The First Mathematical Law of International Relations The Law of Anarchy: Power of Anarchy=0
Nabil Hamidi
PhD Candidate in Diplomacy and IR
Master of Arts in International Studies and Diplomacy
Al Akhawayn University-Arizona State University
+212 708361292
Anarchy and Zero: A Scientific Clarification of World Power
1. Introduction
The study of international relations (IR) has long revolved around one fundamental condition: anarchy. Kenneth Waltz (1979) famously defined anarchy as the absence of a central authority above states. This definition is deceptively simple. It has structured entire schools of thought: realism, which views anarchy as the driver of self-help and competition; liberal institutionalism, which argues that cooperation can mitigate anarchy; and constructivism, which insists that the meaning of anarchy is not given but constructed by state interaction (Wendt, 1992).
Niklas Luhmann (1995), writing from outside IR, conceptualized politics as an autopoietic system: self-reproducing communication without external foundation. His theory suggested that no “final center” exists to ground politics — an idea consistent with the absence of world authority in IR.
Despite decades of debate, one implication of Waltz’s definition remains underdeveloped. If anarchy means the absence of a supreme central authority, then the power of that authority must be zero. And if world power is defined as the power of that central authority, then under conditions of anarchy, world power = 0.
This article develops that claim rigorously. It proceeds scientifically and methodologically, offering precise definitions, hypotheses, logical derivation, and empirical illustrations. The argument is not rhetorical but deductive: a conceptual axiom that clarifies the meaning of world power in an anarchic system.
2. Conceptual Foundations
2.1 Anarchy in International Relations
For Waltz (1979), anarchy is the ordering principle of the international system. Unlike domestic politics, where hierarchy ensures rule by a sovereign, international politics lacks such a structure. There is no global sovereign. This absence conditions state behavior: self-help, security competition, and balancing.
Liberal scholars did not deny this absence but argued that regimes and institutions can provide order under anarchy (Keohane, 1984). Constructivists went further: anarchy is not deterministic, because its consequences depend on how states interpret it (Wendt, 1992). Yet across these perspectives, the definitional point holds: no central authority exists above states.
2.2 Central Authority
A central authority, for precision, means a global sovereign: an actor with universal, legitimate, and enforceable power over all states. Domestic analogies would be the state vis-à-vis individuals. In IR, no such actor exists. Even the United Nations does not qualify: its Security Council is paralyzed by vetoes, and compliance depends on state consent. Hedley Bull (1977), in The Anarchical Society, made this explicit: international order arises not from sovereignty at the global level, but from institutions negotiated among states.
2.3 World Power Defined
This article defines world power (WP) as the effective power of a central authority (P_CA).
WP = P_CA.
This definition excludes state power. States may be powerful, but their influence remains relative and contested. To call the U.S. or China “world powers” is a linguistic shortcut, not a conceptual truth. True world power would mean the binding authority of a global sovereign.
3. Hypotheses
The logic can be expressed in three hypotheses:
- H1: Under anarchy, central authority (CA) does not exist (Waltz, 1979).
- H2: If CA does not exist, then P_CA = 0.
- H3: Since WP = P_CA, then WP = 0.
These hypotheses are deductive, not empirical. They derive from definitions, much like axioms in mathematics.
4. Methodological Framework
4.1 Deductive Logic
This article employs deductive formalization. Instead of induction from cases or statistical inference, the claim follows directly from definitions. If “world power” is the power of a central authority, and if anarchy means no such authority exists, then world power must equal zero.
4.2 Quantification of Absence
In science, absence is represented as “zero.” Physics treats the absence of quantity as zero. Mathematics treats the null set as zero. Here, absence of central authority is represented as P_CA = 0.
4.3 Consistency Check
Although deductive, the argument requires empirical consistency: if world power were not zero, we would expect to find a central sovereign exercising legitimate authority globally. No such actor exists.
5. Logical Demonstration
The logical derivation proceeds as follows:
- Definition: WP = P_CA.
- Condition: Anarchy = absence of CA.
- Implication: If CA does not exist, then P_CA = 0.
- Conclusion: WP = 0.
This is tautologically valid. The conclusion cannot be false if the premises are true.
6. Clarifying Scope
6.1 State Power vs. World Power
States exercise relative power: military, economic, cultural. Great powers shape outcomes disproportionately. But none wields world power because none commands universal authority.
6.2 Great Powers Misnamed
Common usage calls the United States a “world power.” Yet by this article’s definition, this is inaccurate. The U.S. is a great power, but it is not a world power. No state can issue universally binding commands.
6.3 International Institutions
Institutions mimic central authority but remain derivative. The UN, WTO, and IMF depend on state consent. Their authority is partial, conditional, and revocable. They cannot alter the equation: WP = 0.
7. Empirical Consistency
Although derived deductively, the axiom is confirmed by observation.
- United Nations Security Council: Resolutions are often blocked by veto. No universal enforcement exists.
- World Trade Organization: Compliance depends on voluntary adherence. No global police enforce rulings.
- U.S. invasion of Iraq (2003): Attempt to impose systemic authority collapsed under insurgency and international resistance.
- Soviet Afghanistan (1979–1989): Another effort at unilateral domination failed.
- Climate Change Negotiations: Paris Agreement (2015) demonstrates voluntary coordination, not sovereign imposition.
In each case, attempts at “world power” failed or proved conditional.
8. Scientific Consequences
8.1 Analytical Precision
The axiom clarifies a persistent confusion: “world power” is not hegemony or great power status. It is the authority of a non-existent sovereign.
8.2 Reinforcing Structural Realism
Waltz’s structural realism gains methodological clarity. Anarchy is not just absence; it produces a quantifiable result: WP = 0.
8.3 Compatibility with Constructivism
Wendt’s constructivism remains valid: states construct meanings under anarchy. But construction does not create a central authority. The structural zero remains.
8.4 Systems Theory Confirmation
Luhmann’s systems theory (1995) shows politics operates without final closure. This aligns with WP = 0: the absent center cannot exercise power.
9. Toward a Scientific Axiom
This article proposes the Axiom of World Power under Anarchy:
If CA = ∅, then WP = 0.
This axiom is as fundamental to IR as conservation laws are to physics. It clarifies what can and cannot exist in the international system.
10. Conclusion
Waltz identified anarchy as the absence of central authority. Wendt showed its meanings are socially constructed. Luhmann emphasized systemic reproduction without closure. Together, these insights converge on a simple deduction: if the central authority does not exist, its power is zero; therefore, world power is zero.
This conclusion is methodological, scientific, and quantitative. It transforms an old definition into a precise axiom. Anarchy is not chaos. It is the absence of a master. And in that absence, world power does not exist.
References
- Allison, Graham. Essence of Decision: Explaining the Cuban Missile Crisis. Little, Brown, 1971.
- Bull, Hedley. The Anarchical Society. Columbia University Press, 1977.
- Keohane, Robert. After Hegemony: Cooperation and Discord in the World Political Economy. Princeton University Press, 1984.
- Luhmann, Niklas. Social Systems. Stanford University Press, 1995.
- Waltz, Kenneth. Theory of International Politics. Addison-Wesley, 1979.
- Wendt, Alexander. “Anarchy is What States Make of It.” International Organization 46, no. 2 (1992): 391–425.
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