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eBook Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra through the Commentaries on Newton's Universal Arithmetick ePub

by Helena M. Pycior

eBook Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra through the Commentaries on Newton's Universal Arithmetick ePub
Author: Helena M. Pycior
Language: English
ISBN: 0521481244
ISBN13: 978-0521481243
Publisher: Cambridge University Press (May 13, 1997)
Pages: 344
Category: Mathematics
Subcategory: Science
Rating: 4.5
Votes: 370
Formats: lrf txt mobi doc
ePub file: 1312 kb
Fb2 file: 1318 kb

Geometric Algebra Symbolical Algebra Late Seventeenth Mathematical Position British Mathematics. Nagel, Ernest: 1935, ‘Impossible Numbers: A Chapter in the History of Modern Logic’, Studies in the History of Ideas, 3, 429–479.

Geometric Algebra Symbolical Algebra Late Seventeenth Mathematical Position British Mathematics. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves. Viète, François: 1983, The Analytic Art: Nine Studies in Algebra, Geometry and Trigonometry, T. Richard Witner (trans. Kent State University Press, Kent, Ohio.

Symbols, Impossible Numbers, and Geometric Entanglements is the first history of the development and reception of algebra in early modern England and Scotland. Not primarily a technical history, this book analyses the struggles of a dozen British thinkers to come to terms with early modern algebra, its symbolic style, and negative and imaginary numbers.

The first book to position algebra firmly in the Scientific Revolution and pursue Newton the algebraist, it highlights Newton's role in completing the evolution of algebra from an esoteric subject into a major focus of British mathematics. Other thinkers covered include Oughtred, Harriot, Wallis, Hobbes, Barrow, Berkeley, and MacLaurin.

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Cambridge University Press, 13 May 1997 - 328 sayfa

Cambridge University Press, 13 May 1997 - 328 sayfa. The first book to position algebra firmly in the Scientific Revolution and pursue Newton the algebraist, it highlights Newton's role in completing the evolution of algebra from an esoteric subject into a major focus of British mathematics.

mixed mathematical legacy of Newton's Universal Arithmetick; 8. George Berkeley at the . George Berkeley at the intersection of algebra and philosophy; 9. The Scottish response to Newtonian algebra; 10. Algebra 'considered as thelogical institutes of the mathematician'. Dennis H. Rouvray, Endeavour "In her book Symbols, Impossible Numbers, and Geometric Entanglements, Helena Pycior paints a novel picture of British mathematical development in the seventeenth and eighteenth centuries. She has brought together much interesting material whose implications will interest scholars for years to come.

Mathematical Institute – University of Oxford.

Symbols, Impossible Numbers, and Geometric Entanglements is the first history of the development and reception of algebra in early modern England and Scotland. Not primarily a technical history, this book analyzes the struggles of a dozen British thinkers to come to terms with early modern algebra, its symbolical style, and negative and imaginary numbers. Professor Pycior uncovers these thinkers as a "test-group" for the symbolic reasoning that would radically change not only mathematics but also logic, philosophy, and language studies. The book also shows how pedagogical and religious concerns shaped the British debate over the relative merits of algebra and geometry. The first book to position algebra firmly in the Scientific Revolution and pursue Newton the algebraist, it highlights Newton's role in completing the evolution of algebra from an esoteric subject into a major focus of British mathematics. Other thinkers covered include Oughtred, Harriot, Wallis, Hobbes, Barrow, Berkeley, and MacLaurin.
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This is the kind of book in which one reads the block quotes only, the author's padding between them being little but empty calories. Indeed, the author's method is to go through various mathematical treatises and quote any nontechnical statements in them while ignoring their specific mathematical content. Thus she jumps back and forth between basic textbooks on algebra, philosophical texts on the ontology of numbers and geometry, and debates on the foundations of infinitesimal calculus with no discernible goal or purpose other than to tally up the pages until they make a "book-length history of British algebra" (p. 1). Personally I prefer historians who aspire to cogent thought and insight rather than booklengthness, but Pycior is not of this school. She makes life very easy for herself by claiming to "underscore the importance of individuals and individualistic thought in the making of mathematics" (p. 3). This gives her the license to quote people's opinions as if they were the arbitrary output of some inexplicably "individualistic" random-opinion generator. A mathematician's views on algebra, complex numbers, and infinitesimals are not picked out of hat at random, but as far as Pycior's methodology is concerned they might as well be, since she fails to consider the mathematical and philosophical sources of them.

The enjoyment of the book lies in the colourful language used in intellectual discourse in days of old, such as accusing a page of algebra of looking "as if a hen had been scraping there" (Hobbes, p. 147) or calling the geometrical interpretation of a product of four things "a Monster in Nature, and less possible than a Chimera or Centaure" (Wallis, p. 117).

Sadly modern academics have abandoned this lively style. Instead they follow another tendency of the 17th century, namely to take "the symbolical style as a model for terse, scientific expression." Indeed, the Royal Society "exacted from all their members a close, naked, natural way of speaking ... bringing all things as near the Mathematicall plainness as they can." (Sprat, p. 46)

Luckily not everyone fell pray to such "plainness", as a few more delightful exceptions may show.

"Symbols are poor unhandsome, though necessary, scaffolds of demonstration; and ought no more to appear in public, than the most deformed necessary business which you do in your chambers." (Hobbes, p. 145)

Negative numbers. "That which most perplexes narrow minds in this way of thinking, is, that in common life, most quantities lose their names when they cease to be affirmative, and acquire new ones so soon as they begin to be negative: thus we call negative goods, debts; negative gain, loss; negative heat, cold; negative descent, ascent, &c.: and in this sense indeed, it may not be so easy to conceive, how a quantity can be less than nothing, that is, how a quantity under any particular denomination, can be said to be less than nothing, so long as it retains that denomination." (Saunderson, p. 287)

Technology in teaching. "That the true way of Art is not by Instruments, but by demonstration: and that it is a preposterous course of vulgar Teachers, to beginne with Instruments, and not with the Sciences, and so in stead of Artists, to make their Schollers onely doers of tricks, and as it were jugglers." (Oughtred, p. 68)
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The author begins her introduction with the words: "As a history of algebra in England and Scotland from the early seventeenth century to the mid-eighteenth century, this book considers not only technical algebra but also the personal, philosophical, religious, and institutional factors that affected the introduction, elaboration, and reception of the subject." (p. 1)

The following algebra texts are discussed in this book:
Girolamo Cardano - The Great Art (Ars magna) - 1545
Francois Viete - Introduction to the Analytic Art - 1591
William Oughtred - The Key of the Mathematicks - 1631, 1647
Thomas Harriot - Praxis (Artis analyticae praxis) - 1631
Rene Descartes - Geometrie - 1637
Johann Rahn w/John Pell - Algebra - 1659, 1668
John Kersey - Algebra - 1673
John Wallis - Treatise of Algebra - 1685
Isaac Newton - Universal Arithmetick - 1707
Nicholas Saunderson - Elements of Algebra - 1740
Colin MacLaurin - Treatise of Algebra - 1748

The author also discusses views concerning algebra or mathematics by Thomas Hobbes, Isaac Barrow, and George Berkeley. The main concerns and disagreements seem to have been around the status (epistemological, ontological, or otherwise) of negative and imaginary numbers, as well as the relative standings of geometry and algebra within the realm of mathematics.

A reader wishing to know how algebra was presented within these various texts - in other words, what the algebra textbooks of the period looked like, what notation was used, how proofs were presented, what topics were included, and so forth - will need to look elsewhere. The author's main concern seems to be the controversies surrounding negative and imaginary numbers and the struggle of algebra to find respectability in the face of geometry.

The following books are available from amazon.

Cardano - The Rules of Algebra: (Ars Magna)

Viete- The Analytic Art

Thomas Harriot - Artis Analyticae Praxis: An English Translation with Commentary

Descartes - Geometry

For a look at what algebra was going to become in England during the early nineteenth century, see
Thomas Peacock's 1830 Treatise on Algebra -

Treatise on Algebra, Volume I: Arithmetical Algebra

A Treatise on Algebra, Volume II: On Symbolical Algebra, and Its Applications to the Geometry of Position

A century later, in 1930, B. L. Van Der Waerden published the enormously influential:

Algebra: Volume I

Algebra, Volume II

And then in 1941, Garrett Birkhoff and Saunders Mac Lane published:

A Survey of Modern Algebra
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