The transition from school-level mathematics to university-level mathematics is rarely simple. Students face a disconnection between the algorithmic and informal approach to mathematics in school, compared to a new emphasis on proof, logic-based reasoning, and a more abstract development of general concepts based on set theory.

The authors have many years of experience regarding the potential difficulties arising from teaching first-year students and researching students' thinking and mathematical processes. The book explains the motivation behind the abstract foundational material, based on students' experiences with school mathematics, and explicitly suggests ways in which students can comprehend formal ideas.

This second edition marks a significant advancement as it not only facilitates the transition from intuitive to formal methods but also reverses the process—using theorem structures to demonstrate that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is illustrated through a new chapter on group theory.

While the first edition extends counting to infinite cardinals, the second also strictly extends real numbers into greater derived domains. This connects intuitive ideas in calculus with the formal epsilon-delta methods of analysis. This approach is not the conventional "non-standard analysis," but a simpler, graphically based treatment that makes the concept of an infinite natural number natural and easy.

This allows for a further insight into the broader world of mathematical thinking, where formal definitions and proofs lead to surprising new ways of defining, proving, visualizing, and symbolizing in mathematics, surpassing previous expectations.

Pages: 416, Dimensions: 14.1x14.1cm