Mathematics

Saturday April 9, 2022

            Two years ago I began a quest to improve my geometric and physical intuition with the ultimate objective of studying the geometry of the bidisk in two-dimensional complex space. The quest led me to re-examine my mathematics and physics background. I discovered that my proficiency in algebraic and symbolic manipulation explained my aptitude for algorithmic design, but the actual geometric and physical models the algorithms represented was lost in the computational efficiency of the algorithms.

            As I reviewed my high school and college textbooks (yes, I still have them), it became clear that even the proofs of statements were “algorithmic” in the sense that a catalog of previously accepted/established results were prepared for writing a logical sequence of statements to establish the given statement. The proof was nothing more than a “logic algorithm”. The process failed to include the development of mathematical intuition.

            In a forthcoming paper,  Rediscovering Mathematics, The Greek Way, I note the value of mathematics as preparation for philosophy. The Greek insight to the good life centered on the soul and the need for discovering and knowing (i.e., acquiring/possessing) truth. This clearly requires a knowledge of mathematics beyond the arithmetic of daily finance and domestic engineering. In fact, the Greeks clearly distinguished between logistic (applied mathematics) and what they called arithmetic (i.e., what we call number theory).

            So where am I now?

            As I began working through Volume I of A History of Greek Mathematics, I spent some time on The Theory of Proportion and Means, pages 84 – 91. Since the topic of ratios and proportions had surfaced in previous research, it seemed reasonable to examine this in more detail. When Heath referenced Archytas for the definition of the arithmetic, geometric and harmonic means, I searched for a document that contained Archytas’s definitions.

            Selections Illustrating the History of Greek Mathematics, Volume I: From Thales to Euclid, with an English Translation by Ivor Thomas, ©1939, contained a passage from Porphyry’s Commentary on Ptolemy’s Harmonics. The passage contained the definitions and noted the relationship of the harmonic mean to music. Further investigation led to Porphyry’s Commentary on Ptolemy’s Harmonics, A Greek Text and Annotated Translation, Andrew Barker, 2015. This work by Andrew Barker currently captures a large chunk of research time.    In essence, I have fallen into a rabbit hole which will require significant time and energy to escape. Barker’s work is being studied in detail because it not only links music to ratio and proportion, but Porphyry also tied his commentary to the philosophy of the soul. This ties to the larger research objective of developing intuition that forms the soul; the intuition that a body of knowledge contributes to the growth of the intellectual soul.

            Although other studies are also in progress, a series of lectures on Porphyry’s Commentary on Ptolemy’s Harmonics is in preparation. Hopefully updates will be posted more frequently.

Monday March 30, 2020

It seems reasonable to begin documenting my current research today.

I am working to improve my geometric and physical intuition. The following question is driving my investigations: What is the intrinsic geometric structure of the bidisk in two-dimensional complex space?

Note, the bidisk is defined as a set-theoretic product of the unit disk in the complex plane with itself so the Euclidean geometric structure is not intuitively clear. As a result, I need to examine the Euclidean structure of some elementary geometric objects in two-, three- and four-dimensional spaces. The physical structure of models for these geometric objects is fairly straightforward in two and three dimensions. The real challenge is four-space.

To prepare for viewing geometric objects in four dimensions, I’ve been working on perspective geometry for art. A fascinating book is Practical Perspective Drawing by Philip J. Lawson. The ability to sketch objects visible in my mind’s eye is necessary for building my intuition.

Since I’m a two and three dimensional kind of guy, the ability to view projections in four dimensions will be very important.  The Carus Mathematical Monograph Projective Geometry by John Wesley Young is providing initial training in that area.

Of course, all of this is focused on my research in complex geometry. Studying the algebraic properties of these geometric objects is essential. Hermann Weyl’s book, Symmetry, is providing geometric and algebraic insights which guide my investigations.  Although I’ve never worked through Birkhoff and Mac Lane’s A Survey of Modern Algebra or H.S.M. Coxeter’s Introduction to Geometry, I am now.

I’m not sure when I’ll post any papers or lecture notes, but I’m committed to completing some documents this year.