Computer Algebra Systems (CAS): Mathematica. Note Quote.


If we are generous, there is one clear analogue to Leibniz’s vision in the contemporary world, and that is a computer algebra system (CAS), such as Mathematica, Maple, or Maxsyma. A computer algebra system is a piece of software which allows one to perform specific mathematical computations, such as differentiation or integration, as well as basic programming tasks, such as list manipulation, and so on. As the development of CAS’s have progressed hand in hand with the growth of the software industry and scientific computing, they have come to incorporate a large amount of functionality, spanning many different scientific and technical domains.

In this sense, CAS’s have some resemblance to Leibniz vision. Mathematica, for example, has all of the special functions of mathematical physics as well as data from many different sources which can be systematically and uniformly manipulated by the symbolic representation of the Wolfram Language. One could reasonably claim that it incorporates many of the basic desiderata of Leibniz’s universal calculus – it has both the structured data of Leibniz’s hypothetical encyclopedia, as well as symbolic means for manipulating this data. 

However, Leibniz’s vision was significantly more ambitious than any contemporary CAS can make claim to have realized. For instance, while a CAS incorporates mathematical knowledge from different domains, these domains are effectively different modules within the software that can be used in a standalone fashion. Consider, for example, that one can use a CAS to perform calculations relevant to both quantum mechanics and general relativity. The existence of both of these capabilities in a single piece of software says nothing about the long-standing theoretical obstacles to creating a unified theory of quantum gravity. Indeed, as has long been bemoaned in the formal verification and theorem proving communities, CAS’s are effectively a large number of small pieces of software neatly packaged into a single bundle that the user interacts with in a monolithic way. This fact has consequences for those interested in the robustness of the underlying computations, but in the present context, it simply serves to highlight a fundamental problem in Leibniz’s agenda.

So in effect, one way to describe Leibniz’s universal calculus, was an attempt to create something like a modern computer algebra system, but which extended across all areas of human knowledge. This goal itself would be quite an ambitious one, but in addition Leibniz wanted the additional property that the symbolic representation should have a transparent relationship to the corresponding encyclopedia, as well as possess the capacity of mnemonics to be memorized with ease. To quote Leibniz (caution: German) himself,

My invention contains all the functions of reason: it is a judge for controversies; an interpreter of notions; a scale for weighing probabilities; a compass which guides us through the ocean of experience; an inventory of things; a table of thoughts; a microscope for scrutinizing things close at hand; an innocent magic; a non-chimerical cabala; a writing which everyone can read in his own language; and finally a language which can be learnt in a few weeks, traveling swiftly across the world, carrying the true religion with it, wherever it goes.

It difficult to not be swept away by the beauty of Leibniz’s imagery. And yet, from our modern vantage point, there is hardly a doubt that this agenda could not possibly have worked.

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