A Course in the Philosophy and Foundations of Mathematics


An examination of mathematical methods and the search for mathematical meaning.

This article curates a reading list (most sources available freely online1) organized into a set of encounters that lie outside the standard mathematics curriculum. They are intended to enrich the reader’s understanding of mathematics and its place in scientific inquiry, increase her/his connection to the historical and philosophical questions behind the mathematics of the past and present, and gain greater satisfaction from further mathematical study. The reader should come away with a better understanding of the culture of mathematics: what mathematics is, mathematical method and meaning, and the relation of mathematics to the empirical world and to science.

We look at seven topics. These may be covered in any order, to suit your particular interests.

  1. What is Mathematics? (Its Nature and Characteristics)
  2. Reality, Truth, and the Nature of Mathematical Knowledge
  3. What is Proof? and the Problem of Certainty
  4. Some Readings in the History of Mathematics and the Evolution of Its Ideas
  5. The Search for Foundations in Mathematics
  6. Mathematics and Science
  7. Thoughts on Mathematical Practice and Mathematical Style

There is no core body of technical material to master in this course; the important thing is a feel for how, why, and in what context the core ideas of mathematics evolved, getting to the essence of their motivation, and understanding the fruits of these efforts. The course such as the below should appeal to all those who have an itch to scratch beneath the surface of mathematics, who find themselves asking “but why?”. It could be useful in all three tiers of education: secondary, post-secondary (undergraduate), and graduate, appropriately restructured.

  • Secondary school elective: to encourage bright students in mathematics, science and technology to enter the university with a broader perspective on the mathematics they will be rapidly learning there.
  • University elective course: offered as a writing-intensive seminar, intended primarily for students in the sciences and engineer: mathematics, physics, engineering.
  • Graduate level course: offered in the first year of graduate school in mathematics or applied mathematics as a supplementary seminar.

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  1. To ensure that the materials are always available for download, I am serving them from copies held on this site. If you are the author of any of these articles and would prefer to have the primary download originate from your site, please send me an email, and I will make the change.

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Reading List…

Looking for the best long-form articles on this site? Below is a curated list by the main topics covered.

Mathematics-History & Philosophy

  1. What is Mathematics?
  2. Prehistoric Origins of Mathematics
  3. The Mathematics of Uruk & Susa (3500-3000 BCE)
  4. How Algebra Became Abstract: George Peacock & the Birth of Modern Algebra (England, 1830)
  5. The Rise of Mathematical Logic: from Laws of Thoughts to Foundations for Mathematics
  6. Mathematical Finance and The Rise of the Modern Financial Marketplace
  7. A Course in the Philosophy and Foundations of Mathematics
  8. The Development of Mathematics
  9. Catalysts in the Development of Mathematics
  10. Characteristics of Modern Mathematics

Electronic & Software Engineering

  1. Electronics in the Junior School - Gateway to Technology
  2. Coding for Pre-Schoolers - A Turtle Logo in Forth
  3. Experimenting with Microcontrollers - an Arduino development kit for under £12
  4. Making Sensors Talk for under £5, and Voice Controlled Hardware
  5. Computer Programming: A brief survey from the 1940s to the present
  6. Forth, Lisp, & Ruby: languages that make it easy to write your own domain specific language (DSL)
  7. Programming Microcontrollers: Low Power, Small Footprints & Fast Prototypes
  8. Building a 13-key pure analog electronic piano.
  9. TinyPhoto: Embedded Graphics and Low-Fat Computing
  10. Computing / Software Toolkits
  11. Assembly Language programming (Part 1 | Part 2 | Part 3)
  12. Bare Bones Programming: The C Language

Pure & Applied Mathematics

  1. Fuzzy Classifiers & Quantile Statistics Techniques in Continuous Data Monitoring
  2. LOGIC in a Nutshell: Theory & Applications (including a FORTH simulator and digital circuit design)
  3. Finite Summation of Integer Powers: (Part 1 | Part 2 | Part 3)
  4. The Mathematics of Duelling
  5. A Radar Tracking Approach to Data Mining
  6. Analysis of Visitor Statistics: Data Mining in-the-Small
  7. Why Zero Raised to the Zero Power IS One

Technology: Sensors & Intelligent Systems

  1. Knowledge Engineering & the Emerging Technologies of the Next Decade
  2. Sensors and Systems
  3. Unmanned Autonomous Systems & Networks of Sensors
  4. The Advance of Marine Micro-ROVs

Math Education

  1. Teaching Enriched Mathematics, Part 1
  2. Teaching Enriched Mathematics, Part 2: Levelling Student Success Factors
  3. A Course in the Philosophy and Foundations of Mathematics
  4. Logic, Proof, and Professional Communication: five reflections
  5. Good mathematical technique and the case for mathematical insight

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