The Pythagorean Crisis Revisited: A Challenge to the Infinite Decimal Representation of the Reals, #1.
By Joseph Wayne Smith and N. Stocks
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TABLE OF CONTENTS
1. Introduction
2. The Geometric-Arithmetic Divide
3. The Metaphysical Paradox
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 first, contains sections 1–3.
But you can also download and read or share a .pdf of the complete text of this essay, including the REFERENCES, by scrolling down to the bottom of this post and clicking on the Download tab.
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The Pythagorean Crisis Revisited: A Challenge to the Infinite Decimal Representation of the Reals, #1.
1. Introduction
The discovery that revolutionized ancient Greek mathematics continues to pose challenges for contemporary mathematical foundations, we will argue. When the Pythagoreans encountered the incommensurability of the diagonal and side of a square, they confronted more than a computational difficulty — they discovered a fundamental tension between geometric reality and arithmetical representation that persists in modern real analysis.
Consider the elementary construction: a right triangle with legs of unit length. The Pythagorean theorem yields a hypotenuse of length √2. This simple geometric object presents us with a profound philosophical puzzle that illuminates deep problems in how we conceive mathematical existence and numerical representation.
2. The Geometric-Arithmetic Divide
2.1 Geometric Completeness
The hypotenuse of our unit right triangle exists as an unambiguously complete geometric entity. Constructible with compass and straightedge, it presents itself to mathematical intuition as a finished object with determinate endpoints and a specific length. We can manipulate this line segment geometrically, compare it to other lengths, and use it in further constructions. Its geometric reality appears immediate and non-problematic. This hypotenuse can be separated from the other sides of the right-angled triangle and exists as a finite determinate line.
This geometric completeness extends beyond mere constructibility. The hypotenuse occupies a definite position in space, has measurable relationships to other geometric objects, and serves as a foundation for further geometric reasoning. In the realm of geometric intuition, √2 exists as naturally and completely as any rational length.
2.2 Arithmetic Incompleteness
When we attempt to represent this completed geometric object arithmetically, however, we encounter an irreducible infinity. The decimal expansion √2 = 1.414213562373095… never terminates, never enters a repeating cycle, and never reaches completion in any finite sense. The arithmetic representation suggests not a completed object but an endless approximation process.
This creates a striking asymmetry. The geometric hypotenuse is given as a completed whole, while its arithmetic representation unfolds through an infinite sequence of digits that never converges to a final form. The same mathematical object appears complete in one domain and incomplete in another.
The geometric segment representing √2 can be physically constructed and compared to other lengths. Yet no instrument can measure it to infinite precision. The decimal expansion is not merely incomplete — it is unmeasurable in principle. This leads to a stark conclusion: if the decimal representation cannot be measured, and the geometric representation cannot be approximated completely, then there is a disconnect in our very concept of what a “real number” is supposed to be.
3. The Metaphysical Paradox
3.1 Two Modes of Mathematical Existence
This discrepancy reveals what we might call two distinct modes of mathematical existence. Geometric objects appear to possess immediate actuality — they are present to mathematical intuition as completed wholes. The hypotenuse simply is √2 units long, with no remainder, qualification, or approximation required.
Arithmetic representations, by contrast, seem to exist as a process. The infinite decimal 1.414213… exists not as a static object but as a rule for generating an endless sequence of approximations. Its “existence” consists in the possibility of carrying out an infinite computation, not in the presence of a completed numerical entity.
3.2 The Problem of Reference
This raises fundamental questions about mathematical reference. When we write “√2,” to what exactly are we referring? If we mean the geometric length, we indicate a completed, determinate magnitude. If we mean the infinite decimal, we indicate an incomplete process that never terminates in a final numerical value.
The same symbol thus appears to denote both a completed geometric reality and an incomplete arithmetic process. This ambiguity in reference suggests a deeper instability in how we conceptualize real numbers and their relationship to geometric magnitudes.
3.3 The Measurement Problem
The tension between geometric and arithmetic representations becomes acute when we consider measurement. The geometric √2 can be measured directly through construction — we can create the hypotenuse, compare it to unit lengths, and use it as a standard for further measurements. Geometric measurement achieves completion through spatial relationships and constructive procedures.
Arithmetic measurement of √2, however, confronts a fundamental impossibility. The infinite decimal 1.414213… cannot be measured in any complete sense, since measurement requires finite specification. We can approximate √2 to any desired degree of accuracy, but we cannot measure it completely through arithmetic means. The infinite decimal is, in principle, unmeasurable — a curious property for something supposedly representing a measurable geometric magnitude.
This measurement asymmetry reinforces our central claim: geometric and arithmetic representations involve fundamentally different ontological commitments about mathematical existence and determinacy.
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