions aren’t (at least dominantly) dialectical practices. (...) The reason is that, dialectical or not, they are not fallacies. That begging the question, BQ for short, is a fallacy is an idea which originates with Aristotle. Given logic’s already long history, it should not be surprising that Aristotle’s views of these matters have in some ways been superseded. But the traditional view retains the original connection between conception and instantiation. Whereas BQ in Aristotle’s sense is said to be a fallacy in Aristotle’s sense, so too is BQ in the modern sense said to be a fallacy in the modern (i.e., EAUI) sense.[1] As currently conceived of, BQ and fallacies can be characterized in the following ways. (shrink)

This short article pairs the realms of “Mathematics”, “Philosophy”, and “Poetry”, presenting some corners of intersection of this type of scientocreativity. Poetry have long been following mathematical patterns expressed by stern formal restrictions, as the strong metrical structure of ancient Greek heroic epic, or the consistent meter with standardized rhyme scheme and a “volta” of Italian sonnets. Poetry was always connected to Philosophy, and further on, notable mathematicians, like the inventor of quaternions, William Rowan Hamilton, or Ion Barbu, (...) the creator of the Barbilian spaces, have written appreciated poems. We will focus here on an avant-garde movement in literature, art, philosophy, and science, called Paradoxism, founded in Romania in 1980 by a mathematician, philosopher and poet, and on the laboured writing exercises of the Oulipo group, founded in Paris in 1960 by mathematicians and poets, both of them still in act. (shrink)

We investigated the effect of distributed practice and more specifically the “lag effect” concerning the retention of mathematical procedures. The lag effect implies that longer retention intervals benefit from longer inter-study intervals (ISIs). University students ( N = 235) first learned how to solve permutation tasks and then practiced this procedure with an ISI of zero (i.e., massed), one, or 11 days. The final test took place after one or five weeks. All conditions were manipulated between-subjects. Contrary to our expectations, (...) the analyses revealed no effect of distributed practice and therewith also no lag effect, even though the sample size was sufficiently large. The only significant effect was that test performance was poorer after 5 weeks than after 1 week. In view of the present results and those of other studies, we assume that distributed practice works differently for declarative and procedural knowledge, with less robust of even absent effects when procedural skills are practiced with ISIs compared to massed practice. (shrink)

The work of which this is an English translation appeared originally in French as Precis de logique mathematique. In 1954 Dr. Albert Menne brought out a revised and somewhat enlarged edition in German. In making my translation I have used both editions. For the most part I have followed the original French edition, since I thought there was some advantage in keeping the work as short as possible. However, I have included the more extensive historical notes of Dr. Menne, (...) his bibliography, and the two sections on modal logic and the syntactical categories, which were not in the original. I have endeavored to correct the typo graphical errors that appeared in the original editions and have made a few additions to the bibliography. In making the translation I have profited more than words can tell from the ever-generous help of Fr. Bochenski while he was teaching at the University of Notre Dame during 1955-56. OTTO BIRD Notre Dame, 1959 I GENERAL PRINCIPLES § O. INTRODUCTION 0. 1. Notion and history. Mathematical logic, also called 'logistic', ·symbolic logic', the 'algebra of logic', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the last century with the aid of an artificial notation and a rigorously deductive method. (shrink)

The current WSN is vulnerable to a variety of malicious attacks, resulting in the decline of its comprehensive performance. Multihop routing involves a number of safety and privacy issues. Problems such as snooping, sinkhole, manipulation, cloning, wormhole, spoofing, and so on affect the integrity, availability, and confidentiality of the WSNs. Therefore, this paper mainly studies the mathematical modeling of WSN multiattack behavior discrimination based on the incidence matrix. The WSN node model is used to collect relevant data and mark and (...) map the disguised data, so as to determine the characteristics of multiattack behavior and establish the WSN multiattack behavior discrimination model based on the incidence matrix. The experimental results show that the designed multiattack behavior discrimination model can distinguish the attack type according to the characteristics, has high recognition ability, can spend a short time to effectively distinguish the attack behavior, has a low false positive rate, and can effectively improve the antiattack ability of the WSN. (shrink)

This is a short, modern, and motivated introduction to mathematical logic for upper undergraduate and beginning graduate students in mathematics and computer science. Any mathematician who is interested in getting acquainted with logic and would like to learn Gödel’s incompleteness theorems should find this book particularly useful. The treatment is thoroughly mathematical and prepares students to branch out in several areas of mathematics related to foundations and computability, such as logic, axiomatic set theory, model theory, recursion theory, (...) and computability. In this new edition, many small and large changes have been made throughout the text. The main purpose of this new edition is to provide a healthy first introduction to model theory, which is a very important branch of logic. Topics in the new chapter include ultraproduct of models, elimination of quantifiers, types, applications of types to model theory, and applications to algebra, number theory and geometry. Some proofs, such as the proof of the very important completeness theorem, have been completely rewritten in a more clear and concise manner. The new edition also introduces new topics, such as the notion of elementary class of structures, elementary diagrams, partial elementary maps, homogeneous structures, definability, and many more. (shrink)

This article deals with physico-mathematical approaches to anatomy in post-Harveyan physiology. But rather than looking at questions of iatromechanics and animal locomotion, which often attracted this approach, I look at the problem of how blood returned to the heart – a part of the circulation today known as venous return but poorly researched in the early modern period. I follow the venous return mechanisms proposed by lesser-known authors in the mechanization of anatomy, such as Jean Pecquet (1622–1674) and Nicolaus Steno (...) (1638–1686), alongside the more famous Giovanni Alfonso Borelli (1608–1679). Their mechanisms differed only in small details. Yet, these minor differences highlight significant aspects of the mechanization of the life sciences in the seventeenth century. First, they relied more on observations than hitherto acknowledged, even if only indirectly. Second, their mechanisms drew more from the physico-mathematical disciplines than from the trending corpuscularian philosophies of their time. Finally, these mechanisms led to a more accurate understanding of the circulation that remains valid today, thus revealing their cognitive benefits. In short, through the single problem of how blood returned to the heart, this article portrays the increasing complexity of anatomy in the early modern period. (shrink)

My discussion will concern itself with mathematics, medicine and the possible relations between the two. It will be an exercise in logical analysis, a review of some sad, sad facts, and in some sense a promise of glad tidings. In short, it will be an effort to bring the immortal inhabitants of the mathematical heaven into harmonious relations with the mortal ills of man's vale of tears.As to the curious role of mathematics with respect to the natural (...) sciences, several preliminary observations insist on prompt consideration. It is a matter of common knowledge, for example, that mathematics is being constantly applied to physics. With great success, Galileo wedded the two in his study of motion down an inclined plane. The mathematics he used to describe the physical phenomenon involved the equation of a parabola. His daring genius thus got far beyond the conventional circles that were permitted to motion by theology. His courageous use of the then “higher mathematics” was followed by centuries of application of still higher and higher mathematics, till now a small army of mathematicians is busy almost exclusively with the problem of inventing or discovering new varieties of mathematics which could serve physics better and better. The mathematical physicist of to-day is a specialist, nine-tenths mathematician, one-tenth physicist, with the nine-tenths serving the one-tenth. (shrink)

This short paper is meant to be an introduction to the ‘Letter to Alan Turing’ that follows it. It summarizes some basic ideas in information theory and very informally hints at their mathematical properties. In order to introduce Turing’s two main theoretical contributions, in Theory of Computation and in Morphogenesis (an analysis of the dynamics of forms), the fundamental divide between discrete vs. continuous structures in mathematics is presented, as it is also a divide in his scientific life. (...) The reader who is familiar with these notions, and is convinced that they (and their differences) are relevant in the mathematical understanding of phenomena, may skip this introduction and go directly to the Letter. (shrink)

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or (...) objective Bayesianism or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (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)

It is undeniable Poincaré was a very famous and influential scientist. So, possibly because of it, it was relatively easy for him to participate in the heated discussions of the foundations of mathematics in the early 20th century. We can say it was “easy” because he didn't get involved in this subject by writing great treatises, or entire books about his own philosophy of mathematics (as other authors from the same period did). Poincaré contributed to the philosophy of (...) mathematics by writing short essays and letters. (shrink)

The uncountability of $\mathbb {R}$ is one of its most basic properties, known far outside of mathematics. Cantor’s 1874 proof of the uncountability of $\mathbb {R}$ even appears in the very first paper on set theory, i.e., a historical milestone. In this paper, we study the uncountability of ${\mathbb R}$ in Kohlenbach’s higher-order Reverse Mathematics (RM for short), in the guise of the following principle: $$\begin{align*}\mathit{for \ a \ countable \ set } \ A\subset \mathbb{R}, \mathit{\ there (...) \ exists } \ y\in \mathbb{R}\setminus A. \end{align*}$$ An important conceptual observation is that the usual definition of countable set—based on injections or bijections to ${\mathbb N}$ —does not seem suitable for the RM-study of mainstream mathematics; we also propose a suitable (equivalent over strong systems) alternative definition of countable set, namely union over ${\mathbb N}$ of finite sets; the latter is known from the literature and closer to how countable sets occur ‘in the wild’. We identify a considerable number of theorems that are equivalent to the centred theorem based on our alternative definition. Perhaps surprisingly, our equivalent theorems involve most basic properties of the Riemann integral, regulated or bounded variation functions, Blumberg’s theorem, and Volterra’s early work circa 1881. Our equivalences are also robust, promoting the uncountability of ${\mathbb R}$ to the status of ‘big’ system in RM. (shrink)

A mathematical approach to heuristics is proposed, in contrast to Gigerenzer et al.'s assertion that laws of logic and probability are of little importance. Examples are given of effective heuristics in abstract settings. Other short-comings of the text are discussed, including omissions in psychophysics and cognitive science. However, the authors' ecological view is endorsed.

200 years ago, on August 5, 1802, Niels Henrik Abel was born on Finnøy near Stavanger on the Norwegian west coast. During a short life span, Abel contributed to a deep transition in mathematics in which concepts replaced formulae as the basic objects of mathematics. The transformation of mathematics in the 1820s and its manifestation in Abel’s works are the themes of the author’s PhD thesis. After sketching the formative instances in Abel’s well-known biography, this article (...) illustrates two aspects of the transformation which concern the introduction of concept based mathematics and the related shift in standards of mathematical rigor. Furthermore, the article outlines some of the many bicentennial celebrations in Norway and gives a short, thematic introduction to the literature on Abel and his work. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Isaac Barrow on the Mathematization of Nature: Theological Voluntarism and the Rise of Geometrical OpticsAntoni MaletIntroductionIsaac Newton’s Mathematical Principles of Natural Philosophy embodies a strong program of mathematization that departs both from the mechanical philosophy of Cartesian inspiration and from Boyle’s experimental philosophy. The roots of Newton’s mathematization of nature, this paper aims to demonstrate, are to be found in Isaac Barrow’s (1630–77) philosophy of the mathematical sciences.Barrow’s attitude (...) towards natural philosophy evolved from his earnest interest in medicine of around 1650, when a young Cambridge graduate, to natural philosophy (apparently under Henry More’s influence); from his thesis on the insufficiency of the Cartesian hypothesis to geometrical optics and the strong program of mathematization of natural philosophy of the middle 1660s; from Lucasian professor of Mathematics to Chaplain of his Majesty and eminent Restoration divine. Contemporary accounts of Barrow’s life suggest that he grew ever more skeptical about the worth of natural philosophy and mathematics. 1 In his last years he became a prolific [End Page 265] author of sermons and theological works. Published shortly after his death by John Tillotson (1630–94), later archbishop of Canterbury, they occupy over two thousand folio pages. Overloaded with involved philosophical arguments, Barrow’s sermons were apparently not very popular, but they were highly regarded by scholars and the Anglican hierarchy. 2 It is on certain of his sermons, as well as on the philosophical discussions contained in the Mathematical Lectures that our account of Barrow’s philosophy of the mathematical sciences will rest. 3 Barrow’s understanding of the mathematical sciences will allow us to discuss together three issues often analyzed independently: the theological background to English natural philosophy, the changing notion of mixed mathematical sciences during the seventeenth century, and finally the philosophical foundations of modern geometrical optics.It has long been recognized that significant relationships exist between theological voluntarism or intellectualism and views on natural philosophy. In particular Robert Boyle’s theological voluntarism is seen as grounding his experimentalist approach to natural philosophy. It is not quite so clear, however, how well theological voluntarism may relate to a strong program of mathematization such as the one embodied in Newton’s Principia Mathematica. In fact it has been suggested that the relationship is a negative one. This is derived from the necessary character of mathematical laws, which would put unwanted restrictions on God’s absolute dominion over nature, and also from the notion that theological intellectualism is conducive to a deductive, a priori science—the paradigm of which is of course geometry. However, Barrow’s theological voluntarism lead him to heighten the role of mathematics within natural philosophy.During the seventeenth century the so-called mixed or subalternate mathematical sciences changed profoundly. In the Enlightenment mixed mathematics—meaning above all rational mechanics—became one of the most prestigious and influential disciplines. The mixed mathematics of the Enlightenment, however, was markedly different from the Aristotelian [End Page 266] mixed mathematical sciences. The differences are noticeable both in the subject matter and in the substitution of mathematical infinitesimal analysis for geometrical synthesis, but also in the way of grounding mathematical theory on empirical evidence. Barrow’s Mathematical Lectures (delivered at Cambridge from 1664 to 1666) offer a fresh insight into the metamorphosis of these sciences just when Newton’s “mathematical principles” were in the making. Not the least interesting feature of Barrow’s discussion is that God’s omnipotence allows an evaluation of the truth of mathematical theories that do not apply to this world. Therefore, Barrow is led to introduce the distinction between the internal consistency, or mathematical truth, of a mathematical theory and its physical truth. This, in turn, leads him to the notion that theories need testing.Barrow on Matter and GodRecent literature has established significant correlations between theological voluntarism and empiricism, as well as between theological intellectualism and rationalism. As E. B. Davis writes, “the Christian doctrine of creation is a dialogue between God’s unconstrained will, which utterly transcends the bounds of human comprehension, and God’s orderly intellect, which serves as the model for the human mind.” Intellectualist theology considers God’s omniscience His... (shrink)

The aim of Reverse Mathematics (RM for short) is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent to the theorem, working over the base theory of RM, a weak system of computable mathematics. The Big Five phenomenon of RM is the observation that a large number of theorems from ordinary mathematics are either provable in the base theory or equivalent to one of (...) only four systems; these five systems together are called the ‘Big Five’. The aim of this paper is to greatly extend the Big Five phenomenon as follows: there are two supposedly fundamentally different approaches to RM where the main difference is whether the language is restricted to second-order objects or if one allows third-order objects. In this paper, we unite these two strands of RM by establishing numerous equivalences involving the second-order Big Five systems on one hand, and well-known third-order theorems from analysis about (possibly) discontinuous functions on the other hand. We both study relatively tame notions, like cadlag or Baire 1, and potentially wild ones, like quasi-continuity. We also show that slight generalizations and variations of the aforementioned third-order theorems fall far outside of the Big Five. (shrink)

Dedicated to Dana Scott on his sixtieth birthday.It is common knowledge that for a short while Hermann Weyl joined Brouwer in his pursuit of a revision of mathematics according to intuitionistic principles. There is, however, little in the literature that sheds light on Weyl's role and in particular on Brouwer's reaction to Weyl's allegiance to the cause of intuitionism. This short episode certainly raises a number of questions: what made Weyl give up his own program, spelled out (...) in “Das Kontinuum”, how did Weyl come to be so well-informed about Brouwer's new intuitionism, in what respect did Weyl's intuitionism differ from Brouwer's intuitionism, what did Brouwer think of Weyl's views,…? To some of these questions at least partial answers can be put forward on the basis of some of the available correspondence and notes. The present paper will concentrate mostly on the historical issues of the intuitionistic episode in Weyl's career.Weyl entered the foundational controversy with a bang in 1920 with his sensational paper “On the new foundational crisis in mathematics”. He had already made a name for himself in the foundations of mathematics in 1918 with his monograph “The Continuum” [18]; this contained in addition to a technical logical-mathematical construction of the continuum, a fairly extensive discussion of the shortcomings of the traditional construction of the continuum on the basis of arbitrary—and hence also impredicative—Dedekind cuts. (shrink)

Effective conditioning requires a correlation between the experimenter's definition of a response and an organism's, but an animal's perception of its behavior differs from ours. These experiments explore various definitions of the response, using the slopes of learning curves to infer which comes closest to the organism's definition. The resulting exponentially weighted moving average provides a model of memory that is used to ground a quantitative theory of reinforcement. The theory assumes that: incentives excite behavior and focus the excitement on (...) responses that are contemporaneous in memory. The correlation between the organism's memory and the behavior measured by the experimenter is given by coupling coefficients, which are derived for various schedules of reinforcement. The coupling coefficients for simple schedules may be concatenated to predict the effects of complex schedules. The coefficients are inserted into a generic model of arousal and temporal constraint to predict response rates under any scheduling arrangement. The theory posits a response-indexed decay of memory, not a time-indexed one. It requires that incentives displace memory for the responses that occur before them, and may truncate the representation of the response that brings them about. As a contiguity-weighted correlation model, it bridges opposing views of the reinforcement process. By placing the short-term memory of behavior in so central a role, it provides a behavioral account of a key cognitive process. (shrink)

There are a number of ways in which Greek mathematics can be seen to be radically original. First, at the level of mathematical contents: many objects and results were first discovered by Greek mathematicians. Second, Greek mathematics was original at the level of logical form: it is arguable that no form of mathematics was ever axiomatic independently of the influence of Greek mathematics. Finally, third, Greek mathematics was original at the level of form, of presentation: (...) Greek mathematics is written in its own specific, original style. This style may vary from author to author, as well as within the works of a single author, but it is still always recognizable as the Greek mathematical style. This style is characterized by the use of the lettered diagram, a specific technical terminology, and a system of short phrases. I believe this third aspect of the originality—the style—was responsible, indirectly, for the two other aspects of the originality. The style was a tool, with which Greek mathematicians were able to produce results of a given kind, and to produce them in a special, compelling way. This tool, I claim, emerged organically, and reflected the communication-situation in which Greek mathematics was conducted. For all this I have argued elsewhere. (shrink)

Open peer commentary on the target article “Who Conceives of Society?” by Ernst von Glasersfeld. First paragraph: Ernst von Glasersfeld seems to say to social constructivists, “You attribute reality to society, but your society is just another construct, all you know is just the bits of light and shadow and color that your visual system provides to you.” Now, “society” is just a big word for “other people.” I can give to this text of von Glasersfeld’s either a short (...) answer or a long one. (shrink)

This book clearly explains what an infinite number is, how infinite numbers differ from finite numbers, and how infinite numbers differ from one another. The concept of recursivity is concisely but thoroughly covered, as are the concepts of cardinal and ordinal number. All of Cantor's key proofs are clearly stated, including his epoch-making diagonal proof, whereby he proved that that there are more reals than rationals and, more generally, that there are infinitely large, non-recursive classes. In the final section, Kurt (...) Gödel's celebrated Incompleteness Theorem is shown to be equivalent with a number of the various different theorems proved in this short but profound work. (shrink)