December 13th, 2007



Leibniz was well aware of the fact that our philosophical reasoning often lacks the cogency of mathematical reasoning (cf. [15, IV, p. 468]). There are no Euclidists and Archimedians in mathematics, as there are Aristotelians and Platonists in philosophy [16, ser. VI, vol. 4A, p. 695]. Whereas mathematicians have their own means of discovering possible mistakes, philosophers, who do not have such means at their disposal, should adopt rigorous reasoning all the more [16, ser. VI, vol. 4A, p. 705; ser. II, vol. 1, pp. 475, 478; 15, IV, p. 469]. Rationality should be such as to allow for a mathematisation of our thought; just as mathematicians have introduced letters and other symbols to designate mathematical objects and rules for operating with them, Leibniz proposed finding symbols and rules, in order to formalize a considerable part of our thought [20,10]. Then two philosophers with different opinions on a philosophical topic would no longer need to quarrel; they could say to each other “calculemus” (let’s calculate) [16, ser. VI, vol. 4A, p. 493]. Therefore, Leibniz’s invention of a calculating machine had a strong philosophical relevance. And besides, Leibniz tells us, this characteristica universalis would be an efficient means of converting pagans, because the true religion is the most rational religion, and it is impossible to resist rational arguments. In addition, apostasy will no longer occur, just as a mathematical truth, once understood, is never rejected. [Note 2: [16, ser. VI, vol. 4A, p. 269; ser. II, vol. I, p. 491]. Nearly twenty years later Leibniz was more skeptical: there are people who even reject indisputable arguments [16, ser. I, vol. 13, pp. 553-554].]

Leibniz made several efforts to find suitable symbols for the representation of our thinking. A very simple and nonetheless very interesting one was his idea that there are primary or irreducible notions and composite notions. If this is true, a map from notions to natural numbers can be defined, mapping primary notions to prime numbers and the relation “implies” between notions to the relation “is divisible by” between numbers. For illustration’s sake, Leibniz gives the example of the traditional definition of the human being as the rational living being. If the notion “rational” is mapped to the number 2 and the notion “being alive” to the number 3, the notion “being a human being” has to be mapped to the number 6 [16, ser. VI, vol. 4A, pp. 182, 201-202]. As there is an infinite number of prime numbers, the model is more powerful than it might seem at first glance. In other drafts, Leibniz maps every notion to an ordered pair of a positive and a negative integer [16, ser. VI, vol. 4A, pp. 224-256]. As Leibniz is aware, even with such a characteristica universalis the deduction of an individual statement like “Caesar was murdered on the ides of March” would be impossible, because such a statement involves an infinity of causes and an individual notion like Caesar is composed of an infinity of elements.

The use of numbers for a characteristica universalis even has a metaphysical foundation. Leibniz quotes [16, ser. VI, vol. 4A, p. 263; cf. also ser. I, vol.12, p. 72 and 15, VI, p. 604] the well-known statement [Plato, Philebos 55e; Sapientia Salomonis 11, 21] that God made everything according to measure, number and weight. Admittedly, Leibniz continues, some entities do not have weight, and some entities do not have parts and therefore lack measure. But there is nothing which does not allow for a number. So number is “quasi figura quaedam metaphysica” [16, ser. VI, vol. 4A, p. 264], and arithmetic is therefore a doctrine for the exploration of the powers of things, and thereby the perfection of God’s creation.

[13] Leibniz: Mathematische Schriften, ed. by CI Gerhardt, 7 volumes, Berlin, Halle 1849-1863; Reprint: Hildesheim, 1971.
[14] Leibniz: Opera omnia, ed. by L. Dutens, 6 volumes, Genève, 1768.
[15] Leibniz: Philosophische Schriften, ed. by CI Gerhardt, 7 volumes, Berlin 1875-1890; Reprint: Hildesheim, 1965, 1978.
[16] Leibniz: Sämtliche Schriften und Briefe, ed. by the Academies of Göttingen and Berlin, 8 series, Darmstadt 1923 seqq. [39 volumes have been published, but the edition is not yet complete].
[20] E. Scheibe, Calculemus! Das Problem der Anwendung von Logik und Mathematik, Leibniz. Tradition und Aktualitat. Vorträge des V. Internationalen Leibniz-Kongresses, vol. 2, Hannover, 1989, pp. 341-349.

—Herbert Breger, “God and Mathematics in Leibniz’s Thought”, in T. Koetsier, L. Bergmans, editors, Mathematics and the Divine: A Historical Study, Elsevier, 2004, pp 485-498, at pp. 487-488;