ε-δ Definition
TABLE OF CONTENTS
1. Introduction
2. The Formal Definition of Limits: Weierstrass and the Alleged Achievement of Mathematical Rigor
4. Objections to the Standard Theory of Limits
5. The Circularity Objection: A Fundamental Challenge to Epsilon-Delta Logic
6. The Structure of the Circularity
7. Applications Under Threat
8. An Alternative Definition: The Sequence-Based Approach and Its Limitations
9. Conclusion: The Illusion of Rigor?
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5. The Circularity Objection: A Fundamental Challenge to Epsilon- Delta Logic
While Wildberger’s critique focused primarily on pedagogical and intuitive concerns, a more fundamental logical challenge to the epsilon-delta definition emerges from critics like John Gabriel (Gabriel, n.d), who argue that the definition suffers from a basic circularity that undermines its claim to rigorous foundations (Boyer, 1949: p. 281). This objection strikes at the heart of the epsilon-delta approach by questioning whether the definition can coherently establish what it purports to define.
The Logical Problem with Circular Definitions
At the heart of the skeptical challenge to the epsilon-delta definition of limits lies a question of logical legitimacy: what is wrong with a definition that refers to itself? Why is circularity, in a definitional context, considered such a serious flaw?
The answer lies in the fundamental role that definitions play in logical and mathematical systems. A proper definition is meant to provide a reduction: it explains a concept in terms of previously understood or more basic concepts. Definitions establish the foundation from which proofs, theorems, and further reasoning can proceed. If a definition includes the very term it is attempting to define—explicitly or implicitly—it fails to reduce; instead, it merely restates or smuggles in the undefined notion through the back door.
This is not a matter of stylistic purity—it’s a breach of epistemic hierarchy. In formal systems, we rely on well-founded chains of meaning, where terms are built from simpler, already defined components. A circular definition breaks that chain and collapses the hierarchy, rendering the system vulnerable to ambiguity, arbitrariness, or contradiction.
To use a classic analogy: defining “truth” as “a true statement” tells us nothing about what truth is. Similarly, defining the limit L of a function as the value that the function gets arbitrarily close to—by comparing it to L—amounts to saying “the limit is the limit.”
Some forms of circularity may be benign or even necessary in informal language—for example, mutual recursion in computer science or interdependent concepts in natural language. But in foundational mathematics, where the entire edifice of formal reasoning is built on clarity and unambiguous definitions, circularity undermines the integrity of deduction itself.
Moreover, circular definitions are unprovable by design. If a definition presupposes the thing to be defined, any attempted proof becomes a tautology dressed up in technical language. This results in what philosophers of mathematics sometimes call a “pseudo-proof”—a statement that feels rigorous only because it hides its assumptions in the very terms it purports to establish.
If the goal of foundational mathematics is to build upwards from clear and independent primitives, then a circular definition is not just a minor technical lapse. It is a foundational failure—a violation of the very principles mathematical rigor claims to uphold.
6. The Structure of the Circularity
As we’ve seen, the standard epsilon-delta definition of limits states that the limit of f(x) as x approaches a equals L if and only if: for every ε > 0, there exists a δ > 0 such that whenever 0 < |x – a| < δ, then |f(x) – L| < ε. The circularity objection centers on a seemingly obvious but troubling feature of this definition: the limit value L appears within the definition itself. The definition presupposes the existence of the very quantity it claims to define.
This circularity operates at multiple levels. First, the definition requires us to know L in order to verify whether the epsilon-delta conditions are satisfied. We cannot determine whether |f(x) – L| < ε without already having determined what L is. And second, the logical structure suggests that we are defining “the limit L” by reference to L itself—a form of circular reasoning that would be rejected in other contexts as fundamentally flawed.
Consider the parallel with attempting to define “the tallest person in the room” by saying “the tallest person in the room is the person P such that everyone else in the room is shorter than P.” This definition presupposes that we already know who P is, making it useless for actually identifying the tallest person. Similarly, the epsilon-delta definition seems to presuppose knowledge of L while claiming to establish what L is.
The Foundational Implications
If the circularity objection is valid, it suggests profound problems with the mathematical foundations that have been built upon epsilon-delta definitions. The entire edifice of real analysis, with its sophisticated theorems about continuity, differentiability, and convergence, would rest upon a logically flawed foundation. This is not merely a technical quibble but a fundamental challenge to the coherence of modern classical mathematical analysis, although, of course, not alternatives such as nonstandard analysis.
The circularity objection also raises questions about what mathematicians are actually doing when they “prove” that a limit exists. If the definition is circular, then limit proofs cannot establish the existence of limits in the way they claim to. Instead, these proofs might be better understood as demonstrations that certain algebraic manipulations are possible, or that certain patterns hold, without actually establishing the existence of the mathematical objects they purport to define.
Furthermore, the objection suggests that the historical triumph over infinitesimals and intuitive approaches may have been pyrrhic. While the epsilon- delta definition eliminated the alleged logical problems of earlier approaches, it may
have introduced an even more fundamental logical flaw—circularity—that renders the entire enterprise questionable.
Cascading Effects: How Circular Definitions of Limits Undermine Classical Differential Calculus
If the ε-δ definition of limits contains a fundamental logical circularity, the implications extend far beyond abstract mathematical theory. The entire edifice of differential calculus, built upon the foundation of limits, would inherit these logical flaws, creating a cascade of foundational problems throughout mathematics and its applications.
Differential calculus stands on the concept of limits as its primary foundation. The derivative—calculus’s central operation—is defined as the limit of a difference quotient:
f'(x) = lim[h→0] [f(x+h) – f(x)]/h
This definition makes every derivative calculation dependent on the logical soundness of the limit concept. If limits suffer from circular reasoning, then derivatives, as constructs built entirely upon limits, inherit this circularity wholesale.
The circularity pervades calculus’s fundamental concepts. Continuity relies entirely on limit definitions. A function f is continuous at point c if lim[x→c] f(x) = f(c). Any logical problems with limits immediately compromise our understanding of continuity, affecting everything from basic function analysis to advanced topology.
Differentiability compounds the problem by requiring both limits and continuity. Since differentiable functions must be continuous, and derivatives are defined as limits, differentiability inherits circularity from both sources.
Compromise of Major Theorems
The foundational theorems of calculus become logically suspect. The Mean Value Theorem states that for a continuous function on [a,b] that is differentiable on (a,b), there exists some c where f'(c) equals the average rate of change. Since this theorem depends on both continuity and differentiability—both limit-dependent concepts—circular limit definitions would undermine its logical foundation.
Rolle’s Theorem, the Fundamental Theorem of Calculus, and L’Hôpital’s Rule all similarly depend on limit-based definitions. Each would require re-examination if limits prove circular.
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