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Are Numbers Real?: The Uncanny Relationships Between Maths and the Physical World (English Edition) Kindle电子书
Have you ever wondered what humans did before numbers existed? How they organized their lives, traded goods, or kept track of their treasures? What would your life be like without them?
Numbers began as simple representations of everyday things, but mathematics rapidly took on a life of its own, occupying a parallel virtual world. In Are Numbers Real? Brian Clegg explores the way that maths has become more and more detached from reality, yet despite this is driving the development of modern physics.
From devising a new counting system based on goats, through the weird and wonderful mathematics of imaginary numbers and infinity to the debate over whether mathematics has too much influence on the direction of science, this fascinating and accessible book opens the reader's eyes to the hidden reality of the strange yet familiar world of numbers.
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In the end, though, math can only approximate to the actual universe. Whole numbers have real-world equivalents, but fractions don’t, and the shapes of geometry fall down when faced with the physical world, which is constructed of atoms and lines that have width and aren’t perfectly straight. Then there is the concept of zero. It appears that Arab mathematicians got the concept from the mathematicians of India. The oldest Indian example of a placeholder zero is in a stone tablet dated 876 CE, but it was likely in use before then. By the time of the late Alexandrians (c 250 CE) approaches to algebra had reached an intermediate level.
There is a bit of material on a sixteenth-century politician and philosopher named Francis Bacon, who left behind his Opus Majus, which gave an incomparable picture of the understanding of the world at the time. He saw the importance of mathematics in the sciences and that it made the human mind better able to address and understand nature. We now see the work of Bacon and his successors as bridging the natural philosophy of the day and the true science of the future. Also in the sixteenth century, Girolamo Cardano introduces us to the basic idea of an imaginary number. He was ahead of his time, as the usefulness of this concept became more apparent in the nineteenth century via Carl Friedrich Gauss. Also around this time we have the work of Newton and Leibniz concerning the development of calculus, but it took Karl Weierstrass, in the 1850s, to introduce the concept of limits, which allows establishing a final value as something is made infinitesimally small. Calculus, by the way, was derived “not from obscure mathematical considerations but from understanding how things change in nature while taking smaller and smaller views on them.”
Chapter ten delves into the world of statistics. The first statistician was John Graunt, who was, incidentally, a button maker. Earlier Girolamo Cardano wrote the first book on probability before he was thirty. We are provided with a host of various probability exercises to boggle the mind. By the 1800s, James Clerk Maxwell became the first in the field of science to put forth a theory where “the mathematics operated in true abstraction from reality.” Speaking of reality, how about the lemniscate or the symbol for infinity? Introduced by Newton’s contemporary, John Wallis, this concept played an interesting role throughout its history. Then there is the concept of set theory – the theoretical foundation for basic numbers in mathematics. This was a bit confusing to me with the talk of aleph-null, the basic infinite set, and subsets and cardinality. By the twentieth century, Einstein had to shed flat Euclidean geometry in order to deal with curved surfaces. Notably, we have situations with four dimensions where all were curved. Through the work of Emmy Noether, we learn of an unbreakable link between symmetry and conservation.
In concluding, the author argues that the universe is inherently not mathematical. Instead, it is more of a great tool that can build models of the universe. So perhaps numbers are real at their most basic, but most of mathematics is not.
In presenting a topic as difficult as the whole of mathematical endeavor, Clegg does a good job of keeping the language down to Earth. This is definitely a book that can be read and enjoyed by almost anyone interested in the history of science--no college math or science degree required.
In the end, I would recommend this book to anyone interested in understanding how we search for truth in the physical world around us.