Reading list for geometry
1. E. A. Moise, Elementary geometry from an advanced standpoint, Addison-Wesley, 1963. (Useful for an easy understanding of synthetic vs metric geometry, Archimedean postulate, infinities and infinitesimals in a non-Archimedean field of which the appendix accidentally gives a historically important example.)
*2. C. K. Raju, Euclid and Jesus: How and why the church changed mathematics and Christianity across two religious wars, Multiversity, Penang, 2012. (Explains how geometry in Plato’s time related to the soul, and was borrowed from the mystery geometry of Egypt. After the first religious war of Christians against pagans, in the 4th c., the notion of the soul was changed, and mathematics was banned from Christendom. It became the basis of Islamic rational theology. The second religious war (waged by Christians against Muslims) were prolonged military failures. Hence, the church changed its theology to Christian rational theology, copied from Islamic rational theology, the better to persuade Muslims. Overnight, from a supposed doctrine of love, Christianity changed to a doctrine of reason. Since Muslims had been made such hate figures, Christians could not acknowledge learning from Muslims, hence invented “Euclid” to claim ownership of reason. The book Elements, a book on mystery geometry to arouse the soul, was wrongly re-interpeted as a book about deductive reasoning intended to persuade others (a church requirement), and especially to persuade Muslims who accepted reason, but not the Christian scriptures. The book was used to teach reasoning to priests. This interpretation does not fit the book, as was eventually accepted in the 20th c.: there is not a single pure deductive proof in the Elements (and there is nil evidence for Euclid). However, it was now claimed that the mythical Euclid intended such proofs. Hilbert’s apologia kicks in to explain the apparent prolixity of the book, by claiming that it is a book about synthetic (non-metric) geometry. Since the “Pythagorean” proposition is about area Hilbert, while prohibiting length measurement, nevertheless admits area measurement!)
More advanced reading:
3. D. Hilbert, Foundations of Geometry, trans. E. J. Towsend, Open Court, La Salle, 1950. (The definitive text on synthetic geometry. Hilbert replaced the (politically loaded) metric term “equality” in the Elements by the term “congruence”.)
4. G. D. Birkhoff, “A set of postulates for plane geometry (based on scale and protractor)”, Ann. Math. 33 (1932) pp. 329-345. (Shows that all the theorems of the Elements can be proved from a metric set of postulates using scale and protractor, rather than straight edge and (collapsible) compasses. However, the proof of the “Pythagorean theorem” becomes so easy that this essentially trivializes the Elements which proved the Pythagorean proposition in 47 step. Hence, mathematicians have rejected Birkhoff’s axiomatisation since it falsifies a key story important to the church.
5. Bertrand Russell, An essay on the foundations of geometry, Cambridge university press, 1897. (Develops on the philosophical foundations of geometry based on Kant who gave a different interpretation of innate knowledge than Plato. Builds on stories about the parallel postulate and non-Euclidean geometry. These are stories, since geometry on the surface of a sphere was being done at least since the time of Bhaskar 1.)
6. Bertrand Russell, “The Teaching of Euclid”, The Mathematical Gazette 2 (33) (1902), 165-167. (Points out that many of the so-called deductive proofs in the Elements are faulty regarded as deductive proofs. In fact, ALL are faulty.)
7. Bertrand Russell, “Mathematics and the metaphysicians”, in: Mysticism and logic and other essays, Longman Green and Co., London, 1919, pp. 71-96. (An account of formal mathematics, and especially why the initial hypothesis are metaphysical.)
*8. C. K. Raju, Cultural Foundations of Mathematics: the nature of mathematical proof, and the transmission of the calculus from India to Europe in the 16th c. CE Pearson Longman, 2007.
9. T. L. Heath, The thirteen books of Euclid’s Elements: translated from the text of Heiberg, Cambridge university press, 1908, vol. 1. (The stock story in full details, based on Heiberg’s fanciful apologia about “Theonine” texts. [The name of the author found in texts of the Elements is that of Theon, not Euclid.] )
10. Proclus: Commentaries of Proclus surnamed Plato’s successor on the first book of Euclid’s Elements… trans. Thomas Taylor, London, 1788.
*11. Proclus, A Commentary on the First Book of Euclid’s Elements, trans. Glenn R. Morrow,
Princeton University Press, Princeton, New Jersey, 1970. (Another translation.)
12. G. Friedlein, Procli Diadochi Commentarii, B. G. Teubner, Lipschitz, 1873. (The Greek “original”. This late text, discovered in the Vatican by Heiberg, in the 19th c., is the sole source of our knowledge of “Euclid”. The actual passage mentioning Euclid is quite clearly an interpolated passage.)
13. W. W. Rouse Ball, A short account of the history of mathematics, Macmillan and Co., London, 1912. Reprint, Dover New York, 1960. [The standard racist account of “Greek” contributions.]
*14. S. N. Sen and A. K. Bag, The sulbasutras, Indian National Science Academy, Delhi, 1983.
15. C. K. Raju, “Teaching mathematics with a different philosophy. Part 1: Formal mathematics as biased metaphysics.” Science and Culture 77 (7-8) (2011) pp. 274–279. http://www.scienceandculture-isna.org/July-aug-2011/03%20C%20K%20Raju.pdf, arxiv:1312.2099.
16. C. K. Raju, “Towards Equity in Math Education 1. Good-Bye Euclid!”, Bharatiya Samajik Chintan 7 (4) (New Series) (2009) pp. 255–264. http://ckraju.net/papers/MathEducation1Euclid.pdf
17. C. K. Raju, “Towards Equity in Math Education 2. The Indian Rope Trick” Bharatiya Samajik Chintan 7 (4) (New Series) (2009) pp. 265–269. http://ckraju.net/MathEducation2RopeTrick.pdf.
18. C. K. Raju, The Pythagorean controversy, Frontier Weekly, 47(34) March 1-7 (2015) http://www.frontierweekly.com/articles/vol-47/47-34/47-34-The%20Pythagorean%20Controversy.html.
19. C. K. Raju, “Black Thoughts Matter: Decolonized Math, Academic Censorship, and the “Pythagorean” Proposition”, Journal of Black Studies, online first 31 Jan 2017, http://journals.sagepub.com/doi/abs/10.1177/0021934716688311.
* = not available for free download.