An examination of mathematical methods and the search for mathematical meaning.
During your studies of mathematics, physics and engineering, you may find yourself distracted or troubled by meta questions about mathematics — questions that fall outside the syllabi of most of the coursework that you’ll take.
For those for whom this itch is persistent, what follows is an outline and reading list for a Course in the Philosophy and Foundations of Mathematics. Among the topics included are the relation of mathematics to science, the examination of mathematical method, and the search for mathematical meaning.
A Course in the Philosophy and Foundations of Mathematics
An examination of mathematical methods and the search for mathematical meaning.
Overview
This course weaves together two lines of inquiry:
- The first track looks at subjects: Mathematics, Science, and Logic.
- The second track considers characteristics: Certainty and Doubt, Empiricism vs. Theory, Abstraction, Proof, Truth and Knowledge.
Within each track, the following questions are considered: what are these concepts? what are their relations and differences with each other? Are any two of them the same? And can they lead us to a better understanding of the enterprise of mathematics? Our investigations along these tracks are facilitated by studying seven topics:
- Topic 1. What is Mathematics? (Its Nature and Characteristics)
- Topic 2. Mathematics and Science
- Topic 3. Some Readings in the History of Mathematics and the Evolution of Its Ideas
- Topic 4. Reality, Truth, and the Nature of Mathematical Knowledge
- Topic 5. The Search for Foundations in Mathematics
- Topic 6. What is Proof? and the Problem of Certainty
- Topic 7. Thoughts on Mathematical Practice and Mathematical Style
The mathematical material of this course moves fluidly from ancient history to modern times, with much in between, and brings in ideas from different branches of mathematical development. But remember, it is not the mastery of the technical material that is the goal of this course. Rather, the important thing is observing the evolution of the ideas, getting to the essence of their motivation, and understanding the characteristics of their development and the fruits of these efforts.
Risks and Rewards
A meta-mathematics itch, once awoken, can take a lot of scratching to satisfy. You should be aware that the ideas you encounter here may pose distractions for you as you proceed along the regular course of your mathematical or technical studies. However, my hope is that an itch such as this, and the ideas you will encounter through this course will inspire you to study further, more broadly, and with greater urgency in both theoretical mathematics and its applications than you might otherwise do. With persistent inquiry and hard work, you will find at some point that you have a much more intimate understanding of your chosen subject, an understanding that should give you a sense of well-being, of historical and philosophical connectedness to the mathematics of the past, and pleasure at the prospect of continuing to participate in mathematical discovery, whether for yourself as an amateur or as a professional mathematician, engineer, or applied scientist.
Course Materials
To keep the course-materials inexpensive and readily accessible for the interested reader to pursue, I have adapted the original readings list for this course to favor materials that are now available freely online.^{1}
Syllabus and Reading List
Topic 1. What is Mathematics? (Its Nature and Characteristics.)
- Article: What is Mathematics? (Davis/Hersh), pp. 6-8
- Article: A General View of Mathematics (Aleksandrov), in Mathematics: Its Content, Methods, and Meaning (Aleksandrov/Kolmogorov/Lavrentev), pp.1-64
- Book: Proofs and Refutations: The Logic of Mathematical Discovery (Lakatos), pp.6.ff (entire book)
- This book is worth buying for your library.
- [Read online – Partial Preview only (Google Books), pp.6-18]
Topic 2. Mathematics and Science
- Article: The Unreasonable Effectiveness of Mathematics in the Natural Sciences (Wigner)
- [Read online] (Dartmouth)
- [Download PDF] (Originating server is Univ. Wuerzburg)
- Article: The Unreasonable Effectiveness of Mathematics (Hamming)
- [Read online] (Dartmouth)
- [Download PDF] (Originating server is UC Davis)
- Article: Science and Hypothesis: Author’s Preface (Poincare), pp.xxi to xxvii
- [Download Article] (Originating server is Internet Archive (Entire Book))
- Article: Mathematics as an Objective Science (Goodman)
- [Preview Article — first page only] (from JSTOR)
- Article: Why a little bit goes a long way: Logical foundations of scientifically applicable mathematics (Feferman)
- [Download PDF] (Originating server is Author’s webpage)
Topic 3. Some Readings in the History of Mathematics and the Evolution of Its Ideas
- 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)
- Article: Evolution of the Function Concept: A Brief Survey (Kleiner)
- [Download PDF] (Originating server is Mathematical Association of America)
- Article: The Evolution of Group Theory: A Brief Survey (Kleiner)
- [Download PDF] (Originating server is Israel Kleiner’s paper)
- Article: Excursus into the History of Calculus (Kutateladze)
- [Download PDF] (Originating server is arXiv)
- Article: History of the Formulas and Algorithms for (Goyanes)
- [Download PDF] (Originating server is arXiv)
- 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)
- Book: A History of “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 4. Reality, Truth, and the Nature of Mathematical Knowledge
The problem of determinism and error. Is nature continuous of discrete? What is Reality? What is Truth? Are the real numbers useful? Empirics and description vs. axiomatics and deduction?
- Article: The Nature of Truth (Schatz)
- [Download PDF] (Originating server is: (Author’s PhD student’s page)
- Article: Mathematics, Physics, and Information (Brillouin, 1957, Inform. and Control 1, 1-5)
- [Download PDF] (Elsevier)
- Article: Inevitable Experimental Errors, Determinism, and Information Theory (Brillouin, 1959, Inform. and Control 2, 45-63)
- [Download PDF] (Elsevier)
- Article: Poincare’s Theorem and Uncertainty in Classical Mechanics (Brillouin, 1962, Inform. and Control 5, 223-245)
- [Download PDF] (Elsevier)
- Article: Mathematical Intuition vs. Mathematical Monsters (Feferman)
- [Download PDF] (Originating server is Author’s webpage)
Topic 5. The Search for Foundations in Mathematics
- Article: The development of programs for the foundations of mathematics in the first third of the 20th century (Feferman)
- [Download PDF] (Originating server is Author’s webpage)
- Article: Does Mathematics Need New Axioms? (Feferman)
- [Download PDF] (Originating server is Author’s webpage)
Topic 6. What is Proof? and the Problem of Certainty
“You can be certain and be wrong.”
– John Kerry, 2004 Presidential Election against George W. Bush
- Article: Rigor and Proof in Mathematics: A Historical Perspective (Kleiner)
- [Download PDF] (Originating server is MAA collection)
- Article: There’s More to Mathematics than Rigour and Proofs (Tao)
- [Read article online]. Also, his on a “heuristic stage” and a language metaphor for why order matters in one’s formal training]
- Article: How to Write a Proof (Lamport)
An argument for heirarchically structured proof instead of the commonly used prose-proof style.- [Download PDF] (Originating server is Author’s webpage, along with commentary on the context in which he wrote the paper
- Microsoft Research of Lamport’s talk and paper.
- 20 years later — Lamport’s current thoughts
- Article: A Collection of Beautiful Proofs (Dijkstra)
- [Download PDF] (Originating server is Author’s collected works page)
- Article: 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 7. Thoughts on Mathematical Practice and Mathematical Style
- Essays on the Nature and Role of Mathematical Elegance (Dijkstra)
- [Download PDF] (Originating server is Author’s collected works page)
- [Read online]
- On the Quality Criteria for Mathematical Writing (Dijkstra)
- [Download PDF] (Originating server is Author’s collected works page)
- [Read online]
- What is Good Mathematics? (Tao)
- [Download PDF] (Originating server is arXiv:math)
Related Articles:
>> Continue reading: Meta-Questions and their Impact on Successful Mathematics, Science & Technology Education
Share Your Experiences. . .
- . . . 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
drop me a line. I would welcome hearing from you.
Further Work (Collaborators welcomed!)
A course such as that outlined in the preceding would, I believe, be useful in all three areas of education: secondary, post-secondary (undergraduate), and graduate, appropriately restructured:
- as a 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.
- as a university elective course, offered as a writing-intensive seminar, intended primarily for students in the sciences and engineer: mathematics, physics, engineering.
- as a graduate level course, offered in the first year of graduate school in mathematics or applied mathematics as a supplementary seminar.
There is much that can be done.
If collaborating on future work in this direction sounds interesting, your efforts would be welcomed!
Footnotes:
- 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. ↩
After closely examining Beautiful (1921), by M.C. Escher, one would grasp that it holds qualities of both a mathematical and artistic field (more so in mathematical then in artistic). With repetition of some, if not all, shapes comes the need for a somewhat mathematical mind set in order to achieve precise repetition and allow for an even distribution of colors (in this case: black and white). Symmetry plays a huge role with the development of the work due to the fact that it is how the work first takes form. Within the piece there are numerous lines (fourteen if I’m not mistaken) which travel from the border of the canvas to a center focal point. From there a horizontal line is drawn from left to right midway through the page. It then looks as though a circle was drawn around the center followed by the filling of each plane (one section being composed as well as its opposite). Within each section its adverse mirrors the same image only inverted. Once each portion is filled colors are added within the same repeating shapes. This allows the picture to look the same upside down as it does right side up. It is possible that this piece was entitled Beautiful because of Escher’s need to contradict his methods of creation. When one thinks of math the think of long complicated problems filled with numbers and equations; but with this piece, Escher has taken a difficult thing and turned it into something intricate and beautiful. The artistic perspective comes from the design idea itself; the want to make something Beautiful.