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The Nature of Mathematical Knowledge.
... Non nè e risultato tanto un avvicinamento delle tradizioni analitica a continentale, quanto una mescolanza di sensibilità, stili e punti di vista, che si manifesta anche e soprattutto entro la comunità filosofica di lingua inglese. Ne sono degli esempi la critica di Imre Lakatos (1976 e 1978) nei confronti di un orientamento accusato di dare troppo peso all'analisi logico-formale della matematica, e l'elaborazione, da parte di Philip Kitcher (1983), di una concezione anti-apriorista che descrive la matematica come un'impresa storica. A questi e altri orientamenti simili fanno oggi seguito molti indirizzi di ricerca volti all'analisi dettagliata della pratica matematica e tendenti a mostrarne la dimensione storica. ...
... NonèNonè che manchino argomenti per sostenere che la matematica abbia una base empirica. Anche se si tratta di una tesi difficilmente formulabile in modo preciso, chiunque abbia un minimo di consapevolezza dell'evoluzione storica della matematica dovrebbe considerare come perfettamente plausibile l'idea che anche lepì u astratte teorie matematiche si possano ricondurre a un'origine empirica (Kitcher 1983). Il problemà e quello di mostrare come il passaggio da credenze empiricamente giustificate a principi generali come quelli che reggono le nostre teorie matematiche possa a sua volta essere empiricamente giustificato: per quanto le nostre capacità percettive possano contribuire a formare queste teorie, e perfino essere essenziali per condurre una dimostrazione matematica (Panza, 2003), l'esistenza di oggetti matematici non sembra possa venir giustificata su basi puramente empiriche. ...
Forthcoming in: A. Coliva (ed.), ManuaLe di filosofia analitica, Carocci, Roma.
... [57, p. 58]. 45 Uma justificativa de McGee para isso é: "Adoption of a rule permitting us to assert a sentence ϕ only commits us to the ontological commitments of ϕ, whatever they are. So adoption of a rule permitting the assertion of Induction Axioms only commits us to the ontological commitments of Induction Axioms, and Induction Axioms are only committed to numbers." ...
This Thesis aims to contribute to the understanding of two important subjects to the philosophy of mathematics. The notion of truth for mathematical propositions and the notion of existence in arithmetic. To pursue this target two original contributions are presented, one for each of these subjects. With regard to the notion of truth, we explore the consequences of adopting a normative framework to fix the truth value of arithmetic propositions. This normative framework is instituted by mathematical practice. The analysis will establish in what sense the standard model of arithmetic - and hence the truth value of arithmetic sentences - is fixed by the analysis’ hypothesis. Concerned with the notion of existence, a proposal is made to evaluate the existential requirements of arithmetic sentences. This evaluation is based on the assumption that the existential import of these sentences is an attribute of the truth conditions of arithmetic propositions. The analysis of this assumption motivates a precise and well-founded definition, in the arithmetical context, to the concept of existence axiom in arithmetical context. In addition, the analysis fosters a criterion of diferentiation between the axioms of theories that are, from the perspective of interpretations, indistinguishable. (Full text in Portuguese)
... Among the manifestations of this emphasis were an axiomatic presentation of elementary algebra and increased classroom attention to the precise formulation of mathematical notions and to the structure of a deductive system." (Hana, 1989, p. 20) This "new mathematics", as it was usually called, was criticized in the 80's by Hana (1983Hana ( , 1989 Kitcher (1984), Davies (1986), Tymoczko (1986) and others, and educators we forced to modify the curriculum de-emphasizing formalities, rigor and proof, and emphasizing more examples and applications. It has been debatable since then, if that was the right approach that should had been followed, as complains were raised later regarding the coherence of the material taught and the impact of reducing rigor and proof had on the critical skills of the students. ...
As it is already observed by mathematicians and educators, there is a discrepancy between the formal techniques of mathematical logic and the informal techniques of mathematics in regards to proof. We examine some of the reasons behind this discrepancy and to what degree it affects doing, teaching and learning mathematics in college. We also present some college students’ opinions about proofs, and we briefly observe the situation in Greek and Greek-Cypriot high schools in which mathematical logic is part of the curriculum. Finally, we argue that even though mathematical logic is central in mathematics, its formal methods are not really necessary in doing and teaching mathematical proofs and the role of those formalities has been, in general, overestimated by some educators.
... The contemporary view on Euclidean geometry consists of not 5 but 20 axioms, introduced by David Hilbert in 1899. As Kitcher (1984) argues, axioms should not be understood as self-evident basic principles of a theory, but as a method of systematization of a domain of an already existing theory. Axiomatization systematizes the domain and unifies the theory, showing that theorems are derivable from a certain group of basic principles. ...
In this article, I consider the possibility of a theoretical integration of phenomenology and a mechanistic framework. First, I discuss the mechanistic model of explanation and the idea of theoretical integration in science as opposed to unification. I argue that the mechanistic model of explanation is preferable for integrating the cognitive sciences, although it is limited and in the case of consciousness studies should be complemented with phenomenology. Second, I examine three possible approaches to the integration of phenomenology and the mechanistic model of explanation. First, I discuss Integrated Information Theory (IIT) of consciousness and propose a new argument against IIT’s axiomatic method—namely, I argue that IIT misuses the notion of axiom. Next, I discuss two different proposals for the integration of phenomenology with cognitive sciences: front-loaded phenomenology and neurophenomenology. I argue that these proposals cannot be integrated with a mechanistic framework unless requisite modifications are made.
... A somewhat more ambitous goal would be to axiomatize the (absolutely) OTM-realizable statements under intuitionistic provability. Moreover, we will explore in further work the connections between generalized effectivity and idealized agency of mathematics, such as advocated by P. Kitcher ( [9]). ...
We define an ordinalized version of Kleene's realizability interpretation of intuitionistic logic by replacing Turing machines with Koepke's ordinal Turing machines (OTMs), thus obtaining a notion of realizability applying to arbitrary statements in the language of set theory. We observe that every instance of the axioms of intuitionistic first-order logic are OTM-realizable and consider the question which axioms of Friedman's Intuitionistic Set Theory (IZF) and Aczel's Constructive Set Theory (CZF) are OTM-realizable. This is an introductory note, and proofs are mostly only sketched or omitted altogether. It will soon be replaced by a more elaborate version.
... It is definitely hard to say. However, what I would claim is that he might follow P. Kitcher's (1984) criticism of the indispensability thesis by insisting that the Putnam-Quine argument does not explain why mathematics is indispensable for science and for economics in particular. Hence Prof. Malawski's question of whether the economic realm is ontologically mathematical. ...
This paper explores the ways Prof. A. Malawski understood the various kinds of interplay between economics, mathematics, and philosophy. In particular, it addresses the issue of the mathematicity of the economy and what it means for economics to be a mathematical science. Next, it focuses on the nature of economic laws. It concludes by claiming that the interpretative key to Prof. Malawski’s research lies in his deep humilit y
... Matematikte ispatın sosyal bir süreç olarak ele alınması ve matematik eğitiminde ispatın formel ispat kavramının ötesinde anlamları olduğu düşüncesi, bir çok araştırmacı tarafından (Davis, 1986;Hanna, 2002;Kitcher, 1984;Lakatos, 1976;Tymoczko, 1986) Tall'a (1989) göre ispat her ne kadar sosyal bir süreç olarak ele alınsa da, gücü ve genellemesiyle geniş bir bağlamda ele alınmalıdır. ...
Bu çalışma ile ortaokul matematik öğretmeni adaylarının ispatın doğası hakkındaki görüşlerini ortaya çıkarmak amaçlanmıştır. Çalışmada nitel araştırma yöntemlerinden durum çalışması kullanılmıştır. Çalışma kapsamında ölçüt örnekleme yöntemi ile seçilen ve bir devlet üniversitesinin ortaokul matematik öğretmenliği programında öğrenim gören üç öğretmen adayının ispatın doğasına ilişkin görüşleri alınmıştır. Matematik öğretmen adaylarına, araştırmacılar tarafından geliştirilen ve açık uçlu sorulardan oluşan “İspatın Doğasına İlişkin Görüşme Formu” yarı yapılandırılmış görüşmeler aracılığıyla yöneltilmiştir. Görüşme verileri içerik analizi yöntemi ile analiz edilmiştir. İçerik analizi sonrasında öğretmen adaylarının ispatın doğasına ilişkin “genelleme”, “yöntem”, “doğruluğa ulaşma”, “problem çözme”, “biçime odaklanma” temaları altında tepkiler verdikleri belirlenmiştir. Çalışmada en sıklıkla ortaya çıkan tema “doğruluğa ulaşma” ; en az sıklıkla ortaya çıkan temalar ise “problem çözme” ve “biçime odaklanma” olarak belirlenmiştir. Bu çalışmada, ortaokul matematik öğretmeni adaylarının ispatın tanımını yapmada, ispatı ispat yapan şeyleri ve başarılı bir ispat için gerekli olan şeyleri belirlemede, kısacası ispatın doğasını anlamada zorluklar yaşadıkları sonucuna ulaşılmıştır.
... Matematiksel ispatla ilgili kullanılan bu ifadelerin sebepleri incelendiğinde, matematiğe yaptığı katkılar ve öğrencilere matematiksel anlamda kazandırdığı beceriler ön plana çıkmaktadır (Doruk ve Kaplan, 2013). Öğrenciler matematiksel ispatlar yaparak; matematiksel bilginin inşasını keşfedebilirler (Stylianides, 2007), matematiksel kavramları sebepleri ile birlikte öğrenebilir ve anlayabilirler (Hanna, 1991), matematikçiler tarafından yapılanların ne anlama geldiğini öğrenebilirler (İmamoğlu, 2010) ve matematiksel bilgilerini geliştirebilirler (Kitcher, 1984). Bununla birlikte öğrenciler matematiksel ispatlar yaparak formülleri son halleri ile bilmenin yeterli olmadığını ve açıklanması gerektiğini öğrenirler (Güven, Çelik ve Karataş, 2005). ...
The aim of this research is to find out prospective mathematics teachers' conceptions regarding mathematical proof. To this aim, the researchers developed a 5-point Likert Conception Scale for Mathematical Proof (CSFMP) composed of 31 items. The scale was administered to a total of 480 prospective teachers who were studying at the department of elementary mathematics teaching in a state university located in the Eastern Anatolia Region. Exploratory factor analysis was conducted in order to determine the structural validity of the scale. In view of the factor analysis, it was observed that the scale was composed of five factors. In view of the conducted reliability analysis, the reliability coefficient (Cronbach's Alpha) of the scale was found to be .93. The reliability coefficients of the five factors, which were obtained from the scale, range from .70 to .90. These factors explain 54.2% of the scale's total variance. At the end of the study, it was found that the prospective teachers were generally indecisive in their conceptions regarding proof. Furthermore, it was determined that the prospective teachers' evel of self-confidence on performing proof and understanding proofs was low, although the meaning that they attributed to proof was positive. When the prospective teachers' conceptions regarding proof were examined in terms of class levels, it was found that the third-year prospective teachers' conceptions were more negative compared to those of first and second year prospective teachers.
... An important [philosophical and historical] example has been contributed by Kitcher (1984). Kitcher's goal was to develop an epistemology of mathematics. ...
... According to Senk et al, proof, which is the core of mathematics (cited by: Bahtiyarı, 2010), is not only associated to what is correct, at the same time it is also associated to why it is correct. Proof is emphasized for the purpose of knowing and doing mathematics, constituting the foundation of the perception of mathematics, the comprehension, use, and development of mathematical knowledge (Hanna and Jahnke, 1996;Kitcher, 1984;Polya, 1981). All this emphasis demonstrates the importance of proof and establishes a strong relation between proof and mathematics teaching. ...
Mathematical proof is important in mathematics teaching in terms of the comprehension of mathematical knowledge. Thus, proof has critical value in the teaching process in terms of the prevention of memorization in mathematics, the construction of conceptual knowledge, and the realization of meaningful learning. This study aims to determine the perceptions of students towards proof after a teaching process with the objective of developing the perceptions and skills of 7 th grade students towards proof. In line with this, an answer was sought for the question on the extent the concept of proof can be acquired by 7 th grade students. Accordingly, the study was designed as action research, which is one of the qualitative research approaches, and descriptive analysis was employed in the study. Purposive sampling was preferred in the selection of the study group. The study group of the study consisted of a 7 th grade from each of the two schools from the districts of Çankaya and Yenimahalle in the province of Ankara. First of all, a readiness test was applied to the classes in the application process of the study and then proof teaching for 1 hour a week was performed for 14 weeks. After this application, a questionnaire with the objective of determining the level of proof perception of students was utilized and semi-structured in-depth interviews were conducted with 16 students determined as a result of this test. As a result of the study, a development was observed in the perceptions of 7 th grades students towards the concept of proof.
... For example, Fischbein (1993) gave a main role to idealization processes in his theory of figural concepts. The idealization process is also emphasized by Kitcher (1984), who assumed empirical and pragmatic origins for mathematics, and adopted a constructivist position by considering mathematics as a science of idealized operations that people are able to carry out on any kind of object. Another example comes from the research program of -embodied cognition‖ (Lakoff & Núñez, 2000), where a key issue is investigating the way people generate mathematical ideas. ...
... The visions of mathematics held by these two communities seem to differ substantially in more than one respect. To be sure, many mathematicians would admit today, however reluctantly, that mathematics itself changes all the time and the criteria according to which one decides what is 'mathematical' are different now than they were a few centuries, or even a few decades, ago (Lakatos 1976;Davis & Hersh 1981;Kitcher 1984;Tymoczko 1986;Ernest 1994). However, the changes currently taking place within the context of research in mathematics education do not necessarily parallel those that alter the face of 'professional' mathematics. ...
... Very often, the computer is the tool that enables us to bring these examples to the desired level." 43 It is disconcerting to the author for use of a computer to bring, 'examples to the desired level,' seems to indicate that ideas are illustrated but not proven or disproven. A trap of too much reliance on inductive reasoning and not enough deductive reasoning seems to be a possible outcome of said endeavour. ...
... The philosophy of mathematics has relatively recently added a new direction, a focus on the history and philosophy of informal mathematical practice, advocated by Lakatos (1976Lakatos ( , 1978, Davis and Hersh (1980), Kitcher (1983), Tymoczko (1998) and Corfield (1997). This focus challenges the view that Euclidean methodology, in which mathematics is seen as a series of unfolding truths, is the bastion of mathematics. ...
One current direction in AI research is to focus on combining different reasoning styles such as deduction, induction, abduction, analogical reasoning, non-monotonic reasoning, vague and uncertain reasoning. The philosopher Imre Lakatos produced one such theory of how people with different reasoning styles collaborate to develop mathematical ideas. Lakatos argued that mathematics is a quasi-empirical, flexible, fallible, human endeavour, involving negotiations, mistakes, vague concept definitions and disagreements, and he outlined a heuristic approach towards the subject. In this chapter we apply these heuristics to the AI domains of evolving requirement specifi-cations, planning and constraint satisfaction problems. In drawing analogies between Lakatos's theory and these three domains we identify areas of work which correspond to each heuristic, and suggest extensions and further ways in which Lakatos's philoso-phy can inform AI problem solving. Thus, we show how we might begin to produce a philosophically-inspired AI theory of combined reasoning.
... Damit erhält man, angeregt durch die Gegenüberstellung von fachlich üblichen und divergierenden Schülerkonzepten, eine mögliche konkurrierende Theorie der Ableitungen, die die Kontingenz der klassischen Theoriebildung erst richtig bewusst macht (vgl. Kitcher 1984, Prediger 2004): Die klassische Ableitung ist eine unter mehreren Möglichkeiten, das qualitative Konzept der Änderung exakt zu fassen; andere wären auch möglich und sind auch bis zu einem gewissen Grad sogar etabliert. ...
Classical higher secondary calculus courses have often been blamed for their focus on procedures without meaning and understanding. From a constructivist perspective, focusing on meaning and conceptions is to be complemented by the demand to start from students' prior conceptions. By discussing two examples, the article introduces approaches for learning arrangements that affiliate horizontal and vertical conceptual change. Both were based on an interview study on grade 10 stu- dents' prior conceptions. The theoretical work, the empirical study and the design were conducted within the research program of "Educational Reconstruction".
... Thus, if such conventions are violated (by other cultures, or, perhaps, by machines) then shared understanding is lost and -mirroring Kuhnian paradigm shift -new conventions may need to be formed which accommodate the rogue participant. Kitcher [44], a philosopher of mathematics, elaborates what a mathematical practice might mean, suggesting a socio-cultural definition as consisting in a language and four socially negotiated sets: accepted statements, accepted reasonings, questions which are considered to be important and meta-mathematical views such as standards of proof and the role of mathematics in science (agreement over the content of these sets helps to define a mathematical culture). Mackenzie [52] looked at the role of proof, especially computer proof, and his student Barany [12] used ethnographic methods to trace the cycle of development and flow of mathematical ideas from informal thoughts, to seminar, to publication, to dissemination and classroom, and back to informal thoughts. ...
The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend.
The Study of Mathematical Practice is an emerging interdisciplinary field which draws on philosophy and social science to understand how mathematics is produced. Online mathematical activity provides a novel and rich source of data for empirical investigation of mathematical practice - for example the community question-answering system mathoverflow contains around 40,000 mathematical conversations, and polymath collaborations provide transcripts of the process of discovering proofs. Our preliminary investigations have demonstrated the importance of “soft” aspects such as analogy and creativity, alongside deduction and proof, in the production of mathematics, and have given us new ways to think about the roles of people and machines in creating new mathematical knowledge. We discuss further investigation of these resources and what it might reveal.
Crowdsourced mathematical activity is an example of a “social machine”, a new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity, and the subject of major international research endeavours. We outline a future research agenda for mathematics social machines, a combination of people, computers, and mathematical archives to create and apply mathematics, with the potential to change the way people do mathematics, and to transform the reach, pace, and impact of mathematics research.
... 25 A particularly striking example of the tenuousness of the Penrose assumption concerning mathematical intuitions was encountered when one of us (Grush) was discussing these issues with a mathematics graduate student, who admitted, in good faith, the following: "I firmly believe that Zorn's lemma is true, and I'm convinced that Zorn's lemma is equivalent to the axiom of choice, and yet I am certain that the axiom of choice must be false." 26 For a lucid and compelling discussion of a non-Platonist approach to mathematics, see Kitcher, 1984. these and similar cases, care and IQ notwithstanding. ...
Using the Godel incompleteness result for leverage, Roger Penrose has argued that the mechanism for consciousness involves quantum gravitational phenomena, acting through microtubules in neurons. We show that this hypothesis is implausible. First the Godel result does not imply that human thought is in fact non-algorithmic. Second, whether or not non-algorithmic quantum gravitational phenomena actually exist, and if they did how that could conceivably implicate microtubules, and if microtubules were involved, how that could conceivably implicate consciousness, is entirely speculative. Third, cytoplasmic ions such as calcium and sodium are almost certainly present in the microtubule pore, barring the quantum-mechanical effects Penrose envisages. Finally, physiological evidence indicates that consciousness does not directly depend on microtubule properties in any case, rendering doubtful any theory according to which consciousness is generated in the microtubules.
... This point of view has been developed with regards to mathematics in general, including proof, inKitcher (1984) andHersh (1998), among others. ...
Students’ mathematical lives are characterized not only by a set of mathematical ideas and the engagement in mathematical
thinking, but also by social relations, specifically, relations of authority. Watching student actions and speaking to students,
one becomes cognizant of a ‘web of authority’ ever present in mathematics classrooms. In past work, it has been shown how
those relations of authority may sometimes interfere with students’ reflecting on mathematical ideas. However, “…by shifting
the emphasis from domination and obedience to negotiation and consent…” (Amit & Fried, 2005, p.164) it has also been stressed
that these relations are fluid and are, in fact, asine qua non in the process of students’ defining their place in a mathematical community. But can these fluid relations be operative
also in the formation of specific mathematical ideas? It is my contention that they may at least coincide with students’ thinking
about one significant mathematical idea, namely, the idea ofproof. In this talk, I shall discuss both the general question of authority in the mathematics classroom and its specific connection
with students’ thinking about proof in the context of work done in two 8th grade classrooms.
The aim of this paper is to point to the analogy between mathematical and physical thought experiments, and even more widely between the epistemic paths in both domains. Having accepted platonism as the underlying ontology as long as the platonistic path in asserting the possibility of gaining knowledge of abstract, mind-independent and causally inert objects, my widely taken goal is to show that there is no need to insist on the uniformity of picture and monopoly of certain epistemic paths in the epistemic descriptive context. And secondly, to show the analogy with the ways we come to know the truths of (natural) sciences.
Kann Mathematik schön sein? Gibt es Leben in der Mathematik? In der Kritik der Urteilskraft (1790) untersucht Kant Prinzipien der Zweckmäßigkeit, eine subjektive Zweckmäßigkeit für die Ästhetik und eine objektive Zweckmäßigkeit für die Teleologie. Die Mathematik aber fällt bezüglich beider durch. Mathematische Gegenstände und Eigenschaften können nach Kant nicht schön sein und bei Erklärungen müssen wir keine Vorstellung von einem Zweck voraussetzen, denn wir können die Gegenstände konstruieren, meint Kant. Jedoch räumt er ein, mathematische „Demonstrationen“ könnten schön sein. Dies hängt mit seiner Unterscheidung von Anschauung und Begriff zusammen. Ich werde diesbezüglich einige seiner Ausführungen darstellen und problematisieren.
The specific aim of mathematics education as a research field is to study the factors that affect the teaching and learning of mathematics and to develop programs to improve the teaching of mathematics. In order to accomplish this aim mathematics education must consider the contributions of several disciplines: psychology, pedagogy, sociology, philosophy, etc. However, the use of these contributions in mathematics education must take into account and be based upon an analysis of the nature of mathematics and mathematical concepts, and their personal and cultural development. Such epistemological analysis is essential in mathematics education, for it would be very difficult to efficiently study the teaching and learning processes of undefined and vague objects.
As I was thinking about the rise of evolutionary ideas in twentieth century science, I remembered a passage in which the celebrated N.R. Campbell shows his attachment to an old metaphysics of time. It provides a telling measure of how bizarre and unexpected our present ideas must have initially been2.
Since 1974, date of publication of M. Ghiselin’s paper “A radical solution to the species problem”, a vigorous polemic has been raging among philosophers of biology on the question of whether biological taxa, and specially biospecies, are classes or individuals. Obviously they are neither, but the lack of adequate conceptual tools has tended to cloud the issue. D. Hull has convincingly shown that biospecies (entities which originate, evolve, and split or become extinct) are not classes, but has been less successful with his claim that species are individuals. Some philosophers, like A. Rosenberg and M. Williams, have accepted Hull’s position. Others, like P. Kitcher and A. Caplan, have remained unconvinced. Several biologists specially concerned with systematics have also accepted the claim that species are individuals, among them E. Mayr, N. Eldredge and R. Willmann. Many others are somehow bewildered by the use of the word individual (normally reserved for organisms) to refer to species.
The nineteenth century witnessed a gradual transformation of mathematics—in fact, a gradual revolution, if that is not a contradiction in terms. Mathematicians turned more and more for the genesis of their ideas from the sensory and empirical to the intellectual and abstract. Although this subtle change already began in the sixteenth and seventeenth centuries with the introduction of such nonintuitive concepts as negative and complex numbers, instantaneous rates of change, and infinitely small quantities, these were often used (successfully) to solve physical problems and thus elicited little demand for justification.
The invention of calculus is one of the great intellectual and technical achievements of civilization. Calculus has served for three centuries as the principal quantitative tool for the investigation of scientific problems. It has given mathematical expression to such fundamental concepts as velocity, acceleration, and continuity, and to aspects of the infinitely large and infinitely small—notions that have formed the basis for much mathematical and philosophical speculation since ancient times. Physics and technology would be impossible without calculus.
Our understanding of mathematics arguably increases with an examination of its growth, that is with a study of how mathematical theories are articulated and developed in time. This study, however, cannot proceed by considering particular mathematical statements in isolation, but should examine them in a broader context. As is well known, the outcome of the debates in the philosophy of science in the last few decades is that the development of science cannot be properly understood if we focus on isolated theories (let alone isolated statements). On the contrary, we ought to consider broader epistemic units, which may include paradigms (Kuhn 1962), research programmes (Lakatos 1978a), or research traditions (Laudan 1977). Similarly, the first step to be taken by any adequate account of mathematical change is to spell out what is the appropriate epistemic unit in terms of which the evaluation of scientific change is to be made. If we can draw on the considerations that led philosophers of science to expand the epistemic unities they use, and adopt a similar approach in the philosophy of mathematics, we shall also conclude that mathematical change is evaluated in terms of a‘broader’ epistemic unit than the one that is often used, such as, statements or theories.
The paper examines the possibility of a positive role for the educational perspective in progressive philosophy of mathematics. In particular, dialectalism in mathematics, as initiated by Imre Lakatos under the inspiration, among others, of the giant mathematics educator George Pólya, is looked into. Important “humanistic” issues such as fallibility and plausibility are addressed. Although still requiring further scrutiny, the quasi-empiricism presented at the very least proves to be a viable contender to the various mainstream a priori philosophical positions concerning the nature of mathematical knowledge/science
In the social study of science, researchers have only recently begun to pay attention to the importance of the cognitive dimension to describe and explain scientists’ decision-making. By understanding cognitive decision-making mechanisms it is possible to assign an exact causal value to methodological and social variables.
Was ist eine Zahl, daß ein Mensch sie kennen kann, und ein Mensch, daß er eine Zahl kennen kann? Diese Frage, von Warren McCulloch 1965 so großartig formuliert, ist eine der ältesten Fragen in der Philosophie der Naturwissenschaft — eine von jenen, die Platon und seine Schüler in den Wandelgängen der ersten Akademie vor 25 Jahrhunderten immer wieder erörterten. Ich frage mich gelegentlich, was wohl die großen Philosophen der Vergangenheit zu den neuen Befunden der Neurowissenschaft und der Kognitionspsychologie gesagt hätten. Zu welchen Dialogen hätten die Bilder der Positronen-Emissions-Tomographie die Platoniker angeregt? Welche drastischen Revisionen hätten die Experimente zur Arithmetik Neugeborener den englischen Empirikern auferlegt? Was hätte Diderot zu den neuropsychologischen Befunden gesagt, die die extreme Fragmentierung des Wissens im menschlichen Gehirn nachweisen? Welche tiefreichenden Einsichten hätte Descartes gehabt, wenn er nicht nur die gedanklichen Höhenflüge seiner Zeitgenossen, sondern auch die strengen Daten der heutigen Neurowissenschaften gekannt hätte?
Over the past quarter century mathematics education has become established world-wide as a major independent area of knowledge and research. The field now has numerous dedicated journals, book series and conferences serving national and international communities of scholars. Many countries offer specialist master’s and doctoral programs of study in mathematics education, and new entrants often receive their postgraduate education within the field itself. Given this coming of age, as we approach the 21st century it is appropriate to engage in a period of critical reflection and self-scrutiny. Thus the present volume embodying the ICMI-sponsored inquiry into the nature of research in mathematics education and its products provides a welcome opportunity for our field to take stock of itself, and its outcomes and effects.
In this paper I concentrate on two distinctions introduced by Patrick A. Heelan. At stake in the distinction between weak and strong hermeneutics of natural science is the issue of the possibility of an interpretative-ontological approach to the rationality of science. The distinction between cultural praxis-laden meaning and theory-laden meaning has much to do with a philosophico-hermeneutic critique of the account of scientific theory elaborated in the post-positivist philosophy of science. My primary aim is to show that the “hermeneutic turn” in the philosophy of science as informed by the two distinctions allows one to delineate a particular context of scrutinizing science. In opposing both the normative epistemology (the rational reconstruction of science’s cognitive structure) and the deconstruction of epistemology (the denunciation that there are aspects of science’s cognitive structure which have to be approached as non-empirical objects of inquiry), I shall treat this hermeneutic context of constitution as an alternative to the context of justification and the context of discovery. What I am referring to is an attempt to forge a notion of scientific rationality by studying the hermeneutic fore-structure of scientific research.1 It is my aim to show that in the context of constitution one can hold the view that (pace Rorty) the science-nonscience opposition “cuts culture at a philosophically significant joint” without appealing to the uniqueness of epistemological features like a special method, or a special relation to reality.
Being a nominalist convinced of the importance of mathematics is not a state to be recommended for its comfort. (Repeat several times a day: “The objects of mathematics are abstract. There are no abstract objects.”) The oldest ingredient of many remedies aiming to reduce the cognitive tension goes back to Aristotle. Aphairesis refers to the way in which mathematicians may treat of objects which are inseparable from real things, by restricting consideration to only some aspects of sensible things while overlooking others. 1 This term was rendered by Boethius as abstractio, together (unfortunately) with chorismos (“separation”), which referred to the (for Aristotle fictitious) postulation of Platonic ideas.2 While the employment of abstraction to explain how knowledge of the general (universal) Arises of mathematical knowledge was there from the start.
Arithmetical Logic is the title of a constructivist programme which aims to provide a logic of mathematics, rather than a mathematical logic. Specifically, it aims to provide a logic of arithmetic in which arithmetic is taken as the foundation of all mathematics. Such a logic of arithmetic is, in a sense, an arithmetization of logic, since the constructivist wishes to overturn the Fregean logicist perspective and make way for an internal logic of mathematical discourse. Here, formal logic does not have any special status; it is a theory of inference coupled with an arithmetical, rather than an algebraic or set-theoretic, representation theory.
As a mathematical platonist, I hold that mathematical objects are causally inert and exist independently of us and our mental lives. This obliges me to explain how we can refer to such alien creatures and acquire knowledge and beliefs about them.1 This paper is a piece of that larger project.
The core thesis of this contribution is that, if we wish to construct formal-logical models of mathematical practices, taking into account the maximum of detail, then it is a wise strategy to see mathematics as a heterogeneous entity. This thesis is supported by two case studies: the first one concerns a mathematical puzzle, the second one concerns Diophantine equations and belongs to mathematics proper. The advantage of the former is that the connection with logical modeling is pretty clear whereas the latter mainly demonstrates the difficulties one will have to overcome. A link is made with Hintikka’s method of analysis and synthesis.
In the 1880s, two men who had both been trained as mathematicians wrote short books defending the idea that arithmetic has an intimate relation with logic. Neither work was exactly a commercial or intellectual success. Frege’s Grundlagen (Frege 1884) went virtually unnoticed, and Frege recorded his disappointment and frustration in the introduction to his Grundgesetze (Frege, 1893, p. xi; Furth, 1967, p. 8). Dedekind’s monograph, Was Sind und was Sollen die Zahlen? (Dedekind, 1888), fared only a little better. Before Dedekind published it, he had been encouraged by the interest of other mathematicians in his project. In 1878, for example, Heinrich Weber urged him not to postpone his planned study on the number concept (Dugac, 1976, p. 273). However, when the book appeared, it made comparatively little stir: certainly, Dedekind’s discussion of the natural numbers aroused nothing like the interest excited by his study of the real numbers (Dedekind, 1872). Although many of Dedekind’s contemporaries viewed him as an important mathematician, they did not rank Was Sind und was Sollen die Zahlen? among his major achievements.1
Why should the question in my title be asked in a book concerned with assessment? Perhaps because it seems a matter of common sense that if we want to assess the mathematical knowledge of students, we need to be able to recognize that knowledge. How can we assess what we do not know? But as soon as we ask the question in its general form we notice that there has been no consensus among the mathematicians and philosophers who have tried to answer it. Common sense also dictates, then, that if we want to assess mathematical knowledge, we cannot afford to wait until the question of its global nature is settled.
This article deals with the question of how to teach conceptual didactical constructs
underlying actual core curricula, „nature of science“ (NOS) and „fundamental mathematical
ideas“ in pre-service teachers education. Contemporary NOS teaching, especially
in pre-service teacher education, is explicit, highly instructive, problem- and contextoriented,
and emphasizes reflective and meta-cognitive aspects. The question is posed in
how far a closer look onto such teaching concepts can be fruitful for teaching on fundamental
mathematical ideas, with a focus on pre-service teacher education. In a first step, NOS
and fundamental mathematical ideas are concretized and compared on a meta-conceptual
level. In a second step, this concretized understanding of fundamental mathematical ideas is
confronted with highly instructive ways of making NOS an explicit object of consideration
in pre-service teachers’ didactical seminars. The conclusion is to refrain from the instructive
character of explicitly teaching conceptual constructs for the case of fundamental mathematical
ideas in teachers’ education. In the last part of this paper, a short description of a preservice
teachers’ course on fundamental mathematical ideas is given which follows a different
explicit-reflective paradigm, keeping the orientation to problems and contexts, and the
consideration of meta-cognitive aspects.
This research explores how 48 seventh graders' ability to do proofs have changed as a result of a treatment consisting of 14 hours of instruction, aimed at developing formal approach to proofs, spread to 13-weeks. In this study, first activities focusing on formal proofs were carried out, in a classroom setting, followed by a Proof Test, designed to measure students' ability to do proofs. After the test, clinical interviews were carried out with the students. After the interventions, an improvement on students' ability to do proofs was observed. Students were more successful doing proofs by contradiction than other methods. Students clearly had difficulty in writing proofs by cases.
"Behavior which is effective only through the mediation of other persons has so many distinguishing dynamic and topographical properties that a special treatment is justified and indeed demanded" (Skinner, 1957, p. 2). Skinner's demand for a special treatment of verbal behavior can be extended within that field to domains such as music, poetry, drama, and the topic of this paper: mathematics. For centuries, mathematics has been of special concern to philosophers who have continually argued to the present day about what some deem its "special nature." Two interrelated principal questions have been: (1) Are the subjects of mathematical interest pre-existing in some transcendental realm and thus are "discovered" as one might discover a new planet; and (2) Why is mathematics so effective in the practices of science and engineering even though originally such mathematics was "pure" with applications neither contemplated or even desired? I argue that considering the actual practice of mathematics in its history and in the context of acquired verbal behavior one can address at least some of its apparent mysteries. To this end, I discuss some of the structural and functional features of mathematics including verbal operants, rule-and contingency-modulated behavior, relational frames, the shaping of abstraction, and the development of intuition. How is it possible to understand Nature by properly talking about it? Essentially, it is because nature taught us how to talk.
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Philosophers frequently link their discussions of progress in science and mathematics to the issues of scientific and mathematical realism. I don’t dispute that these connections can be made, but I think that questions of progress in mathematics and science are more complicated than this, and that perhaps the more important measures of progress are independent of questions of realism. So I want to begin by distinguishing several senses in which we might measure progress in mathematics. My investigation on this front ends on a pessimistic note: perhaps we can establish that mathematics as a whole makes progress, but it is unlikely that we can measure progress in one branch of mathematics or in one historical period against that in another.
The aim of this paper is two-fold: 1) to explore a particular variety of mathematical development that exemplifies progress; 2) to examine what bearing, if any, it has on the issue of mathematical realism. Often in the history of mathematics we find mathematicians employing concepts and algebraic techniques that produce a body of successful results, but their procedures only become properly intelligible in the context of definitions, concepts, and structures developed much later. My discussion of this progress to later intelligibility, particularly its bearing on the issue of realism, will take place against the background of analogous discussions in philosophy of science. Steady growth views of scientific progress, according to which scientific development is cumulative — a gradual accretion (without loss) of scientific truths and refinement (without jettisoning) of scientific methods — are taken to support scientific realism. Since Kuhn’s (1962), however, such views of progress, and with them scientific realism, have been constantly challenged. If Kuhn is right, the history of science exhibits such radical conceptual discontinuities that the image of science as revealing more and more truths about the world must be given up.
In 1977 when Appel, Haken and Koch used a computer to mathematically solve the century old four-color-problem philosopher Thomas Tymoczko thought that the epistemic justification in mathematics had been changed. Essentially, Tymoczko, and others, argue we can now have mathematical epistemic justification through a posteriori means. This has obvious implication in philosophy of mathematics and epistemology because this would be the first case where mathematics isn’t justified through a priori means of investigation. However, I ultimately disagree with Tymoczko. I argue that computer-aided-proofs still warrant an a priori means of justification. In order to show this, I refer to advances in philosophy of mind, mainly, the extended mind thesis. ). I will argue that our mind has evolved to enter into symbiotic relationships with non-organic entities in order to offload certain internal capacities. I believe that this is what constitutes humans amazing gift of rationality and intelligence. Thus, when we use a computer-aided-proof to solve unsurveyable proofs, we are really extending our minds into these cognitive tools and extending our method of proof checking to be more efficient and quicker. Thus, the a priori is saved because the computer is just a part of the causal cognitive loop that constitutes our mind.
This study has been prepared with the purpose to get the views of senior class Elementary Education Mathematics preservice teachers on proving. Data have been obtained via surveys and interviews carried out with 104 preservice teachers. According to the findings, although preservice teachers have positive views about using proving in mathematics teaching, it is seen that their experiences related to proving is limited to courses and they think proving is a work done only for the exams. Furthermore, they have expressed in the interviews that proving is difficult for them, and because of this reason they prefer memorizing instead of learning.
I will be discussing not Kant, but a Kantian issue. The issue is the classical question of how a priori truths are possible.
I will outline an approach to this issue, an approach the availability of which seems to me to have been overlooked in the
discussion of these matters in our century. The account I have to offer does, though, bear on several Kantian concerns and
Kantian projects, and I will try to indicate these links.
School mathematics reflects the wider aspect of mathematics as a cultural activity. From the philosophical point of view,
mathematics must be seen as a human activity both done within individual cultures and also standing outside any particular
one. From the interdisciplinary point of view, students find their understanding both of mathematics and their other subjects
enriched through the history of mathematics. From the cultural point of view, mathematical evolution comes from a sum of many
contributions growing from different cultures.
“Mathematics in education: Is there room for a philosophy of mathematics in school practice?” That was the central question
at the conference from which this volume grew. My answer to the question is yes—absolutely! In the article, I argue why and
how philosophical reflections should be included in mathematics classrooms. The general ideas will be explained by three examples
from classrooms.
Logical thinking as an expression of human reason grasps the actual reality by the basic forms of thinking: concept, judgment, and conclusion.
Mathematical thinking abstracts from logical thinking to disclose a cosmos of forms of potential realities hypothetically. Mathematics as a form of mathematical thinking can therefore support humans within their logical thinking about realities which, in particular, promotes sensible actions. This train of thought has
been convincingly differentiated by Peirce’s philosophical pragmatism and concretized by a “contextual logic” invented by members of the mathematics department at the TU Darmstadt.
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