Beautiful mathematics: a section of the Mandelbrot set. (Image taken from the Wikipedia article on this set.)
Short answer, phrased in three ways:
- The science and logic of patterns.
- A pattern club and tool box.
- The game of abstraction.
Long answer:
What follows is a fairly loose description of mathematics without much support given. Maybe later I could fill these thoughts out more to address some of the deeper issues discussed, for example, in the Stanford Encyclopedia of Philosophy article on the Philosophy of Mathematics.
I’d say that math is the science and logic of patterns. It is inspired by what we observe in the real world but is ultimately based upon what we can imagine and the implications of our imaginings through our reasoning. The patterns can be in space, or time, or sequence, or any realm you can imagine, including the patterns of logic or reasoning itself. You can do mathematics by recognizing, creating, or imagining patterns, categorizing them, finding the relationships between and within them, creating language to describe all this, and making your concepts and language more precise and exact. If your results are sensible enough, they may be accepted by professional mathematicians as qualifying as mathematics.
(A few notes and caveats: You will probably have to use a certain amount of commonly accepted mathematical language or jargon before anyone will pay attention to your work, since it takes a lot of effort to translate idiosyncratic expressions into familiar ones. Also, it seems exceedingly rare, from my knowledge, for something to be accepted as rigorous mathematics and later to be rejected, so the bar for “sensible” has historically been quite high. Furthermore, there is a chance that your work will not be recognized as correct until after your death, so you may be ahead of your time. And, apparently, your work doesn’t even have to be consistent to be accepted nowadays, but it probably has to be “paraconsistent”.)
Amazingly, if you take interest in this kind of activity and get into the swing of doing it and expressing creativity, you can spend a lifetime generating new mathematical concepts and honing and tweaking existing ones. The richness of mathematics just keeps growing as the years pass.
Mathematics can be done alone, by a single person, but people tend to interact and so mathematics tends to be shared too. Taken cumulatively across the world, mathematics is a global social enterprise and a growing collection of concepts, procedures, tricks, and arguments for reasoning about patterns — a kind of pattern club and tool box. Learning mathematics can also be more enlightening when learning the history, lore, myths, people, and context of mathematical development. So mathematics comes with a collection of stories as well. As a human endeavor it has a colorful history with the full range of emotion that comes with any human endeavor.
Perhaps even more amazingly, as mathematics has grown from its crude and physical beginnings and become more abstract and esoteric, it has frequently and nearly continuously helped to advance the human pursuit of the physical sciences, to understand and manipulate the world around us. Granted, it seems necessary that the surface-level patterns of our world must be fairly understandable, because otherwise how could humans have evolved to have the capacity for understanding in the first place? But it could have been the case that the underlying patterns in our world were extremely complex and required tremendous progress in mathematics before much progress could be made in science. It’s possible that we do encounter such ruts now and then, or perhaps we are even in such a rut right now, but it is impressive how much progress has been made in the physical sciences with the use of mathematics. This “effectiveness” of mathematics has sometimes even been called unreasonable. I should note too that advances in physical science as well as computer science and other fields have provided inspiration for advances in mathematics, so there is a kind of virtuous symbiosis amongst these fields.
Does math ever lead people astray? I would say not the math itself, but its misapplication can. (I believe more errors are made in various branches of science, and it is more likely that “science” leads people astray sometimes.) Perhaps you’ve heard the phrase, “There are three kinds of lies: lies, damned lies, and statistics.” Erroneous reasoning wrapped up in mathematical jargon can seem much more respectable and convincing than the reasoning stripped bare. Mathematics can help to give legitimacy to a bogus economic theory as it supports a deceptive political agenda. A similar thing happens when science is misapplied or mimicked to support pseudoscientific beliefs. But, hopefully, those with good mathematical, scientific, and philosophical understanding can help to publicly expose and discredit such theories and beliefs.
I would say that math, considered separately from its applications, is the game of abstraction. If you dream up a pattern of abstraction to play with, then you are doing mathematics (even if as an amateur), and there’s a ridiculously significant chance that even if it seems useless, some day, someone will find your game a useful one. (Or, maybe it’s already been dreamed up and used!) Mathematics is a joy in itself, but combined with its applications it is a source of great power and effectiveness. Let’s have fun and use it wisely.