Mathematical Proof and the Principles of Mathematics/Preliminaries/What is mathematics
As we mentioned at the start of this book, mathematics as a whole is very different from what most people learn as children. So it won't hurt to have a bit of a discussion about what we mean by mathematics. It's actually a difficult thing to define, not the least because the definition changes over time. But it may be useful to focus on the different aspects of mathematics, though the reader may feel such an approach is like the blind men describing an elephant.
Mathematics as a body of knowledgeEdit
As a body of knowledge, mathematics holds a place somewhere between philosophy and science. Unlike science, it relies on deductive rather than inductive reasoning as the test of truth. Unlike philosophy, it concerns specific entities such as number, space, and relationships. Though abstract, these entities seem to take on an existence of their own and behave in unexpected ways. Like science, mathematics seeks to find and describe patterns, but patterns which don't necessarily connect to anything in the real world. Like philosophy, it attempts to use reason to solve fundamental problems, but only problems concerning those whose nature it considers mathematical.
In more specific terms, mathematics is the study of things like quantity (arithmetic, algebra), space (geometry), change (calculus), and abstract structures (set theory, combinatorics). But what really sets mathematics apart is the method of study rather than what is studied. Mathematics attempts to form a bridge to expand knowledge. It does not use experiment or observation to do this, but logical deduction. This is not to say that mathematics does not use experiments, but they are used to guide the direction of research rather than to confirm theories.
Mathematics as a processEdit
Another aspect of mathematics is that of an activity or process. You might think this includes calculating square roots or evaluating derivatives, but that is only a tiny part. As an activity, mathematics plays a role which, though important, you rarely see or even think about. Yet you may often do or use mathematics without realizing it. Solving a Sudoku puzzle is essentially a mathematical process, but nearly any game or puzzle, from Baseball to World of Warcraft has its mathematical aspects. When you watch a video on YouTube or order something on Amazon, an algorithm goes to work to suggest a similar video or product you may be interested in. These algorithms are extremely sophisticated (still imperfect but improving) and self-modifying in response to new data, but they are ultimately based on mathematical models of viewers or customers individual preferences. On a personal level, you may use a type of mathematical reasoning to make everyday decisions in your own life. For example, if you're in the market for a new home you might be thinking about such factors as the length of your commute, the quality of schools, and how much crime there is in different areas. Weighing the importance of these issues and making a decision which you predict will produce the best outcome is, at least in part, a mathematical process.
As a process, the place of mathematics might be shown in the following diagram.
Data collection Existing Mathematical theory ↘ ↓ Model → Mathematical reasoning → Predictions ↗ ↑ Simplifying assumptions Logic
This diagram shows the place of mathematics in scientific theories as well as in the examples given above.
Mathematics as a languageEdit
It has been said (paraphrasing Galileo) that the nature speaks to us in the language of mathematics. If so, then what are the vocabulary and grammar of this language? Clearly, mathematical symbols and equations are part of the vocabulary, but so are abstract concepts such as number and triangle. If grammar is the way that vocabulary is assembled to make something meaningful, then the grammar of mathematics is deductive reasoning. Mathematicians wrap this reasoning up in theorems and proofs, but it is essential to understand its rules to truly understand mathematics.
Mathematics as an art formEdit
While images of the Mandlebrot set to the right are commonplace and visually striking, there is an additional layer of mathematical beauty which can't be immediately seen. This is the fact that all the intricate, lace-like patterns and the endlessly repeated motifs are generated using a single, simple formula:
There is quite a bit of creativity which goes into making an image such as this; colors are chosen to bring out details and enhance the aesthetic appeal and the image is framed to give the best presentation. But these are analogous to the lighting and framing of a photograph, and the object itself is the star.
The Mandlebrot set is particularly photogenic, but more importantly it's an example mathematical beauty. Quite often, marvelously complex and interesting phenomena arise from what seem like very simple assumptions, just as in this case where the swirling patterns of the Mandlebrot set arise from the simple formula above. At other times, deceptively simple patterns are found, but the reasons for these patterns are unexpectedly complex. In this way, mathematical concepts seem to take on a life of their own, behaving in unexpected ways yet, being mathematical in nature, behaving in the only way they can. This is, perhaps, at the heart of mathematical beauty, but the art of mathematics goes still a bit further. Sometimes a solution to a mathematical problem is unexpectedly simple, requiring a twist that seems obvious only after it's seen. In this case, what might be called mathematical elegance is created by the appreciation of human artisanship.
The essence of mathematicsEdit
As with the elephant, no single aspect of mathematics captures the real essence of the subject. But, hopefully we've given you enough to show that mathematics is a living entity, not just something hard and smooth as you might think if you've only ever laid hands on the tusks.