Customer Reviews

The Development of Mathematics,

5 star:		(4)
4 star:		(1)
3 star:		(0)
2 star:		(1)
1 star:		(0)

 

 

As a retired professional mathematician eager to learn more about the history of my subject, I found this book absolutely fascinating. Bell writes very forcefully, sometimes expressing his personal judgments in a manner that some might find offensive but which I found provocative (he frequently gives references in his notes to other scholars who disagree with his views). He doesn't hesitate to report on the dark side of mathematicians' battles (both philosophical and personal) with one another.

I recommend that one read a more conventional history of mathematics (such as Boyer, Kline or Gratton-Guinness) before attempting this controversial one.

Be forewarned that Constance Reid, in her biography of Bell, points out errors in this book. I forgive Bell for those because no one person could possibly comprehend in detail all the abstruse mathematics which he covers relatively well. I recommend this book only to readers already somewhat knowledgeable in mathematics.

This is the most engaging and entertaining history of mathematics ever written. If you think that math history is dry, then this is the book that will change your mind. This masterpiece traces early evolution through the Greeks and Hindus, but spends most of its time on modern math since 1600 up to about 1940. The mathematical insight is alive and sparkles with clarity and depth. The exposition draws you into the struggle between competing ideas, and is a tribute to man's creativity. If you are the type of person for whom math is beautiful art, then this is the book for you.

"Once we venture beyond the rudiments," says Bell, "we may agree that those who cultivate mathematics have more interesting things to say about it than those who merely venerate." No more eloquent substantiation of this assertion could be wished for than this book in which it appears. A cultivator himself, Bell requires no introduction to mathematicians. He knows mathematical creation--its trials and its rewards--at first hand. Nor does he need introduction to the wider reading public. It seems, however, that in this work he has risen to a new level of accomplishment, which merits the genuine appreciation of all those who regard mathematics and its related sciences as a vital field of human activity, and find interest in the history of their development. This is an eminently readable book, written in an engaging and graceful style. At the same time it is a scholarly work with a wholly serious purpose, full of information and fact, and covering much material which is otherwise not easily accessible.

As the keynote of the book Bell sounds an old quotation: "There is probably no other science which presents such different appearances to one who cultivates it and to one who does not, as mathematics. To [the noncultivator] it is ancient, venerable, and complete; a body of dry, irrefutable, unambiguous reasoning. To the mathematician, on the other hand, his science is in the purple bloom of vigorous youth, everywhere stretching out after the attainable but unattained, and full of the excitement of nascent thoughts; its logic beset with ambiguities, and its analytic processes, like Bunyan's road, have a quagmire on one side and a deep ditch on the other, and branch off into innumerable by-paths that end in a wilderness."

To the student of mathematics the historical development of his subject appears all too inevitably as a wilderness, and moreover as an almost impenetrable one when the last century or two are approached. With research pressed in this time and at the present on many fronts by a vast number of investigators, with many different groups of these pursuing apparently quite diverse objectives, and with all of them changing their tactics and goals disconcertingly often, the residue of their attainments is a sweltering jungle indeed. Through this the present book lays a very welcome road. The typical and more significant trends and episodes are isolated, the genesis, growth and efflorescence of some of the concepts and methods, whose survival to the present is their guarantee of significance, are traced, and often their decadence in periods of sterile overelaboration is observed.

The book is not of the "popular" kind, as this term is generally understood, since it makes small effort to be intelligible to readers wholly uninitiated mathematically. Indeed, its appeal will probably be found to vary almost directly with the reader's mathematical attainments. The less trained will find much that is entirely narrative and non-technical, and will some-times find quite enlightening the concise but generally clear technical surveys that are given. The advanced student of mathematics and science will find much more to interest him, and will value the orientations which the book supplies. Professional mathematicians, even those who are themselves momentarily engaged in extending mathematical theories and their applications, will find the book a thoroughly worth-while reading of mathematical evolution. This is not to say by any means that they will in all instances read from the noted trends and related episodes precisely the same inferences as does the author. The better, perhaps, that in some cases they should not.

For the purposes of this review it is convenient to regard the book as falling into two parts, consisting respectively of the first six chapters, which treat of mathematics up to the year 1637, and the remaining seventeen chapters which terminate the discussion at the present time. The first part, which begins with a general prospectus, is given over thereafter to a review of mathematics in the ancient Babylonian and Egyptian eras, in the Greek period, in the dark age of Europe, through the Arabian epoch and the Renaissance. While completely nontechnical, even these chapters are not to be regarded as a historical text. There is not the customary cataloguing of names and facts, but rather a sort of running narrative commentary, of which a full appreciation will be somewhat conditioned upon the reader's previous knowledge of the history. Bell acknowledges these pages to hold in the main a collation of material from more or less familiar and classical works. These chapters appear to be by far the weaker part of the book; to be in fact a trifle pedestrian, though not always unprovocative. As is well known, iconoclastic tendencies are not invariably eschewed by Bell. The so-called debunking of tradition is often salutary. An excess of it, however, though it adds a sensational element to the reading, may in the case of immature or otherwise undiscriminating readers leave impressions that are not wholly fortunate or just. Enjoyable or regrettable, as the reader may find them, he will find here, and throughout the book, a sprinkling of the quips and sophistications which those who know Bell would rather expect, and some will perhaps deplore his occasional momentary lapses from a generally prevalent high scholarly objectiveness to the inclusion of less happy and rather discordant contemporary comment.

The peculiar contribution of the book is by all odds to be found in its second part. Here Bell's excellent qualifications for his task, which include a technical equipment beyond the range of the usual historian, and a literary facility far beyond the range of the usual mathematician, really come to bear. The wide gamut of topics discussed is perhaps best suggested by the chapter headings, which are the following: The beginnings of modern mathematics 1637- 1687; Extension of number; Toward mathematical structure; Arithmetic generalized; Emergence of structural analysis; Cardinal and ordinal to 1902; From intuition to absolute rigor, 1700-1900; Rational arithmetic after Fermat; Contributions from geometry; The impulse from science; From mechanics to generalized variables; Differential and difference equations; Invariance; Certain major theories of functions; Through physics to general analysis and abstractness; Uncertainties and probabilities.

It would be entirely impossible to abstract these chapters briefly. They should be read in their completeness. Mathematics and mathematicians live in them, and not infrequently lend themselves to genuine drama. The presentation of the whole is admirable. It is flowing and graceful and often characterized by a genuine and delightful humor. A feature which will be prized is Bell's almost invariable practice of labeling all investigators and notable publications with their nationality and dates.

The publishers of the book are to be thanked for an attractive and legible volume. Bell deserves recognition and high praise for such a significant work. Many the scientist who has come to realize, to his humility, that his vaunted work would in his absence have soon been accomplished by another. One may safely venture that no other would soon have written this book had Bell not done so.

Speaking as someone with a degreee in math, plus some graduate work, I found this book frustrating and disappointing. I wanted to fill in some historical perspective to all the math I had learned, but I was little illuminated by this book. While the author clearly knows a lot about math, his writing is vague, poorly organized, and unfocused. He answered few of my questions and does not present a coherent picture of his topic. Rather than this book, I recommend Morris Kline "Mathematical Thought from Ancient to Modern Times", which is infinitely better.

I like to compare mathematical concepts to an Arch(so did Jacob Bronowski; but, I think I've figured out the true comparison between the two). The arch represents the unification of concepts, the delecate balance between too little and too much that all mathematical concepts are. The arch also can't be built like a crossbeam(two vertical columns with a crossbeam laid on top haphazardly); it's like the way an airliner, or a racecar can't be turned on by the flip of a switch. It takes an understanding of the underlying structural relations.

All mathematical concepts are abstractions. Abstractions are the general form that many structures can take on. Take a given relation, and many different content can make sense with that relation. I love you; you love me; two apples and two oranges make up the number two. One could say the arch is composed of a verb, the keystone, and the contents that make sense with that keystone; each elemnt of the arch is differnt in some ways(some being higher or lower on the arch), but they have a similar relation with respect to the keystone.

The elements of the arch that make sense with respect to the keystone is similar once again to the way the various subject and object nouns of a verb need to make sense. This making sense with respect to a given verb(see Suzanne K Langer's "Introduction to Symbolic Logic" for a good discussion of how symbolic logic relates to the abstract nature of mathematics) brings up something else I think I've found about the mathematical activity.

I've been watching James Burke's "Connections" and "The Day the Universe Changed" a lot, and I can't help noticing certain technological groupings. James Burke tries to show the connections between science and all aspects of the human condition through history. Some connections are just kind of outside influences things. But, some connections between the developments of technologies seem to me to have a far more profound 'connections.' Like in episode one of Connections(the first series is the only one i've seen), he shows how agriculture led to a host of technologies. He shows how agriculture leads to like the specialization of jobs; the farmers do the farming allowing others to do government, military duty, maybe further down the road some can do science freely. I point out that 'clearing the fields' suddenly makes mankind need to take on the jobs of irrigation, fertilization, and pest control. These concepts 'make sense' in relation to the verb/act of 'clearing of the fields.' James Burke mentions other possible related technologies that makes sense with respect to agricultural civilization - like storage containers for the surplus grain. In episode three, he goes into how the horse changed the economies and created a bunch of associated technologies just like agriculture did - the horseshoe, the sturip, the metallergy of the knight's armour. In episode two, he mentions how the vacuum leads to noting how mice suffocate in it; flames go out, bells don'r ring in vacuums; and he notes that with the vacuum, following the history after can take many different paths - "we could go from the vacuum pump to the investigation of air to the discovery of oxygen, or vacuum pump to steam engine, or to the cathode ray tube and so on. See, by 'clearing of the fields' so to speak, certain concepts just come about; they make sense with respect to daming nature up so to speak. I associate this daming up of nature(creating a water dam 'generates' electricity) with idealization in mathematics. Idealization leads to the definitions in axiomatic mathematics. This all probably came to my mind because I've read Jacob Bronowski's "Origins of Knowledge and Imagination".

In Jacob Bronowski's "Origins of Knowledge and Imagination", he points out that we figure out the universe from our current perspective; and, our current perspective is always finite while the universe is this infinitely connected whole. To get at nature, we have to introduce an artificiality(an idealization/ in mathematics, a definition); this definition creates a vacuum in nature that we need to introduce the keystone and the elements of the arch to keep that vacuum from collapsing back in on us again. Jacob Bronowski argues that when we first evolved from our non-human selves, we were one with nature; our language was command sentences. And that we decodes the words from the sentence by a process of generalization and specialization. His ideas are nice, but until I watched James Burke's "Connections and The Day the Universe Changed" enough, and noticed these 'clearing of the fields' and how fundamental the vacuum has been to the investigation of new concepts and technologies, even I sometimes wondered how seriously to take his little book.

For a quick mathematical example; introduce a line on to two parallel lines; what makes sense is that opposite angles are equal. It's pure language. Physicists love to point out Romer's discovery of the finite speed of light. He observed the motions of the Galilean moons and couldn't help notice an inconsistency in the times of the moons motions; the only thing that made sense was that the light must have a finite speed of light; establishing a boundary creates problems; the establishing of a boundary asks what is the opposite shape of the boundary? The Newtonian laws had a certain boundary established; Romer applied them to the motions of the moons of Jupiter and found a problem; the opposite shape of the consistent boundary established meant the finite speed of light.

Point is that a certain noun words 'work' or 'make sense' with a given verb. Similarly, the finite speed of light 'made sense' with respect to newtonian mechanics; a consistent 'daming up of nature.'

More mathematical examples would be how the irrational numbers were derived out of a problem of the pythagorean theorem; take the case where the sides equal one and bring down the hypotenuse; this line cannot be measured! Similarly, negative numbers, imaginary numbers in quadratic equations, group and field theory in Galois solution(or disproof) of the general equation of the fifth degree all 'spring out' of daming up nature in just the right way.

In logical proof, you split up the conjecture in terms of hypotheses and conclusion; you know your rules of how to transform one side of another to get things on one side and you find the keystone the bridges the vacuum made up.

Einstein's theories of Relativty, Quantum theory, and Chaos theory didn't throw away Newtonian theory; they generalized it and derived it. E.T. Bell notes the Hilbert problem solved by Dehn which generalizes the Egyptian solution to the volume of the truncated pyramid. E.T. Bell stresses these phenomenon of abstraction and generalization throughout the history of mathematics.
Mathematics is arches within arches. It is a cathedral of the mind alright. I feel like I've been through a cathedral when I read E.T. Bell's "The Development of Mathematics." E.T. Bell shows the unity and extent of mathematics of all mathematics; he also shows the connections between modern mathematics and the great accomplishments of mathematics of as far back as he could get when he wrote this book.

There's been great artists of various arts; but, outside of passing down to future generations new techniques, a given artist creates their art, and the next creates their art(in the blues, they build up on one anothers songs actualy!). But, in mathematics each generation opens up the previous generations mathematics, generalizes and finds new unities. Mathematics is by far now the deepest human construct. It's possiblities are infinit(as proved by Kurt Godel; Godel's theormes say an inconsistent finite set of axioms can prove an infinity of truths; a consistent set of finite axioms cannot prove an infinity of truths; mathematicians pick the second possibility; hence their activity will always continue).

Jacob Bronowski essentialy argues that humanity is the science and technologicaly dependent species. We came from ignorance. Mathematics is the ultimate expression of our effort to comprehend the universe. E.T. Bell shows this development from as early times as he could when he wrote it. He shows the connections between all the great accomplishments of the past and all the latest mathematics. It's kindof remarkable how a study of mathematics shows the main outlines of human history!

--------------------------------------------------------------------------------------------

E.T. Bell seems to have anticipated me a little bit in my connections between Jacob Bronowski's 'inferred units' and damming up nature. In chapter 8, "Extentsions of Number" right before 'from manipulation to interpretation' he pretty much says what I'm saying in my link above(this refers to my Jacob Bronowski "Scientific Humanism" blog).

I'd like to further remark that despite his anticipation, he says it more in passing; he stresses the role of abstraction much more throughout his "Development of Mathematics" than my points and Jacob Bronowskis. I don't think he realized the significance quite as much as I've stressed.

While the style could be better, this is a general and good outline of the history of mathematics.

I would point out that the little time spent on Al-Magi Al-Khawarizmi (literally "The Magus (Zorastarian, not Muslim) from Khazar (near the Caspian Sea)") is justified. Al-Khawarizmi merely translated the formulaic Algebra (which the Indians developed from the Greeks and systemized it from its rhetorical origin) from a Hindu text (brought to the court of the Caliph by Indian Ambassadors seeking trade, they were soon rewarded with Moghul Jihad) and translated the Hindu word for 'reorganisation' or 'rebalancing' into Arabic (Al-Jabr). From there, Spanish scholars were able to access the work. As a translator, AL-Khawarizmi certainly provided a service, but he was a Zorastarian from the biggest population of Zorastarians outside of Persia, Khazar. No self-respecting Muslim would keep "al-Magus" as part of his name after conversion.