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.
The 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.
This 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.
While 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.
This 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.
從學校數學到大學數學的過渡往往並不簡單。學生面臨著學校中算法和非正式數學態度之間的脫節，與此同時，大學數學則強調基於邏輯的證明，以及基於集合論的更抽象的概念發展。
作者擁有多年教授一年級本科生的經驗，並研究學生和數學家思考的方式，因此對潛在的困難有深刻的理解。本書解釋了基於學生學校數學經驗的抽象基礎材料背後的動機，並明確建議學生如何理解正式的概念。
這一版的第二版在不僅使直觀方法轉變為正式方法方面邁出了重要的一步，還反轉了這一過程——使用結構定理來證明正式系統具有增強數學思維的視覺和符號解釋。這一點在關於群論的新章節中得到了體現。
第一版將計數擴展到無窮基數，而第二版則嚴格地將實數擴展到更大的有序域。這將微積分中的直觀概念與分析中的正式ε-δ方法聯繫起來。這裡的方法並不是傳統的“非標準分析”，而是一種更簡單、基於圖形的處理，使得無窮小的概念變得自然且簡單明瞭。
這使得我們能夠進一步展望數學思維的更廣闊世界，在這個世界中，正式的定義和證明導致了驚人的新方法來定義、證明、可視化和符號化數學，超越了以往的期望。
Ian Stewart,
Emeritus Professor, University of Warwick, David Tall,
Emeritus Professor, University of Warwick Ian Stewart is Emeritus Professor of Mathematics at the University of Warwick. He remains an active research mathematician and is a Fellow of the Royal Society. Famed for his popular science writing and broadcasting, for which he is the recipient of numerous awards, his bestselling books include: Does God Play Dice?, Nature's Numbers, and Professor Stewart's Cabinet of Mathematical Curiosities. He also co-authored The Science of Discworld series with Terry Pratchett and Jack Cohen
David Tall is Emeritus Professor of Mathematical Thinking at the University of Warwick. Internationally known for his contributions to mathematics education, his most recent book is How Humans Learn to Think Mathematically (2013).
伊恩·斯圖爾特（Ian Stewart），華威大學名譽教授，大衛·陶爾（David Tall），華威大學名譽教授
伊恩·斯圖爾特是華威大學的數學名譽教授。他仍然是一位活躍的研究數學家，並且是英國皇家學會的院士。他以其流行科學寫作和廣播而聞名，並因此獲得了多項獎項，他的暢銷書包括：《上帝擲骰子嗎？》（Does God Play Dice?）、《自然的數字》（Nature's Numbers）以及《斯圖爾特教授的數學奇觀》（Professor Stewart's Cabinet of Mathematical Curiosities）。他還與特里·普拉切特（Terry Pratchett）和傑克·科恩（Jack Cohen）共同撰寫了《碟世界科學》（The Science of Discworld）系列。
大衛·陶爾是華威大學的數學思維名譽教授。他因對數學教育的貢獻而享譽國際，他最近的著作是《人類如何學會數學思考》（How Humans Learn to Think Mathematically，2013年）。
