Table of Contents
History of Mathematics
Overview
This SMSTC supplementary module introduces postgraduate mathematics students to the methods and perspectives of the history of mathematics. We will examine mathematical sources from the oldest surviving written texts (tallies on bones and clay tablets) to the near-present alongside a variety of historical works to develop a critical approach to the history of mathematical theory and practice in its social, cultural, political, and other contexts.
Instructors
This module is co-taught by Michael Barany (Edinburgh) and Deborah Kent (St Andrews). You can email us at our respective university emails (ed.ac.uk and st-andrews.ac.uk), which are easily sleuthed.
Logistics
Meeting time: 11:15-12:45 Tuesdays, starting 16 January.
Location: You will receive a Zoom link from SMSTC. If you are in Glasgow, Dundee, or Edinburgh you are encouraged to join with peers (in Edinburgh, with 0-2 of the instructors, depending on the day) in a local SMSTC video conferencing room. In-person participants should still bring a device to connect (on mute/silent) to Zoom if possible, for a better experience interacting with those at other locations.
Preparation
Class meetings will be discussion-based, and students are expected to prepare by completing weekly readings. We ask you to engage with each assigned reading at a basic level and to prepare one of them in detail. Please sign up for the reading you will prepare in detail at least a week ahead of time so that you know which peers to talk with as you prepare and so that we have a balance of students focusing on each reading. The sign up page requires a login that will be shared with students in class and on the SMSTC page for this module.
A major focus of this course will be how to read like a historian. The process of preparing each week will make you a more confident and effective reader, not just of history. In our experience, students are often surprised by how rapidly and noticeably they build these skills with a little effort. We also do not want the module to be onerous, and our focus on reading efficiently and effectively will help you get the most out of the time you are able to commit to it.
Links to readings will be posted on the course outline below. Some will require you to figure out how to access them using your university's library resources, which is a good skill to practice anyway! (If you encounter difficulties, let the instructors know as soon as possible and we will connect you with copies.) Certain hard-to-source materials will be posted on the SMSTC page for this module under Resources.
Preparing a reading means, at a minimum, being prepared to contribute actively to explaining and discussing the following aspects of the text:
- basic information: by whom was it written, where and when was it published, what kind of source is it?
- the major subjects or topics
- the geography and periodisation
- the sources and evidence
- the main arguments and how they intervene in the literature
- any other notable points
We will discuss what these mean and extensively practice how to determine them.
Assessment
There is an optional assessment for students requiring a mark for credit. Please contact the instructors about this as soon as possible if you plan to submit something.
Weekly Outline
1. Long ago division (16 Jan, MB+DK)
- Welcome
- Warmup/introduction exercise: who and what are part of history of mathematics?
- Module overview
- Reading historical mathematics (exercises and discussion)
- We will focus on long division in Robert Recorde's 1543 Ground of Artes.
- (further reading) Michael has a recent book chapter on this in the context of the long history of algorithms (featuring historical fart jokes): https://doi.org/10.1093/oso/9780197502426.003.0003
- (further reading) A recent piece by Deborah (and Alisz Reed) involves an algorithm for manual root extraction (and the first maths book printed in England): https://www.lms.ac.uk/sites/default/files/inline-files/NLMS_506_for%20web2.pdf
- Reading history of mathematics (discussion)
2. Runes and reconstruction (23 Jan, DK)
- Megalithic Evidence and Comment, chapter 1.C of The History of Mathematics: A Reader (Open University 1987), scan posted to SMSTC Resource page.
- Eleanor Robson, Mathematics in Ancient Iraq (Princeton 2008) chapters 1 and 9 (focus on 9) https://doi.org/10.1515/9780691201405
- Jim Ritter, Reading Strasbourg 368, History of Science, History of Text (Springer 2004) https://doi.org/10.1007/1-4020-2321-9_9
- Sabetai Unguru, On the need to rewrite the history of Greek mathematics, AHES 1975 https://doi.org/10.1007/BF00327233
- Cambridge History of Ancient Science, chapters 5 + 8 (Egypt), 26 (Early India), 28 (Early Imperial China) https://doi.org/10.1017/9780511980145
- (further reading) Michael Barany, Savage numbers and the evolution of civilization in Victorian prehistory, BJHS 2014, https://doi.org/10.1017/S0007087413000356
3. Counters and canons (30 Jan, MB)
- Karine Chemla, Canons and Commentary in Ancient China, https://www.mpiwg-berlin.mpg.de/Preprints/P344.PDF
- Chemla, Proof, Generality and the Prescription of Mathematical Action, https://doi.org/10.1111/1600-0498.12111
- Reviel Netz, Counter Culture, https://doi.org/10.1177/007327530204000303
- Netz, Deuteronomic Texts, http://www.numdam.org/article/RHM_1998__4_2_261_0.pdf
- Jens Høyrup, Leonardo Fibonacci and Abbaco Culture, http://www.numdam.org/article/RHM_2005__11_1_23_0.pdf
- Høyrup, The formation of a myth: Greek mathematics – our mathematics, Mathematical Europe (pdf on SMSTC site)
- (further reading) Roger Hart, The Chinese Roots of Linear Algebra https://press.jhu.edu/books/title/10232/chinese-roots-linear-algebra
4. Geometry and nature (6 Feb, DK)
- Michael Mahoney, The Mathematical Realm of Nature, https://doi.org/10.1017/CHOL9780521307635.024
- Judith Grabiner, Maclaurin among the molasses barrels, https://doi.org/10.1177/030631298028001005
- Ivor Grattan-Guinness, Solving Wigner's mystery, https://doi.org/10.1007/BF02985373
- Massimo Mazzotti, From Genius to Witch, https://lareviewofbooks.org/article/rise-fall-filosofessa/
- Domenico Bertoloni Meli, Equivalence and Priority (Oxford 1993), chapter tbd, https://doi.org/10.1093/oso/9780198539452.001.0001
- Deborah Kent, The curious aftermath of Neptune's discovery,https://doi.org/10.1063/PT.3.1363
- Deborah Kent, The North American eclipse of 1869, https://doi.org/10.1063/PT.3.4271
5. Intermediate values (13 Feb, MB)
- Judith Grabiner, Is mathematical truth time-dependent? https://doi.org/10.2307/2318997
- Michael Barany, Stuck in the middle, https://www.ams.org/notices/201310/rnoti-p1334.pdf
- Joseph Dauben, Abraham Robinson and Nonstandard Analysis, in History and Philosophy of Modern Mathematics (Minnesota, 1988) https://www.jstor.org/stable/10.5749/j.cttttp0k.10 pdf
- Primary sources:
- Bolzano doi:10.1016/0315-0860(80)90036-1
- Cauchy (remember there are two proofs, in the main text and in the appendix!) google books
- Pastor archive.org
- Robinson https://doi.org/10.2307/3038223
- Rudin pdf
- Find and share others!
6. Stats and Probability (20 Feb, DK)
- Rosaleen Love (1979) ‘Alice in Eugenics-Land’: Feminism and Eugenics in the scientific careers of Alice Lee and Ethel Elderton, Annals of Science, 36:2, 145-158, https://doi.org/10.1080/00033797900200451
- Chapter 1, Karl Pearson and the Cambridge Economists in Stigler, Stephen M. Statistics on the Table: The History of Statistical Concepts and Methods. Harvard University Press, 1999. https://doi.org/10.2307/j.ctv1pdrpsj
- Chapter 8, The History of Statistics in 1933 in Stigler, Stephen M. Statistics on the Table: The History of Statistical Concepts and Methods. Harvard University Press, 1999. https://doi.org/10.2307/j.ctv1pdrpsj
- Lorraine Daston, Rational Individuals versus Laws of Society: From Probability to Statistics, Chapter 13 in The Probabilistic Revolution, v. 1. Ideas in history / edited by Lorenz Krüger, Lorraine J. Daston, and Michael Heidelberger. This is available on archiv.org urn:oclc:record:1280755537
7. Bodies and minds (27 Feb, MB)
- Mary Terrall, Metaphysics, mathematics, and the gendering of science in eighteenth-century France (we will post a pdf on the smstc module resource page)
- Andrew Warwick, A Mathematical World on Paper, ch 3 of Masters of Theory (Chicago 2003), https://doi.org/10.7208/chicago/9780226873763.003.0003
- Michael Barany, A Young Man's Game, ch 2 of Gender, Embodiment, and the History of the Scholarly Persona (Palgrave 2021), https://doi.org/10.1007/978-3-030-49606-7_2
- Matthew Jones, Reckoning with Matter, chapters 1 and 2 (and the First Carry between them), focus on chapter 2 https://press.uchicago.edu/ucp/books/book/chicago/R/bo24836963.html
- Lorraine Daston, Enlightenment calculations https://www.jstor.org/stable/1343891
- Stephanie Dick, Models and machines, https://doi.org/10.1086/683527
8. Foundations and roots (5 Mar, DK)
- Bruce Hunt, Heaviside versus the mathematicians, https://doi.org/10.9783/9781512801590-005
- David Bloor, Polyhedra and the abominations of Leviticus, https://www.jstor.org/stable/4025779
- Graham/Kantor, Two cultures https://doi.org/10.1086/501100
- (OPTIONAL) Leo Corry, The origins of eternal truth in mathematics, https://doi.org/10.1017/S0269889700002659
9. Worlds of mathematics (12 Mar, MB)
- Alma Steingart, A group theory of group theory, https://doi.org/10.1177/0306312712436547
- Michael Barany, Integration by parts, https://doi.org/10.1525/hsns.2018.48.3.259
- Andrea Breard, Nine Chapters on Mathematical Modernity, chapters 1 and 9 (and pick at least one additional chapter from in between to read quickly) https://link.springer.com/book/10.1007/978-3-319-93695-6
- Arunabh Ghosh, Sino-Indian statistical exchanges, https://doi.org/10.1017/bjt.2016.1
- Slava Gerovitch, Parallel worlds, https://web.mit.edu/slava/homepage/articles/Gerovitch-Parallel-Worlds.pdf
- Two books for further reading on this topic: Karen Parshall and Adrian Rice, Mathematics Unbound; Norbert Schappacher, Framing Global Mathematics
10. History and Mathematicians (19 Mar, DK+MB)
Everyone please try to read all three of these this week:
- Grattan-Guinness, History or Heritage? https://doi.org/10.1080/00029890.2004.11920041
- Catherine Goldstein, Long-term history and ephemeral configurations, https://doi.org/10.1142/9789813272880_0021
- Karine Chemla, All roads come from China, https://euromathsoc.org/magazine/articles/5
- Some further readings:
- Anachronisms, https://doi.org/10.1017/9781108874564
- Historiography of mathematics in the 19th and 20th centuries, https://link.springer.com/book/10.1007/978-3-319-39649-1
- Bourbaki, Éléments d'histoire des mathématiques
Additional Resources
- Jacqueline Stedall, The History of Mathematics: A Very Short Introduction https://academic.oup.com/book/601
- Benjamin Wardhaugh, How to Read Historical Mathematics https://press.princeton.edu/books/hardcover/9780691140148/how-to-read-historical-mathematics