Ernst Cassirer's Neo-Kantian Philosophy of Geometry

One of the most important philosophical topics in the early twentieth century – and a topic that was seminal in the emergence of analytic philosophy – was the relationship between Kantian philosophy and modern geometry. This paper discusses how this question was tackled by the Neo-Kantian trained philosopher Ernst Cassirer. Surprisingly, Cassirer does not affirm the theses that contemporary philosophers often associate with Kantian philosophy of mathematics. He does not defend the necessary truth of Euclidean geometry but instead develops a kind of logicism modeled on Richard Dedekind's foundations of arithmetic. Further, because he shared with other Neo-Kantians an appreciation of the developmental and historical nature of mathematics, Cassirer developed a philosophical account of the unity and methodology of mathematics over time. With its impressive attention to the detail of contemporary mathematics and its exploration of philosophical questions to which other philosophers paid scant attention, Cassirer's philosophy of mathematics surely deserves a place among the classic works of twentieth century philosophy of mathematics. Though focused on Cassirer's philosophy of geometry, this paper also addresses both Cassirer's general philosophical orientation and his reading of Kant.     

Hilbert,logicism, and mathematical existence

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom).     

A Formalist Philosophy of Mathematics Part I: Arithmetic

In this paper I present a formalist philosophy mathematics and apply it directly to Arithmetic. I propose that formalists concentrate on presenting compositional truth theories for mathematical languages that ultimately depend on formal methods. I argue that this proposal occupies a lush middle ground between traditional formalism, fictionalism, logicism and realism.     

The aim of Russell’s early logicism: a reinterpretation

I argue that three main interpretations of the aim of Russell’s early logicism in The Principles of Mathematics (1903) are mistaken, and propose a new interpretation. According to this new interpretation, the aim of Russell’s logicism is to show, in opposition to Kant, that mathematical propositions have a certain sort of complete generality which entails that their truth is independent of space and time. I argue that on this interpretation two often-heard objections to Russell’s logicism, deriving from Gödel’s incompleteness theorem and from the non-logical character of some of the axioms of Principia Mathematica respectively, can be seen to be inconclusive. I then proceed to identify two challenges that Russell’s logicism, as presently construed, faces, but argue that these challenges do not appear unanswerable.     

A theory of truth based on a medieval solution to the liar paradox

After the 1930s, the research into the foundations of mathematics changed.None of its main directions (logicism, formalism and intuitionism) had any longer the pretension to be the only true mathematics.Usually, the determining factor in the change is considered to be Gödel’s work, while Heyting’s role is neglected.In contrast, in this paper I first describe how Heyting directly suggested the abandonment of the big foundational questions and the putting forward of a new kind of foundational research consisting in the isolation of formal, intuitive, logical and platonistic elements within classical mathematics.Furthermore, I describe how Heyting indirectly influenced the abandon‐ment of the old directions of foundational research by making out some lists of degrees of evidence that exist within intuitionism     

Bad company tamed

Neo-Fregean logicism seeks to base mathematics on abstraction principles. But the acceptable abstraction principles are surrounded by unacceptable (indeed often paradoxical) ones. This is the “bad company problem.” In this introduction I first provide a brief historical overview of the problem. Then I outline the main responses that are currently being debated. In the course of doing so I provide summaries of the contributions to this special issue.     

The influence of Platonism on mathematics research and theological beliefs

An age-old debate in the philosophy of mathematics is whether mathematics is discovered or invented. There are four popular viewpoints in this debate, namely Platonism, formalism, intuitionism, and logicism. A natural question that arises is whether belief in one of these viewpoints affects the mathematician’s research? In particular, does subscribing to a Platonist or a formalist viewpoint influence how a mathematician conducts research? Does the area of research influence a mathematician’s beliefs on the nature of mathematics? How are the beliefs regarding the nature of mathematics connected to theological beliefs? In order to investigate these questions, five professional research mathematicians were interviewed. The mathematicians worked in diverse areas within analysis, algebra, and within applied mathematics, and had a combined 160 years of research experience. Although none of the mathematicians wanted to be pigeonholed into any one category of beliefs, the study revealed that four of the mathematicians leaned towards Platonism, which runs contrary to the popular notion that Platonism is an exception today. This study revealed that beliefs regarding the nature of mathematics influenced how mathematicians’ conducted research and were deeply connected to their theological beliefs. The findings are presented in the form of vignettes that give an insight into the mathematical and theological belief structures of the mathematicians.     

Mathematics instruction in Head Start: Nature,extent, and contributions to children's learning

This study employed the most recent (2006) cohort of the nationally representative Family and Child Experiences Survey (FACES) to explore the nature of mathematics instruction in Head Start and the child, family, and teacher factors that contribute to children's mathematics learning over the preschool year. In total, 2501 preschoolers and their families, as well as their teachers (n = 335), participated in the study from fall 2006 to spring 2007. Results showed that teachers reported frequent mathematics instruction, although direct observations did not entirely confirm this frequency. A variety of factors predicted children's mathematics knowledge at Head Start entry, and several – including instructional quality – were linked to learning over time. No thresholds in instructional quality emerged. Overall, this study provides new information about classroom mathematics instruction and child learning among the nation's most vulnerable early learners.     

HOW DID FREGE FALL INTO THE CONTRADICTION?

Quine made it conventional to portray the contradiction that destroyed Frege's logicism as some kind of act of God, a thunderbolt that descended from a clear blue sky. This portrayal suited the moral Quine was antecedently inclined to draw, that intuition is bankrupt, and that reliance on it must therefore be replaced by a pragmatic methodology. But the portrayal is grossly misleading, and Quine's moral simply false. In the person of others – Cantor, Dedekind, and Zermelo – intuition was working pretty well. It was in Frege that it suffered a local and temporary blindness. The question to ask, then, is not how Frege was overtaken by the contradiction, but how it is that he didn't see it coming. The paper offers one kind of answer to that question. Starting from the very close similarity between Frege's proof of infinity and the reasoning that leads to the contradiction, it asks: given his understanding of the first, why did Frege did not notice the second? The reason is traced, first, to a faulty generalization Frege made from the case of directions and parallel lines; and, through that, to Frege's having retained, and attempted incoherently to combine with his own, aspects of a pre‐Fregean understanding of the generality of logical principles.     

Introduction

Neo-Fregean logicism seeks to base mathematics on abstraction principles. But the acceptable abstraction principles are surrounded by unacceptable (indeed often paradoxical) ones. This is the “bad company problem.” In this introduction I first provide a brief historical overview of the problem. Then I outline the main responses that are currently being debated. In the course of doing so I provide summaries of the contributions to this special issue.     

Shaping the Enemy: Foundational Labelling by L.E.J. Brouwer and A. Heyting

The use of the three labels (logicism, formalism, intuitionism) to denote the three foundational schools of the early twentieth century are now part of literature. Yet, neither their number nor their adoption has been stable over the twentieth century. They were not introduced by the founding fathers of each school: namely, neither Frege nor Russell spoke of ‘logicism’; and even Hilbert did not use the word ‘formalism’ to introduce his foundational programs. At a certain point, only Brouwer used the label ‘intuitionism’ in his scientific production to personify his philosophy of mathematics and he used the label ‘formalism’ for Hilbert’s foundational viewpoint. Starting with Brouwer, the origin of the use of the three labels to represent a foundational meaning, will be analysed in this paper. Thereafter, the role that Brouwer’s pupil Arend Heyting had in the production and use of foundational labels will be considered. On the basis of the comparison of the attitudes of these two scholars I will finally advance the thesis that not only the creation but also the use of labels, far from being a mere gesture of academic reference to literature, can be a sign of the cultural operation each scholar wanted to do.     

Mathematics,science and ontology

According to quasi-empiricism, mathematics is very like a branch of natural science. But if mathematics is like a branch of science, and science studies real objects, then mathematics should study real objects. Thus a quasi-empirical account of mathematics must answer the old epistemological question: How is knowledge of abstract objects possible? This paper attempts to show how it is possible. The second section examines the problem as it was posed by Benacerraf in ‘Mathematical Truth’ and the next section presents a way of looking at abstract objects that purports to demythologize them. In particular, it shows how we can have empirical knowledge of various abstract objects and even how we might causally interact with them. Finally, I argue that all objects are abstract objects. Abstract objects should be viewed as the most general class of objects. The arguments derive from Quine. If all objects are abstract, and if we can have knowledge of any objects, then we can have knowledge of abstract objects and the question of mathematical knowledge is solved. A strict adherence to Quine's philosophy leads to a curious combination of the Platonism of Frege with the empiricism of Mill.     

Erkenntnistheorie der Zahldefinition und philosophische Grundlegung der Arithmetik unter Bezugnahme auf einen Vergleich von Gottlob Freges Logizismus und platonischer Philosophie (Syrian, Theon von Smyrna u.a.)

The epistomology of the definition of number and the philosophical foundation of arithmetic based on a comparison between Gottlob Frege's logicism and Platonic philosophy (Syrianus, Theo Smyrnaeus, and others). The intention of this article is to provide arithmetic with a logically and methodologically valid definition of number for construing a consistent philosophical foundation of arithmetic. The – surely astonishing – main thesis is that instead of the modern and contemporary attempts, especially in Gottlob Frege's Foundations of Arithmetic, such a definition is found in the arithmetic in Euclid's Elements. To draw this conclusion a profound reflection on the role of epistemology for the foundation of mathematics, especially for the method of definition of number, is indispensable; a reflection not to be found in the contemporary debate (the predominate ‘pragmaticformalism’ in current mathematics just shirks from trying to solve the epistemological problems raised by the debate between logicism, intuitionism, and formalism). Frege's definition of number, ‘The number of the concept F is the extension of the concept ‘numerically equal to the concept F”, which is still substantial for contemporary mathematics, does not fulfil the requirements of logical and methodological correctness because the definiens in a double way (in the concepts ‘extension of a concept’ and ‘numerically equal’) implicitly presupposes the definiendum, i.e. number itself. Number itself, on the contrary, is defined adequately by Euclid as ‘multitude composed of units’, a definition which is even, though never mentioned, an implicit presupposition of the modern concept ofset. But Frege rejects this definition and construes his own - for epistemological reasons: Frege's definition exactly fits the needs of modern epistemology, namely that for to know something like the number of a concept one must become conscious of a multitude of acts of producing units of ‘given’ representations under the condition of a 1:1 relationship to obtain between the acts of counting and the counted ‘objects’. According to this view, which has existed at least since the Renaissance stoicism and is maintained not only by Frege but also by Descartes, Kant, Husserl, Dummett, and others, there is no such thing as a number of pure units itself because the intellect or pure reason, by itself empty, must become conscious of different units of representation in order to know a multitude, a condition not fulfilled by Euclid's conception. As this is Frege's main reason to reject Euclid's definition of number (others are discussed in detail), the paper shows that the epistemological reflection in Neoplatonic mathematical philosophy, which agrees with Euclid's definition of number, provides a consistent basement for it. Therefore it is not progress in the history of science which hasled to the a poretic contemporary state of affairs but an arbitrary change of epistemology in early modern times, which is of great influence even today. This revised version was published online in August 2006 with corrections to the Cover Date.     

Synchronicity,the infinite unrepressed,dissociation and the interpersonal

This paper uses the logic derived by Matte Blanco to provide an Archimedean point and a mathematics, both of which Jung complained of lacking, with which to validate the notion of synchronicity and to demonstrate that it is one of the inevitable properties of an unconscious which is unrepressed such as Jung's collective unconscious, and that such an unconscious will also be affective and interpersonal as well as intrapersonal. These have important clinical implications. After an exposition of Matte Blanco's thinking, some clinical material is presented of an episode in which patient and author both suffered the same psychosomatic symptom some time just prior to a session. Correspondences between Matte Blanco's logically derived ideas and Jung's phenomenological observations are made.     

Classroom context,teacher expectations,and cognitive level: Predicting children's math ability judgments

Classroom practices that make ability differences salient communicate differential teacher expectations for students. This study reports on a new observational tool for measuring Classroom Ability-based Practices (CAP) and explores how young children's self-perceptions of ability in mathematics are related to their teachers' expectations for them and to their cognitive reasoning skills in classrooms that vary in usage of such ability-differentiated practices. The sample consisted of 193 children and their teachers in 15 first grade classrooms. The CAP was a reliable measure of ability-based differential teacher treatment and showed criterion validity as a moderator variable in predicting children's ability perceptions. In highly ability-differentiating classrooms, children's self-ratings were more congruent with teachers' expectations of students' mathematics ability. Also in highly ability-differentiating classrooms, higher cognitive levels predicted lower self-ability ratings in math. These findings extend our understanding of the social and cognitive factors that shape young children's self-judgments of math competence.     

Math Anxiety,Mother's Education,and the Mathematics Performance of Adolescent Boys and Girls: Evidence from the United States and Thailand

ABSTRACT

In this study, I investigated the relationship of mathematics performance to math anxiety, mother's education, and gender. A secondary analysis was conducted using nationally representative samples of 13-year-old children in the United States (N = 4,091) and Thailand (N = 3,613) collected as a part of the Second International Mathematics Study (Garden, 1987). Separate ANOVAs (Math Anxiety × Mother's Education × Gender) were run within each country using a 40-item math performance test as the dependent variable. Math anxiety has an inverse relationship with mathematics performance in the United States (r = ?.24) and in Thailand (r ?. 14). The relationship between math anxiety and mathematics performance is significant in both countries after controlling for previous achievement, mother's education, and gender, although the data suggest that there is a three-way interaction between math anxiety, mother's education, and gender in Thailand.     

VOLKER PECKHAUS (ed.), Oskar Becker und die Philosophie der Mathematik.

It has been contended that it is unjustified to believe, as Weyl did, that formalism's victory against intuitionism entails a defeat of the phenomenological approach to mathematics. The reason for this contention, recently put forth by Paolo Mancosu and Thomas Ryckman, is that, unlike intuitionistic Anschauung, phenomenological intuition could ground classical mathematics. I argue that this indicates a misinterpretation of Weyl's view, for he did not take formalism to prevail over intuitionism with respect to grounding classical mathematics. I also point out that the contention is false: if intuitionism fails, in the way Weyl thought it did, i.e. with respect to supporting scientific objectivity, then one should also reject the phenomenological approach, in the same respect.     

Abstraction in Fitch's Basic Logic

Fitch's basic logic is an untyped illative combinatory logic with unrestricted principles of abstraction effecting a type collapse between properties (or concepts) and individual elements of an abstract syntax. Fitch does not work axiomatically and the abstraction operation is not a primitive feature of the inductive clauses defining the logic. Fitch's proof that basic logic has unlimited abstraction is not clear and his proof contains a number of errors that have so far gone undetected. This paper corrects these errors and presents a reasonably intuitive proof that Fitch's system K supports an implicit abstraction operation. Some general remarks on the philosophical significance of basic logic, especially with respect to neo-logicism, are offered, and the paper concludes that basic logic models a highly intensional form of logicism.     

ACTION,AGENCY, AND EMPATHY: SCHAFER ON THE ANALYST'S DILEMMA

Throughout his career, a central feature of Roy Schafer's theorizing has been to highlight the role of activity and personal agency in every facet of human experience. This theme has remained at the forefront of Schafer's work, despite being embedded within different frames of reference. In this paper, the author highlights Schafer's focus on activity, notes some clinical problems to which it can give rise, and suggests the way that Schafer has attempted to deal with these difficulties.