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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Syllabus and Reading List

Topic 1. What is Mathematics? Its Nature and Characteristics.

To be a mathematician meant to participate in a great common effort, to continue work begun thousands of years ago, to shed new light on old discoveries, and to seek new ways to prepare the developments of the future. — Richard Brauer (mathematician), “Emil Artin”, Bulletin of the American Mathematical Society, Vol. 73, No.1, January 1967

1. [Ebrahim, 2023] – What is Mathematics?, Assad Ebrahim, 2023, Mathematical Science & Technologies
2. [Hoyrup, 2017] – What is Mathematics? Perspectives Inspired by Anthropology, Jens Hoyrup, 2017
3. [Klein, 1908, 77-85] – Concerning the General Structure of Mathematics, from Elementary Mathematics from an Advanced Standpoint, Vol.1, Arithmetic, Algebra, Analysis, by Felix Klein, 1908, (Online)
4. [Davis/Hersh, 1981] – What is Mathematics?, Philip Davis, Reuben Hersh, pp.6-8, 1981 from The Mathematical Experience, 1981, (partial preview in Amazon-Look Inside feature, pp.6-8)
5. [Aleksandrov, 1956] – A General View of Mathematics, (article by Aleksandrov pp.1-64, partial preview in Look Inside feature: pp1-6), in Mathematics: Its Content, Methods, and Meaning [Aleksandrov/Kolmogorov/Lavrentev), 1963, 1969 (2nd ed)]

Topic 2. Reality, Truth, and the Nature of Mathematical Knowledge

One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than we originally put into them. — Heinrich Hertz (physicist)

1. [Grabiner, 1974] – Is Mathematical Truth Time-Dependent, by Judith Grabiner, 1974
2. [Lakatos, 1979] – Proofs and Refutations: The Logic of Mathematical Discovery by Imre Lakatos, 1979
3. [Feferman] Mathematical Intuition vs. Mathematical Monsters – [Download PDF] (Originating server is Author’s webpage)
4. [Schatz], The Nature of Truth – [Download PDF] (Originating server is: (Author’s PhD student’s page)

Topic 3. What is Proof? Proof and the Problem of Certainty

“It’s one thing to be certain, but you can be certain and you can still be wrong.”
– John Kerry, 2004 Presidential Election against George W. Bush

1. [Kleiner] – Rigor and Proof in Mathematics: A Historical Perspective, [Download PDF] (Originating server is MAA collection)
2. [Tao, 2009] – There’s More to Mathematics than Rigour and Proofs.
   • Elaboration on the “heuristic stage” as an alternative path on the route to post-rigorous mathematics.
3. [Lamport, 1993] How to Write a Proof, [Download PDF] (Originating server is Author’s webpage, along with commentary on the context in which he wrote the paper
4. [Lamport, 1999] How to Write a Proof, Millenium Edition [Download PDF] How to Write a Proof – Millenium Edition (20 years later, 2011) Original server – An argument for heirarchically structured proof instead of the commonly used prose-proof style.
5. [Leron, 1983] Structuring Mathematical Proofs (Leron, 1983) [Download PDF]
6. [Dijkstra] A Collection of Beautiful Proofs (Dijkstra) – [Download PDF] (Originating server is Author’s collected works page)
7. [Dijkstra] The Mathematical Divide (Dijkstra) – An argument for why computational proof methods should be encouraged – [Download PDF] (Originating server is Author’s collected works page) [Read online]

Topic 4. Some Readings in the History of Mathematics and the Evolution of Its Ideas

The interplay between generality and individuality, deduction and construction, logic and imagination—this is the profound essence of live mathematics. … In a far reaching development all of them will be involved … start from the “concrete” ground, then discard ballast by abstraction and rise to the lofty layers of thin air where navigation and observation are easy; after this flight comes the crucial test of landing and reaching specific goals in the newly surveyed low plains of individual “reality”. In brief, the flight into abstract generality must start from and return again to the concrete and specific. — Richard Courant (mathematician)

1. Article: Pre-modern algebra: A concise survey of that which was shaped into the technique and discipline we know (Hoyrup) [Download PDF] (Originating server is Author’s webpage)
2. Article: Evolution of the Function Concept: A Brief Survey (Kleiner) [Download PDF] (Originating server is Mathematical Association of America)
3. Article: The Evolution of Group Theory: A Brief Survey (Kleiner) [Download PDF] (Originating server is Israel Kleiner’s paper)
4. Article: Excursus into the History of Calculus (Kutateladze) [Download PDF] (Originating server is arXiv)
5. Article: History of the Formulas and Algorithms for (Goyanes) [Download PDF] (Originating server is arXiv)
6. Book: A History of Astronomy Ch.1–5, pp.1-62 (Pannenoek)
   Astronomy as a catalyst driving the increasing sophistication of arithmetic, algebra, geometry and calculational methods in Ancient Mathematics (Babylonia, Assyria, Egypt, Vedic India, Greece, and Alexandria), Muslim mathematics (Arabia, Persia, Central Asia), and in Renaissance Europe. [Read Ch.1–5 online (Ancient Mathematics)] (Google Books)

7. Book: A History of “Early Modern” Mathematics (from the start of the 1700s to the end of the 1800s) (Smith) [Download entire eBook as PDF] (Originating server is Project Gutenberg)

Topic 5. The Search for Foundations in Mathematics

I have yet to see any problem, however complicated, which when you looked at it in the right way, did not become still more complicated. — Paul Anderson, New Scientist

1. [Feferman, 1999] – Solomon Feferman. Does mathematics need new axioms? American Mathematical Monthly, Vol. 106, No. 2, pages 99-111, Feb 1999. [Download PDF] (Originating server is Author’s webpage) – This paper provides an expository look at the questions of axiomatic foundations of mathematics.
2. [Feferman] – The development of programs for the foundations of mathematics in the first third of the 20th century, by Solomon Feferman, [Download PDF] (Originating server is Author’s webpage)

Topic 6. Mathematics and Science

It is true that Fourier has the opinion that the principal object of mathematics is the public utility and the explanation of natural phenomena; but a scientist like him ought to know that the unique object of science is the honor of the human spirit and on this basis a question of the theory of numbers is worth as much as a question about the planetary system. — C.J. Jacobi (mathematician)

1. Article: Why a little bit goes a long way: Logical foundations of scientifically applicable mathematics (Feferman) [Download PDF] (Originating server is Author’s webpage)
2. Article: The Unreasonable Effectiveness of Mathematics in the Natural Sciences (Wigner) [Read online] (Dartmouth) [Download PDF] (Originating server is Univ. Wuerzburg)
3. Article: The Unreasonable Effectiveness of Mathematics (Hamming) [Read online] (Dartmouth) [Download PDF] (Originating server is UC Davis)
4. Article: Science and Hypothesis: Author’s Preface (Poincare), pp.xxi to xxvii [Download Article] (Originating server is Internet Archive (Entire Book))
5. Article: Mathematics as an Objective Science (Goodman)
   [Preview Article — first page only] (from JSTOR)

Topic 7. Thoughts on Mathematical Practice and Mathematical Style

The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them. — Sir William Lawrence Bragg, Beyond Reductionism

1. Essays on the Nature and Role of Mathematical Elegance (Dijkstra) [Download PDF] (Originating server is Author’s collected works page) [Read online]
2. On the Quality Criteria for Mathematical Writing (Dijkstra)
   [Download PDF] (Originating server is Author’s collected works page) [Read online]

3. What is Good Mathematics? (Tao) [Download PDF] (Originating server is arXiv:math)

8. Other Readings

The following 3 articles of Brillouin address the problem of determinism and error. Is nature continuous of discrete? Are the real numbers useful? Which should dominate: empirics and description vs. axiomatics and deduction?
1. Great Moments in History of Logic – 30 of the Key People who developed Modern Logic (2004)
2. Article: Mathematics, Physics, and Information (Brillouin, 1957, Inform. and Control 1, 1-5) [Download PDF] (Elsevier)
3. Article: Inevitable Experimental Errors, Determinism, and Information Theory (Brillouin, 1959, Inform. and Control 2, 45-63) [Download PDF] (Elsevier)
4. Article: Poincare’s Theorem and Uncertainty in Classical Mechanics (Brillouin, 1962, Inform. and Control 5, 223-245) [Download PDF] (Elsevier)

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Share your thoughts

Share Your experiences in the Comments section below if you have enjoyed the materials or have suggestions, if you’re a student and have taken a course such as this, or if you have taught a similar course.

If collaborating on future work in this direction sounds interesting, drop a note in the comments section.

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Footnotes: