\(\newcommand{\Strut}{\rule{0pt}{12pt}}\newcommand{\True}{\text{T}}\newcommand{\False}{\text{F}}\)The book Sets, Groups, and Mappings was written for a one-semester algebra-flavored introduction-to-proofs course. This short post, essentially the preface for students, indicates the book’s expository tone and pedagogical stance.
Mathematics is unique among human intellectual endeavors; it is not art, philosophy, or science, but it shares certain features with each. The example of digital data storage will help convey the nature and uses of mathematics and the flavor of the material covered in this book.
Computers store, manipulate, and transmit data as bits or binary digits. Physically, bits have been represented and conveyed in a vast array of schemes from historical to modern, including
- Shaking or nodding one’s head.
- Dots and dashes in Morse code.
- Holes and “no-holes” in a paper strip: punch cards, ticker tape.
- Magnetic domains: floppy and ZIP disks, PC hard drives.
- Light and dark spots or bands: compact disks, UPC symbols, QR codes.
- Charged and uncharged capacitors: flash memory, RAM.
A mathematician or theoretical computer scientist sees no essential difference between these schemes: The central mathematical “object” is a pair of contrasting states. Depending on context, the states might be called (and, in actual practice, are called) “zero and one”, “false and true”, “white and black”, “no and yes”, “open and closed”, “low and high”, or “off and on”.
Isomorphism
Mathematical abstraction extends beyond data, however, encompassing the operations performed on objects.
Binary arithmetic Think of \(0\) as representing an arbitrary even integer, and \(1\) as representing an arbitrary odd integer; namely, identify an integer with its remainder on division by \(2\). The sum of two odd integers is even (“\(1 + 1 = 0\)”), the product of an even and an odd integer is even (“\(0 · 1 = 0\)”), and so forth. These relationships may be tabulated as \[ \begin{array}{c|cc} + & 0 & 1 \\ \hline \Strut 0 & 0 & 1 \\ 1 & 1 & 0 \end{array} \qquad \begin{array}{c|cc} · & 0 & 1 \\ \hline \Strut 0 & 0 & 0 \\ 1 & 0 & 1 \end{array} \]
Boolean logic Think of \(\False\) as representing an arbitrary “false” assertion (such as “\(2 + 2 = 5\)”) and \(\True\) as representing an arbitrary “true” sentence (such as “\(1 + 1 = 2\)”). Since, e.g., “\(2 + 2 = 5\) or \(1 + 1 = 2\), but not both” is true, we write “\(\False \operatorname{xor} \True = \True\).” (“xor” stands for “exclusive or”: one statement is true, but not both.) Since “\(2 + 2 = 5\) and \(1 + 1 = 2\)” is false, we write “\(\False \operatorname{and} \True = \False\).” Tabulating the “truth value” of the statements made by conjoining two statements, according to whether or not each statement is true or false, \[ \begin{array}{c|cc} \text{xor} & \False & \True \\ \hline \Strut \False & \False & \True \\ \True & \True & \False \end{array} \qquad \begin{array}{c|cc} \text{and} & \False & \True \\ \hline \Strut \False & \False & \False \\ \True & \False & \True \end{array} \]
Abstract implementation The truly mathematical observation is the entries of these pairs of tables correspond: Under the correspondence even-False and odd-True, “addition (mod \(2\))” corresponds to “xor”, and “multiplication (mod \(2\))” corresponds to “and”. The tables above are different “implementations” of the same abstract structure, which we might denote by \[ \begin{array}{c|cc} \vee & \circ & \bullet \\ \hline \Strut \circ & \circ & \bullet \\ \bullet & \bullet & \circ \end{array} \qquad \begin{array}{c|cc} \wedge & \circ & \bullet \\ \hline \Strut \circ & \circ & \circ \\ \bullet & \circ & \bullet \end{array} \]
These three pairs of tables exemplify an abstract relationship, known as “isomorphism”, between operations on pairs of contrasting states. Any logical consequence that holds for one implementation necessarily holds for other implementations. For example, if we let variables \(x\) and \(y\) stand for either of two contrasting states, and we denote by \(x’\) the state unequal to \(x\), then the identities \[ (x’)’ = x,\qquad x \vee y = (x \wedge y’) \vee (x’ \wedge y) = x’ \vee y’ \] hold regardless of what values are assigned to \(x\) and \(y\), and, more significantly, no matter which implementation is used. In this fashion, mathematical structures can be studied and organized with their extraneous details removed. More subtle patterns can sometimes be discerned in the abstraction, leading to a deeper understanding of the original structures.
Among of the most remarkable internal features of mathematics is its absoluteness: the perfect and intricate logical meshing of truth even when drawn from widely separated sub-disciplines, the universality of theorems across cultures, the sense among mathematicians that their work describes an objective (if non-physical) reality.
As a language, mathematics is unparalleled in its ability to express features of the natural world, often with astounding accuracy. At the same time, mathematics has no known a priori connection to the real world. The objects of mathematics are idealized concepts (such as “a pair of contrasting states”), and do not have physical existence in the same way stars, molecules, or people do. Conversely, stars, molecules, and people are not mathematical objects, though they do possess attributes that can be modeled by mathematical concepts.
This book was written to help you bridge the gap between informal intuition and the more formal language and framework of modern mathematics. Learning mathematics requires active preparation and participation from you, but offers continual rewards, including deepened comprehension of the natural world and the sheer enjoyment of mathematical beauty.
Practice reading actively, with a pencil and scratch paper. When you encounter a new definition, try to construct examples and non-examples before reading further, and ask yourself how you might test an object to see if it satisfies the definition.
Develop the habit of filling in the missing steps of calculations and omitted “standard” steps of proofs. When you first read the statement of a theorem, pause to think about what the theorem claims, and whether or not you believe the assertion. Try to sketch out an argument on your own before reading the book’s proof.
Situate new general concepts and examples in the context of your existing mathematical knowledge. Pay attention to the overall structure of proofs, not merely to the details. Look for commonalities in arguments, and be sure you are able to use these strategies yourself. Your repertoire of proof techniques and other mathematical idioms will grow steadily.
Work on mathematics outside of class every day, rather than in one or two long “marathon sessions” per week. Don’t become discouraged if new ideas don’t immediately “click”. Re-read confusing passages after a day or two. Speak with classmates and your instructor for clarification as necessary. At the same time, develop intellectual self-reliance. The more mathematics you have made your own, the easier learning new mathematics becomes.
Above all, cultivate the enjoyment of thinking about new ideas, solving problems, and finding meaningful connections between seemingly disparate concepts. The greatest reward of your mathematical studies will, ideally, be a deeper experience of life itself.