The Language of Calculus
The Story of Leibniz, his notations and ideas
Gottfried Wilhelm Leibniz was a prominent German polymath of the 17th century. He made contributions not only in the field of mathematics, which his most prominent discoveries on the language of symbolic logic, and calculus were established, but also in philosophy, which he formally studied, and theology where he argued a rational stance for the existence of God. While Leibniz works in different fields were found to be foundational he was involved in controversies that although he was a member of the Royal Society and the Berlin Academy of Science, his grave went unmarked for 50 years.
This is the story of how Leibniz dreamt of expressing our thoughts through symbols that formally captures the meaning of sentences. A link to that of his works in Infinitesimal Calculus.
Among his numerous contributions, the language of Calculus is most suited for our discussion. Leibniz believed that much of human reasoning can be reduced to calculations of some sort[2].
The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate[calculemus], without further ado, to see who is right.
Leibniz held the view that symbols were important for human understanding. His belief about the essential role of symbols and notations to a well-running logic, and mathematics, made him a precursor of semiotics[1]. He believed that formal operations such as logic and arithmetic can automate knowledge by a rule-based combination of symbols [2].
Leibniz took his idea further by constructing a character which he ascribed as real characters that represent an idea directly and not simply as the word embodying the idea. He proposed the creation of a characteristica universalis (universal characteristic) which was constructed on an alphabet of human thought. He imagined a formal language that can able to express abstract mathematical, scientific, and metaphysical ideas. By combining simple representations of thoughts, one can build some level of complexity (through abstraction).
Leibniz's idea of reasoning from symbolic representations remarkably foreshadows the later developments in modern logic which made him a prominent logician of his time. He is noted as one of the most important logicians between the times of Aristotle and Gottlob Frege[4].
In relation to Leibniz’s works on calculus, these ideas were remarkable for his choice of notations that enunciate the underlying concepts of the Infinitesimal Calculus.
Leibniz’s Notation
Leibniz introduced the integral sign, we know and love ∫ representing an elongated S, from its Latin word summa, and the d-operator used for differentials, from the Latin word differentia. Remarkably, Leibniz did not publish anything about his calculus until 1684. It is also worth noting that the notations introduced by Leibniz made the concepts of calculus more transparent.
The Limit Operator
The limit of a function is the value that a function approaches as in the input of Δx approaches. Below is the definition of a derivative with respect to x.
A derivative, of a function, tells us about the rate of change of that function as the limit of Δx approaches 0, or simply the tangent of a function; we can notice that Leibniz’s definition of a derivative is an approximation of a tangent by defining a secant which approaches to 0.
Notations for Higher-order derivatives and integrals
In general, derivatives can be expressed using the d-operator which is placed below with the Langrangian differential operator f ⁿ(x).
This is a suggestive notational device that comes from formal manipulations of symbols, as in
It is important to note that these equations are not theorems: they are simply definitions of notation. Leibniz’s notation allows one to specify the variable for differentiation (in the denominator). This is especially helpful when considering partial derivatives for higher-dimensions as in
where it denotes the second partial derivative of the function f with respect to y. And finally for denoting higher-order integrals with the Summa symbol which describes the volume under a surface. Leibniz’s notation reminds the notion of infinitesimal changes under curved surfaces which is then compiled as the infinitesimal sum of the summa operator.
Derivatives and Integrals are related by the fundamental theorem of Calculus which is also suggestive of the above notation with the d-operator sitting on the other side of the summa.
References
[1]. Marcelo Dascal, Leibniz. Language, Signs and Thought: A Collection of Essays (Foundations of Semiotics series), John Benjamins Publishing Company, 1987, p. 42.
[2]. O. Schwartz (2019). In the 17th Century, Leibniz Dreamed of a Machine That Could Calculate Ideas. IEEE Spectrum.
[3]. The Art of Discovery 1685, Wiener 51
[4]. Lenzen, W., 2004, “Leibniz’s Logic,” in Handbook of the History of Logic by D. M. Gabbay/J. Woods (eds.), volume 3: The Rise of Modern Logic: From Leibniz to Frege, Amsterdam et al.: Elsevier-North-Holland, pp. 1–83.
