The Limits of Limits: A Skeptical Inquiry into the Foundations of the Calculus, #1.

ε-δ Definition

TABLE OF CONTENTS

1. Introduction

2. The Formal Definition of Limits: Weierstrass and the Alleged Achievement of Mathematical Rigor

3. The Triumph of 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?

The essay below will be published in four installments; this one, the first, contains sections 1-2.

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.

The Limits of Limits: A Skeptical Inquiry into the Foundations of the Calculus

1. Introduction

In this essay, we critically examine the standard ε–δ definition of limits as formulated by Karl Weierstrass (1815-1897), arguing that its claim to foundational rigor masks a series of unresolved conceptual problems. While the formal apparatus successfully avoided vague notions of infinitesimals (for the time, but now formalized in treatments such as nonstandard analysis (Robinson, 1966; Bell, 1998; Stroyan and Luxemburg, 1976; Davis, 1977) and Smooth Infinitesimal Analysis (SIA) (Moerdijk and Reyes,1991)) and dynamic approximation, it does so at the cost of introducing new forms of abstraction that are no less philosophically contentious. We explore two central objections: (i) the “infinite tasks” problem, which questions the operational meaning of universal quantification over all ε > 0, and (ii) the “circularity” objection, which highlights the definitional dependence on a pre-assumed limit value L. Drawing on critiques by N. J. Wildberger and J. Gabriel (Gabriel, n.d.), we investigate whether these issues reflect deep structural flaws in the epistemic status of limits, continuity, and differentiation. An alternative approach, such as that based on convergent sequences, is assessed and found to either replicate or reframe the same foundational concerns. Our critical analysis suggests that modern limit theory, while indispensable in practice, may rest on conceptual foundations that are provisionally coherent rather than intrinsically rigorous. We conclude by considering whether the pursuit of formal precision in analysis has, paradoxically, obscured the intuitive and ontological ambiguities it sought to resolve.

2. The Formal Definition of Limits: Weierstrass and the Alleged Achievement of Mathematical Rigor

The development of calculus in the 17^th and 18^th centuries produced powerful computational tools, but it rested on philosophically troubling foundations (Cauchy, 1821; Boyer, 1949; Kline, 1972; Grabiner, 1981; Lakoff and Núñez, 2000; Ehrlich, 2006). The widespread use of infinitesimals—quantities treated as both zero and non-zero depending on mathematical convenience—drew sharp criticism from observers like Bishop Berkeley, who derided these “ghosts of departed quantities” as logically incoherent. Even when mathematicians abandoned infinitesimals in favor of more intuitive language about limits “tending toward” values, fundamental questions remained unanswered. What did it mean, precisely, for a mathematical quantity to “approach” something? How could mathematics claim logical rigor while relying on metaphorical descriptions borrowed from motion and time (Błaszczyk et al., 2013)?

Karl Weierstrass revolutionized mathematical analysis by providing an answer that eliminated all appeals to intuition and established calculus on a foundation of pure logical precision, allegedly (Weierstrass, 1894). His epsilon-delta definition of limits transformed the field not merely through technical innovation, but also by demonstrating how mathematical concepts could achieve complete rigor without sacrificing their essential meaning, supposedly.

The Weierstrass definition, as per the diagram at the top of this essay, 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. This deceptively simple statement contains profound philosophical implications embedded within its logical structure.

The definition operates by means of what might be called a “challenge-response framework.” The epsilon represents a challenge: no matter how small a positive number someone chooses as a measure of acceptable accuracy, the limit claim must be defensible. And the delta provides the response: given any such challenge, there exists a corresponding restriction on the input variable that guarantees the desired level of accuracy in the output. Crucially, this relationship must hold universally—for every possible epsilon, no matter how small.

Let’s consider the geometric interpretation of this logical structure. The epsilon creates a horizontal band of width 2ε centered around the proposed limit value L, extending from (L – ε) to (L + ε). This represents the acceptable “tolerance” for how close function values must be to the limit. Correspondingly, δ creates a vertical band of width 2δ centered around the approach point a, extending from (a – δ) to (a + δ), while explicitly excluding the point x = a itself through the condition 0 < |x – a|. The definition requires that whenever the input x falls within this δ-neighborhood of a, the corresponding function value f(x) must fall within the ε-neighborhood of L.

What makes this definition philosophically revolutionary is its complete elimination of dynamic or temporal language. There is no reference to variables “approaching” values, no description of “tending toward” limits, no invocation of processes unfolding over time. Instead, Weierstrass replaced all such metaphorical language with precise logical relationships between numerical quantities. The limit is not something that happens; it’s a property that either holds or fails to hold based on the existence of appropriate δ values for all possible ε challenges.

This transformation addressed the fundamental philosophical problem that had plagued earlier approaches to limits. When mathematicians spoke of variables “approaching” limits, they imported intuitions from physical motion into pure mathematics. But mathematical objects have no location in space and undergo no processes in time. By recasting limits as static logical relationships, Weierstrass demonstrated that mathematical analysis could achieve complete precision without relying on potentially misleading physical analogies.

The logical structure also clarifies exactly what must be proven to establish the existence of a limit. To demonstrate that a limit exists, one must show that for any proposed ε—no matter how small—a suitable δ can be found that satisfies the required relationship. Contrapositively, to prove that a limit does not exist, one need only identify a single epsilon value for which no corresponding delta can guarantee the necessary relationship. This transforms limit proofs from vague arguments about “getting arbitrarily close” into precise logical demonstrations.

The Weierstrass definition represents more than just a technical improvement in mathematical analysis; it exemplifies a fundamental shift in the philosophical understanding of mathematical rigor. Prior approaches to limits, whether through infinitesimals or intuitive “tending” language, retained connections to non-mathematical intuitions about physical processes or spatial relationships. Weierstrass demonstrated that mathematics could achieve autonomy from such external supports.

This achievement established a template that influenced the development of modern mathematical standards throughout analysis and beyond. The definition showed how careful logical formulation could eliminate conceptual confusion while preserving the essential mathematical content that made calculus so powerful. Rather than weakening mathematical intuition, the rigorous approach actually strengthened it by providing a solid foundation upon which geometric and physical insights could be confidently built.

The epsilon-delta definition also answered Berkeley’s critique by showing that calculus required no “ghosts of departed quantities” or other logically problematic entities. Every element of the definition refers to ordinary real numbers and their relationships. The apparent mystery of limits—how could something approach a value without reaching it?—dissolved once limits were understood as logical properties rather than dynamic processes.

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