Infinite Series of Value: Money as Mathematical Abstraction

Money as Infinite Series: Chasing Convergence

Consider the geometric series: 1 + 1/2 + 1/4 + 1/8… Each term approaches zero, yet the sum converges to exactly 2. This paradox—infinity contained within a finite limit—reveals something fundamental about abstraction itself. Simmel observed that modern societies transform social relations into symbolic objects, with money becoming the purest expression of this process. Money is not wealth itself but a symbol representing all possible exchanges, all potential values collapsed into numerical form.

Protestant doctrine created a peculiar anxiety: salvation predetermined by God’s inscrutable will, yet evidence demanded by tormented believers. This psychological trap required outward proof of inner grace, and money provided the elegant solution. By standardizing diverse human achievements into comparable units, monetary accumulation became measurable evidence of divine favor. The wine experiment illustrates this perfectly—when prices attach to bottles, sudden consensus emerges about quality and worth. Money collapses ambiguity into hierarchy, transforms spiritual uncertainty into numerical certainty.

But here is the mathematical problem: capitalism promises infinite accumulation (1 + 2 + 3 + 4…) while operating in a finite world. Each dollar earned is another term in an infinite series, each investment another fraction approaching some imagined limit. Protestant anxiety drives this pursuit—hoarding wealth as proxy for salvation, accumulation as evidence against existential dread. Yet no finite sum satisfies an infinite hunger. The abstraction cannot resolve concrete needs.

The Partition Problem: Decomposing Value

My work on partitions explores this question: how many ways can a number be expressed as a sum? The number 5 decomposes as: 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1. Seven distinct partitions for a single value. Money functions similarly—$100 decomposes into commodity bundles, investment portfolios, exchange rates, speculative futures. The underlying value remains abstract, never directly measurable.

This reveals money’s fundamental instability. A partition is merely one representation among many possible decompositions. When money replaced God after the Enlightenment—becoming the supreme organizing principle of reality—it inherited divine authority while retaining mathematical abstraction. Fiat currency makes this explicit: governments print unlimited paper that people accept as real wealth. The system persists through collective belief, not intrinsic value. The partition can change at any moment, yet we treat current decomposition as eternal truth.

Divergent Series: When Growth Cannot Converge

Mathematics distinguishes carefully between convergent and divergent series. The harmonic series (1 + 1/2 + 1/3 + 1/4…) appears to approach a limit but actually diverges to infinity. Capitalism’s logic resembles this deception—perpetual growth promises eventual stability while requiring infinite expansion.

Does capitalism fail because it applies infinite logic to finite reality? Resources deplete, time exhausts, lives end, yet the system demands unlimited accumulation. The abstraction becomes more real than material constraints. Protestant anxiety transformed private hoarding into moral virtue, elevating frugality and accumulation as evidence of salvation. But mathematics shows not all series converge. Some chase infinity without ever reaching a limit.

The question emerges clearly: if capitalism is a divergent series—pursuing growth without convergent bound—must the system eventually collapse under its own mathematical impossibility? The partition of value becomes infinitely complex, the terms multiply beyond comprehension, yet we continue adding, hoping somehow that infinity will resolve into a finite, satisfying sum. Perhaps it cannot.