Making and Breaking Mathematical Sense

Mathematical Thinking and Learning, 2019

Foundations of Science, 2006

This was a talk at "Theology in Mathematics?" (Kraków, Poland, June 8-10, 2014). The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern is nineteenth-century mathematics. Theology was present in modern mathematics not through its objects or methods, but mainly through popular philosophy, which absolutized mathematics. Moreover, modern pure mathematics was treated as a sort of quasi-theology; a long-standing alliance between theology and mathematics made it habitual to view mathematics as a divine knowledge, so when theology was discarded, mathematics naturally took its place at the top of the system of knowledge. It was that cultural expectation aimed at mathematics that was substantially responsible for a great resonance made by set-theoretic paradoxes, and, finally, the whole picture of modern mathematics.

Semiotic Review, 2020

Starting from the profound impact of Kenneth Arrow's Impossibility Theorem on the social sciences of the postwar twentieth century, this essay engages with the ways in which mathematics can be seen as a language-ideologically inflated notational system. In the mid-twentieth century, a profound belief in mathematics as a purely objective and non-ideological organization of knowledge took hold, and mathematical proof became the most authoritative type of statement on reality. When something was ruled 'logically impossible', real-world occurences could be seen as transgressions and exceptions. Hidden inside this belief is a set of irrational, metaphysical assumptions about humans and social behavior that can be laid bare by means of linguistic-anthropological analysis.

Journal for Research in Mathematics Education, 2012

Imagine you have woken from a dream, a dream brimming with meaning, with passion, with mystery. You try to sustain the feeling, recount the details, share the experience. You fail. Your powers of reconstruction too meager, your tongue too clumsy. Mathematics is such a dream, dreamed by individuals, personal, yet remarkably in a waking state, and provoking sufficient commonality in its recounting to bring individuals together, to create a community of shared passion. The first dreamer of the dream, shrouded in history and myth, perhaps was Thales of Miletus, who later advised Pythagoras of Samos who subsequently founded an order of adherents holding knowledge and property in common while pursuing philosophical and mathematical studies as a moral basis for the conduct of life

Archimedes Series 66, 2023

This colective book, edited by Karine Chemla, J Ferreirós, Lizhen Ji, Erhard Scholz & Chang Wang, is a unique introduction to historiographical questions concerning the history of mathematics, with essays by many leading scholars, aimed at guiding newcomers to the field. It provides multiple perspectives on mathematics, its role in culture, development, connections with other sciences, with philosophy, etc.

The Mathematics Enthusiast, 2012

In this work we present some reflections on mathematics and mathematicians. Special emphasis is placed on the questions (1) what is mathematics? And (2) what is a mathematician? Some reflections and open questions are posed at the end of the work. 0. Introduction. Professions have played a key role in the development of disciplinarily, and vice versa. Within some disciplines the direct binding to a profession or a field have over time been loosened and (re)searching knowledge for its own sake has become a main driving force of a new, advanced kind of disciplinarily. For mathematics these historical shifts are symptomatic in the debates over the discipline's true nature. While the relationship between science, technology and mathematics historically the last 200 years has been rather symbiotic, mathematics today serve so many different professions and fields, that a unified, valid definition of its nature is hard to find.

THE CONSTRUCTION OF MATHEMATICAL KNOWLEDGE: CURIOSITIES IN A LACON AND PHILOSOPHICAL PERSPECTIVE (Atena Editora), 2022

When he stopped being a nomad and started to live a principle of society, man began to feel the need to count. The first idea gave rise to one-to-one correspondence (basis for working with concepts involving function). The fingers of the hands, stones (from the Latin, calculus) become instruments used in this correspondence, whose decimal traces are preserved and considered to this day. However, groups of stones have ephemeral characteristics that make it impossible to preserve numerical information in the long term. The first innovative idea was to register values ​​through marks on bones and sticks. Thus begins a language that was developed within principles, whose objective was to facilitate the work of the man who calculated (even without him knowing it). “The tendency of language to develop from the concrete to the abstract can be seen in many of the measures of length in use today. A horse's height is measured in “spans” and the words “foot” and “ell” (elbow, elbow) also derive from body parts” (BOYER, 1996, p. 04). All this contributed to the development of mathematics, transforming it into something much bigger than just counting and measuring. The origin of the first indications of a structure, which would later be called mathematics, is lost in times without records. Sometimes it is due to practical necessity (in Herodotus' view - the “rope-drawers” ​​of ancient Egypt, for example), sometimes to priestly and ritual leisure (in Aristotle's view). Therefore, our chronological observation will have as a principle to highlight important points according to personal motivations, arranged on a firmer ground in the history of mathematics, recorded in documents that have been preserved.

Journal of Urban Mathematics Education, 2011

Skovsmose’s previous work, digging into differing ways that mathematics operates, orients the reader to the significance of his most recent book In Doubt—About Language, Mathematics, Knowledge and Life-Worlds (2009). This new book dissects the important questioning of the legitimacy of knowledge, in particular mathematics, in the context of a just, democratic, and equitable education. Skovsmose stakes a strong claim that the modern view of knowledge, and in particular mathematics, must be taken in doubt; that “the ideas of objectivity, certainty, transparency, progress, and neutrality should be considered myths, and that the modern conception of knowledge is an illusion” (p. 73). To trouble the ways in which mathematics, as a form of knowledge, governs (Foucault, 1980; Popkewitz, 2004) mathematics education, puts at the fore the socio-political nature of the teaching and research practices of the field of mathematics education. Throughout In Doubt, Skovsmose works further to address a fundamental question that locates his work regarding “the relationship of traditionally established areas of know- ledge: What is the status of [mathematics] in critical education” (Skovsmose, 1994, p. 24)?

Springer/Nature, Heidelberg, 2022

This book is a philosophical study of mathematics, pursued by considering and relating two aspects of mathematical thinking and practice, especially in modern mathematics, which, having emerged around 1800, consolidated around 1900 and extends to our own time, while also tracing both aspects to earlier periods, beginning with the ancient Greek mathematics. The first aspect is conceptual, which characterizes mathematics as the invention of and working with concepts, rather than only by its logical nature. The second, Pythagorean, aspect is grounded, first, in the interplay of geometry and algebra in modern mathematics, and secondly, in the epistemologically most radical form of modern mathematics, designated in this study as radical Pythagorean mathematics. This form of mathematics is defined by the role of that which beyond the limits of thought in mathematical thinking, or in ancient Greek terms, used in the book’s title, an alogon in the logos of mathematics. The outcome of this investigation is a new philosophical and historical understanding of the nature of modern mathematics and mathematics in general. The book is addressed to mathematicians, mathematical physicists, and philosophers and historians of mathematics, and graduate students in these fields.