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

By David Reed

Hardly ever has the heritage or philosophy of arithmetic been written approximately through mathematicians, and the research of mathematical texts themselves has been a space nearly totally unexplored. Figures of notion seems to be at ways that mathematical works will be learn as texts, examines their textual options and demonstrates that such readings offer a wealthy resource of philosophical matters concerning arithmetic: concerns which conventional methods to the heritage and philosophy of arithmetic have ignored. David Reed, a certified mathematician himself, deals the 1st sustained and important 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 historical past, 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 corresponding to Grothendeick. He additionally lines the results of this method of the knowledge of the historical past and improvement of arithmetic.

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Example text

This is simultaneously a reflection and consequence of the role of figures in his argument. On the one hand, a figure is a specific instance of boundary and being bounded so that by constructing figures through problems or QEF propositions Euclid is constructing ‘solutions’ to the general problem located by Definitions 6 and 7. On the other hand, a figure is also defined by its boundary so that by analysing their boundaries and the parts thereof Euclid can prove theorems or QED propositions about figures.

Whatever doubts and controversies have surrounded the larger Hilbert programme, GdG has remained a model of the axiomatic method and its publication has been hailed as the dawn of modern ‘abstract’ mathematics. Once again, in common with Descartes, Hilbert begins his text with a‘problem’, but his ‘problem’ does not in the slightest resemble a Cartesian construction and from this point forward the reader notices 32 FIGURES OF THOUGHT a curious set of points of contact and divergence between the two texts.

3. A closer look at the text once again provides the clue. e. to demonstrate that no one of the axioms can be deduced as a logical consequence of any of the others. His proof of independence involves the construction of geometries— systems whose elements satisfy some but not all of the original axioms. As the total original collection was consistent, any subcollection will also be consistent so its ‘geometry’ will ‘exist’. By then constructing models of these geometries, Hilbert demonstrates that the axiom sets can be assumed to apply or not to apply independently of one another.

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