nLab Gottfried Leibniz

Gottfried Leibniz (1646-1716)

• Wikipedia entry

is responsible for a great many constructions and concepts of current interest:

• differential calculus and integral calculus, especially the basic notation that we use today;

• the product rule of calculus may be called the Leibniz rule or Leibniz law;

• differentiation under the integral sign? follows the Leibniz integral rule;

• identity of indiscernibles may be called the Leibniz law;

• a version of infinitesimals sometimes regarded as a precursor to synthetic differential geometry;

• a philosophy of ‘monads’ (monadology)

  (compare also in Hegel, Science of Logic the section The monad of Leibniz)

  from which it certainly seems (but remains implicit) that Robinson 1966 (p. 57) motivated the terminology of monads in nonstandard analysis (infinitesimal neighbourhoods)

  and which may or may not have motivated Bénabou 1967 (§5.4) to coin the term monad in category theory (see there for more).

Metaphysics

An instructive starting point distinction made in the contemporary literature … is between three ‘levels of reality’ in Leibniz’s mature metaphysics. At the most basic level, what is real for Leibniz are simple substances which alone have true unity. These are the famously obscure monads. Next, we have the ‘phenomenal level’ that is made up of phenomena bene funda – well founded phenomena – that, due to pre-established harmony, are accurate reflections of the real and actual monadic states. Finally, we have the ideal level which, by contrast, is made up of entia rationis – abstract or fictional things – that include ‘phenomena’ founded upon possible but non-actual monadic states. Crucially, although both the phenomenal and the ideal levels can include things which are infinite, all concepts that depending upon the continuum are only applicable to the ideal realm. Thus, if we were to define time as the real line, $\mathbb{R}$, then this concept of time could only be represented for Leibniz as an entia rationis and thus ideal. Furthermore, phenomenal things for Leibniz can only acquire their status as phenomena bene funda by their grounding upon the actual. They must always be understood as representations or perceptions of the monads of the actual world. (Thébault 19, p. 2)

References

• Georg Hegel, Lectures on the History of Philosophy – Leibniz

On infinitesimals:

• Mikhail Katz, David Sherry, Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond, D. Erkenn (2013) 78: 571 (arXiv:1205.0174, doi:10.1007/s10670-012-9370-y)

• Karim Thébault, The Problem of Time, (PhilSci Archive)

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