Further reading

  • Euclid. The Elements. Portuguese translation by Irineu Bicudo. São Paulo: Editora Unesp, 2009. The common notions and the structure of the propositions.
  • Proclus. A Commentary on the First Book of Euclid’s Elements. Translated by Glenn R. Morrow. Princeton: Princeton University Press, 1970. The six parts of a proposition.
  • Frege, Gottlob. Über Sinn und Bedeutung. Zeitschrift für Philosophie und philosophische Kritik, 100, 1892, pp. 25–50. Brazilian translation in: Lógica e filosofia da linguagem. São Paulo: Cultrix/Edusp, 1978.
  • Frege, Gottlob. Die Grundlagen der Arithmetik. Breslau: Koebner, 1884. The quotation from Leibniz is in §65.
  • Wittgenstein, Ludwig. Tractatus Logico-Philosophicus. London: Kegan Paul, 1922. On identity, propositions 5.53 to 5.5303.
  • Gauss, Carl Friedrich. Disquisitiones Arithmeticae. Leipzig: Gerhard Fleischer, 1801. The definition of congruence and the sign ≡ are in articles 1 and 2.
  • Pascal, Blaise. Traité du triangle arithmétique. Paris: Guillaume Desprez, 1665 (written in 1654). Induction appears in consequence 12.
  • Dijkstra, Edsger W.; Scholten, Carel S. Predicate Calculus and Program Semantics. New York: Springer, 1990. The proof format with hints.
  • de Bruijn, Nicolaas G. Asymptotic Methods in Analysis. Amsterdam: North-Holland, 1958. The one-way equalities of big $O$.
  • Halmos, Paul R. I Want to Be a Mathematician: An Automathography. New York: Springer, 1985. The origin of $\blacksquare$.
  • Kieran, Carolyn. Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 1981, pp. 317–326.
  • Healy, Lulu; Hoyles, Celia. A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 2000, pp. 396–428.
  • Fischbein, Efraim. Intuition and proof. For the Learning of Mathematics, 3(2), 1982, pp. 9–18 and 24.
  • Cajori, Florian. A History of Mathematical Notations. 2 vols. Chicago: Open Court, 1928–1929. On the ≡ of Gauss and of Riemann and on Rahn’s $\therefore$.
  • Earliest Uses of Symbols of Relation, Jeff Miller.

Image sources

Image Source Licence
Portrait of Gottlob Frege, by Emil Tesch (c. 1905) Wikimedia Commons public domain
Title page of Gauss’s Disquisitiones Arithmeticae (1801) Wikimedia Commons public domain
First page of Pascal’s Triangulus Arithmeticus, Bibliothèque du patrimoine de Clermont Auvergne Métropole Wikimedia Commons CC BY-SA 4.0

The diagram of the sum of odd numbers and the interactive figures (the congruence clock and the steps of a solution) were produced for this book and work in all four languages of the site. The record of the downloaded images, with the hash of each file, is in data/semiotica_imagens.json.

Chapter index · ← How to read a proof: a guide and a glossary · Start of the book