Chapter 2: The True method(1677) Gottfried Wilhelm Leibniz

This is my commentary and summary on Chapter 2 of Harry R. Lewis’ book Ideas That Created the Future: Classic Papers of Computer Science which sheds light on The True method written by Gottfried Wilhelm Leibniz around 1677.

Gottfried Wilhelm Leibniz was a polymath who made contributions in the field of math, physics, politics, law, and philosophy. While Aristotle laid out a system of logic, it was not easy to understand and apply it. On the other hand, Leibniz recognized the importance of good notation and how it could contribute towards clear thought. After studying Aristotle’s work he developed formal notations for logical reasoning in his early work which would later go on to become one of the foundational concepts for Computer Science. Even to this day, Leibniz’s notation for an infinitesimal change in x, dx, is preferred over many other notations such as 𝝙x or ẋ.

Perhaps, Leibniz’s direct contribution to the field of Computer Science was his invention of binary arithmetic. Although binary arithmetic was in use before Leibniz’s formalization, it wasn’t investigated well enough to understand the advantages it provided in its use in certain applications. He goes on to explain in detail binary counting, addition, subtraction, multiplication, and division. In doing so his work also sheds light on interesting properties numbers possess. As an example, he describes binary numbers to possess a “celebrated property of the geometric progression by twos.” Let’s say we have a set A of n binary numbers of form 12, 102, 1002, 10002 … (100..n)2. Any positive binary number below the kth number can be expressed as a combination of numbers from set A which are below k. So if we take 10002 then it would mean that all binary numbers below 10002 -: 12, 102, 112, 1002, 1012, 1102, 1112 can be expressed as combinations of 12, 102, 1002. With the first 10 numbers from A, we can express 1023 binary numbers. Since computers don’t understand words and numbers like humans do, Leibniz showed with minimal expression how it was possible to cover a vast set of information through binary.

Leibniz had deep-rooted motivations to invent a computation device. One of which was to cast away ignorance in the sciences. He believed that there was yet an invention to be made which would test the veracity of claims by a ‘certain method’. This would be done through the art of what he called near-perfect demonstration. During his time, Mathematics was one of the only areas where this kind of demonstration existed. Claims could be verified either by repeating calculations or trying some tests. Other fields had not adopted such rigor in demonstrating their claims which left open room for ambiguity. He thought that in physics the cost of setting up experiments could be one reason why such demonstrations may have been difficult to carry out. However, he does mention that the existence of the ‘certain method’ is not yet known to everyone and is in a way encouraging his readers to have an open mind towards the demonstrability of seemingly difficult things. Today, we have computers that can simulate complex body motions with great accuracy and we don’t need to physically set up those experiments.

Leibniz was also an optimist who believed in the existence of these ‘certain methods’ to such an extent that he believed all our problems could be resolved by plugging and chugging, resulting in an unequivocally better world. He expresses his fascination by telling us that if we want to measure the perimeter of a huge circle with a known diameter we could do so much more accurately by using a formula rather than physically measuring the circle. Thus we need to find variables that express thought accurately which would go on to easily verify claims which would be otherwise difficult to demonstrate. In True method(1677), Leibnitz’s mentions, “I dare say that this is the highest effort of the human mind, and when the project is accomplished it will merely be up to men to be happy since they will have an instrument which will serve to exalt reason no less than the telescope serves to perfect vision.” He went on to invent the first binary digital mechanical calculator called Step Reckoner which could perform all four arithmetic operations. Even as he was recognized for his contributions he was ridiculed for his optimism in the idea that the world would be a better place if reasoning could be captured in a logical system and all we would need to do is to plug and chug, but, yet to this day his logical reductionism appears in every automated decision support system.

References

1. The True Method(1677), Gottfried Wilhelm Leibniz
2. Ideas That Created the Future: Classic Papers of Computer Science, Harry R. Lewis