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?
The essay below has been published in four installments; this, the fourth and final one, contains sections 7-9.
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.
7. Applications Under Threat
The practical consequences extend beyond pure mathematics. Physics relies heavily on derivatives for concepts like velocity, acceleration, and electromagnetic field calculations. If derivative definitions are circular, it raises questions about the mathematical foundations underlying physical laws. Engineering applications from structural analysis to control systems depend on calculus-based models. Circular foundations could compromise the theoretical basis for these critical applications. Economics uses calculus for optimization problems, marginal analysis, and dynamic modelling. Logical circularity in limits would affect the mathematical rigor of these economic theories.
The Circularity Inheritance Problem
The core issue is that mathematical definitions should be built from more basic, well-defined concepts. If limits are defined circularly, then every concept built upon them—derivatives, integrals, continuity, differentiability—inherits this circularity. This creates a house of cards whereby the entire structure depends on a conceptually flawed foundation.
The mathematical community has generally dismissed circularity concerns about limits, but if such concerns prove valid, they would necessitate a complete reconstruction of analysis. This would require either (i) developing alternative, non- circular definitions of limits, as constructionists have done, or (ii) finding entirely different approaches to derivatives and continuity, which for example, nonstandard analysis and Smooth Infinitesimal Analysis (SIA) allegedly do.
Accepting that current classical calculus operates on logically imperfect foundations is the worst of these options. The implications of circular limit definitions extend far beyond abstract mathematical philosophy. They strike at the heart of one of mathematics’ most successful and widely-applied theories. While calculus undeniably works in practice, the question of whether its theoretical foundations are logically sound remains critical for mathematical rigor and the long-term development of analysis.
Rejoinders: The Cultural Practice Defense and Its Limitations
Classical mathematicians typically respond to the circularity objection by arguing that the definition’s widespread acceptance and practical utility demonstrate its validity. They contend that mathematical definitions need not conform to strict logical purity as long as they enable productive mathematical work. The epsilon-delta definition has facilitated centuries of successful mathematical development, and this pragmatic success allegedly justifies its use despite potential logical concerns.
However, this defense faces serious philosophical challenges. The appeal to cultural practice and historical success commits what might be called the “pragmatic fallacy”—the assumption that usefulness implies truth or logical validity. Many historically successful but ultimately false theories in science and mathematics demonstrate that practical utility does not guarantee logical soundness. The geocentric model of planetary motion, for instance, was enormously useful for astronomical calculations while being fundamentally incorrect about the structure of the solar system.
Moreover, the cultural practice defense fails to address the specific logical problem that has been identified. If the definition is genuinely circular, then its acceptance by the mathematical community might reflect collective oversight rather than collective wisdom. Mathematical communities have, historically, accepted definitions and approaches that were later recognized as problematic or inadequate.
The Deeper Skeptical Challenge
The circularity objection connects to broader philosophical skepticism about mathematical knowledge and foundations. Drawing on David Hume’s problem of induction, mathematical skeptics can argue that the acceptance of epsilon-delta definitions represents a form of inductive reasoning that cannot be logically justified. The mathematical community has inductively concluded that these definitions are sound based on their past utility, but this inductive inference cannot provide the kind of logical certainty that mathematics claims to achieve.
This skeptical perspective suggests that mathematical definitions and concepts are ultimately human constructions subject to the same uncertainties and fallibilities that characterize other human endeavors. The search for absolute logical rigor in mathematics may be fundamentally misguided, and different definitional approaches may coexist without any being inherently superior to others.
The existence of alternative approaches to limits—such as sequence-based definitions or Cauchy’s formulation—supports this skeptical view. If multiple, potentially incompatible approaches can serve as foundations for calculus, this suggests that the choice among them is more a matter of convention and convenience than logical necessity. The epsilon-delta approach’s dominance might reflect only historical contingency rather than logical superiority.
Implications for Mathematical Education and Practice
If the circularity objection is taken seriously, it has significant implications for how mathematics should be taught and understood. Rather than presenting the epsilon-delta definition as the uniquely rigorous foundation for calculus, educators might need to acknowledge its limitations and present it as one approach among several possible alternatives. This would require a more modest and less dogmatic approach to mathematical foundations.
The objection also suggests that mathematical skepticism serves a valuable critical function by exposing potential weaknesses in accepted approaches. Even if the circularity objection does not ultimately undermine the epsilon-delta definition, it forces mathematicians to examine and defend their foundational assumptions more carefully.
The circularity objection thus presents a fundamental challenge that cannot be easily dismissed through appeals to practical utility or historical precedent. If the objection is valid, it entails that the epsilon-delta definition fails to provide the logical rigor it claims to establish. If the objection is invalid, classical mathematicians need to provide more convincing logical arguments for why the apparent circularity does not constitute a genuine problem.
This unresolved tension points to deeper questions about the nature of mathematical definition, the relationship between logical rigor and practical utility, and the extent to which mathematics can achieve the kind of certainty and foundation that mathematicians have traditionally claimed. Our critique, pushed to its full skeptical conclusion, suggests that mathematical foundations may be far more provisional and problematic than the mathematical community has generally acknowledged.
8. An Alternative Definition: The Sequence-Based Approach and Its Limitations
In response to concerns about the ε-δ definition of limits, mathematicians have developed several alternative approaches (Rudin, 1976; Abbott, 2001; Bartle and Sherbert, 2011). While nonstandard analysis using infinitesimals represents one significant alternative that merits separate detailed critical examination, the sequence- based definition has also gained particular attention and traction as a potentially more intuitive foundation for limit theory.
The sequence-based approach defines limits through convergent sequences rather than the ε-δ framework. Under this definition, a function f has a limit L at point c if, for every sequence {xₙ} converging to c (where xₙ ≠ c for all n), the sequence {f(xₙ)} converges to L.
This definition appears to offer several advantages: it can be more intuitive for students familiar with sequence convergence; it avoids the quantifier complexity of the ε-δ definition; and it connects function limits directly to the more elementary concept of sequence limits.
But despite its apparent advantages, the sequence-based definition faces significant objections that suggest it may not resolve the foundational issues plaguing the ε-δ approach.
The Circularity Problem Persists
The most serious objection concerns circularity. The sequence-based definition relies fundamentally on the concept of sequence convergence, which itself requires a rigorous definition. When we examine how sequence convergence is typically defined, we find it depends on the same ε-δ framework we sought to avoid: a sequence {xₙ} converges to L if for every ε > 0, there exists N such that for all n > N, |xₙ – L| < ε.
This creates a circular dependency: we define function limits using sequence limits, but sequence limits are defined using the same ε-δ logic. Rather than eliminating the foundational problems, the sequence-based approach merely shifts them to a different level of abstraction.
Loss of Geometric Intuition
The ε-δ definition, whatever its logical flaws, provides a clear geometric interpretation by means of an appeal to neighborhoods and distances. The sequence- based definition lacks this intuitive spatial understanding. While sequences can be visualized, the connection between sequence convergence and function behavior at a point becomes more abstract, potentially making the concept less accessible rather than more so.
Complications with Directional Limits
The sequence-based definition becomes cumbersome when handling one-sided limits or directional approaches. While the ε-δ framework naturally accommodates left-hand and right-hand limits through appropriate modifications, the sequence- based approach requires additional constructs and qualifications that complicate rather than simplify the definition.
Multivariable Complexity
In higher dimensions, the sequence-based definition becomes significantly more complex and less intuitive. The ε-δ approach generalizes naturally to functions of multiple variables, but the sequence-based approach struggles with the variety of ways sequences can approach a point in multidimensional spaces. This limitation makes it less suitable as a foundational framework for advanced analysis.
Philosophical and Foundational Concerns
From a foundational perspective, the sequence-based definition faces similar philosophical objections to those raised against the ε-δ approach. The concept of sequence convergence still relies on the notion of “arbitrarily close” or “approaching” a limit, which contains the same conceptual circularities that plague other definitions.
Moreover, constructivist and finitist mathematicians (Bridger and Richman, 1987; Bishop, 1967) raise additional concerns: the sequence-based definition relies on infinite sequences and their completion, concepts that may not be constructively valid. If one objects to the ε-δ definition on grounds of infinite processes or non-constructive elements, the sequence-based approach faces similar objections.
Pedagogical Challenges
Contrary to claims that the sequence-based approach is more intuitive, many students find the abstract nature of sequences and their convergence difficult to grasp. The connection between sequence behavior and function limits can be less obvious than the direct geometric relationship expressed in the ε-δ definition. This suggests that the purported pedagogical advantages of the sequence –based approach might be overstated.
The Inadequacy of Alternative Approaches
The examination of the sequence-based definition reveals a troubling pattern: alternative approaches to limits often relocate rather than resolve the fundamental logical issues. Whether we use sequences, nets, filters, or other topological constructs, we consistently encounter similar problems of circularity, infinite processes, and foundational assumptions.
This in turn suggests that the problems with limit definitions may be more fundamental than they initially seemed. Rather than representing flaws in a particular definitional approach, they may reflect deeper issues with how we conceptualize the notion of mathematical “approach” or being “arbitrarily close.”
Implications for Foundational Reform
The failure of alternative definitions to resolve circularity concerns has significant implications. If multiple independent approaches to limits suffer from similar logical problems, it suggests that the issue lies not with specific definitions but with our fundamental conceptual framework for understanding limiting behavior.
This points toward the need for more radical foundational reforms rather than definitional modifications. Such reforms might require either abandoning traditional approaches to limits entirely, or developing completely new mathematical frameworks that avoid the problematic concepts of “approaching” or “arbitrarily close,” which has been done with nonstandard analysis, for example.
The sequence-based definition, while offering some practical advantages in specific contexts, ultimately fails to provide the foundational clarity needed to resolve the logical concerns surrounding limit theory. Its examination reveals that the problems with limits may be more deeply rooted in our mathematical conceptual framework than has been previously recognized.
9. Conclusion: The Illusion of Rigor?
The epsilon-delta definition of limits has long been enshrined as the pinnacle of mathematical precision—a triumph of logic over intuition, and rigor over ambiguity. But as we’ve shown in this this essay, that triumph may be less secure than it appears. What is celebrated as a foundation may instead be a carefully concealed circle: a definition that assumes what it claims to define, a standard that requires the impossible execution of infinite tasks, and a framework that obscures mathematical meaning behind layers of abstraction.
Critics like Wildberger have exposed the philosophical cracks in this classical mathematical structure, but the mathematical community has largely responded with silence, dismissal, or appeals to tradition and utility. Yet pragmatic success is not the same as conceptual coherence. If we accept definitions on the basis of habit or effectiveness rather than clarity and justification, then mathematics risks becoming a kind of ritualized formalism—fluent in symbols but detached from meaning.
Attempts to reform limit theory, via sequences, or constructivist logic, have typically replicated the same foundational problems they aimed to solve. This suggests that the difficulty lies not in the choice of formalism, but in the very concept of “approach” itself—a concept suspended uneasily between static logic and dynamic intuition. Perhaps it is not calculus that needs rescuing, but our philosophy of mathematics that needs rethinking, as an important minority have done?
The path forward is unclear, but necessary. We may need to rethink what we mean by rigor, to re-engage with geometric intuition, or to develop finitist or constructive alternatives that avoid the metaphysical sleights of hand buried in classical analysis. What cannot continue is the uncritical veneration of a definition that refuses to define without borrowing what it seeks to prove.
If the price of rigor is circularity, then what exactly are we proving?
REFERENCES
(Abbott, 2001). Abbott, S. Understanding Analysis. New York: Springer-Verlag.
(Apostol, 1967). Apostol, T.M. Calculus, Volume I: One-Variable Calculus with an Introduction to Linear Algebra. 2nd edn., New York: John Wiley & Sons.
(Bartle and Sherbert, 2011). Bartle, R.G. and Sherbert, D.R. Introduction to Real Analysis. 4th edn., New York: John Wiley & Sons.
(Bell, 1998). Bell, J.L. A Primer of Infinitesimal Analysis. Cambridge: Cambridge Univ. Press.
(Bishop, 1967). Bishop, E. Foundations of Constructive Analysis. New York: McGraw- Hill.
(BL, 2023). BL. “When Topology Meets Calculus: Epsilon-Delta.” Medium. 4 April. Available online at URL = <https://medium.com/@mathgames/when-topology-meets- calculus-5090c53989dc>.
(Błaszczyk et al., 2013). Błaszczyk, P., Katz, M., and Sherry, D. “Ten Misconceptions from the History of Analysis and Their Debunking.” Foundations of Science 18, 1: 43-74.
(Boyer, 1949). Boyer, C.B. The History of the Calculus and Its Conceptual Development. New York: Dover.
(Bridges and Richman, 1987). Bridges, D. and Richman, F. Varieties of Constructive Mathematics. Cambridge: Cambridge Univ. Press.
(Cauchy, 1821). Cauchy, A.L. Cours d’Analyse de l’École Royale Polytechnique. Paris: Debure frères.
(Davis, 1977). Davis, M. Applied Nonstandard Analysis. New York: John Wiley & Sons.
(Ehrlich, 2006). Ehrlich, P. “The Rise of Non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of Non-Archimedean Systems of Magnitudes.” Archive for History of Exact Sciences 60, 1: 1-121.
(Gabriel, n.d). Gabriel, J. An Introduction to the Single Variable New Calculus. Available online at URL= <https://www.thenewcalculus.weebly.com>.
(Grabiner, 1981). Grabiner, J.V. The Origins of Cauchy’s Rigorous Calculus. Cambridge MA: MIT Press.
(Hrbacek and Jech, 1999). Hrbacek, K. and Jech, T. Introduction to Set Theory. 3rd edn., New York: Marcel Dekker.
(Katz and Katz, 2012). Katz, M.G. and Katz, U.K. “Stevin Numbers and Reality.” Foundations of Science 17, 2: 109-123.
(Kleene, 1952). Kleene, S.C. Introduction to Metamathematics. Amsterdam: North- Holland.
(Kline, 1972). Kline, M. Mathematical Thought from Ancient to Modern Times. New York: Oxford Univ. Press.
(Lakatos, 1976). Lakatos, I. Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge: Cambridge Univ. Press.
(Lakoff and Núñez, 2000). Lakoff, G. and Núñez, R.E. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.
(Laugwitz, 1989). Laugwitz, D. “Definite Values of Infinite Sums: Aspects of the Foundations of Infinitesimal Analysis around 1820.” Archive for History of Exact Sciences 39, 3: 195-245.
(Machover, 1996). Machover, M. Set Theory, Logic and Their Limitations. Cambridge: Cambridge Univ. Press.
(Moerdijk and Reyes, 1991). Moerdijk, I. and Reyes, G. Models for Smooth Infinitesimal Analysis. New York: Springer.
(Robinson, 1966). Robinson, A. Non-Standard Analysis. Amsterdam: North-Holland.
(Rudin, 1976). Rudin, W. Principles of Mathematical Analysis. 3rd edn, New York: McGraw-Hill.
(Spivak, 2008). Spivak, M. Calculus. 4th edn., Houston TX: Publish or Perish.
(Stewart, 2015). Stewart, J. Calculus: Early Transcendentals. 8th edn., Boston: Cengage Learning.
(Stroyan and Luxemburg, 1976). Stroyan, K.D. and Luxemburg, W.A.J. Introduction to the Theory of Infinitesimals. New York: Academic Press.
(Tall, 1980). Tall, D. “The Notion of Infinite Measuring Number and Its Relevance in the Intuition of Infinity.” Educational Studies in Mathematics 11, 3: 271-284.
(Weierstrass, 1894). Weierstrass, K. Mathematische Werke. Berlin: Mayer & Müller. (White, 2005). White, P. “What is a Limit? A Study of Students’ Concept Images.” Proceedings of the 28th Annual Conference of the Mathematics Education Research Group of Australasia 2: 590-597.
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