Friday, May 19, 2017

Gottfried Leibniz and the philosophy of mathematics


Gottfried Leibniz (1646-1716) was an influential mathematician and philosopher best known for developing differential and integral calculus independently from Isaac Newton. Leibniz was also known for building calculating machines. Mathematician Michael Beeson said,
Gottfried Leibniz is famous... for his slogan 'Calculemus', which means 'Let us calculate'. He envisioned a formal language to reduce reasoning to calculation, and he said that reasonable men, faced with a difficult question of philosophy or policy, would express the question in a precise language and use rules of calculation to carry out precise reasoning. This is the first reduction of reasoning to calculation ever envisioned... he actually designed and built a working calculating machine, the Stepped Reckoner... inspired by the somewhat earlier work of Pascal, who built a machine that could add and subtract. Leibniz's machine could add, subtract, divide, and multiply, and was apparently the first machine with all four arithmetic capabilities.  
This post is a list of quotes from Leibniz regarding his philosophy of mathematics.

Support for mathematics


"...if controversies were to arise, there would be no more need of disputation between two philosophers than between two calculators. For it would suffice for them to take their pencils in their hands and to sit down at the abacus, and say to each other (and if they so wish also to a friend called to help): Let us calculate." (Dissertatio de Arte Combinatoria , 1666)

"Only geometry can hand us the thread [which will lead us through] the labyrinth of the continuum’s composition, the maximum and the minimum, the infinitesimal and the infinite; and no one will arrive at a truly solid metaphysic except he who has passed through this [labyrinth]." (Dissertatio Exoterica De Statu Praesenti, 1676)

"Music is a hidden arithmetic exercise of the soul, which does not know that it is counting." (Letter to Christian Goldbach, 1712)

"This miracle of analysis, this marvel of the world of ideas, an almost amphibian object between Being and Non-being that we call the imaginary number." (Quoted in Singularités by Christiane Frémont)

Epistemology


"I am convinced that the unwritten knowledge scattered among men of different callings surpasses in quantity and in importance anything we find in books, and that the greater part of our wealth has yet to be recorded." (Discours touchant la méthode de la certitude et de l'art d'inventer, 1688)

"There are two kinds of truths: those of reasoning and those of fact. The truths of reasoning are necessary and their opposite is impossible; the truths of fact are contingent and their opposites are possible." (The Monadology, 1714)

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