Gottfried Wilhelm Leibniz

Gottfried Wilhelm Leibniz's intellectual system is governed by four sovereign metaphysical and formal principles:

• Principle of Contradiction (PC): Every proposition whose negation implies a formal contradiction is necessarily false; every proposition whose negation does not imply a contradiction is contingent. Governs all necessary mathematical truths.
• Principle of Sufficient Reason (PSR): Nothing exists or occurs without a reason why it should be so rather than otherwise. There is an intelligible explanation for every fact, state of affairs, and contingent truth.
• Identity of Indiscernibles (Leibniz's Law): No two individual substances can differ solely in number (solo numero). If two entities share every intrinsic property in common, they are identical: $\forall F (F(x) \leftrightarrow F(y)) \implies x = y$.
• Relational Spacetime: Space is not an absolute physical container (contra Newton), but the order of possible coexisting entities. Time is the order of successive possibilities.

                    [ METAPHYSICAL FOUNDATION ]
                                 |
         +-----------------------+-----------------------+
         |                                               |
[ Principle of Contradiction ]              [ Sufficient Reason ]
(Governs Relations of Ideas)                (Governs Contingent Worlds)
         |                                               |
[ Formal Logic & Calculus ]                 [ Pre-Established Harmony ]
(Universal Characteristic)                  (Monadic Optimization)

2.1 The Differential and Integral Calculus (Nova Methodus, 1684)

Leibniz defined the differential $dx$ as an infinitesimal increment, and $dy$ as the corresponding change along a curve $y = f(x)$.

The Product Rule (Leibniz's Rule):

Consider the product of two variables $u$ and $v$: $$d(uv) = (u + du)(v + dv) - uv = u \cdot dv + v \cdot du + du \cdot dv$$ Neglecting the second-order infinitesimal product $du \cdot dv \approx 0$: $$d(uv) = u \, dv + v \, du$$

The Fundamental Theorem of Calculus:

Leibniz recognized integration (summation of infinitesimal strips $\int y \, dx$) and differentiation (tangent slope calculation $dy/dx$) as mutually inverse operations: $$d \left( \int y \, dx \right) = y \, dx \quad \iff \quad \int \left( \frac{dy}{dx} \right) dx = y$$

Leibniz General Product Rule for $n$-th Derivatives:

$$(u \cdot v)^{(n)} = \sum_{k=0}^{n} \binom{n}{k} u^{(n-k)} v^{(k)}$$

2.2 Formalization of Binary Arithmetic (1703)

Leibniz established base-2 positional representation using the set $\{0, 1\}$: $$N = \sum_{i=0}^{m} b_i 2^i, \quad b_i \in \{0, 1\}$$

Binary addition rules: $$0 + 0 = 0, \quad 0 + 1 = 1, \quad 1 + 0 = 1, \quad 1 + 1 = 10_2 \text{ (0 with carry 1)}$$

Binary multiplication rules: $$0 \times 0 = 0, \quad 0 \times 1 = 0, \quad 1 \times 0 = 0, \quad 1 \times 1 = 1$$ He observed that binary arithmetic eliminates complex multiplication tables, reducing all arithmetic to shifting and conditional additions.

2.3 Conservation of Vis Viva (Kinetic Energy)

Leibniz contested Descartes' claim that the total quantity of motion conserved in mechanics is $m |v|$. He demonstrated through falling body experiments that the true living force (vis viva) conserved in elastic collisions is proportional to mass times velocity squared: $$\text{Vis Viva} = m v^2$$ This formed the direct historical ancestor of kinetic energy ($E_k = \frac{1}{2} m v^2$).

3.1 The Stepped Drum Mechanism (Staffelwalze)

The mechanical core of the Stepped Reckoner (1673), enabling mechanical multiplication via variable-length gear teeth:

                  [ DRIVE AXLE ]
                        |
            +-----------+-----------+
            |  Stepped Cylinder     |
            |  (Staffelwalze)       |
            |                       |
 Tooth 9    |  |=================|  |  Length 9
 Tooth 8    |  |================ |  |  Length 8
 Tooth 7    |  |===============  |  |  Length 7
 Tooth 6    |  |==============   |  |  Length 6
 Tooth 5    |  |=============    |  |  Length 5
 Tooth 4    |  |============     |  |  Length 4
 Tooth 3    |  |===========      |  |  Length 3
 Tooth 2    |  |==========       |  |  Length 2
 Tooth 1    |  |=========        |  |  Length 1
            +-----------------------+
                        |
           [ Slidable Pinion Gear ]  <--- Shifted along axis to select digit (0-9)
                        |
            [ Dial Accumulator / Register ]

3.2 The Relational Monadic Network vs. Absolute Space

NEWTONIAN ABSOLUTE SPACE:
   +---------------------------------------------------+
   | Rigid Empty Stage (Sensorium of God)              |
   |   [Body A] ---- d ---> [Body B]                   |
   | (Space exists independently of existing objects)  |
   +---------------------------------------------------+

LEIBNIZIAN RELATIONAL NETWORK:
   ( No empty background space )

   [Monad 1] <.... Perceptual Perspective ....> [Monad 2]
       :                                            :
       :             Pre-Established                :
       :                 Harmony                    :
       :                                            :
   [Monad 3] <................................> [Monad 4]
   (Space is nothing other than the order of coexisting perceptions)

• Operational Dominance of Leibnizian Notation: The $dx$, $dy$, and $\int$ notation completely superseded Newton's dot-notation across continental Europe and eventually Britain (via the Analytical Society of Cambridge in 1812), enabling the development of advanced differential geometry and analysis.
• Physical Realization of the Stepped Reckoner: Constructed functional working prototypes in 1673 and 1694; validated by the Royal Society of London and the Académie des Sciences in Paris.
• Binary Computing Architecture: Leibniz's 1703 treatise provided the explicit theoretical blueprint utilized by Claude Shannon (1937) and John von Neumann (1945) to build electronic digital computers.
• Relational Field Equations in General Relativity: Albert Einstein credited Leibniz's relational view of spacetime as a conceptual precursor to the diffeomorphism invariance of general relativity, where spacetime possesses no independent existence apart from the metric tensor.

5.1 Infinitesimals and the Lack of Rigorous Limits

Leibniz treated infinitesimals ($dx$) ambiguously as both non-zero quantities that can divide expressions and zero-equivalent quantities that can be dropped from sums ($du \cdot dv = 0$). This heuristic looseness provoked George Berkeley's celebrated 1734 critique (The Analyst) mocking infinitesimals as 'the ghosts of departed quantities', unresolved until Cauchy and Weierstrass formalized limits in the 19th century.

5.2 The Problem of Pre-Established Harmony and Free Will

By positing that each monad contains its entire past, present, and future within its initial internal programming, Leibniz's system approaches absolute determinism. Reconciling human moral responsibility and free choice with monads that cannot be influenced by any external cause remains a profound tension within the Theodicy.

5.3 Over-Optimism in the Universal Character

The dream that all conceptual disputes could be resolved through symbolic calculation (Calculemus) underestimated the semantic ambiguity of human language, the context-dependence of qualitative values, and the formal boundaries of axiomatic systems established by Gödel's Incompleteness Theorems (1931).

• De Arte Combinatoria (On the Combinatorial Art), 1666. Early vision of formal logic and universal symbolics.
• Nova Methodus pro Maximis et Minimis, 1684. First published treatise on differential calculus in Acta Eruditorum.
• Brevis Demonstratio, 1686. Refutation of Cartesian conservation of momentum and introduction of vis viva.
• Discours de métaphysique (Discourse on Metaphysics), 1686. Foundations of individual substance, grace, and pre-established harmony.
• Explication de l'Arithmétique Binaire, 1703. Formal introduction of binary arithmetic.
• Théodicée (Theodicy), 1710. Metaphysical defense of God's justice and the best of all possible worlds.
• La Monadologie (The Monadology), 1714. Definitive summary of the metaphysical system of monads.
• Leibniz-Clarke Correspondence, 1715–1716. Historic debate on relational vs. absolute spacetime and Newtonian mechanics.