Download PDF by David Reed: Figures of Thought: Mathematics and Mathematical Texts

By David Reed

ISBN-10: 0415081467

ISBN-13: 9780415081467

Hardly ever has the historical past or philosophy of arithmetic been written approximately by means of mathematicians, and the research of mathematical texts themselves has been a space virtually fullyyt unexplored. Figures of idea appears to be like at ways that mathematical works may be learn as texts, examines their textual techniques and demonstrates that such readings offer a wealthy resource of philosophical matters relating to arithmetic: matters which conventional ways to the historical past and philosophy of arithmetic have overlooked. David Reed, a qualified mathematician himself, deals the 1st sustained and significant try to discover a constant argument or narrative thread in mathematical texts. In doing so he develops new and interesting interpretations of mathematicians' paintings all through background, from an in-depth research of Euclid's components, to the math of Descartes and correct as much as the paintings of up to date mathematicians equivalent to Grothendeick. He additionally strains the results of this method of the knowledge of the heritage and improvement of arithmetic.

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Euclid is not free to select a set of postulates according to philosophical predisposition, pedagogical efficiency or a subjective sense of beauty in mathematics. 19 The Common Notions do not deal with the measurement of specific defined things, but with measured things in general, providing the ability to relate measured things to each other and thus to create propositions. 20 Common Notions 1–4 provide a syntax for the use of the word ‘equality’ which Euclid never defines as such. As noted above, Euclid takes ‘equality’ to be a fundamental concept or principle of measurement in contrast to the modern preference for looking at ‘what is equated’.

Suffice it to say that Descartes’ geometry is a realm of problem constructions within a larger and so far undefined mathematical domain. Within the realm of geometry, however, it is clear that all demonstrations are constructions and the subject matter of this realm can be identified with ‘the geometrically constructible’. Descartes the philosopher is most famous for his ‘cogito’ in which he demonstrates the indubitable existence of his thinking self. Ultimately, as an example of ‘method’ in Descartes’ philosophy, his Géométrie will reside in the mind of the geometer who has absorbed it.

The negative definition of parallel lines indicates the lack of bounds to the size of planes or plane figures. Postulate 5 provides a determination of the existence of triangles based solely on the data of transverse lines and angles without limit as to ‘size’. Together, Definition 23 and Postulate 5 eliminate the possibility that external data will be required to decide on the possible limits to figures. Euclid’s material is completely self-determining and is not subject to constraints outside of its own definition.

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Figures of Thought: Mathematics and Mathematical Texts by David Reed

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