History and Philosophy of Mathematics
Course infoSchedule
Course moduleNWI-WM302
Credits (ECTS)3 - 6
CategoryMA (Master)
Language of instructionEnglish
Offered byRadboud University; Faculty of Science; Wiskunde, Natuur- en Sterrenkunde;
C.C. Kooloos
Other course modules lecturer
prof. dr. N.P. Landsman
Other course modules lecturer
prof. dr. N.P. Landsman
Other course modules lecturer
prof. dr. N.P. Landsman
Other course modules lecturer
Contactperson for the course
prof. dr. N.P. Landsman
Other course modules lecturer
Academic year2021
KW3-KW4  (31/01/2022 to 31/08/2022)
Starting block
Course mode
Registration using OSIRISYes
Course open to students from other facultiesYes
Waiting listNo
Placement procedure-
This course provides an introduction to the history and philosophy of mathematics (HPM), in interaction with each other as well as with mathematics education via: (i) the possible use of HPM in teaching; (ii) the influence of HPM on teaching; (iii) the history of math education itself. Our running theme will be that mathematics is both an axiomatic-deductive activity and an approximate description of reality. The tension between these two complementary aspects of mathematics goes back to antiquity and has deeply affected math education: think of “New Math” (based on abstraction) versus “Realistic Math” (based on applications).  

Students will be encouraged to look at original sources in the context of their times, starting with Euclid, down to the present. Students will understand that, despite superficial formal analogies, axiomatic-deductive reasoning in the style of Euclid profoundly differs from modern mathematics. They will learn that the transition to the modern era gradually took place in the 19th century via developments in (non-Euclidean) Geometry, Analysis, and Algebra. Students will learn how axiomatic thinking was revived and was and given a new, formal interpretation especially by Hilbert around 1900. Simultaneously, modern mathematical logic was invented. Students will understand how these developments led to a highly abstract view of mathematics, ultimately based on axiomatic set theory. Students will see how applied mathematics and mathematical physics unexpectedly and deeply benefited from this turn to abstraction. Teacher trainees will learn how to develop classes and more generally deepen their teaching based on these insights.
Basic early history of science. The beginnings of (axiomatic-deductive) mathematics in Greece, ~500-300 B.C. Plato versus Aristotle about the nature of mathematical objects. Origins of logic and deduction. Euclid. Archimedes. Jump to Newton. From Calculus to Analysis (Euler). Analysis made rigorous (Cauchy, Dirichlet, Weierstrass). Non-Euclidean geometry (Gauss, Bolyai, Lobachevsky, Riemann). Origins of modern algebra. Loss of visualisation. From calculations to concepts. New axiomatization of Euclidean geometry (Hilbert). Frege-Hilbert correspondence. Creation of set theory (Riemann, Dedekind, Cantor, Zermelo, von Neumann). Development of mathematical logic (Frege, Peano, Russell, Skolem, Hilbert, Gödel). Platonism, structuralism, and constructivism. Lakatos. History and philosophy of mathematics education, as outlined above (with emphasis on the Netherlands). Students are welcome to add their own interests to this list!

Instructional modes
Lectures, assigning texts to read and interpret, small research assignments, discussions and debates, presentations by students.

Presumed foreknowledge
bachelor degree in mathematics
Test information
Testing is partly done by monitoring active participation in discussions based on previous reading material, partly through individual presentations, and partly through a final assignment (which for teacher trainees consists in designing a class relevant to the history and philosophy of mathematics, and for other students consists of an essay).
Literature will be made available during the course. Books from which chapters will be read and discussed may include
D.C. Lindberg, The Beginnings of Western Science (2007), 
C. Dutilh Novaes, The Dialogical Roots of Deduction (2021),
R. Hartshorne, Geometry: Euclid and Beyond (2000),
J. Grey, Plato’s Ghost: The Modernist Transformation of Mathematics (2008),
J. Ferreiros, Labyrinth of Thought: A History of Set Theory and its Role in Mathematics (2007),
L. Haaparanta (ed.), The Development of Modern Logic (2009),
O. Linnebo, Philosophy of Mathematics (2017),
I. Hacking, Why is there Philosophy of Mathematics at All? (2014),
F. Goffree, M. van Hoorn, B. Zwaneveld, Honderd Jaar Wiskundeonderwijs (2000),
P. Ernest, The Philosophy of Mathematics Education (1991),
H. Freudenthal, Revisiting Mathematics Education: China Lectures (2002),
W.P.  Berlinghoff &  F.Q.  Gouvêa, Math through the Ages: A Gentle History for Teachers and Others (2002), vertaald als Wortels van de Wiskunde, Een historisch overzicht voor leraren en anderen (2019).
Instructional modes
Course occurrence

Tentamen 3 ec
Test weight1
OpportunitiesBlock KW3, Block KW4

Tentamen 6 ec
Test weight1
OpportunitiesBlock KW4, Block KW4