Claudius Ptolemy (c 100 - c 160 AD) is best known for his Almagest, a collection of 13 books that formed a treasure trove for astronomers and historians because of the historical collection of astronomical data they contain. He also wrote the Tetralibris consisting of four books about astrology, and in his Planetary Hypotheses he describes the universe as a set of spheres defining the movements of the celestial bodies either via an epicyclic or an eccentric model. Circular motion was essential as it represented mathematical and hence divine-like perfection. He was not only an mathematician/astronomer since he also left us his Geographia with geographic coordinates of the world as it was known in his days, and in the Optics he studies properties of light.
All these texts are used as sources by Feke although her book is not about the astronomy or the mathematics, but about the philosophy that Ptolemy is conveying in his books. He lived in the midst of what is often considered to be a period of philosophical eclecticism where elements were freely imported from Aristotle, Plato, Epicurus, or Stoics. Elements of all these are indeed found in Ptolemy's vision, however with a twist of his own, as Feke clearly explains. In that respect the beginning of the Almagest and even more so his book Harmonics, in which he mainly discusses the application of mathematics to music, are crucial elements to understand the role of mathematics for his philosophical and ethical views. These are the ones that Feke cites most. The perfect ratios of the harmonic pitches, are reflected in the harmony of the stars in heaven and that divine commensurable organized constancy should also be a guide for the human soul. Mathematics is the only way to gain knowledge and mathematical ethics will furnish a good life since it will transform the soul into a godlike condition by mimicking the perfection of the stars. Let me quote the strong statement by which Feke summarizes Ptolemy's vision in the last sentences of her book:
[...] according to Ptolemy, the benefits mathematics provides are epistemological and ethical. Mathematics is the only field of enquiry through which human beings can acquire knowledge, and it is the only path to the good life. In Ptolemy's philosophy, the best way an individual can live his life is to live the mathematical way of life.
To come to such a conclusion, some explanation is in order.
According to Ptolemy, theoretical philosophy has three parts: theology, physics, and mathematics. The object of theology is the immaterial "Prime Mover" who governs the motion of the stars but who is imperceptible. Physical objects are of course perceptible but mathematical objects are in between, since the latter can be observed with your senses or thought of as abstractions. Mathematics is however the only way to generate knowledge because the Prime Mover is too far out and cannot be perceived, while in physics, one has to measure and make observations, which are necessary to define the parameters of the mathematical model but observations are always inaccurate. Thus theology and physics can only lead to conjectures, while mathematics generates knowledge. Moreover, mathematics is useful since if can help to predict the outcome of astronomical and physical phenomena.
Feke claims that the ethical aspects of Ptolemy's vision are the result of his solution for a contemporary debate over the relation between theoretical and practical philosophy. Ptolemy wrote a pamphlet about this, known as On the Kritêrion and Hêgemonikon. It gives a criterion for truth, explains how knowledge can be acquired and describes the structure of the soul, of which the hêgemonikon is the ruling part, located in the brain. His authorship is somewhat controversial since it is not completely in line with what he wrote elsewhere. So Feke assumes that it was written at an earlier stage before his Harmonics in which his mathematical methodology was fully developed. Ptolemy considers the soul to be something mortal and physical, with faculties in different parts of the body, only it consists of much finer particles than the body, so that it evaporates much faster after death. The same harmony of perfect ratios of the musical pitches, and the mathematical study of this harmony will generate the beauty in mathematical objects. This has ethical and practical applications because mathematical models will transform physical matter into a perfect form, and likewise, when applied to the soul, it will lead to an orderly, calm, commensurable good life that, as was quoted above, resembles the astronomical properties of the stars, and hence approximate a perfect divine status.
What Ptolemy describes in his Harmonics are the rational and unavoidable rules that govern musical pitches, the heavenly bodies and the human soul alike. These harmonic relationships and the role of mathematics is what Feke explores in further detail in the subsequent chapters discussing Ptolemy's astronomy, the human soul, astrology, and cosmology. I will not report on all her fine dissections, but I will just mention how she sees Ptolemy's view on the role, the interplay, and the relative importance of the classical subjects of the quadrivium: arithmetic, geometry, music, and astronomy.
According to Ptolemy, geometry and arithmetic are not mathematics but they are just methods. Geometry can be applied to study optics via observations, arithmetic can handle the harmonic ratios that are provided by mathematical theory. Astronomy relies on both geometry and arithmetic: it requires geometry for the optical observations, while arithmetic can deal with the harmonics that are intrinsically derived from the theoretical model.
Cosmology (the study of the heavenly bodies) and astrology (the effect of the movement of the stars on sublunary bodies) are both physical sciences that Ptolemy bases on astronomy which he thus considers superior to cosmology and astrology. Therefore the results of the latter two are predictive but remain just like in physics only conjectural,
In a book like this, the precise meaning and nuances of a word are important and for that Feke gives precise references and quotations from the work of Ptolemy. Often the translation is in English in the text, while the Greek text from the source is in a footnote. She also relates meticulously the interpretation that Ptolemy attached to some concept and she discusses the difference or the similarity to corresponding notions of Aristotle, Plato or other of the ancient philosophers. Occasionally she also refers to interpretations given by more modern authors who wrote on Ptolemy's work and ancient Greek philosophy. That does not happen very often because her book is mostly historical. Only in her concluding chapter she briefly discusses Ptolemy's influence in later centuries. Even the way she is harvesting an idea of Ptolemy from his different books is impressive. One can read sentences like "Ptolemy used this word at only one other place namely...". Ptolemy's philosophical ideas are indeed scattered all over his mathematical books and it requires a thorough knowledge of all the texts to crystallise them into clear English sentences.
As I said before, Feke shows that mathematics is for Ptolemy the science that rules everything and to which humans should submit to lead a good life, but her book is far from dealing with mathematics as such. This is a book written by a philosopher mainly for fellow philosophers. Thus if the reader is a mathematician who is not so familiar with the philosophical jargon and writing style, it may require some pages to adapt, but Feke does a good job to make it a comfortably readable text also for an interested non-philosopher. I am not a philosopher myself, so I may not be aware of all the professional philosophical literature, but my search on the Web seems to confirm that this is indeed the first fully-fledged book bringing the philosophy of Ptolemy to the foreground (as it is announced in the blurb on the back cover). Neither have I the background to agree or disagree with the conclusions that Feke distils from the available sources, so I leave that to her peers. As a mathematician, I can only be warmed by a tingling ASMR experience while reading the conclusion that she arrived at in the quote formulated above. I am so pleased by Ptolemy's tap on my shoulder reassuring me that it has not been a waste of time to have devoted the larger part of my life to mathematics.