Preface
Introduction to Mathematics: Number, Space, and Structure is a gate, an architect, and a docent.
Mathematics offers a wealth of riches for understanding the world. Unfortunately, many of these riches are hidden in fortresses impenetrable by outsiders. Indeed, mathematics' two biggest strengths-formal, precise language and abstraction—are also the highest walls surrounding the subject. This text is a gate in those walls.
Mathematics has accumulated a tremendous variety of techniques, concerns, concepts, and methods. These are unified by the common concern of studying number and space. This text is the architect who reveals the blueprint exhibiting how the building blocks are assembled in pursuit of a coherent structural vision.
Mathematics is an architectural wonder of the world. It combines skyscrapers and sprawling palatial complexes. Whether climbing the skyscraper of abstraction or curiously exploring room after room of treasure, it helps to have a guide. This text is the docent that gives you your first tour, explaining the underlying structure that governs not just the unity of architectural vision but also the girders that allow us to build strong, flexible mathematical arguments.
Introduction to Mathematics: Number, Space, and Structure aims to:
• increase your ability to absorb new mathematical definitions, understand theorem statements, and construct proofs of those statements using the definitions;
• provide ample opportunity for writing formal arguments to ensure (to the best of our ability) the intellectual soundness of our arguments;
• provide a variety of examples demonstrating how formal arguments can be written to ensure the greatest possible understanding in the mind of the reader;
•use a variety of images and metaphors to convey an informal and intuitive understanding of the topics;
Introduction to Mathematics: Number, Space, and Structure is a gate, an architect, and a docent.
Mathematics offers a wealth of riches for understanding the world. Unfortunately, many of these riches are hidden in fortresses impenetrable by outsiders. Indeed, mathematics' two biggest strengths-formal, precise language and abstraction—are also the highest walls surrounding the subject. This text is a gate in those walls.
Mathematics has accumulated a tremendous variety of techniques, concerns, concepts, and methods. These are unified by the common concern of studying number and space. This text is the architect who reveals the blueprint exhibiting how the building blocks are assembled in pursuit of a coherent structural vision.
Mathematics is an architectural wonder of the world. It combines skyscrapers and sprawling palatial complexes. Whether climbing the skyscraper of abstraction or curiously exploring room after room of treasure, it helps to have a guide. This text is the docent that gives you your first tour, explaining the underlying structure that governs not just the unity of architectural vision but also the girders that allow us to build strong, flexible mathematical arguments.
Introduction to Mathematics: Number, Space, and Structure aims to:
• increase your ability to absorb new mathematical definitions, understand theorem statements, and construct proofs of those statements using the definitions;
• provide ample opportunity for writing formal arguments to ensure (to the best of our ability) the intellectual soundness of our arguments;
• provide a variety of examples demonstrating how formal arguments can be written to ensure the greatest possible understanding in the mind of the reader;
•use a variety of images and metaphors to convey an informal and intuitive understanding of the topics;
• give a variety of useful applications showing how mathematics is useful in both science and art;
balance mathematics that will or might be encountered later in a student's undergraduate mathematics career (such as in real analysis or abstract algebra) with material that the student might otherwise not encounter;
• explore sets and operations pertaining to both number systems and geometric/topological spaces;
• introduce important themes of mathematical culture, especially the role of axiom systems and notions of infinity.
We pay particular attention to motivating topics and demonstrating their genuine usefulness. Courses acting as a bridge from introductory calculus to upper-level math courses must convince students that abstract mathematics is worth studying. This is true regardless of whether you are more inclined to applied mathematics or to pure mathematics. Some of us find abstraction very difficult. If that's you, I hope that seeing the applicability of the abstractions to concrete ideas is helpful in making the abstract concrete. Others of us may relish abstraction for its own sake. If that's you, I hope that you see how abstraction, in addition to being fun, is a method for understanding aspects of the world. Furthermore, all students of mathematics should be able to explain to nonmathematicians why mathematics is worthwhile. Although a single text or a single course will not transform a person into the super-hero of mathematics communication, I hope this text will help you on your journey.
Particular features of this text include:
⚫ specific guides on how to structure the different kinds of proofs; •
⚫ an emphasis on modern mathematics, including:
- references to recent high profile mathematical successes,
- advanced views of elementary mathematics, uncovering subtleties previously hidden,
- applications showcasing the relevance of mathematics to computer science, the natural sciences, the arts and humanities,
nuanced handling of foundational issues in mathematics, including comparisons of the Zermelo-Fraenkel axioms and category theoretic axioms for set theory,
- use of topics from more advanced courses, presented at an appropriate level, to demonstrate the relevance of the concepts beyond the current course;
the use of analogies to explain mathematical topics, including a discussion of the limitations of the analogies;
• proofs of significant results which are left to the reader, as well as outlines of proofs with important steps left to be filled in by the reader;
• early introduction of interesting mathematical results. In some cases this necessitates delaying important steps to later in the text. All such interdependencies are carefully noted. This models the way professional mathematicians are willing to
balance mathematics that will or might be encountered later in a student's undergraduate mathematics career (such as in real analysis or abstract algebra) with material that the student might otherwise not encounter;
• explore sets and operations pertaining to both number systems and geometric/topological spaces;
• introduce important themes of mathematical culture, especially the role of axiom systems and notions of infinity.
We pay particular attention to motivating topics and demonstrating their genuine usefulness. Courses acting as a bridge from introductory calculus to upper-level math courses must convince students that abstract mathematics is worth studying. This is true regardless of whether you are more inclined to applied mathematics or to pure mathematics. Some of us find abstraction very difficult. If that's you, I hope that seeing the applicability of the abstractions to concrete ideas is helpful in making the abstract concrete. Others of us may relish abstraction for its own sake. If that's you, I hope that you see how abstraction, in addition to being fun, is a method for understanding aspects of the world. Furthermore, all students of mathematics should be able to explain to nonmathematicians why mathematics is worthwhile. Although a single text or a single course will not transform a person into the super-hero of mathematics communication, I hope this text will help you on your journey.
Particular features of this text include:
⚫ specific guides on how to structure the different kinds of proofs; •
⚫ an emphasis on modern mathematics, including:
- references to recent high profile mathematical successes,
- advanced views of elementary mathematics, uncovering subtleties previously hidden,
- applications showcasing the relevance of mathematics to computer science, the natural sciences, the arts and humanities,
nuanced handling of foundational issues in mathematics, including comparisons of the Zermelo-Fraenkel axioms and category theoretic axioms for set theory,
- use of topics from more advanced courses, presented at an appropriate level, to demonstrate the relevance of the concepts beyond the current course;
the use of analogies to explain mathematical topics, including a discussion of the limitations of the analogies;
• proofs of significant results which are left to the reader, as well as outlines of proofs with important steps left to be filled in by the reader;
• early introduction of interesting mathematical results. In some cases this necessitates delaying important steps to later in the text. All such interdependencies are carefully noted. This models the way professional mathematicians are willing to
delay the proofs of important steps until they have thought through the entire argument.
The emphasis of this text is on writing mathematics. Just as playing a musical instrument enables the player to be a better listener at a concert, so writing mathematics enables students to be better readers of mathematics. Only after we have wrestled with the organization of a proof or the proper order of quantifiers can we understand and appreciate the meticulous writing of professional mathematicians and technical professionals. Similarly, the act of converting intuition into formal mathematics helps us also develop the ability to convert formal mathematics into intuition. Becoming adept at this is, for most people, many years of work, but my hope is that this text provides an excellent starting point.
Finally, in the midst of difficult mathematical work, it can be easy to lose sight of the fact that mathematics is fun, creative, and inventive. I've incorporated a lot of quotes and references reflecting my own interests and explorations. I hope they'll inspire you to read, explore, and make connections between mathematics and your life and culture and then to share those connections with others. My friend Michael Scholz generously provided whimsical illustrations of key ideas. When you see these, be reminded that there are many different ways to conceive of mathematical objects, and that we can draw on all of our senses (including our sense of humor!) to understand them. Indeed, I challenge you to find your own original images or stories that embody mathematical ideas.
Who is this book for?
This book has been written for students who want to develop the ability to read, write, and construct mathematical proofs with a view to developing the mental flexibility, intellectual rigor, and diligence necessary to enter into advanced mathematics. On a few occasions, the book uses examples from multivariable calculus, but on the whole the book should be accessible to anyone with some small amount of calculus, computer science, discrete math, or formal logic background. Most students with one year of mathematics, statistics, or computer science at the university level should be amply prepared to use this text.
The text is designed to be very readable; more advice on how to read it is given in the chapter "To Students". That said, I do assume that there is a teacher to give you feedback on the proofs that you write and that you have someone with whom you can discuss mathematical ideas. The book should work well for both lecture-style classrooms and flipped classrooms. My classes are a hybrid of the two methods.
Although not much college level mathematics is a prerequisite, this book is intentionally challenging. You are expected to relish the challenge and work hard, while being patient with yourself and exhibiting a growth mindset. The palaces and gardens of advanced mathematics are not far away. This book will help you enter and enjoy. Through your efforts, with the help of your teacher and classmates, you will be able to create and understand far more mathematics than you ever thought possible. You will be empowered to independently learn and create new abstract mathematics. You will become a mathematician.
The emphasis of this text is on writing mathematics. Just as playing a musical instrument enables the player to be a better listener at a concert, so writing mathematics enables students to be better readers of mathematics. Only after we have wrestled with the organization of a proof or the proper order of quantifiers can we understand and appreciate the meticulous writing of professional mathematicians and technical professionals. Similarly, the act of converting intuition into formal mathematics helps us also develop the ability to convert formal mathematics into intuition. Becoming adept at this is, for most people, many years of work, but my hope is that this text provides an excellent starting point.
Finally, in the midst of difficult mathematical work, it can be easy to lose sight of the fact that mathematics is fun, creative, and inventive. I've incorporated a lot of quotes and references reflecting my own interests and explorations. I hope they'll inspire you to read, explore, and make connections between mathematics and your life and culture and then to share those connections with others. My friend Michael Scholz generously provided whimsical illustrations of key ideas. When you see these, be reminded that there are many different ways to conceive of mathematical objects, and that we can draw on all of our senses (including our sense of humor!) to understand them. Indeed, I challenge you to find your own original images or stories that embody mathematical ideas.
Who is this book for?
This book has been written for students who want to develop the ability to read, write, and construct mathematical proofs with a view to developing the mental flexibility, intellectual rigor, and diligence necessary to enter into advanced mathematics. On a few occasions, the book uses examples from multivariable calculus, but on the whole the book should be accessible to anyone with some small amount of calculus, computer science, discrete math, or formal logic background. Most students with one year of mathematics, statistics, or computer science at the university level should be amply prepared to use this text.
The text is designed to be very readable; more advice on how to read it is given in the chapter "To Students". That said, I do assume that there is a teacher to give you feedback on the proofs that you write and that you have someone with whom you can discuss mathematical ideas. The book should work well for both lecture-style classrooms and flipped classrooms. My classes are a hybrid of the two methods.
Although not much college level mathematics is a prerequisite, this book is intentionally challenging. You are expected to relish the challenge and work hard, while being patient with yourself and exhibiting a growth mindset. The palaces and gardens of advanced mathematics are not far away. This book will help you enter and enjoy. Through your efforts, with the help of your teacher and classmates, you will be able to create and understand far more mathematics than you ever thought possible. You will be empowered to independently learn and create new abstract mathematics. You will become a mathematician.
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