Adam Tetlow on Mathematics Useful for Understanding Plato

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The sixth-century CE Christian Platonist Elias reports that the entrance to Plato’s Academy bore the inscription: ‘Let none who is un-geometric enter’. But, reading scholarly literature on Plato, one could be forgiven for thinking that geometry plays no real role at all in his thought. In this episode we discuss some of the most important numerical and geometrical concepts necessary for a deeper appreciation of dialogues like the Timæus and Republic.

The discussion opens with the basic consideration of what the ancients meant by number, which, it turns out, is in some ways very different from the conceptualisation current in modern mathematics. Number is a principle of reality itself. Ratios – numbers set in relation with each other, but not collapsed into irrational, decimalised single figures – play a major role in this ancient mindset.

Macrobius’ Commentarii in somnium Scipionis, Lambda diagram of the world-soul from Plato’s Timaeus, Walters Manuscript W.22, fol. 66r

We then move on to a wide-ranging discussion of:

  • The theory and practice of musical harmony and the role it plays in Plato’s works,
  • The five ‘Platonic solids’,
  • The Pythagorean tetraktys and its relationship to the ‘lambda’ figure of Plato’s Timæus,
  • And the Golden Section, and its relationship to the Divided Line passage of Plato’s Republic.

The discussion also includes

  •  A discursus into ancient metrology, debunking the metric system and exploring the fascinating world of ancient measure more generally,
  • And discussion of the state of geometry in the modern-day ‘Plato industry’, and the need for scholars of Plato to pick up a compass and straight-edge and get drawing.

Works Discussed in this Episode:

  • Critchlow, K., 2007. Time Stands Still: New Light on Megalithic Science. Floris Books, Edinburgh. Critchlow discusses the Platonic solids and their ancient reception.
  • Kayser, H. Godwin, J. & Godwin, A. (Ed.), 2006. Textbook of Harmonics. Sacred Science Institute.
  • Martineau, J., 2002. A Little Book of Coincidence. Wooden Books, Glastonbury. For the extraordinary harmonic relationship of Venus with the Earth.
  • McClain, E., 1978b. The Pythagorean Plato: Prelude to the Song Itself. Nicolas-Hays, York Beach, MN.
  • Partch, H., 1979. Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments. De Capo. The great Partch’s own account of his music. Technical and philosophical, a very tough read for those daunted by enormous tables of fractions. But check out his music; it’s so hip it’s unbelievable.

Recommended Reading:

  • Adam, J., 1985. The Nuptial Number of Plato. Kairos, Wellingborough.
  • Critchlow, K., 1976. Islamic Pattern: An Analytical and Cosmological Approach. Thames and Hudson, London.
  • Critchlow, K., 2011. The Hidden Geometry of Flowers: Living Rhythms, Form, and Number. Floris Books, Edinburgh.
  • Lawlor, R., 2000. Sacred Geometry. Thames and Hudson, London.
  • McClain, E., 1978a. The Myth of Invariance: The Origin of the Gods, Mathematics, and Music from the Rg Veda to Plato. Shambhala.
  • Michell, J., 1988. The Dimensions of Paradise: The Proportions and Symbolic Numbers of Ancient Cosmology. Thames and Hudson, London.
  • Michell, J., 2009. How the World Is Made: The Story of Creation According to Sacred Geometry. Inner Traditions, New York, NY.
  • Mountford, J. (1923). ‘The Musical Scales of Plato’s Republic’, Classical Quarterly 17 : 125-36.
  • Schneider, M., 1995. A Beginner’s Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science. Harper Collins, New York, NY.
  • Stewart, M., 2009. Patterns of Eternity: Sacred Geometry and the Starcut Design. Floris Books, Edinburgh.
  • Tetlow, A., 2013. Celtic Pattern: Visual Rhythms of the Ancient Mind. Wooden Books, Glastonbury.
  • Waterfield, R. (Ed.), 1988. The Theology of Arithmetic. Phanes Press, Grand Rapids, MI.

And for some good practical help with geometric constructions in the old-school way:

  • Allen, J., 2007. Drawing Geometry: A Primer of Basic Forms for Artists, Designers and Architects. Floris Books, Edinburgh.
  • Sutton, A., 2009. Ruler and Compass. Wooden Books, Glastonbury.

Watch this space from many helpful geometric diagrams!