Standard histories of mathematics and of analytic philosophy contend that work on the foundations of mathematics was motivated by a crisis such as the discovery of paradoxes in set theory or the discovery of non-Euclidean geometries. Recent scholarship, however, casts doubt on the standard histories, opening the way for consideration of an alternative motive for the study of the foundations of mathematics—unification. Work on foundations has shown that diverse mathematical practices could be integrated into a single framework of axiomatic systems (...) and that much of mathematics could be expressed in a single language. The new framework was the product of an interdisciplinary coalition whose ideas resemble those later adopted by the Vienna Circle and logical empiricists. (shrink)

This historiographical study aims at introducing the category of “situated mathematics” to the case of Ancient Egypt. However, unlike Situated Learning Theory, which is based on ethnographic relativity, in this paper, the goal is to analyze a mathematical craft knowledge based on concrete particulars and case studies, which is ubiquitous in all human activity, and which even covers, as a specific case, the Hellenistic style, where theoretical constructs do not stand apart from practice, but instead remain grounded in it.The historiographic (...) interpretation that we will give of situated mathematics is inscribed in a characterization of mathematical styles that focuses on the role of mathematical practice. This categorization describes three types of mathematization, where, on the one hand, type I represents the classical and dominant Hellenocentric approach, which seeks to generate a body of principles that could then be applied in other fields. On the other hand, types II and III represent two kinds of situated mathematics, a parametrized and a concrete one. Type II proceeds in the opposite direction from Type I describing an application of a previously obtained theory. That is, given a practice in any domain, it seeks to build a mathematical systematization a posteriori to explain said practice. Finally, type III starts from a concrete practice and develops another similar practice that explains analogically the relationship.Based on the typology adopted, we seek to describe a case study within ancient Egyptian mathematics, which reveals how it is possible to subsume it in the two types of situated mathematization II and III. The foregoing will allow to bridge the gap between theory and practice. (shrink)

Some classic historical vignettes depict scientists achieving breakthroughs without effort: Archimedes grasping the principles of buoyancy while bathing, Galileo Galilei discovering the isochrony of the pendulum while sitting in a cathedral, James Watt noticing the motive power of steam while passing time in a kitchen, Alexander Fleming finding penicillin in Petri dishes that he had omitted to clean before going on holiday. These stories suggest that, to establish important findings in science, hard work is not always necessary. In this article, (...) I suggest that such stories capture an important aspect of scientific.. (shrink)

In the previous article (Found. Phys. Lett. 16:325–341, 2003), we showed that a reciprocity of the Gauss sums is connected with the wave and particle complementary. In this article, we revise the previous investigation by considering a relation between the Gauss optics and the Gauss sum based upon the recent studies of the Weil representation for a finite group.

This paper develops some respects in which the philosophy of mathematics can fruitfully be informed by mathematical practice, through examining Frege's Grundlagen in its historical setting. The first sections of the paper are devoted to elaborating some aspects of nineteenth century mathematics which informed Frege's early work. (These events are of considerable philosophical significance even apart from the connection with Frege.) In the middle sections, some minor themes of Grundlagen are developed: the relationship Frege envisions between arithmetic and geometry and (...) the way in which the study of reasoning is to illuminate this. In the final section, it is argued that the sorts of issues Frege attempted to address concerning the character of mathematical reasoning are still in need of a satisfying answer. (shrink)

Seventeenth-century “chance combinatorics” was a self-contained theory. It had an objective notion of chance derived from physical devices with chance properties, such as casts of dice, combinatorics to count chances and, to interpret their significance, a rule for converting these counts into fair wagers. It lacked a notion of chance as a measure of belief, a precise way to connect chance counts with frequencies and a way to compare chances across different games. These omissions were not needed for the theory’s (...) interpretation of chance counts: determining which are fair wagers. The theory provided a model for how indefinitenesses could be treated with mathematical precision in a special case and stimulated efforts to seek a broader theory. (shrink)

یکی از مهم‌ترین ویژگی‌های علم نوین روش کمّی و ریاضیاتی آن است. باور رایج این است که این مبنای علم نوین در قرن شانزدهم و در انقلاب علمی به همراه خود علم نوین به وجود آمده است، لکن بررسی‌های تاریخی حاکی از آن است که روش کمّی علم نه در بحبوحه ظهور علم نوین در انقلاب علمی، بلکه در منازعات الهیاتی قرون وسطای متأخر متولد شد. این نکته از جهت روشن کردن رابطه‌ای که علم و دین در طول تاریخ مغرب‌زمین (...) داشته‌اند حائز اهمیت است. شواهدی حاکی از آن است که متألهان مسیحی، با انگیزه‌های الهیاتی، مسائلی را مطرح کردند که بستر مناسبی را برای پیدایش تجربه‌گرایی به وجود آورد. در نوشتار حاضر عوامل پیدایش روش کمّی را در قرون وسطای متأخر و در میان متألهان مسیحی مورد بررسی قرار خواهیم داد. از خلال این بررسی آشکار خواهد شد که در این عصر سه عامل مهم که با انگیزه‌های الهیاتی همراه بودند در به وجود آمدن روش کمّی موثر واقع شدند: توجه به نورشناسی با انگیزه‌های الهیاتی، افلاطون‌گرایی در الهیات، و وجود انگیزۀ الهیاتی در استفاده از ریاضیات. در ضمنِ این تحلیل، رابطۀ تجربه‌گرایی و الهیات مسیحی در قرون وسطای متأخر روشن خواهد شد. (shrink)

Elements of notation are variables and sentences are sequences of different variables. Both listening and reading are processes, which makes a sentence a stream of variations of a single variable. Thus, a simple sentence is a one-dimensional object, measured along the stream of variation. A sentence with coordinated or subordinated material effectively encodes multiple streams which makes a complex sentence a two-dimensional object with that second dimension measured across the multiple streams. A single symbol does not vary and is therefore (...) a zero-dimensional object. Processes exist in time and the dimensions of notation are therefore abstract dimensions. (shrink)

There is not a clear consensus on the categorization framework of the educational values of the history of mathematics. By analyzing 20 Chinese teaching cases on integrating the history of mathematics into mathematics teaching based on the relevant literature, this study examined a new categorization framework of the educational values of the history of mathematics by combining the objectives of high school mathematics curriculum in China. This framework includes six dimensions: the harmony of knowledge, the beauty of ideas or methods, (...) the pleasure of inquiries, the improvement of capabilities, the charm of cultures, and the availability of moral education. The results show that this framework better explained the all-educational values of the history of mathematics that all teaching cases showed. Therefore, the framework can guide teachers to better integrate the history of mathematics into teaching. (shrink)

We analyze the connections of Lavoisier system of nomenclature with Leibniz’s philosophy, pointing out to the resemblance between what we call Leibnizian and Lavoisian programs. We argue that Lavoisier’s contribution to chemistry is something more subtle, in so doing we show that the system of nomenclature leads to an algebraic system of chemical sets. We show how Döbereiner and Mendeleev were able to develop this algebraic system and to find new interesting properties for it. We pointed out the resemblances between (...) Leibniz program and Lavoisier legacy, particularly regarding the lingua philosophica for understanding and thinking Nature, in this particular case, chemistry. In the second part we discuss, from the linguistic viewpoint, how Lavoisian algebraic system may be taken further to build a language. We study the constituents of such a chemical language. Finally, we formalize some of the ideas here presented by using elements of network theory and discrete mathematics. (shrink)

In this note we reproduce the book reviews that De Morgan wrote on Boole's and Jevons's first logical works. The most notable property of these documents is the mere fact of their existence and the absence of any reference to them in the specialized literature.

The paper discusses to what extent the conceptual issues involved in solving the simple harmonic oscillator model fit Wigner’s famous point that the applicability of mathematics borders on the miraculous. We argue that although there is ultimately nothing mysterious here, as is to be expected, a careful demonstration that this is so involves unexpected difficulties. Consequently, through the lens of this simple case we derive some insight into what is responsible for the appearance of mystery in more sophisticated examples of (...) the Wigner problem. (shrink)

(1995). In the shadow of giants: The work of mario pieri in the foundations of mathematics. History and Philosophy of Logic: Vol. 16, No. 1, pp. 107-119.

Invention by Design: How Engineers Get from Thought to Thing by Henry PetroskiBut Is It Science? The Philosophical Question in the Creation/Evolution Controversy by Michael RuseImpure Science: Aids, Activism and the Politics of Knowledge by Steven EpsteinA purposeless history and a ‘ Brave New World’ for animalsCity of Bits: Space, Place and the Infobahn by William J. Mitchell and Telecommunications and the City: Electronic Spaces, Urban Places by Stephen Graham and Simon MarvinExpecting Trouble: Surrogacy, Fetal Abuse & New Reproductive Technologies (...) edited by Patricia BolingWhose science? Which justice?Beyond the diffusionist history of colonial science. (shrink)