The Geometric Mindset
The concepts of philosophy and number have been intertwined with one another for close to 2500 years within the Western world. The idea that there could be a Philosophy of Number was firmly embraced by the Platonists, the later Neoplatonic philosophers, and even those hailing from more modern times.
Without the efforts of one philosopher in particular, much of what I've written here would never have transpired. The Neoplatonic philosopher Thomas Taylor, most famous for being the first person to translate the works of Aristotle and Plato into English, also took great pains to aggregate what information he could find on ancient ancient number theory.
Along with his Theoretic Arithmetic of the Pythagoreans, Taylor also translated Proclus's Commentaries on the First Book of Euclid's Elements while adding his own unique flair. This book, first published in 1788, contains a most unusual quote taken from Sir Isaac Newton's Universal Arithmetic, which was published in 1707:
…geometry was invented that we might expeditiously avoid, by drawing lines, the tediousness of computation. Therefore, these two sciences ought not to be confounded.1
Even with my engineering background, this statement confounded me to no end, as did the words that followed:
The ancients so industriously distinguished them from one another, that they never introduced arithmetical terms into geometry. And the moderns, by confounding both, have lost the simplicity in which all the elegancy of geometry consists.
These few sentences, along with excerpts from a handful of other sources, paved the way for the Geometric approach providing the foundation for a true Philosophy of Number.2, 3, 4, 5, 6, 7
According to my interpretation, even numbers can be viewed as arising from a Geometric process. It is my belief that this Geometric approach fills a much-needed gap — potentially bridging the perceived divide between the "real" world and the language of mathematics.
1 Taylor, Thomas, The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements, 1788, p. cviii.
2Taylor, Thomas, Theoretic Arithmetic of the Pythagoreans, 1816.
3Schwaller de Lubicz, R.A., The Temple of Man, Inner Traditions, 1998, Vol. I.
4Vandenbroeck, Andre, Philosphical Geometry, Inner Traditions, 1972.
5Vandenbroeck, Andre, Al-Kemi: Hermetic, Occult, Political, and Private Aspects of R.A. Schwaller de Lubicz, Inner Traditions, 1987, pp. 6-7, 117. This particular book, with its statement, "nothing stamps the mind more surely than does number", as well as the quote, "All science lies between the numbers One and two." that initially sparked my interest in ancient mathematics.
6Byrne, Oliver, The First Six Books of The Elements of Euclid, Taschen, 1847 reproduction. This is the only other book besides Taylor's quoting Newton that I have found reference to the importance of separating Geometry from calculations. If the commentary is accurate, then even in 1847 there was still a discussion as to whether numbers and symbols belonged within the discipline of Geometry.
7Waterfield, Robin, The Theology of Arithmetic, Phanes Press, 1988.