• 1. Plato’s Republic, trans. G.M.A. Grube, 2nd edition revised by C.D.C. Reeve (Indianapolis: Hackett Publishing Company, 1992), 527a-b, 199.
  • 2. The two diagrammatic proofs are reproductions of proofs appearing in Roger B. Nelsen’s Proofs Without Words II: More Exercises In Visual Thinking (Washington DC: The Mathematical Association of America, 2000). The first proof is credited to Rick Mabry and the second to Frank Burk.
  • 3. Ludwig Wittgenstein, Remarks on the Foundations of Mathematics, eds. G.H. von Wright, R. Rhees, and G.E.M. Anscombe, trans. G.E.M. Anscombe (Oxford: Blackwell, 1956), Part VI, Section 72.
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