Gottfried Wilhelm Leibniz

Philosophy — The Monadology & Sufficient Reason

Leibniz rejected Descartes' interactionist dualism. Instead, reality is composed of monads — simple, immaterial, windowless substances. Each monad perceives the universe from its own point of view, and changes according to an internal law. There is no causal influx between monads; their agreement is due to a pre-established harmony instituted by God, like two perfect clocks keeping time without influencing each other.

Monads: Perception and Appetition

Monads have two internal qualities: perception (the multitude in the unity) and appetition (the tendency to pass to new perceptions). Human souls are monads with memory and apperception; God is the original Monad, perceiving all distinctly.

Mathematics — Calculus, Binary, and Infinity

Leibniz published calculus first (1684), with notation designed to make reasoning mechanical. While Newton used fluxions $\dot{x}$, Leibniz used differentials $dx$, $dy$, making the chain rule transparent.

Leibniz Notation

• Derivative: $\displaystyle \frac{dy}{dx} = \lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}$
• Integral: $\displaystyle \int y\,dx$ as sum of infinitesimals ($\int$ from summa)
• Chain rule: $\displaystyle \frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$
• Product rule: $d(uv)=u\,dv+v\,du$

Leibniz Integral Rule

Differentiating under the integral sign:

Used by Leibniz for optics, by Feynman for integrals.

His binary arithmetic was not mere calculation. Leibniz saw in 0 and 1 a metaphysics: creation from God (1) and nothing (0). Sent to the Jesuits in China, it matched the I Ching hexagrams — evidence for his universal characteristic.

Logic — Characteristica Universalis & Calculus Ratiocinator

Leibniz dreamed of a universal symbolic language where concepts are written like numbers, and reasoning becomes calculation. “Calculemus!” — let us compute.

Prime Concept Encoding

Assign each primitive concept a prime: MAN=2, RATIONAL=3, ANIMAL=5, MORTAL=7. Then a composite concept is the product: HUMAN = 2·3·5·7 = 210. Inclusion is divisibility: $A$ contains $B$ iff $\text{code}(B)$ divides $\text{code}(A)$. This anticipates Gödel numbering by 250 years.

Four Logical Systems (1679–1690)

Unfinished, but it became modern symbolic logic (Frege, Russell), computer science, and knowledge graphs.

Proofs and Demonstrations

1. Leibniz Rule (General Product Rule)

Proof. Induction. For $n=1$, $(uv)'=u'v+uv'$. Assume true for $n$, differentiate: $\sum \binom{n}{k}[u^{(n-k+1)}v^{(k)}+u^{(n-k)}v^{(k+1)}]$. Re-index and use $\binom{n}{k}+\binom{n}{k-1}=\binom{n+1}{k}$. ∎

2. Alternating Series Test (Leibniz Test)

If $a_n \downarrow 0$ and $a_n>0$, then $\sum (-1)^{n}a_n$ converges.

Proof sketch. Partial sums $S_{2n}$ increase, $S_{2n+1}$ decrease, and $S_{2n+1}-S_{2n}=a_{2n+1}\to0$. By continuity principle, they sandwich a unique limit.

3. Fundamental Theorem (Leibniz's infinitesimal view)

Let $A(x)=\int_a^x f(t)dt$. Increase $x$ by $dx$: $dA = f(x)dx$. Hence $dA/dx=f(x)$. Integration sums, differentiation divides — inverse operations.

4. Binary Uniqueness

Every $N\in\mathbb{N}$ has a unique expansion $N=\sum b_i2^i$, $b_i\in\{0,1\}$.

Proof. Repeated division: $N=2q_0+r_0$, $0\le r_0<2$. Continue with $q_0$. Remainders give bits. Uniqueness from $2^{k+1} > \sum_{i=0}^k 2^i$.

GNU Octave — Leibniz in Code

1. Leibniz π series (slow but beautiful)

% leibniz_pi.m
N = 1e6;
k = 0:N-1;
pi_est = 4*sum((-1).^k ./ (2*k+1));
fprintf('π ≈ %.10f, error = %.2e\n', pi_est, abs(pi - pi_est));
% Vectorized: converges as O(1/N)

2. Binary conversion à la 1703

function s = dec2bin_leibniz(n)
  if n==0, s='0'; return; end
  s='';
  while n>0
    s = [char('0'+mod(n,2)), s];
    n = floor(n/2);
  endwhile
endfunction
dec2bin_leibniz(42)  % 101010

3. Leibniz integral rule check

f = @(x,t) exp(-x.*t);
dfdx_num = @(x) integral(@(t) -t.*exp(-x*t), 0, 1);
x0 = 2;
% analytic: (1 - exp(-x)*(x+1))/x^2 derivative
dfdx_num(x0)

4. Prime concept encoding

pr = primes(50);
encode = @(ids) prod(pr(ids));
MAN=1; RATIONAL=2; ANIMAL=3;
human = encode([MAN,RATIONAL,ANIMAL])  % 2*3*5=30
animal = encode([ANIMAL])              % 5
mod(human, animal) == 0               % true: human ⊆ animal

5. Pre-established harmony simulation

t = 0:0.01:10;
monad1 = sin(t);           % internal law
monad2 = sin(t);           % same law, no interaction
plot(t,monad1, t,monad2,'--');
title('Two monads in harmony without causation');

Interactive Demos

Legacy

Leibniz's notation won because it is calculational. His binary became the machine code of civilization. His characteristica became predicate logic, type theory, and AI ontologies. His monads reappear in process philosophy, panpsychism, and decentralized computation where agents act locally yet globally cohere.