The Concept of Culture in Critical Mathematics Education

© Springer International Publishing AG, part of Springer Nature 2018. This is a post-peer-review, pre-copyedit version of a chapter published in The Philosophy of Mathematics Education Today. The final authenticated version is available online at: http://dx.doi.org/10.1007/978-3-319-77760-3A well-known critique in the research literature of critical mathematics education suggests that framing educational questions in cultural terms can encourage ethnic-cultural essentialism, obscure conflicts within cultures and promote an ethnographic or anthropological stance towards learners. Nevertheless, we believe that some of the obstacles to learning mathematics are cultural. ‘Stereotype threat’, for example, has a basis in culture. Consequently, the aims of critical mathematics education cannot be seriously pursued without including a cultural approach in educational research. We argue that an adequate conception of culture is available and should include normative/descriptive and material/ideal dyads as dialectical moments

The effect of railway network evolution on the Kaliningrad region's landscape environment

This article addresses methodology of modern landscape studies from the perspective of natural and man-made components of a territory. Railway infrastructure is not only an important system-building element of economic and settlement patterns; it also affects cultural landscapes. The study of cartographic materials and historiography made it possible to identify the main stages of the development of the Kaliningrad railway network in terms of its territorial scope and to describe causes of the observed changes. Historically, changes in the political, economic, and military environment were key factors behind the development of the Kaliningrad railway network. Nature was less important. The existing Kaliningrad railway network is to a great degree the legacy of the earlier, pre-war times. Today, its primary function is to provide international cargo and passenger transportation. Two types of railway infrastructure are identified in the Kaliningrad region - modern (functioning) and relic (abandoned) ones. In the Kaliningrad region, the process of land reclamation of the railway system starts when the maintenance of railroads is discontinued, which is followed by the formation of primitive soils and emerging biocenoses enhanced by fill soils and artificial relief

On the Mathematical Constitution and Explanation of Physical Facts

The mathematical nature of modern physics suggests that mathematics is bound to play some role in explaining physical reality. Yet, there is an ongoing controversy about the prospects of mathematical explanations of physical facts and their nature. A common view has it that mathematics provides a rich and indispensable language for representing physical reality but that, ontologically, physical facts are not mathematical and, accordingly, mathematical facts cannot really explain physical facts. In what follows, I challenge this common view. I argue that, in addition to its representational role, in modern physics mathematics is constitutive of the physical. Granted the mathematical constitution of the physical, I propose an account of explanation in which mathematical frameworks, structures, and facts explain physical facts. In this account, mathematical explanations of physical facts are either species of physical explanations of physical facts in which the mathematical constitution of some physical facts in the explanans are highlighted, or simply explanations in which the mathematical constitution of physical facts are highlighted. In highlighting the mathematical constitution of physical facts, mathematical explanations of physical facts deepen and increase the scope of the understanding of the explained physical facts. I argue that, unlike other accounts of mathematical explanations of physical facts, the proposed account is not subject to the objection that mathematics only represents the physical facts that actually do the explanation. I conclude by briefly considering the implications that the mathematical constitution of the physical has for the question of the unreasonable effectiveness of the use of mathematics in physics

Is There A Monist Theory of Causal and Non-Casual Explanations? The Counterfactual Theory of Scientific Explanation

The goal of this paper is to develop a counterfactual theory of explanation (for short, CTE). The CTE provides a monist framework for causal and non-causal explanations, according to which causal and non-causal are explanatory by virtue of revealing counterfactual dependencies between the explanandum and the explanans. I argue that the CTE is applicable to two paradigmatic examples of non-causal explanations: Euler's explanation and renormalization group explanations of universality

O znanych i mniej znanych relacjach Leonharda Eulera z Polską

In this work we focus on research contacts of Leonhard Euler with Polish scientists of his era, mainly with those from the city of Gdańsk (then Gedanum, Danzig). L. Euler was the most prolific mathematician of all times, the most outstanding mathematician of the 18th century, and one of the best ever. The complete edition of his manuscripts is still in process (Kleinert 2015; Kleinert, Mattmüller 2007).Euler’s contacts with French, German, Russian, and Swiss scientists have been widely known, while relations with Poland, then one of the largest European countries, are still in oblivion. Euler visited Poland only once, in June of 1766, on his way back from Berlin to St. Petersburg. He was hosted for ten days in Warsaw by Stanisław II August Poniatowski, the last king of Poland. Many Polish scientists were introduced to Euler, not only from mathematical circles, but also astronomers and geographers. The correspondence of Euler with Gdańsk scientists and officials, including Carl L. Ehler, Heinrich Kühn and Nathanael M. von Wolf, originated already in the mid-1730s. We highlight the relations of L. Euler with H. Kühn, a professor of mathematics at the Danzig Academic Gymnasium and arguably the best Polish mathematician of his era. It was H. Kühn from whom Euler learned about the Königsberg Bridge Problem; hence one can argue that the beginning of the graph theory and topology of the plane originated in Gdańsk. In addition, H. Kühn was the first mathematician who proposed a geometric interpretation of complex numbers, the theme very much appreciated by Euler.Findings included in this paper are either unknown or little known to a general mathematical community.W tej pracy skupiamy się na kontaktach badawczych Leonharda Eulera z polskimi naukowcami jego epoki, głównie z Gdańska (wtedy Gedanum, Danzig). L. Euler był najbardziej płodnym matematykiem wszystkich czasów, najwybitniejszym matematykiem osiemnastego wieku i jednym z najlepszych w historii. Kompletne wydanie jego rękopisów nie zostało dotąd zakoń- czone (Kleinert 2015; Kleinert, Mattmüller 2007).Kontakty Eulera z francuskimi, niemieckimi, rosyjskimi i szwajcarskimi naukowcami są powszechnie znane, a stosunki z Polską, wtedy jednym z największych krajów europejskich, są nadal zapomniane. Euler odwiedził Polskę tylko raz, w czerwcu 1766 roku, w drodze powrotnej z Berlina do Petersburga.Ostatni król Polski Stanisław August poniatowski gościł Eulera w Warszawie przez dziesięć dni. Wielu polskich naukowców przedstawiono Eulerowi, nie tylko z kręgów matematycznych, ale również astronomów i geografów. Korespondencja Eulera z gdańskimi naukowcami i urzędnikami, w tym Carlem L. Ehlerem, Heinrichem Kühnem i Natanaelem M. von Wolfem zaczęła się już w połowie lat 30. XVIII wieku. Wyróżniamy relacje L. Eulera z H. Kühnem, profesorem matematyki w Gimnazjum Akademickim w Gdańsku i prawdopodobnie najlepszym polskim matematykiem tamtej epoki. To od H. Kühna Euler dowiedział się o problemie mostów królewieckich. Dlatego można argumentować, że początek teorii grafów i topologii płaszczyzny wywodzi się z Gdańska. Ponadto, H. Kühn był pierwszym matematykiem, który zaproponował interpretację geometryczną liczb zespolonych, bardzo cenioną przez Eulera.Ustalenia zawarte w niniejszym artykule są albo nieznane lub mało znane ogólnej społeczności matematyków

Lietuvos Didžiosios Kunigaikštystės pašto kelių struktūra XVI–XVIII amžiuje

This paper analyzes the dynamics of the postal route system of the Grand Duchy of Lithuania in the 16th–18th centuries. The first postal route connected Vilnius and Krakow in 1562 – weekly postal services were rendered. In 1669, postal carriages started running from Moscow to Vilnius and then further on through Tilsit to Königsberg. The GDL postal route network underwent its largest expansion in the 18th century. An important postal line proceeded along the route of Warsaw – Grodno – Kaunas – Jelgava – Riga. Part of it coincided with the Warsaw – Vilnius route; at Ratnyčia, the postal carriage would turn northeast and continue via Merkinė and Varėna to Vilnius. From Vilnius, one postal route led to Königsberg via Kaunas, and another to Moscow; there are also data about a postal line to Polotsk. Another crossroads of the GDL’s postal routes was Grodno. The routes leading from Warsaw to Kaunas, Vilnius, and Riga intersected there. Separate lines to Lublin and Slonim were in operation; one of the postal routes led to the border of the Russian Empire. The network of the GDL’s postal routes also consisted of other roads. The location of some of them were subject to change due to the political situation, natural disasters, and seasonal practicability.Tiriamas Lietuvos Didžiosios Kunigaikštystės pašto kelių sistemos susidarymas ir dinamika XVI–XVIII a. Pirmas pašto kelias Vilnių ir Krokuvą sujungė 1562 m., juo kartą per savaitę važiavo pašto karieta. 1669 m. ėmė kursuoti paštas tarp Maskvos ir Vilniaus, o iš pastarojo per Tilžę jis vežtas į Karaliaučių. Labiausiai LDK pašto kelių tinklas išsiplėtojo XVIII a. Reikšminga linija buvo Varšuva–Gardinas–Kaunas–Jelgava–Ryga. Dalis to paties maršruto nuo Varšuvos atitiko ir pašto kelią į Vilnių, tik prie Ratnyčios buvo pasukama į šiaurės rytus ir pro Merkinę bei Varėną vykstama į LDK sostinę. Iš Vilniaus vedė keliai į Karaliaučių, per Kauną; kita kryptimi – į Maskvą; žinomas maršrutas į Polocką. Dar viena LDK pašto kelių kryžkelė buvo Gardinas. Pro jį važiuota iš Varšuvos į Kauną, Vilnių, Rygą. Iš jo ėjo atskiros linijos į Liubliną, Slanimą. Driekėsi maršrutas net iki Rusijos imperijos sienos. LDK pašto kelių tinklą sudarė ne tik regioniniai, bet ir labiau lokalūs maršrutai. Kai kurių kelių išsidėstymas galėdavo keistis, tam įtakos turėdavo politinės aplinkybės, stichinės nelaimės, sezoninis kelių pravažumas

On the Role of Mathematics in Scientific Representation

In this dissertation, I consider from a philosophical perspective three related questions concerning the contribution of mathematics to scientific representation. In answering these questions, I propose and defend Carnapian frameworks for examination into the nature and role of mathematics in science. The first research question concerns the varied ways in which mathematics contributes to scientific representation. In response, I consider in Chapter 2 two recent philosophical proposals claiming to account for the explanatory role of mathematics in science, by Philip Kitcher, and Otavio Bueno and Mark Colyvan. My novel and detailed critique of these accounts shows that they are too limited to encompass the diverse roles of mathematics in science in historical and contemporary scenarios. The conclusion is that any such philosophical account should aim to faithfully capture the structure of our theories and their use in applied contexts. This insight prompts the second question guiding this dissertation that I consider in Chapter 3, regarding a viable philosophical account of the role of mathematics in scientific theories. I respond by proposing a modified form of the reconstructive frameworks for philosophical analysis developed by Rudolf Carnap for theoretical entities. I propose three amendments to Carnap’s account: i) a semantic view for the representation of theories, ii) a careful consideration of instances of the use of theory in representing target systems, and iii) consideration of the practical complexity of relating theory to experimental data. The final research question for this dissertation asks what, if anything, we can legitimately conclude about the nature of theoretical entities invoked by a theory in light of its success in representing phenomena. In the backdrop of the Carnapian frameworks proposed in Chapter 3, I argue that contemporary ontological debates in the philosophy of science are largely premised on an acceptance of Willard Quine’s epistemological outlook on the world and a dismissal of Carnap’s approach, which can be used to offer a satisfactory deflationary resolution. This is in the service of my contention that a Carnapian attitude to central issues in the philosophy of science is decidedly preferable to the route championed by Quine