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A Mathematical Mystery Tour: Discovering the Truth and Beauty of the Cosmos

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Four journeys, four hosts, one traveler and an infinite mystery. Where does mathematics end and physics begin? Or is it the other way around? What is the language of ultimate reality, and is this invention a chimera, or does reality reflect inself in the language used to understand it?

Take the journey with Dewdney and come away a little better for it. You'll share beauty under the immense heavens in the desert night, or felt in conversation along a garden path. I'll bet you never thought that metaphysics, physics and math could ever taste or feel so wonderful.

A rarity among texts in the field, sampling these delights will be immediately enjoyable to those with little exposure to math or physics or philosophy, but can also satisfy those with more experience and education. As with other books that dare to be written for the "layman" reader, purists may object to the lack or rigor or occassional whimsy. To them I say, "just go back to your ivory tower." Compared to other texts which may prove more comprehensive or rigorous, Dewdney's offering has it's own unique charm and is probably much more accessible than others. There is nothing wrong with bringing charm to science writing. We need more books like this.

Tour is certainly a rare book, one on mathematical philosophy. After all, it is commonly accepted that physics describes the real world, while metaphysics grapples with the questions of what is real? With that being said, Dewdney turns his attention the tools of mathematics that Physics uses in its descriptions, and ponders if there could be the "meta-mathematical." In Tour, he undertakes to answer two questions. First, why is mathematics so amazingly successful in describing the structure of physical reality, and second, is mathematics discovered or is it created? These are not trifling questions. Consider the discovery of the planet Neptune. In 1845, the 23-year old British John Couch Adams completed calculations pinpointing a new planet that he believed was perturbing the orbit of Uranus. That same year, French astronomer Urban Jean LeVerrier independently published his prediction of the position of the new planet, within a degree of Adams. Alone, this proves almost nothing. Given the same set of data from observations, regardless of how complex the celestial mechanics are, the formulae do not change and accurate algebraic answers should agree (although they do provide standalone verification for each other). What happened next though transforms a merely mathematical exercise into Dewdney's quest to understand the true nature of math. On the same night they read of it, Johann Galle and Heinrich d'Arrest at the Berlin Observatory search and see an eighth magnitude "star" right where LeVerrier envisaged it. Dewdney dares asks, "Why is the physical universe determined (or accurately describable) to so great an extent by Mathematical ideas?"
To answer, Dewdney employs what in the preface he describes as "a fictional narrative," that leads from Greece to Arabia to Italy to England. However, his level of detail and his concluding notes in the postscript lead me to question how fictional the four characters are who elucidate on the subject. I suppose that is to be expected in a work of philosophy, blurring the distinction of who is real. There is one other "character," dead some 2,500 years, but whose mystical and mathematical spirit is still felt: Pythagoras. Carl Sagan credits Pythagoras as the first to "use the word Cosmos to denote a well-ordered and harmonious universe, a world amenable to human understanding" (hence the title for Sagan's series; and now you know the rest of the story!) The followers of Pythagoras developed an entire mathematical cult, a sect that sincerely believed that through math they were glimpsing a perfect reality, a nonmaterial higher realm, of which the physical world was a manifestation. The dwellers of Plato's cave were inheritors indirectly of the Pythagoreans: "The stars that decorate the sky, though we rightly regard them as the finest and most perfect of visible things, are far inferior, just because they are visible, to the true realities; that is, to the true relative velocities, in pure number and perfect figures, of the orbits and what they carry in them, which are perceptible to reason and thought but not visible to the eye. (The Republic, Plato, book VII, 529-E)." In his first stop, along the coast of the Aegean, Dewdney encounters the "holos," the place where all of mathematics, known and unknown, exists. The Cosmos is the manifestation, but the holos is the source, so much so that the Tour is actually Discovering the Truth and Beauty of the Holos. But be cautioned though that a proper frame of mind is a prerequisite. Just as Galileo's journals show he observed Neptune in 1612 but failed to recognize it for what it was, so to Tour benefits from a second or third revisit for complete comprehension.

Not a difficult read, but you'll need some background understanding of mathematics to stay interested. The basic question of whether mathematics is discovered or created runs throughout the book as Dewdney travels around, meeting interesting characters who try to help him answer the question. A few vivid descriptions of Dewdeny's travels relieve some of the theoretical discussions and the character development of the people he meets are interesting, but some end abruptly. Overall, a thought provoking little book.

Why does nature seem to follow certain laws that can be accurately described by mathematics? Is it because nature itself is a mathematical construction? On this imaginary tour Dewdney "talks" to mathematicians and tries to determine why the cosmos is so darn mathematical. The format is unusual, and I think it works. The disappointment is that we are not any closer to the answer after the tour than before we have even embarked.

To explore the great beauty of mathematics, it is always necessary to go back to the ancient Greeks. It was there that the great intellectual breakthrough of abstract mathematics was made. Dewdney begins his mathematical journey at the logical place, Athens, Greece. His first discussions are with Petros Pygonopolis, a specialist in ancient Greek mathematics. Quite fittingly, Pygonopolis is found measuring the stones of an ancient structure with a ruler. This proves to be an excellent starting point for the explanation of the discovery of irrational numbers. Before the great proof that the square root of two was irrational, it was believed that all values were commensurate. This means that by repeating one length a specific number of times and the other length a different number of times, two distances of the same length could be created. The most interesting part of this discussion is the descriptions of how numbers were represented in ancient Greece. In many ways, it is incredible to realize how cumbersome their notation was.

In keeping with the historical development of mathematics, Dewdney then travels to Amman, Jordan to examine the development of mathematics in the Arab world. After the collapse of the Roman Empire, the development of mathematics essentially ceased in Europe. All mathematical progress for centuries after the collapse took place in the Arab world. He meets with Jusuf al-Flayli, an Egyption astronomer who is an expert in the Arab view of the heavens. The naming of the stars is important in Arab culture, and it is clear when you read Dewdney's account. For where else but in the desert would one see the greatest, clearest spectacle of stars.

Venice, Italy is the next stop, for discussions with Maria Canzoni. These discussions are about the development of numerical operations, the new representations and the notations used in the operations. It is difficult to overstate the significance of positional notation. It is hard to see how modern commerce could have ever developed without it. One thing that it is difficult for math students to appreciate is how clean modern mathematical notation is. Even the simple symbols of addition, subtraction and so on abbreviate some very advanced mathematical concepts.

The last stop on the tour is Oxford, England where Dewdney has discussions with Sir John Brainard. The conversation uses nonsense words such as gadzooks, blorgs, semiblorgs, zooks and horping tables. These terms are actually replacements for the words of group theory. Brainard uses this as an example to illustrate the idea put forward by David Hilbert for geometry. Namely that geometry should be constructed so that the terms point, line and plane can be replaced by chair, table and beer mugs. It was quite fun to read and reminded me of some of the writings of Lewis Carol.

There is great beauty in mathematics, sometimes even practitioners fail to appreciate it. Dewdney does an excellent job in describing some of the more significant events in the history of mathematics. Read it and appreciate what some of the great minds have done.

I started reading this book like all others: with anticipation to learn something new and exiting but instead mistakes hit me over my head. On page 85 the author calls the angle between the zenith and a star its declination. Wrong name: It is (90 degrees minus altitude), also called the complementary angle. The declination is the distance from the equator. On page 97 the author gives the impression that the polar star (Polaris) is moving with the seasons; that is not so. Polaris is fixed on the celestial sphere except for a very small daily circle of about 1.5 degree radius. Page 101 makes Kepler a priest which he was not. He just studied Lutheran theology before turning completely to natural philosophy and teaching for a living. Kepler never ignored Copernicus as claimed by Dewdney and neither is the Earth in the middle of the Platonic Solids but the Sun as described in the Mysterium Cosmographicum. On page 103 the author makes us believe that the geographical latitude of Baghdad as measured by the altitude of Polaris is 33 degrees 19 minutes at the beginning of spring; the altitude of Polaris for Baghdad is the same all the time except for that small daily variation mentioned before. On page 110 the author gives the title of Fibonacci's main work as "Algorismus"; there is no work of this title, the common name of his major work is given as "Liber abaci (or abbacy)" - The Book of Abacus. On page 113 the author attributes the spectral lines of the Balmer Series to "vibration" of the atoms. Not so. In Bohr's model the spectral lines are caused by transition of electrons between different levels of energy or different orbits. Vibrations are involved in the spectra of molecules. Finally, why has the author forgotten Mr. Balmer while attributing the different Hydrogen series with changing parameter n to (1) Lyman, (2) Paschen, (3) Brackett and (4) Pfund? The correct attribution is (1) Lyman, (2) Balmer, (3) Paschen, (4) Brackett and (5) Pfund.
I think that I lost my patience reading this book at that point.

Mr. Dewdney's book is a valiant if not completely sucessful effort to address the problem of how much of the world we live in is objectively real and how much is a product of our own minds. Is mathematics describing nature or is it creating it? For what modern physics has shown is that the world we live in is much more a product of our own minds than we think. This has come to the point where it is a question of whether anything can be said to exist independently of that mind. Mathematics has been put forward as that solid ground of reality and here we have an attempt to answer this question. Flying on the wings of his imagination Mr. Dewdney returns to the temple of Miletus in ancient Greece to study the works of Pythagoras before returning to the present to meet with more modern ideas in the Jordanian desert, the palazzos of Venice and the hallowed halls of Oxford, which comes off sounding very much like the setting for a Dorothy Sayers novel.

Well written with the help of a lively imagination it manages to explore the history and meaning of mathematics without a heavy dose of mathematics itself. If you have trouble balancing your check book you can read this book without fear. The book may not be deep thinking but it is charming thinking that makes the discovery of the ideas of math accessible to the layman. In the end however it is more history than either math or philosophy. If you are interested in the history of mathematics this book is certainly worth a look. Yet in the final analysis the question of whether mathematics has an objective reality independent of the world it is describing has not been answered. The author should not be falted for this since no other philosopher thoughout history has been able to answer this question either. And perhaps this is an answer in itself.

This virtual odyssey, exploring the historical and cultural roots of mathematics and the mystery of its timeless questions, stimulates learning and wonder in a way no standard math textbook can do.

The book combines the charm of travel adventures with the mystery of ancient mathematics. Interspersed in the entertaining narrative are thoughtful questions on the nature of the universe, questions entertained by Pythagoras himself. The author talks with Greek, Arab, Italian, and English mathematicians, each of whom ponders the question of Pythagoras in his or her own intriguing way.

At the temple of Apollo, Dr. Pygonopolis introduces the fundamental idea of the holos and ponders whether the discoveries of the Pythagorean School could have been made in another culture.

Under the desert sky, Prof. al-Flayli gives a fascinating account of Arabic influence on astronomy and mathematics.

In Venice, under the sound of "Missa Sancta," physicist Maria Canzoni contemplates the holos, the cosmos, and the ultimate reality of menos, or a consciousness beyond the quantum curtain.

In Oxford, England, Sir John Brainard lectures on the intrinsic simplicity of mathematics and the evidence that computers provide for its independent existence.

Accessible and thought-provoking, this is the most fascinating math book I have read to date.

The book tries to explore some of the more philisophical aspects of mathematics, and cannot be faulted for its failure to reach any real conclusions there. If an answer were easy to come by, it would indicate that the questions were not so philisophical after all.

Unfortunately, the book also fails to achieve secondary goals. Both the historical and mathematical topics it covers could be covered just as well in 1/5 the pages. The extra length comes from trying to work the material into a journey, a technique possibly used in an attempt to make the topics less intimidating or more interesting, but which in fact just makes them boring. If one travels to see and talk to somebody, the travel itself (plane ride, taxi, carrying of the bags) is oftentimes a boring chore needed to get where you are going. Why describe it here? It is not more interesting to read about than to experience.

Instead, this book seems to combine all of the filler material needed to make a fictional story consistant and realistic, but there is not really any story being told here. If one wants a story with some math worked in, try A. K. Doxiadis's "Uncle Petros and Goldbach's Conjecture" instead.

When I saw the title of the book, it reminded me of Ivars Peterson's "The Mathematical Tourist", which I have only looked at briefly, but which explores interesting areas of mathematics. If one wants an overview of some interesting math topics, this is a much more appropriate book.

In Dewdney's book, we get a "tourist" book with few points of interest, mathematical, historical, or geographical. Given the interesting people, places, and material covered, it really misses the mark.

A very endearing, well written book. The story of ancient mathematics is fascinating, albeit nothing new here, but unfortunately from a philosophical and historical point of view it is very desapointing. To tell the truth that's philosophy for high school teenagers. Old math prejudice, fascination for the doing of one's self, misconception about independant culture, and so on and on and on. Hundreds of objections for each of author's proposition could be imagined. His holos is mainly hollow. A very desapointing book, but very well written indeed.