Useful Notes / Mathematics

If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.
Roger Bacon

*takes a deep breath*

Mathematics is one of the oldest and most well-developed disciplines. It has some of the same features as science: there are multiple branches that all build on each other, old theories are not discarded, just absorbed into "more correct" or "more general" theories, and it has very rigorous notions of what is true and what can be claimed to be correct. Mathematics also avoids some of the major problems of science: you don't need huge amounts of background material to get into a topic, and you don't need lots of expensive equipment to do new mathematics.

It is also hated by roughly 99.9% of all people. This is the major reason that there aren't more scientists, doctors, engineers, or Wall Street workers. This is probably due to the way it is taught: 6 to 8 years of number crunching, most of which a calculator can do faster and with less chance of error, followed by massive, abstract generalizations (algebra and calculus). To make things worse, students are rarely told why they need to learn the current topic: science teachers tend to wait until after you've learned the math to show you the uses for it, and a lot of math teachers focus on the techniques rather than the applications. (Or they try to demonstrate the applications by assigning word problems—without realizing that "translate this word problem into math" is also a skill that needs to be taught, and that assigning said problems to students who don't have that skill won't help.) Worst of all, many elementary school teachers are poorly trained in mathematics, so they don't know good problem-solving techniques, they don't know the particular real-world applications of any given topic, and they don't like it enough to teach math for its own sake.

If you want to know what the other 0.1% of us see in it, we can suggest a few places to find fun or cool mathematics:
  • Francis Su's fun facts website. These are short web pages detailing some "fun fact" about mathematics; the maintainer likes to spend the first five minutes of his calculus classes explaining one of these facts.
  • Vihart and Numberphile are youtubers who show awesome math stuff for people who aren't scientists, engineers, or mathematicians.
  • Martin Gardner's books of collected Scientific American columns. These go into a bit more detail than the fun facts above. Each chapter in these books was originally a magazine article about some topic in recreational mathematics, so each chapter stands alone; reading one (or a few) is not a huge time investment.

The most common depiction of mathematics in media is of dry lectures in high school, where the teacher is 100% unaware of the class and assigns tons of homework. If the setting is college, then the professor is insane, the lectures are only a little more interesting, the homework is a lot less but very hard, and it still is an unenjoyable experience. Occasionally, a mathematician character will be shown working, but the math is rarely explained.

These depictions tend to include a lot of random equations (or meaningless pieces of equations) scrawled across blackboards. A lot of these equations actually come from physics, as advanced mathematics is more prone to forming Walls of Text than walls of symbols. (See E = MC Hammer for examples.)

There are many fields and subfields of mathematics:
  • Algebra studies objects with some sort of inherent structure. Usually, this structure is a way to put things together: for example, the real numbers (where you can add or multiply any two numbers), modular arithmetic ("clock arithmetic", in which you can still add numbers, but now 8 plus 9 is 5, because nine hours after eight o'clock is five o'clock), and symmetries. (To a mathematician, a "symmetry" is something you can do to an object without changing it: "rotate 90 degrees", "rotate 180 degrees" and "flip left to right" are all symmetries of a square, but "rotate 30 degrees" is not. You can "add" two symmetries by doing one followed by another.) Algebra also includes finding smaller structures inside bigger ones: for example, the symmetries of an octagon include all the symmetries of a square.
  • Analysis starts with calculus done rigorously and goes on from there. Analysis includes measure theory ("how big is this?") real numbers, complex numbers, and most familiar functions (sine, cosine, logarithms, and so on). Analysis also includes the idea of approximations: Archimedes didn't know exactly what π was, but he knew it was close to 22/7, and that he wasn't off by more than 1/497. A lot of applied mathematics comes under the heading of analysis.
    • Measure theory is, perhaps, misnamed. It does not so much answer the question "how big is this?", as verify the main ideas of calculus for the situations where "big" is defined very unusually. It is a final example of a trope that is extremely familiar to students of mathematics: "Painfully Working Out the Obvious".
  • Geometry started out as a way to measure the shapes of tracts of land on the earth (hence the name) and went on to include the study of all sorts of odd shapes and objects in the plane, in space, and in higher dimensions. (Modern mathematicians generally try not to think of "a ball in space", but the ball as an object unto itself.)

These three fields are probably the biggest ones, but there are numerous smaller fields that do not fit into any of the big three, including:

  • Number theory is the study of the integers, and other structures that are very similar to integers. One of the main problems is to understand the distribution of the primes, and the most famous unsolved problem in all of mathematics, the Riemann hypothesis, is related to the distribution of the prime numbers. Nowadays, number theory is the basis of cryptography. (Thanks to the aforementioned unsolved problems, many of which can be phrased as "If I scramble up these numbers thus, is there an easy way for someone else to undo it?")

    In ages past, some (mostly number theorists) considered number theory to be the most "pure" discipline, as it had absolutely no practical applications until cryptography came along: mathematicians did it because they thought it was cool and for no other reason. (There are contrasting branches of mathematics—fluid dynamics come to mind—that even mathematicians tend not to like, but which they do anyway because it's useful.)
    • Note that fluid dynamics is actually a field of physics, but it uses such complex and sophisticated models that whole branches of mathematical analysis were developed purely to support these models and solve these problems. That's actually how mathematics and physics are usually related: physicists find some problems or processes and develop a mathematical model of it, while mathematicians (who are quite often the same people, these groups tends to intersect a lot) work them out.
  • Graph theory studies the ways in which discrete things are connected to each other. The objects used to represent this are call graphs and consist of points (called nodes or vertices) connected by lines (called edges). A family tree is a graph (people are vertices, and parents are connected to their children). The pictures on the Triang Relations page are directed multi-graphs. The applications of graph theory are enormous: It is used to study the movement of traffic in cities, how power grids work, social relationships between people, the design of highways and railroads, to determine routes for shipping packages, and many more things that clearly relate to having various points that are connected to each other. More surprisingly graph theory is also used in compression algorithms like MP3 and JPEG which use graphs to calculate the way they will encode the file in order to reduce its size.
    • Topology is cousin to graph theory but is interested in how continuous things are connected rather than discrete things. For instance the points on the surface of sphere are connected in the same way as the points on the surface of a cube. (This is usually represented out of context as questions about transforming shapes into other shapes using seemingly arbitrary rules.) Topology finds uses in molecular biology where it is used to help understand the manipulation of complex molecular structures. Many fields of physics rely heavily on topological descriptions of objects.
  • Combinatorics is the art of counting things. (Yes, there's actually a discipline of counting.) Since mathematicians like to count really weird things, it is much harder than it sounds. For example, how many ways are there to write 4953 as the sum of smaller numbers if we allow repetition and order doesn't matter? 72941390690430942437223889128779314587984674067394751139900443450045622445 ways or so.

Number theory is an extreme example, but a lot of the mathematics above was done originally because some mathematician thought it was interesting; applications were discovered later. However, a lot of mathematics is done with the intention of being useful in the real world.
  • Probability lets you know things about a system and calculate how likely outcomes are. For example, if you know that a coin is fair (50% chance of producing heads), then you can calculate that if you flip that coin five times, the chances of seeing at least four heads is 18.75%, and if you know that the coin is biased and has a 70% chance of producing a head, then the chances of seeing exactly four heads is 52.822%. Probability began in the gambling hall (probabilities in card games are easy enough to calculate, but tricky enough to be interesting), but some ideas (expected value, variance, conditional probability) are useful outside of the gambling hall.
  • Statistics goes the other way: you know the results of a lot of experiments, and you want to determine something about the underlying system. For example, if you flip a coin five times and get four heads, you know that it's a little weird to see that coming from a fair coin, and not really weird at all to see it coming from a biased coin. To a lot of people, statistics is probability's Boring, but Practical cousin: you need to understand statistics to do any sort of science, but probability is often a lot more fun.
  • Another direct application of mathematics is modeling: you let x be something you care about, write down a system of equations that x satisfies (or almost satisfies), and solve for x. These equations are often differential equations: for example, an object falling near the earth accelerates at 10 meters/second2, so its velocity is changing all the time; solving the differential equation lets you know the velocity at any given time. In school, these are the dreaded word problems; outside of school, this is how mathematics is commonly used.

There are some fields of mathematics that act as a basis, or foundation, of the other fields. Mathematicians like to base their work on as few assumptions as possible; this is both an aesthetic ideal of mathematics, and minimizes the chances of two of those assumptions contradicting each other. Thus, the basis of mathematics involves brutally simple things and things that we really do need for everything.
  • Mathematical logic is the study of formal logic (logic that follows a suitable degree of rigor) and its applications to other areas of mathematics.
  • Set theory is the study of collections (sets) of objects, and ways that these collections can interact. So you can have the set of all integers, a subset (the set of all positive integers, or the set of all even integers); you can have the union of two sets (the set of all positive integers which are either even or positive or both) or the intersection of two sets (the set of all integers which are both positive and even). Set theory also includes a notion of a bigger set: for example, {1,2,3,4} is a bigger set than {2,3,4}.
  • Category theory, which describes mathematical structures—that is, objects that attach to other elements in a set—and their relationships between them.note  Category theory is perhaps the most abstract kind of math on this page; it's a bit hard to describe, and even mathematicians tend to find those who specialize in it off. Additionally, it has given birth to one of the stranger bits of mathematical jargon: "abstract nonsense".

While these foundations were already solid enough back in the 19th century, their close relationship with computer science has spurred a lot of research in the subject since the beginning of the 20th century.

Most math researchers work in a subfield of the areas listed above, and have a special type of problem that they work on. The biggest field is probably algebraic geometry, as there are many ways to get at it, and there are many theories that can be useful in solving problems there.