The Pythagorean Crisis Revisited: A Challenge to the Infinite Decimal Representation of the Reals, #2.
By Joseph Wayne Smith and N. Stocks
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TABLE OF CONTENTS
1. Introduction
2. The Geometric-Arithmetic Divide
4. Historical Resonances: The Zenoesque Structure
5. Implications for Real Analysis
6. Toward a Critical Examination
7. Conclusion
The essay below will be published on three installments; this one, the second, contains sections 4–5.
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4. Historical Resonances: The Zenoesque Structure
4.1 Parallels with Zeno’s Paradoxes
The √2 problem exhibits the same logical structure as Zeno’s celebrated paradoxes. In both cases, we begin with an apparently unproblematic whole — Achilles’s journey from start to finish, the flight of an arrow, the completion of a geometric construction. When we subject this whole to mathematical analysis, however, it dissolves into infinite complexity.
Zeno showed that motion, seemingly continuous and natural, becomes paradoxical when analyzed through infinite division. The runner must complete infinitely many sub-journeys, each taking some positive time, yet somehow accomplishes this impossible task in finite duration. Similarly, the geometric hypotenuse presents itself as a simple, unified magnitude that becomes mathematically intractable when we attempt arithmetic representation.
4.2 The Strategy of Infinite Approximation
Modern mathematics addresses Zeno’s paradoxes through the theory of limits, treating infinite processes as converging to finite results. The calculus provides formal machinery for handling infinite sums and infinite subdivisions. Yet this strategy, when applied to real numbers, may create more problems than it solves.
The infinite decimal representation of √2 relies on a similar limiting process — the decimal expansion “approaches” √2 as we include more digits. But this raises the question: if the geometric √2 is already complete, what does it mean to approach it through an infinite sequence of rational approximations? The limit process seems to treat as incomplete what geometric intuition presents as already finished.
5. Implications for Real Analysis
5.1 The Foundation Problem
If the simplest irrational number creates this tension between geometric completion and arithmetic incompletion, what implications follow for the entire real number system? Modern real analysis builds the continuum on infinite decimal representations, treating real numbers as equivalence classes of Cauchy sequences, Dedekind cuts, or infinite decimal expansions.
Each of these approaches relies fundamentally on infinite processes. Yet our analysis of √2 suggests that such processes may be metaphysically inadequate to capture the reality of geometric magnitudes. The completed geometric line conflicts with the incomplete arithmetic process meant to represent it.
5.2 The Consistency Challenge
This tension points toward potential inconsistencies in real analysis itself. If geometric intuition provides genuine insight into mathematical reality — as suggested by the constructive completeness of √2 — then arithmetic methods that conflict with this intuition may be systematically misleading.
The infinite decimal approach treats real numbers as processes rather than objects, limits rather than completed magnitudes. But if geometric magnitudes exist as completed objects, then representing them through incomplete processes introduces a fundamental mismatch between mathematical reality and mathematical formalism.
5.3 The Transcendental Intensification
The problems we observe with √2 become even more acute when we consider transcendental numbers like π and e. While √2 at least emerges from algebraic relationships (x² = 2), transcendental numbers resist even algebraic specification. The geometric reality of π as the ratio of circumference to diameter in any circle conflicts even more dramatically with its infinite decimal representation π = 3.14159265358979…
If a simple algebraic irrational like √2 creates foundational tensions, transcendental numbers reveal the full inadequacy of infinite decimal representation. The geometric completeness of circular relationships stands in stark contrast to the arithmetic incompleteness of infinite, non-repeating, non-algebraic decimal expansions.
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AGAINST PROFESSIONAL PHILOSOPHY REDUX 1037
Mr Nemo, W, X, Y, & Z, Monday 15 September 2025
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