![]() Part I, set in sixteenth- and seventeenth-century Italy, pits the anti-infinitesimalist Jesuits, led by mathematician Christopher Clavius, renowned for his mathematical work on the then-new Gregorian calendar, against the infinitesimalist Galileo Galilei and his disciples, most notably Bonaventura Cavelieri and Evangelista Torricelli. By contrast in England, which had been “a wild and semibarbarous country at the northern edge of European civilization” (292), the subsequent triumph of the infinitesimalists was critical to its transformation into “a leading center of European culture and science and a model for political pluralism and economic success” (292).Īlexander structures the book into two main parts, along with an introduction and an epilogue, with each part telling a similar story but in different settings and with different protagonists and antagonists. Moreover, it is his contention that in Italy, which had been at the forefront of mathematics and science in the time of Galileo, the victory by the anti-infinitesimalists led to scientific, mathematical, and cultural stagnation. But when the struggle raged in the seventeenth century, the combatants on both sides believed that the answer could shape every facet of life in the modern world that was then coming into being. Whether the continuum is made up of infinitesimals seems like the quaintest of questions, and it is hard for us to fathom the passions it unleashed. What Alexander offers us is a look at how fraught the question was with cultural significance.Īlexander’s thesis is that the question of indivisibles, which may seem to be of interest only to mathematicians and philosophers, became ground zero for the ideological battles over the nature of European society: The infinitesimalists were eventually vindicated by those such as Newton and Leibniz, in whose hands the method led to what we now know of as Calculus. This revival drew battle lines between those who argued for their inclusion into mathematics because of their demonstrable usefulness and those who argued against them on the grounds that they threatened the ordered and reasonable method of mathematics. It was not until the sixteenth century that there was a revival of interest in infinitesimals among European mathematicians. For instance, if the indivisibles that compose a line segment have zero length, then how can the line segment have non-zero length? On the other hand, if the indivisibles have some non-zero length, why is the length of the line segment not infinite?įollowing the early successes with infinitesimals by such giants as Archimedes, concern for logical precision meant that for two millennia infinitesimals were shunned by mathematicians who created a Euclidean edifice of mathematics free from paradox and ambiguity that they believed “was fixed, orderly, and eternally true” (78). However, the paradoxes articulated by Zeno and others demonstrated that accepting infinitesimals leads to what seem like logical contradictions. Ancient Greek mathematicians who accepted this assertion did groundbreaking work on areas and volumes. The doctrine of infinitesimals states that the continuum is composed of indivisibles, that is, that “every line is composed of a string of points, or ‘indivisibles,’ which are the line’s building blocks, and which cannot themselves be divided” (9). Alexander’s synthesis of interpretive history makes a compelling case, and his book provides an enjoyable and non-technical read for those interested in the history of ideas. ![]() As the subtitle to Infinitesimal suggests, Amir Alexander makes the startling assertion that ground zero of the battle over the shape of the modern world was a seemingly innocuous and abstruse mathematical claim about the nature of lines. Such changes were contested, and from them emerged a new way of perceiving the world. The transition to modernity was shaped by changes in science, politics, religion, economics, and culture. ![]() Matthew DeLong is Professor of Mathematics at Taylor University.
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