From time immemorial, logic has been at the back of several facets of knowledge—humanities, science, technology, engineering and math. But wait! What is “logic”? Is there a way to measure the extent of our logical thinking? Are there principles of valid logic? If by the latter one means coherence, deductive inferences and syllogism, several philosophers and mathematicians, from Aristotle, von Neumann, to al-Khwārizmī, to Gottfried Leibniz, to Alan Turing, used such concept to decipher and manipulate the symbols of a mechanical language of thought. But, “logic”, as a principle of arithmetic, could also mean the construction of large edifices from small bricks (divided categories) using a binary language which consists of just two digits 0 and 1. But it seems, with the breakneck rapidity of computing and mathematical language, the role of logicians remains questionable.
Regardless, the use of logical concepts has been central not only in answering big questions and regulating daily matters, but also in designing and creating new technology like the map, the printer, digital computers, etc. It would suffice to refresh one’s memory of high school math drills and a number of nostalgic moments would be recalled. I was taught to equate people’s intelligence with solving math problems. The math-nerd your are, the better logician you become! It went on to frame my perception of assessing people’s logical thinking and computational literacy. At that stage, one would have assumed that there is a strong connection not only between logic and math, but also between logic and modern digital machines.
Tracing the intellectual lineage of digital computers is not as recent as one might think. In fact, as Martin Davis argues, computers are in many ways the culmination of the glorious and powerful mathematical tradition we now call logic. Nevertheless, Davis traces biographical sketches of the lives of several contributors, a la Shakespearian tragedy, where each suffered an ill-fated dream. While logicians, such as Leibniz, grasped the broader significance of systematic logic and mechanizing calculation in fixing difficult problems and completing Aristotle’s project of codifying syllogisms, his doctrine of optimism of creating an encyclopedic compilation and reducing human reasoning to a purely mechanical and symbolic task, lingered myopic.
That being said, their conceptual marvel of logic and how it laid the ground for the design of modern technology is impeccable. Take the computer, for example and how it can perform so many different things simultaneously. It is remarkable that my ingenuous gadget on which I am typing this blog is equally adept at generating solutions to partial differential equations—the various input we enter and the kind of output we expect to obtain. So, the ultimate test of a theory of logic that aims at completeness is whether it encompasses all mathematical reasoning is relative and finite. But, at the end one still wrestles with the “engineers” vs. “logicians” dialectic or “who shaves the barber?” This dilemma demonstrates the existence of infinities that are higher than the infinity of the integers.