G. W. Leibniz

Gottfried Wilhelm Leibniz

Arithmetic, symbolic reasoning, and mechanical calculation · 1646–1716

Leibniz developed binary arithmetic, designed calculating machinery, and explored calculi of reasoning. These are related interests, but they are not one completed machine. In particular, his decimal stepped reckoner was not a binary computer executing a universal logical language.

Life, projects, and evidence

Born in Leipzig in 1646, Leibniz studied philosophy and law and worked in a range of scholarly and administrative roles. His Paris period in the 1670s was important to his mathematics. He developed calculus independently of Newton; the subsequent dispute should not be replaced by a story in which one man's notation simply won everywhere. Newton's dot notation remains useful for time derivatives. Leibniz died in Hanover in 1716. Stanford Encyclopedia biography

His calculator used a stepped-cylinder mechanism and a movable arrangement for different decimal positions. Multiplication could be organized as repeated addition, and division as repeated subtraction with bookkeeping. A model was demonstrated in 1673, while subsequent construction and development continued. Mechanism, fabrication, carry handling, and operation all matter; declaring that a flawless design was defeated only by poor manufacturing oversimplifies the engineering history. INRIA and ACONIT, machine description

The binary paper associated with the Paris Academy's 1703 volume explains positional notation with 0 and 1 and gives arithmetic examples. It also discusses Chinese figures through Leibniz's interpretation. That interpretation is not evidence that those figures historically served as a positional arithmetic system. The original paper's title and contents can be consulted in a scholarly presentation of the text. MAA, binary arithmetic and Leibniz

His proposals for a universal characteristic and calculi of reasoning remained a collection of projects rather than a finished language covering all knowledge. Much of the technical logical material appeared in print long after his death. Historical influence on nineteenth-century logicians must therefore be established through actual access and reception, not an arrow drawn between similar ideas. Stanford Encyclopedia, Leibniz's Influence on 19th Century Logic

Worked mechanism: prove a binary multiplication algorithm

The following algorithm is a modern teaching construction. For nonnegative integers a and b, set x=a, y=b, and r=0. Repeatedly add x to r when y is odd, double x, and replace y with its integer quotient by two. Stop when y=0.

Multiply 13 by 11:

State before step	Action	Next state
x=13, y=11, r=0	y odd, add 13	x=26, y=5, r=13
x=26, y=5, r=13	y odd, add 26	x=52, y=2, r=39
x=52, y=2, r=39	y even, add nothing	x=104, y=1, r=39
x=104, y=1, r=39	y odd, add 104	x=208, y=0, r=143

The invariant is r + xy = ab. If y=2q, the next state has r+(2x)q=r+xy. If y=2q+1, the next state has (r+x)+(2x)q=r+xy. Initially the invariant holds; when y=0, it proves r=ab. Integer halving strictly decreases positive y, establishing termination.

The binary expansion 11=1011₂=8+2+1 explains the selected additions: 13×8 + 13×2 + 13×1 = 143. Binary representation exposes which powers of two contribute. It does not require that physical components themselves have only two possible states, nor does it make every arithmetic operation a single elementary action.

Arithmetic is not automatically logic

Using digits 0 and 1 does not identify ordinary addition with Boolean OR. For truth values, 1 OR 1=1, whereas integer 1+1=2. A circuit implementation requires a specified interpretation and suitable operations. A binary adder also distinguishes its output bit from its carry bit.

An equally important distinction concerns symbolic deduction. Consider an original toy language with facts P(a) and a rule "for every object x, if P(x), then Q(x)." Applying the rule derives Q(a). The inference is valid because the rule's semantics preserve truth when its premise holds. Merely enumerating combinations of the symbols P, Q, and a would not establish that result.

Even a sound formal system does not automatically decide every meaningful question. Some systems are deliberately finite and decidable; sufficiently expressive systems face stronger limitations. A slogan saying incompleteness makes every formal system incomplete would itself be false. Leibniz's historical ambitions and modern results need distinct statements and assumptions.

Exercises

1. Trace the doubling-and-halving algorithm for 19×14. State the invariant after each step and explain how the proof changes, if at all, when one input is zero.
2. A decimal calculator multiplies 347 by 82 using shifted repeated addition. Count the additions using decimal digits, then compare with 82 repetitions of 347. Why does a lower addition count not by itself establish a faster physical machine?
3. In the toy logic, facts are P(a) and Q(b), and the only rule is P(x) implies Q(x). List all new atomic facts derivable. Does P(b) follow? Construct a model to justify your answer, then explain why Leibniz's metaphysical monads should not be identified with programming-language monads on the basis of a shared word.

Compare your reasoning with the worked solutions

1. The states after each step are (38,7,0), (76,3,38), (152,1,114), and (304,0,266). In every state r+xy=266; the initial state has 19×14=266 as well. If b=0, the loop is skipped and r=0 is already the product. If a=0, every addition adds zero and the loop still terminates by halving b. The proof requires nonnegative integer y; unrestricted negative integers need a different termination argument.
2. At the units position, add 347 twice. At the tens position, add 3470 eight times. This uses ten additions rather than 82, giving 694+27760=28454. Shifting, setting digits, turning a crank, resetting, and propagating carries all have physical costs. Counting one abstract operation equally across different mechanisms is an assumption that needs justification before making timing claims.
3. The only new fact is Q(a). Nothing supplies a premise P(b). A model with P={a} and Q={a,b} satisfies both initial facts and the rule while making P(b) false, proving it is not a logical consequence. Similar names likewise do not establish identical definitions or transmission: philosophical monads concern substances in Leibniz's metaphysics, whereas programming-language monads use a mathematical structure with specified operations and laws. A historical connection needs evidence beyond terminology.