InfinitesimalsEdit
Infinitesimals are small quantities that are not zero but are smaller than any positive real number. In the history of mathematics they served as the intuitive backbone of early calculus, enabling the description of instantaneous rates of change and infinitely small quantities that were treated as if they could be manipulated in equations. Today the notion survives in two closely related but formally distinct ways: as actual infinitesimals in nonstandard analysis, and as the limit-based foundation underlying standard calculus. In both traditions, infinitesimals play a central role in how we model change, motion, and the geometry of smooth objects, even as the modern language of mathematics makes precise the ideas that once lived in the realm of intuition.
The core idea is simple to state, yet historically controversial in its interpretation. An infinitesimal is typically envisioned as a quantity that is nonzero but smaller than every positive real number. In practice, this meant that quantities like dx and dy, representing an infinitesimal change in x or y, could be treated as if they were legitimate objects in algebraic manipulations. This approach helped mathematicians encode the idea of instantaneous change and the notion of a tangent or a differential without resorting to a limiting process in every calculation. The notational economy and geometric intuition of this view were powerful, and they influenced not only mathematics but also the development of physics and engineering.
In contemporary mathematics there are two mainstream ways to formalize infinitesimals. One is through nonstandard analysis, in which infinitesimals are genuine, albeit nonstandard, numbers in an enlarged number system called the hyperreals. The second is through the limit-based, or epsilon-delta, framework of standard analysis, where derivatives and differentials are defined by limiting procedures rather than by actual infinitesimals. The transition from the older, intuitive language to the modern, rigorous language was a major project of 19th and 20th-century mathematics, and it reshaped how people think about continuity, differentiation, and the foundations of calculus. See Nonstandard analysis for the infinitesimals-as-quantities viewpoint, and see Calculus for the limit-based picture that became dominant in most undergraduate and graduate curricula.
History
Early uses and the calculus of infinitesimals
The use of infinitesimals goes back to the origins of calculus in the work of Gottfried Wilhelm Leibniz and his contemporaries, who introduced differential symbols like dx and dy as part of a symbolic calculus for rates of change. In this tradition, infinitesimals were treated as if they were actual quantities that could be added, subtracted, and multiplied according to familiar algebraic rules. The Newtonian formulation of calculus, centered on fluxions and accelerations, shared this intuition about small quantities moving the mathematical story forward. The phrase “infinitely small quantities” was part of the language of the subject from the outset.
Critiques and the rise of rigor
The intuitive use of infinitesimals faced persistent objections, most famously from the philosopher George Berkeley, who argued that the calculus relied on quantities that could neither be clearly defined nor consistently manipulated. In his critiques, Berkeley warned that the arguments bore the appearance of reasoning about “ghosts of departed quantities.” These criticisms contributed to a push toward a more rigorous foundation of calculus, culminating in the arithmetization of analysis with the development of the limit concept by figures such as Augustin-Louis Cauchy and Karl Weierstrass. This shift solidified a framework in which derivatives and integrals could be defined and manipulated without appealing to actual infinitesimals.
The 20th century and the revival of infinitesimals
A major development in the 20th century was the formal reconstruction of infinitesimals within a rigorous system, most notably by Abraham Robinson with Nonstandard analysis. In this framework, infinitesimals are real mathematical objects within the hyperreal number system, and familiar calculus can be developed in a way that mirrors the original intuition. The transfer principle provides a bridge between the standard real numbers and their nonstandard counterparts, allowing many statements about real numbers to be extended to infinitesimals and infinities in a precise way. See Hyperreal numbers and Standard part for related concepts.
Foundations and philosophical perspectives
Beyond technical developments, infinitesimals sit at the crossroads of foundational philosophies in mathematics. While the standard epsilon-delta approach emphasizes limits, constructivist and intuitionist lines of thought have questioned the existence of completed totalities that include actual infinitesimals. In some approaches, infinitesimals are avoided or reinterpreted in order to align with particular philosophical commitments. See Intuitionism for discussions of alternative foundations, and see Archimedean property for a precise mathematical statement that characterizes when infinitesimals cannot exist in a given number system.
Mathematical foundations
Standard analysis and limits
In the standard framework, derivatives are defined as limits of difference quotients, and differentials are interpreted via linear approximations derived from those limits. This approach makes no use of actual infinitesimals as standalone objects, but it preserves the practical success and notation that tradition associated with infinitesimals. See Calculus for an overview of how the limit concept underpins differentiation and integration.
Nonstandard analysis and hyperreal numbers
Nonstandard analysis introduces hyperreal numbers, an extended number system that contains infinitesimals and infinite numbers alongside the ordinary real numbers. In this setting, dx can be treated as a nonzero infinitesimal, and the derivative can be defined as the standard part of a ratio of infinitesimals. This formalization preserves the computational intuition of infinitesimals while ensuring consistency and rigor. See Nonstandard analysis and Hyperreal numbers for more on this approach, including the transfer principle that connects standard and nonstandard statements.
Differentials as notational devices
Even within standard analysis, differentials such as dx and dy often function as notational conveniences that represent linear approximations rather than concrete numbers. In many texts, the differential is understood as a linear map that, when applied to a velocity or tangent vector, yields an infinitesimal change in the dependent variable. This usage preserves calculus’s practical language while attaching it to a solid, limit-based foundation.
Philosophical and educational dimensions
The battleground over infinitesimals has both mathematical and philosophical dimensions. Proponents of nonstandard analysis emphasize that infinitesimals can illuminate intuition and simplify reasoning about certain problems, while critics point to the sufficiency of limits and the discomfort of introducing a larger, more abstract number system. Educational practice varies accordingly, with some curricula presenting infinitesimals as a heuristic, and others adopting a strictly limit-based exposition.
Controversies and debates
Infinitesimals have long been a site of intellectual dispute. The early debates centered on whether infinitesimals could be said to exist as well-defined quantities, or whether they were merely convenient fictions. The Berkeley critique highlighted worries about coherence and foundations, helping to spur a more careful articulation of what calculus could claim to be mathematically.
In the modern era, the revival of infinitesimals in nonstandard analysis resolved many old concerns by providing a coherent and rigorous framework in which infinitesimals are legitimate elements of an enlarged number system. Critics who prefer a strictly epsilon-delta universe sometimes view nonstandard analysis as unnecessarily complex or as a departure from the historical intuition. Proponents argue that nonstandard analysis offers a cleaner and more transparent account of differentiation and integration in many contexts, and it connects with other areas of mathematics, including model theory and probability.
Other debates involve the role of infinitesimals in educational settings. Some educators and mathematicians favor explicit infinitesimal reasoning as a means to convey the intuition behind derivatives and integrals, while others emphasize the precision and stability of limit-based introductions. Across these discussions, the central issue is not a disagreement about usefulness alone but about the most effective and coherent way to build mathematical understanding—whether by preserving intuitive infinitesimals in a robust formal system or by adhering to a framework that formerly required more careful handling of limits and sequences.
Applications and connections
Infinitesimals appear in many mathematical contexts. In differential geometry, they underpin ideas about tangent spaces and smooth maps, where infinitesimal changes are encoded in differentials and linear approximations. In physics, differential concepts provide a language for describing motion, forces, and fields, where infinitesimal changes in space and time form the backbone of equations of motion. In probability and stochastic calculus, notions of infinitesimal increments arise in Itô calculus and related theories, where small, random fluctuations are analyzed through differential notation. See Calculus, Differential geometry, Itô calculus, and Getzler-style approaches for related topics and perspectives.
The foundational choices surrounding infinitesimals influence other areas of mathematics as well. Nonstandard analysis links to model theory and logic, while standard analysis connects to numerical methods and rigorous analysis. The concept also appears in discussions of the continuum and the philosophical questions about the nature of the real numbers, the infinitude of the line, and the ways in which mathematics represents motion and change.