The Pythagorean Crisis Revisited: A Challenge to the Infinite Decimal Representation of the Reals, #3.
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, the third and final one, contains sections 6–7.
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.
6. Toward a Critical Examination
6.1 Addressing Standard Responses
Objection 1: Orthodox mathematicians typically respond to these concerns through what we might call “completion by fiat” — declaring that infinite decimal expansions simply ARE complete real numbers, not incomplete processes. Dedekind cuts, Cauchy sequences, and equivalence classes of infinite decimals are pronounced “complete” by definition, supposedly resolving the tension between geometric and arithmetic representation.
Claim 1: This objection fails to address the underlying ontological problem. Declaring an infinite process “complete” does not eliminate the processual character that creates the original difficulty. A Cauchy sequence remains an infinite sequence of rational approximations, regardless of how we classify it formally. A Dedekind cut remains a division of the rational numbers into infinite sets, regardless of whether we call it a “real number.” The completion-by-fiat strategy amounts to linguistic stipulation rather than mathematical resolution. It changes how we talk about infinite processes without changing their fundamental character as processes rather than completed objects.
Reply 1: Treating an infinite decimal as a completed whole smuggles in an actually-infinite totality; this re-creates the Pythagorean crisis at a higher level because the infinite totality is both indispensable and incoherent.
Objection 2: Standard set theory (ZFC) is consistent relative to remarkably weak systems (and by Gödel’s relative-consistency proofs has no known contradiction). The “incoherence” is psychological, not logical.
Claim 2: Consistency-proofs are relative; they assume precisely the abstract infinite we are questioning. Moreover, Gödel’s incompleteness theorems show that consistency cannot be proven from within any reasonable system that models arithmetic. Hence the demand for a non-circular justification of the actual infinite is still unmet. Set theory relies on actual infinity (e.g., infinite sets), which assumes what it must prove. The geometric critique highlights that completed infinities remain philosophically contentious and fail to capture the intuitive “givenness” of magnitudes like √2.
Reply 2: We can never verify the equality √2 =1.4142… by a finite inspection of digits, so the identity statement lacks empirical and computational content; decimal equality is non-effective.
Objection 3: In classical mathematics truth is not limited to effective verifiability. Identity is defined analytically (e.g., “the unique real whose square is 2”) and proofs establish it once-and-for-all. Computability constraints belong to constructive mathematics, not to analysis as normally practiced.
Reply 3: The critique is epistemic, not merely constructivist: if no finite datum can distinguish √2 from an adversary sequence that first agrees for a trillion digits then diverges, what grounds the identity? A purely analytic definition pushes the mystery back a step — why should the analytic definition latch onto the geometric magnitude rather than to any other model satisfying the same first-order axioms (Löwenheim-Skolem theorem/Skolem “paradox”)?
Claim 3: Real analysis models the continuum as a set of points, yet any set of points is measure-zero; ergo, the model misrepresents extension.
Objection 4: The σ-algebra/measure apparatus recovers extension; the Lebesgue measure of [0,1] is 1. “Measure-zero” applies only when you view points in isolation; when united in a set they generate length through outer measure.
Reply 4: The reply presumes precisely what is questioned, namely, that length emerges from aggregating non-extended elements. The fact that measure theory postulates additive axioms does not explain how zero-measure elements combine to yield positive measure. Intuitively, summing zeros forever ought still to be zero; treating it otherwise is a formal trick that hides metaphysical discomfort.
Claim 4: Because mathematics is just manipulation of symbols in a formal system, there is no “object” whose nature can conflict with geometry.
Objection 5: If one adopts formalism the crisis disappears; there is nothing to be “in crisis” about.
Reply 5: Formalism secures consistency at the price of mathematical truth. Working mathematicians treat theorems as discovering facts about a shared subject matter (e.g., prime distribution), not merely as permissible moves in a calculus. A philosophy that contradicts actual practice solves the crisis by denying mathematics itself.
Objection 6: You’re conflating a representation with the object itself. Mathematicians know that the decimal 1.414… is just one way of expressing it. The real number is an equivalence class — say, of Cauchy sequences or Dedekind cuts — not a literal sequence of digits.
Reply 6: Yes, but this substitution only shifts the problem. Cauchy sequences and Dedekind cuts are still infinite procedures — limiting processes of rational approximation. The claim that these are “completed” objects is not an outcome of construction but a declaration by fiat. The tension we highlight lies not in misuse of notation, but in the ontological mismatch between what is said to exist (a complete real number) and the means by which it is defined (an infinite process).
Calling an infinite process a “completed object” does not make it so in any constructive or intuitive sense. It merely reclassifies it within a formalist framework that declines to engage with metaphysical consistency.
Objection 7: Why should we care about metaphysical tension if the formalism is coherent and the math works? Calculus, physics, and engineering rely on real analysis with great success.
Reply 7: Indeed, the machinery works. But Zeno’s paradoxes of motion also describe things that obviously occur — like Achilles overtaking the tortoise — yet do so in a way that renders the process paradoxical. Functional success does not entail conceptual coherence.
Moreover, if mathematics aims to describe or reflect reality — even idealized reality — it must not simply function, but also make sense, and be rationally justified. Metaphysical scrutiny may seem academic, until we consider the limits of our models. For example, physical theories involving the continuum (like general relativity) may be misleading if built upon flawed mathematical intuitions.
Objection 8: We never need infinite precision in practice. We work to tolerances.
Reply 9: True. But tolerances presuppose a completed value toward which we converge. Our critique is not of approximation, but of the idea that an infinite approximation is the thing being approximated. That metaphysical leap is never justified — it is assumed. Tolerances presuppose a completed value toward which we “converge.” The critique is not of approximation, but of the idea that an infinite approximation is the thing being approximated. This metaphysical leap is never justified — it is assumed.
Objection 9: The infinite decimal is merely a specification of a real number, not a “temporal process.” Modern mathematics, post-Dedekind and Cantor, defines real numbers as completed objects (e.g., equivalence classes of Cauchy sequences or Dedekind cuts), where all digits are “given at once by a single law.” There’s no gap; geometric segments and these abstract structures denote the “same object inside a single abstract structure.”
Reply 9: This substitution only shifts the problem because Cauchy sequences and Dedekind cuts are still infinite procedures — limiting processes of rational approximation. Declaring them “completed” is a declaration by fiat, not an outcome of construction. This merely reclassifies it within a formalist framework that declines to engage with metaphysical consistency.
Objection 10: Many mathematicians would argue that the “asymmetry” highlighted here between geometric completeness and arithmetic incompleteness is precisely why formal definitions of real numbers (Dedekind cuts, Cauchy sequences) were developed. They bridge this gap by providing a rigorous, formal definition that completes the rational numbers in a way that aligns with geometric intuition, without relying on the messy, infinite decimal representation as the definition itself. The infinite decimal is merely a representation of that formally defined complete object. The “crisis” was resolved by providing a consistent axiomatic system.
Reply 10: Even granting for the sake of argument that formal definitions of real numbers were devised to deal with these and other problems, it does not follow that these accounts are successful, there being objections to both the Dedekind cuts and Cauchy sequences, summarized by us in another paper (Smith and Stocks, 2025). These formal definitions still rely upon “infinite processes.” The completeness is still stipulated, not inherently derived from the nature of the infinite process itself.
Objection 11: While you acknowledge that functional success does not entail conceptual coherence, the degree of success in describing and predicting the physical world, from quantum mechanics to general relativity, which heavily relies on the real number continuum, is a powerful empirical argument for its underlying coherence. If there were deep, unresolvable metaphysical contradictions, wouldn’t they manifest as fundamental breakdowns in these applications?
Reply 11: Physical theories may be misleading if built on flawed intuitions. These applications often deal with approximations and finite measurements, and the “crisis” only appears at the infinite, foundational level, which is rarely directly confronted in practical applications.
Objection 12: You propose constructive mathematics, finite mathematics, and nonstandard analysis as alternatives. An objector might ask if these alternatives introduce their own metaphysical or practical problems. For example, constructive mathematics often lacks certain powerful theorems of classical analysis, and nonstandard analysis, while elegant, can be seen as introducing its own “infinities” (infinitesimals). Do these alternatives truly resolve the core tension, or just shift it?
Reply 12: It’s granted these alternatives have their own trade-offs, but they do confront the metaphysical discomfort more directly rather than hiding it through formal tricks or linguistic stipulation. The goal isn’t necessarily a “perfect” system, but a more honest one.
Objection 13: The paper seems to lean towards a Platonistic view where geometric objects have an immediate actuality. A formalist or constructivist might simply deny this, arguing that all mathematical objects are constructs of the human mind or formal systems, and therefore, there’s no independent “geometric reality” for the arithmetic representation to “conflict” with. The conflict dissolves if reality itself is seen as a construct.
Reply 13: Our reply to the formalism objection already touches on this by asserting that mathematicians do treat theorems as discovering facts about a shared subject matter. Further that geometric intuition, as a rigorous geometric construction, provides a powerful and consistent mode of mathematical insight that should not be dismissed simply because it clashes with a purely symbolic, arithmetic approach.
Objection 14: Non-Euclidean geometry undermines geometric primacy.
If geometry is framework-dependent (e.g., √2 varies in curved spaces), why privilege Euclidean constructibility?
Reply 14: Euclidean constructions remain the minimal context for discovering irrationals. Non-Euclidean geometries extend rather than invalidate this; the tension between completeness and incompleteness persists in any continuous space. The objection from non-Euclidean geometry misses the point. The discovery that Euclidean geometry is not the only possible geometry does not undermine the mathematical validity of Euclidean constructions within Euclidean space. The √2 construction remains valid and illuminating within its proper geometric context.
Objection 15: category theory generalizes smoothly: Modern foundations (e.g., topoi) define reals without decimals or geometry, thereby resolving the paradox.
Reply 15: Alternative foundations (e.g., synthetic differential geometry) retain actual infinity. The paper’s core insight — that geometric intuition reveals limitations of infinitistic reasoning — applies universally.
6.2 The Geometric Construction Alternative
In response to charges that geometric intuition is unreliable, we must distinguish between (i) mere spatial intuition and (ii) rigorous geometric construction. The compass-and-straightedge construction of √2 provides precise, rule-governed procedures that yield determinate mathematical objects. This is not psychological intuition subject to error, but constructive proof yielding mathematical entities with well-defined properties.
Geometric construction offers genuine mathematical insight that infinite decimal representation systematically obscures. The constructive character of √2 reveals mathematical relationships (proportion, incommensurability, constructibility) that disappear when we focus solely on infinite decimal approximation. A compass-and-straightedge hypotenuse is a finished object, whereas its decimal expansion √2 =1.4142… is an unending, never-arrived-at process; therefore the two “cannot be the same” and the arithmetic picture is inadequate.
Objection 16: Mathematics after Dedekind and Cantor treats the decimal expansion as a specification of a real number, not a temporal “process”. All digits are given at once by a single law (e.g. a Cauchy sequence or Dedekind cut). No gap exists: the geometric segment and the equivalence class of Cauchy sequences denote the same object inside a single abstract structure (ℝ, +, ×, <).
Reply 16: Our worry targets the metaphysical identification itself: a geometric individual that is present in intuition is being identified with an unending syntactic schema whose totality can only be fixed “by fiat.” The fact that we define a real to be that equivalence class does not show that the class has the unity a genuine individual should have. Dedekindian completion dissolves the problem by stipulation; it does not solve it.
6.3 Opening Up Mathematical Possibilities
Our critique does not advocate mathematical conservatism but rather points toward alternative approaches that could resolve the tensions we identify. Constructive mathematics, for instance, approaches real numbers through algorithmic procedures rather than infinite decimal representations, potentially avoiding the metaphysical problems we have outlined.
Finite mathematics explores mathematical structures that avoid problematic infinities altogether. Nonstandard analysis provides alternative foundations for analysis using infinitesimals rather than limits. These approaches suggest that our critique opens new mathematical possibilities rather than merely closing down existing ones.
The goal is not to eliminate irrational numbers, but instead to find more adequate representations that respect both their geometric reality and their arithmetic properties without falling into the contradictions that plague infinite decimal approaches.
7. Conclusion
The discovery of incommensurable magnitudes created more than a historical crisis for ancient mathematics — it revealed a permanent fault line between geometric intuition and arithmetic formalization. The case of √2 demonstrates that this fault line persists in contemporary mathematics, where the same tensions between completeness and incompleteness continue to generate philosophical and potentially mathematical difficulties.
The standard responses from the orthodox theory of real analysis — completion by fiat, appeals to formal definitions, dismissal of geometric insight — all fail to resolve the underlying ontological tensions. If anything, they reveal how deeply these tensions are embedded in the foundations of contemporary mathematics.
As we proceed to examine specific inconsistencies in infinite decimal representation, the Pythagorean crisis provides both historical context and philosophical motivation. If the foundations of real analysis cannot adequately handle the simplest cases of irrationality, we have reason to question whether they can provide a consistent foundation for the mathematical continuum.
The ancient problem of √2 thus opens up onto contemporary questions about mathematical existence, infinite processes, and the relationship between geometric intuition and formal arithmetic. These questions are not merely philosophical: they bear directly on the consistency and coherence of real analysis as a mathematical discipline. The time has come to take seriously the possibility that our current foundations may be inadequate to the mathematical reality they purport to describe.
The crisis of incommensurability did not end with the invention of real analysis. It was merely formalized out of view. The tension between geometric and arithmetic representations — between completed magnitudes and infinite approximations — remains unresolved at the heart of modern mathematics. A system that treats infinity as a definitional shortcut is not metaphysically neutral. It is inherently fragile.
We must revisit the Pythagorean crisis not as a solved curiosity but as an enduring challenge. If the simplest irrational number still resists coherent representation, then what confidence can we have in a system that claims to encompass the continuum?
REFERENCES
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(Smith & Stocks, 2025). Smith, J. W. & Stocks, N. “Mathematical Skepticism: Infinity and Real Numbers.” 2–22 June. Available online at URL = <https://againstprofphil.org/2025/06/22/mathematical-skepticism-infinity-and-real-numbers-4/>.
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