{"product_id":"the-foundations-of-mathematics","title":"The Foundations of Mathematics","description":"The transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory.\n\nThe authors have many years' experience of the potential difficulties involved, through teaching first-year undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas.\n\nThis second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process- using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups.\n\nWhile the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilon-delta methods of analysis. The approach here is not the conventional one of 'nonstandard analysis', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward.\n\nThis allows a further vision of the wider world of mathematical thinking in which formal definitions and proof lead to amazing new ways of defining, proving, visualising and symbolising mathematics beyond previous expectations.\u003cbr\u003eASIN: 019870643X\u003cbr\u003eVSKU: DBV.019870643X.A\u003cbr\u003eCondition: Acceptable\u003cbr\u003eAuthor\/Artist:Stewart, Ian|Tall, David\u003cbr\u003eBinding: Paperback\u003cbr\u003e\u003cb\u003eNote:\u003c\/b\u003e Any images shown are stock photographs and product may differ from what is shown.  \u003cbr\u003e\u003cb\u003eCondition Notes\u003c\/b\u003e: This copy has clearly been enjoyed—expect noticeable shelf wear and some minor creases to the cover. Binding is strong, and all pages are legible. May contain previous library markings or stamps.  \u003cbr\u003e","brand":"Dream Books Co.","offers":[{"title":"Default Title","offer_id":41444967186490,"sku":"DBV.019870643X.A","price":12.32,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0555\/6011\/0138\/files\/019870643X-0.jpg?v=1780684191","url":"https:\/\/shop.dreambooksco.com\/products\/the-foundations-of-mathematics","provider":"Dream Books Co.","version":"1.0","type":"link"}