Are Numbers Real?

October 10, 2016
Even the most math-phobic have nothing to fear in the latest from English science writer Clegg (Ten Billion Tomorrows): a lighthearted yet far-reaching look at the history of numbers and how we use them. From their earliest days, numbers haven’t been seen as “real,” but they represent real things remarkably well, helping people keep track of livestock and produce, for instance. As Clegg explains, it wasn’t long before the ancient Greeks began using geometry to describe patterns such as the shape of objects and the motion of the sun overhead. Euclid’s geometry described a type of elegant, perfect world that Plato had imagined centuries prior, but that didn’t quite mesh with the curvilinear nature of the universe. Clegg also digs into the development of a way to represent nothing with zero, and absolutely everything with infinity. When Clegg isn’t reveling in such “mathematical toys” as imaginary numbers, he’s sharing stories about the calculus feud between Isaac Newton and Gottfried Leibniz, Galileo’s deviousness at sneaking heretical ideas past the Inquisition, and a legal wrangle over exactly what the number one means. Whether he’s counting sheep or measuring warped space around black holes, Clegg offers an entertaining and accessible look at the numbers we take for granted every day.

October 15, 2016
The emphasis is on "real" in the latest by the prolific British science writer, who questions the extent to which mathematics truly reflects the workings of nature.Clegg (Ten Billion Tomorrows: How Science Fiction Technology Became Reality and Shapes the Future, 2015, etc.) has a degree in physics from Cambridge, and he uses that knowledge to discourse on the relationship between math and science. Math builds its own universe based on given (i.e., not proven) axioms and rules of operation to derive facts (theorems) that are true in that system. Science, on the other hand, builds theories based on observations and experiments, and the theories become conventional wisdom until questioned by new observations and data. Nevertheless, over time, there has been an eerie congruence between abstruse developments in math--e.g., non-Euclidean geometry--and the equations that govern Einstein's theory of general relativity. Clegg suggests that math increasingly has come to rule the roost in physics. Nobody has ever seen a black hole he notes; the objects are "more the product of mathematics than of science," the evidence for their existence being indirect. Likewise the Higgs boson and superstring theory. The author urges caution and a step back rather than obedience to a questionable math authority. Before reaching this conclusion, Clegg treats readers to an orderly history of math. He begins with counting on fingers or marks on sticks to match the amount of a physical object, leading to symbols for numbers. These numbers are really real, he says, because they are based on matches with objects in nature. But as math evolved, that connection blurred. By the 19th century, with set theory and concepts of orders of infinity, and 20th-century proofs on the incompleteness of mathematical systems and other logical conundrums, the relation to reality has faded--as will some readers' attention, because this is not easy stuff. Solid as a straightforward chronology of how mathematics has developed over time, and the author adds a provocative note urging scientists to keep it in its place.

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October 1, 2016

The title question is one that has been pondered by philosophers and mathematicians for centuries. In an attempt to lead us to a conclusion, Clegg (Ten Billion Tomorrows; Final Frontier) investigates the historical growth of the number system and the development of the field into more abstract and abstruse areas. His emphasis is on the interplay of mathematics and science, particularly physics, as each area of investigation in turn takes the lead. However, it is in the book's last two chapters that he develops the major point. That is, as mathematical models of physical theories increase and improve, we must avoid the pitfall of confusing model with reality. For example, when the model predicts the existence of subatomic particles never previously encountered, research efforts are frequently directed toward finding these objects, although they may be simply mathematical artifacts. And when experiments produce results at odds with mathematical theory, we try to patch up the model rather than look for physical explanations. VERDICT Clegg is an outstanding science writer, and this book lives up to his usual standard. Highly recommended for those interested in math or science.--Harold D. Shane, mathematics emeritus, Baruch Coll. Lib., CUNY

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