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In Search of Interior Riches

Reuben Hersh

Geometrical Landscapes: The Voyages of Discovery and the Transformation of Mathematical Practice. Amir R. Alexander. xviii + 293 pp. Stanford University Press, 2002. $65.

Geometrical Landscapes—in part a history of English exploration of the Americas, in part a history of precursors of the infinitesimal calculus—is an original and challenging contribution to the history of ideas. Author Amir R. Alexander acknowledges several well-known historians of mathematics and several centers for cultural studies, humanities, European studies and international studies. But his claims for his methodology leave me wondering how much he consulted with actual mathematicians.

The book makes and documents the claim that in the 17th century, geographical and mathematical explorers shared a "standard narrative of exploration and discovery." This narrative

posited a wondrous land of riches in the interior, surrounded by hazardous terrain of forests, mountains, and, occasionally, icebergs. The enterprising explorer who arrived on its shore would find hidden passages and break through all obstacles in his way to the fabled land of the interior. For this he was rewarded with fabulous riches and the possession of a wondrous land.

Alexander gives many examples of this narrative, presented literally in reports of explorations of the Atlantic coast of North America and figuratively in accounts of the mathematics of rhumb lines and equiangular spirals.

At left is a modern depiction of an equiangular spiral . . .Click to Enlarge Image

The exploration stories are fascinating: Martin Frobisher repeatedly convinced himself that he had found the Northwest Passage to the Orient, and Walter Raleigh vainly attempted to colonize Virginia and Guyana. But Alexander focuses mainly on advertisements for and reports about these explorations. Regardless of geographical reality, they conform to the standard narrative.

Mathematicians were closely involved in these explorations. Thomas Hariot, a leading Elizabethan mathematician, was actually a member of Raleigh's first Virginia colony. His contemporary Henry Briggs, the first Savilian Professor of Mathematics at Oxford, constructed "elaborate mathematical tables, which were to be used by mariners in determining their location according to the magnetic dip." Briggs was an "advisor and promoter" of two competing voyages seeking the Northwest Passage in 1631.

In an appendix on "The Mathematical Narrative," Alexander notes that Hariot and his colleagues played important roles in expeditions, designing nautical instruments, preparing astronomical tables, composing promotional pamphlets, drawing maps of discoveries and sometimes even going along for the journey.

Navigators needed mathematical help in following the "rhumb line"—the curve on the surface of the Earth that leads a ship along a constant bearing to its desired destination. A natural planar simplification of the rhumb line is the equiangular spiral, which Alexander describes as "a curve that revolves endlessly around a central point, approaching it ever more closely but never actually reaching it. . . . [I]f straight lines were drawn emanating from the central point, the spiral would cross each and every one of them at the same angle." To calculate either the rhumb line or the equiangular spiral, the natural method was approximation by straight line segments. Hariot theorized that a curved line is made of connected infinitesimal straight line segments. Alexander says Hariot was not just describing the continuum, he was exploring its inner essence.

In a chapter on "Navigating Mathematical Oceans," Alexander describes the better-known work done with infinitesimal methods in the 16th century by Continental mathematicians Bonaventura Cavalieri, Evangelista Toricelli and Simon Stevin. These mathematical explorations by means of infinitesimals were different in spirit from what one finds in Euclid's Elements. Paradoxes were relished. They involved searching the geometrical unknown, rather than rigorously deriving known facts. And the descriptive rhetoric that accompanied this mathematical work often used the same images and metaphors as did the proposals and reports of geographic exploration, all following the standard narrative of discovery.

On the basis of this historical material, Alexander makes a grandiose claim for a new historical methodology: "Certain mathematical techniques, developed by Elizabethan mathematical practitioners, were shaped by a ubiquitous cultural narrative." He maintains that "In mathematics, as also in cartography, Hariot's work was guided by the standard narrative of exploration and discovery."

Mathematicians will see the shaping and guiding in a different light. Practical needs focused attention on the rhumb line and its model, the equiangular spiral. The only tool available to study such curves was approximation by straight line segments. Once a mathematician had made some progress in analysis and calculation, it would be natural to talk about these successes with the rhetoric and metaphors available from geographical exploration. To a mathematician it would seem a blatant non sequitur to claim that those metaphors and rhetorics "shaped" or "guided" the mathematical work.

Alexander offers this book as a new paradigm for the history of mathematics. He finds that standard history of mathematics, which "emphasizes the progressive unveiling of universal truths, rather than the contingencies of human existence, has little to do with 'history' as it is commonly understood."

"I find a narrative approach to be a most promising avenue for historicizing mathematics," writes Alexander.

Mathematical work does, I argue, contain a narrative. Once this narrative is identified, it can be related to other, nonmathematical cultural tales that are prevalent within the mathematicians' social circles. A clear connection between the "mathematical" and the "external" stories would place a mathematical work firmly within its historical setting.

No problem so far. But then Alexander goes on to claim that "If a strong relationship can be established between an historically specific nonmathematical tale and the narrative of a mathematical work that originated within its social sphere, then mathematics can indeed be said to be fundamentally shaped by its social and cultural setting." By using a narrative approach to the history of mathematics, he maintains, "complex mathematical techniques are shown to be dependent on cultural narratives prevalent in their wider social setting."

No such thing is or could be shown.

Mathematics progresses by the recognition or invention of problems and the struggle to solve them. The choice of problems for Hariot and Stevin came from navigation and engineering. Social reality shapes mathematics, not by its narratives, but by its practical needs. The means to solve the problem are then "shaped" or "guided" by the problem itself, by the mathematical knowledge and technique available at that time and place, and by the ingenuity of the mathematicians who work on it.

If there existed a prevalent social or cultural story that was analogous or parallel to the mathematical story, it by no means follows that such a story "shaped" or "guided" the mathematics. Such a social or cultural story may have simply served as a model for how one talked about or advertised the mathematics.

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