While Garfinkel’s early work, captured in Studies in Ethnomethodology, has received a lot of attention and discussion, this has not been the case for his later work since the 1970s. In this paper, we critically examine the aims of Garfinkel’s later ethnomethodological studies of work programme and evaluate key ideas such as the ‘missing what’ in the sociology of work, ‘the unique adequacy requirements of methods’, and the notion of ‘hybrid studies’. We do so through a detailed engagement with a (...) study that has frequently been singled out as exemplary by Garfinkel for his studies of work programme, namely Livingston’s The Ethnomethodological Foundations of Mathematics. We show how Livingston uses the proof of Gödel’s Incompleteness Theorem as a way to exhibit the work involved in understanding mathematical proofs. We then discuss how Livingston uses this example to introduce a distinction between the written ‘proof account’ and the associated ‘lived work’ of working through that proof, which we argue allows Livingston to provide a powerful critique of a formalist understanding of the objectivity of mathematics. We then discuss three aspects that we find problematic in Livingston’s and Garfinkel’s claims about Livingston’s study. Firstly, we question whether written proofs are best conceived of as descriptions or accounts. Secondly, we interrogate whether an ethnomethodological study could teach mathematicians how to make discoveries. Thirdly, we throw doubt on Garfinkel’s claim that Livingston’s results are results in mathematics. We conclude this paper with a discussion of how Livingston’s study Gödel’s proof highlights key tensions in Garfinkel’s later work. Firstly, we argue that there exists an ambiguity in Garfinkel’s treatment of texts as ‘incompetent’. Secondly, we show that Garfinkel’s attempt to extend his idea of classical studies from sociology to other professions and disciplines is problematic. Finally, we question Garfinkel’s proposals to reorient ethnomethodological studies away from sociological audiences and ask whether ethnomethodological studies promise the delivery of a ‘large prize’ or, rather, provide something like ‘helpful therapy’. (shrink)

This paper claims that the awareness of crucial philosophical questions and controversies, which have arisen during the historical evolution of fundamental concepts, ideas and processes in mathematics, should be an essential component of the professional knowledge of student teachers who intend to teach children mathematics.

There are many ontologies of the world or of specific phenomena such as time, matter, space, and quantum mechanics1. However, ontologies of information are rather rare. One of the reasons behind this is that information is most frequently associated with communication and computing, and not with ‘the furniture of the world’. But what would be the nature of an ontology of information? For it to be of significant import it should be amenable to formalization in a logico-grammatical formalism. A candidate (...) ontology satisfying such a requirement can be found in some of the ideas of K. Turek, presented in this paper. Turek outlines the ontology of information conceived of as a part of nature, and provides the ‘missing link’ to the Z axiomatic set theory, offering a proposal for developing a formal ontology of information both in its philosophical and logicogrammatical representations. (shrink)

We propose a novel, ontological approach to studying mathematical propositions and proofs. By “ontological approach” we refer to the study of the categories of beings or concepts that, in their practice, mathematicians isolate as fruitful for the advancement of their scientific activity (like discovering and proving theorems, formulating conjectures, and providing explanations). We do so by developing what we call a “formal ontology” of proofs using semantic modeling tools (like RDF and OWL) developed by the computer science community. In this (...) article, (i) we describe this new approach and, (ii) to provide an example, we apply it to the problem of the identity of proofs. We also describe open issues and further applications of this approach (for example, the study of purity of methods). We lay some foundations to investigate rigorously and at large scale intellectual moves and attitudes that underpin the advancement of mathematics through cognitive means (carving out investigationally valuable concepts and techniques) and social means (like communication, collaboration, revision, and criticism of specific categories, inferential patterns, and levels of analysis). Our approach complements other types of analysis of proofs such as reconstruction in a deductive system and examination through a proof-assistant. (shrink)

Although mathematical philosophy is flourishing today, it remains subject to criticism, especially from non-analytical philosophers. The main concern is that even if formal tools serve to clarify reasoning, they themselves contribute nothing new or relevant to philosophy. We defend mathematical philosophy against such concerns here by appealing to its metaphysical foundations. Our thesis is that mathematical philosophy can be founded on the phenomenological theory of ideas as developed by Roman Ingarden. From this platonist perspective, the “unreasonable effectiveness of mathematics in (...) philosophy”—to adapt Wigner’s phrase—is analogous to that of mathematical explanations in science. As success-criteria for mathematical philosophy, we propose that it should be correct, responsive, illuminating, promising, relevant, and adequate. (shrink)

The aim of the presented article is to provide an in-depth analysis of the adequacy of designating Penrose as a complex Pythagorean in view of his much more common designation as a Platonist. Firstly, the original doctrine of the Pythagoreans will be briefly surveyed with the special emphasis on the relation between the doctrine of this school and the teachings of the late Platonic School as well as its further modifications. These modifications serve as the prototype of the contemporary claims (...) of the mathematicity of the Universe. Secondly, two lines of Penrose’s arguments in support of his unique position on the ontology of the mathematical structures will be presented: their existence independent of the physical world in the atemporal Platonic realm of pure mathematics and the mathematical structures as the patterns governing the workings of the physical Universe. In the third step, a separate line of arguments will be surveyed that Penrose advances in support of the thesis that the complex numbers seem to suit these patterns with exceptional adequacy. Finally, the appropriateness of designation Penrose as a complex Pythagorean will be assessed with the special emphasis on the suddle threshold between his unique position and that of the adherents of the mathematicity of the Universe. (shrink)

Aesthetics in philosophy of mathematics is too narrowly construed. Beauty is not the only feature in mathematics that is arguably aesthetic. While not the highest aesthetic value, being interesting is a sine qua non for publishability. Of the many ways to be interesting, being explanatory has recently been discussed. The motivational power of what is interesting is important for both directing research and stimulating education. The scientific satisfaction of curiosity and the artistic desire for beautiful results are complementary but both (...) aesthetic. (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)

. The circumstance, that the text of Imre Lakatos' doctoral thesis from the University of Debrecen did not survive, makes the evaluation of his career in Hungary and the research of aspects of continuity of his lifework difficult. My paper tries to reconstruct these newer aspects of continuity, introducing the influence of László Kalmár the mathematician and his fellow student, and Sándor Karácsony the philosopher and his mentor on Lakatos' work. The connection between the understanding of the empirical basis of (...) exact ideas—which is a common feature in the papers of the members of the Karácsonycircle—and Lakatos' way of thinking regarding mathematics is more direct and can be documented through his connection to Kalmár. The central element of Lakatos' philosophy of mathematics is criticism of formalism and his tendency is to use the empirical view. Discussions at the 1965 International Colloquium in the Philosophy of Science in London were very helpful in clarifying the quasi-empirical conception. Kalmár's lecture in London, based on one of his papers published by Karácsony in 1942, emphasized the empirical character of mathematics. After this colloquium some elements of the heritage of the Karácsony-circle were integrated again in the development of Lakatos' way of thinking. First I will analyze the Kalmár lecture of 1965 at the Colloquium of Philosophy of Science and Lakatos's reflections on the problem of the foundation of mathematics. Then I will present their common Hungarian background, their education and the beginning of their career, which have many important common features; third I draw attention to the network of contacts of Karácsony-disciples. (shrink)

On the basis of a wide range of historical examples various features of axioms are discussed in relation to their use in mathematical practice. A very general framework for this discussion is provided, and it is argued that axioms can play many roles in mathematics and that viewing them as self-evident truths does not do justice to the ways in which mathematicians employ axioms. Possible origins of axioms and criteria for choosing axioms are also examined. The distinctions introduced aim at (...) clarifying discussions in philosophy of mathematics and contributing towards a more refined view of mathematical practice. (shrink)

Some viewpoints on the foundations of mathematics and its philosophy are more connected to scientific practice and its heuristics, mainly with the construction of physical theories and the search for the best explanations of physical phenomena by means of abduction or the solution of problems by the analytical method. Some researchers have introduced the importance of human cultural activities into the cognitive aspects of the mental processes of scientists, proposing an embodied approach in the bridge between mathematics and reality. Fluid (...) mechanics is an interesting area in this sense due to its position if the network of mathematics. By means of an historical example on vortex motion by Helmholtz, we show that the intuitive idea of eddy contains cognitive properties of a mental schema and that it gives many heuristic options for an embodied mathematical explanation about vortex dynamics. (shrink)

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

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

The use of a reporter gene in transgenic mice indicates that there are many local mutations and large genomic rearrangements per somatic cell that accumulate with age at different rates per organ and without visible effects. Dissociation of the cells for monolayer culture brings out great heterogeneity of size and loss of function among cells that presumably reflect genetic and epigenetic differences among the cells, but are masked in organized tissue. The regulatory power of a mass of contiguous normal cells (...) is expressed in its capacity to normalize the appearance and growth behavior of solitary homophilic neoplastic cells, and to redirect differentiation of solitary heterophilic stem‐like cells. Intimate contact between the interacting cells is required to induce these changes. The normalization of the neoplastic phenotype does not require gap junctional communication between cells, though transdifferentiation might. These varied relationships are manifestations of the unifying biological principle of “order in the large over heterogeneity in the small”. BioEssays 28: 515–524, 2006. © 2006 Wiley Periodicals, Inc. (shrink)

In the complex teaching paradigm constructed and celebrated in classical Greek philosophy, geometry was the gateway to knowledge. Historically, mathematics provided the generational basis of education in Western civilization. Its impact as a disciplining subject was philosophically served by Plato’s most influential metaphysical involvement with the dialectical interplay of form and content, ideas and images, and the formal, hierarchic divisions of reality. Mathematics became a key--perhaps the key--for the establishment of natural, social and intellectual hierarchies in Plato’s work, and mathematical (...) capacities became synonymous with power--the power of abstraction needed to effect and control change. It was the provenience of the academic demands, levels, and achievements celebrated as culturally Western. From ancient Athens to medieval Europe, it became the principle of interactional/hierarchic selection: Similia similibus cognoscuntur--that is, only those things alike can interact cognitively; this launched the notion that mathematical ability is the best test for objectively determining who does and who does not qualify for a social, political, and intellectual meritocracy. (shrink)

This paper explores how a pluralist view can arise in a natural way out of the day-to-day practice of modern set theory. By contrast, the widely accepted orthodox view is that there is an ultimate universe of sets V, and it is in this universe that mathematics takes place. From this view, the purpose of set theory is “learning the truth about V.” It has become apparent, however, that the phenomenon of independence—those questions left unresolved by the axioms—holds a central (...) place in the investigation. This paper introduces the notion of independence, explores the primary tool for establishing independence results, and shows how a plurality of models arises through the investigation of this phenomenon. Building on a familiar example from Euclidean geometry, a template for independence proofs is established. Applying this template in the domain of set theory leads to a consideration of forcing, the tool par excellence for constructing universes of sets. Fifty years of forcing has resulted in a profusion of universes exhibiting a wide variety of characteristics—a multiverse of set theories. Direct study of this multiverse presents technical challenges due to its second-order nature. Nonetheless, there are certain nice “local neighborhoods” of the multiverse that are amenable to first-order analysis, and set-theoretic geology studies just such a neighborhood, the collection of grounds of a given universe V of set theory. I will explore some of the properties of this collection, touching on major concepts, open questions, and recent developments. (shrink)

The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment of the method of mathematics, the notion of demonstration, the questions of discovery and justification, the nature of mathematical objects, the character of mathematical definition, the role of intuition, the role of diagrams in mathematics, and the (...) effectiveness of mathematics in natural science. (shrink)