Infinitesimal - Philosophical Concept | Alexandria
Infinitesimal: an entity smaller than any assignable quantity, yet not quite zero—a concept that dances on the edge of nothingness, beckoning mathematicians and philosophers alike into a realm where intuition falters and the very foundations of number tremble. Are infinitesimals truly nonexistent, or do they merely elude our conventional grasp?
The seeds of the infinitesimal concept may be traced back to ancient Greek philosophy, particularly the method of exhaustion employed by Archimedes (c. 287–212 BCE) to calculate areas and volumes. While not explicitly defining infinitesimals, Archimedes’ approach involved dividing shapes into increasingly smaller segments, prefiguring later developments. Zeno's paradoxes, particularly those regarding motion and division, questioned the fundamental nature of continuity and infinite divisibility. These early explorations laid the groundwork for a more formalized treatment of the infinitely small, though a precise definition remained elusive for well over a millennium. Thinkers within Aristotelian traditions, especially those concerning physics and motion, wrestled with concepts akin to infinitesimals. The "great conversation" of humanity echoed with the murmurs of thinkers trying to contain this elusiveness into a cogent idea.
The calculus developed independently by Isaac Newton (1643–1727) and Gottfried Wilhelm Leibniz (1646–1716) in the late 17th century relied heavily on infinitesimals, with Leibniz even using the notation dx to represent an infinitely small increment. This approach, though immensely powerful, faced criticism due to its lack of rigorous justification and led to the ethical paradox that a tool so effective could be conceptually ambiguous. Bishop George Berkeley (1685–1753), a staunch critic of the new calculus, famously derided infinitesimals as "ghosts of departed quantities." The 19th century saw the rigorous reformulation of calculus by mathematicians such as Augustin-Louis Cauchy (1789-1857) and Karl Weierstrass (1815-1897), who replaced infinitesimals with the concept of limits, providing a solid foundation. The cultural impact of infinitesimals extends beyond mathematics: it has become a byword for that which is immeasurably small, the barely perceptible yet significant, the seed that can grow into the monumental. The concept also found purchase in discussions of the ethics of persuasion, because it deals with an effect so subtle it is barely perceptible.
Yet, the story of infinitesimals does not end with their apparent banishment from mainstream calculus. Non-standard analysis, developed by Abraham Robinson in the 20th century, gave infinitesimals a rigorous logical foundation, demonstrating that they could be incorporated into a consistent mathematical system. This reintroduction of infinitesimals, once considered paradoxical, revealed the enduring legacy and continuing mystique surrounding this elusive concept. Today, infinitesimals find applications in various fields, from physics to economics, reminding us that even the smallest of things can hold immense power and that our grasp on reality is constantly evolving. What other seemingly paradoxical ideas might hold hidden truths waiting to be unearthed?