Descartes & Leibniz

René Descartes (1596–1650)

French philosopher, scientist, and mathematician, Descartes refused authority and sought certainty through method. He connected geometry and algebra into analytic geometry, and gave modern philosophy its starting point: "cogito, ergo sum" — I think, therefore I am.

Philosophy — Method and Foundations

Method of Doubt: In Discourse on Method (1637) he lays four rules: accept only what is clear and distinct; divide difficulties into parts; proceed from simple to complex; review completely.

Cogito: Even if deceived by an evil demon, the act of doubting proves a thinking thing exists. From this res cogitans (thinking substance) he argues to God (via trademark argument) and then to res extensa (extended substance). This is mind–body dualism.

Mathematics — From curves to coordinates

La Géométrie (1637) introduced algebraic techniques for geometrical problems, blending algebra and geometry to revolutionize practice. The Cartesian coordinate system is named after him.

• Analytic geometry: A curve is the set of points satisfying an equation. Lines, circles, conics become algebra.
• Method of normals: Find tangent by constructing normal via circle intersection — precursor to derivative.
• Descartes' Rule of Signs: Number of positive real roots of polynomial ≤ number of sign changes in coefficients, differing by an even number.
• Folium of Descartes: $x^3 + y^3 = 3 a x y$, a loop with asymptote, used to test new methods.

Logic — Mathesis Universalis

Descartes sought a universal mathematics of order and measure, where clear and distinct ideas function like axioms. Truth is recognized by natural light, not syllogism alone.

Gottfried Wilhelm Leibniz (1646–1716)

German polymath credited, together with Newton, with inventing calculus, and pioneering binary arithmetic. He also built a metaphysics of simple substances.

Philosophy — Principles and Monads

Leibniz worked from seven core principles:

• Identity/Contradiction: A proposition cannot be true and false.
• Identity of Indiscernibles: Two distinct things cannot share all properties.
• Sufficient Reason: "There must be a sufficient reason for anything to exist, for any event to occur, for any truth to obtain."
• Pre-established Harmony: Substances correspond without causal interaction, like synchronized clocks.
• Continuity: Natura non facit saltus — nature makes no jumps.
• Optimism: God chooses the best of all possible worlds.
• Plenitude: The best world actualizes all genuine possibilities.

Monads: The universe is made of infinite simple substances, "ultimate units of existence." Each monad is windowless, changes internally by appetition, mirrors the whole universe from its perspective, and is synchronized by God.

Mathematics — Calculus and Binary

• Calculus (1684): Published Nova Methodus. Notation $dy/dx$ and $\int$ survives because it reveals structure: chain rule, product rule are syntactic.
• Binary: Devised modern binary system, basis of computing.
• Series: Leibniz formula $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + \cdots$
• Determinants, matrices, combinatorics: Early work on determinants and Ars Combinatoria (1666).

Logic — Calculate Disputes

Leibniz dreamed of a characteristica universalis — a universal symbolic language — and a calculus ratiocinator to resolve disputes by calculation: "Let us calculate, without further ado, to see who is right." He anticipated prime encoding (later Gödel numbering) and symbolic logic.

Compare: Descartes vs Leibniz

GNU Octave — Working Examples

Copy these into Octave (.m files). They directly illustrate Descartes and Leibniz.

1. Descartes Folium

% folium.m — plot x^3 + y^3 = 3 a x y
a = 1;
t = linspace(-0.5, 5, 400);
% param form: x = 3 a t / (1+t^3), y = 3 a t^2 / (1+t^3)
x = 3*a*t ./ (1 + t.^3);
y = 3*a*t.^2 ./ (1 + t.^3);
plot(x, y, 'b-', 'LineWidth', 2); axis equal; grid on;
title('Folium of Descartes'); xlabel('x'); ylabel('y');

2. Descartes Rule of Signs

function r = descartes_rule(p)
  % p = coefficients highest to lowest
  s = sign(p(p~=0)); % drop zeros
  changes = sum(abs(diff(s))>0);
  r.pos_max = changes;
  r.pos_possible = changes:-2:mod(changes,2);
  q = p .* ((-1).^(length(p)-1:-1:0)); % p(-x)
  s2 = sign(q(q~=0));
  changes2 = sum(abs(diff(s2))>0);
  r.neg_max = changes2;
end
% example: p(x)=x^3 - 3x + 2
p = [1 0 -3 2]; descartes_rule(p)

3. Leibniz π series

% leibniz_pi.m
N = 1e6;
k = 0:N-1;
s = sum((-1).^k ./ (2*k+1));
pi_est = 4*s;
printf('N=%d → pi≈%.10f error=%.2e\n', N, pi_est, abs(pi-pi_est));

4. Leibniz derivative (numerical)

% leibniz_derivative.m — dy/dx via limit
f = @(x) x.^3 - 3*x;
x0 = 2; h = logspace(-1,-12,12);
d = (f(x0+h)-f(x0))./h;
[ h' d' ] % approaches 9

5. Binary conversion (Leibniz)

function b = to_binary(n)
  b = dec2bin(n); % Octave built-in mirrors Leibniz idea
end
to_binary(42) % → '101010'

6. Pre-established harmony simulation

% two "monads" with internal clocks, no interaction
t = 0:0.01:10;
m1 = sin(t); % monad 1
m2 = sin(t + 0); % monad 2 pre-programmed same phase
plot(t,m1,'r',t,m2,'b--'); legend('Monad A','Monad B');
title('Pre-established Harmony — synchronized without causal link');

Interactive Examples

Proofs to Remember

Descartes — Cogito as intuition

Not a syllogism but immediate: thinking entails existence because existence is contained in the concept of thinking. Deny it and you perform it. From clear and distinct perception, Descartes builds mathematics as the model of certainty.

Leibniz — Calculus from continuity

Using law of continuity, $dy/dx$ is the ratio of infinitesimals where error is smaller than any assignable quantity. Product rule: $d(uv)=u dv+v du+du dv$, last term negligible by transcendental law of homogeneity → $d(uv)=u dv+v du$. This symbolic reasoning is why Leibniz notation won.

Common ground

Both sought a universal method: Descartes' Mathesis Universalis and Leibniz's Characteristica Universalis. Both believed reason discovers structure that God guarantees — Descartes through clear ideas, Leibniz through sufficient reason and best-world selection.