Defining Math

Most people in modern society have been trained in arithmetic and algebra from childhood, but don’t often know what math is.

The Basis of Math

Every concept in math is a precisely parsed concept in an imaginary space. While some numbers represent real things (e.g., 2, 3) and are therefore called “real numbers”, others can only exist in the philosophical realm of mental ideals (e.g., √2, π) and are called “imaginary numbers”.

The actual existence of math is subject to value theory, specifically the question over whether it exists in nature around us or in the minds that perceive nature. However, for the sake of working with math, it doesn’t really matter where it exists, or even if it does. All we need is the basis that it’s perfectly consistent and something we can be certain over.

Math represents itself symbolically, and can best be described as the science of structure and order, which represents as patterns across our perception. Unlike other sciences, though, repetition doesn’t prove a mathematical concept: the only way we can tell is with deductive certainty that it can’t be anything else. Proving these patterns require generating proofs, and that is most of the work mathematicians spend time on.

Most people find math too difficult to work with, but it is not a trivial discipline in many STEM careers, and even within some domains of art. All the patterns of nature, including science and engineering, have the patterns of mathematical order behind them, and those patterns have allowed us to make very efficient factories, get to the moon, have well-designed traffic circles, and build better cities.

Math education often overlooks most of this, and many standard college-educated math teachers will focus on rote memorization and formulas instead of conveying the core concepts that give mathematics any use whatsoever in the first place. This level of ignorance has magnified math anxiety more than it should be.

Math Anxiety

Math anxiety is a legitimately real problem, and most people who don’t specialize in math-based occupations simply imagine it’s too difficult for them to understand.

However, this is a self-reinforced falsehood driven by several converging factors:

  1. Modern Western education, especially in the USA, has handed off most of the calculation work to computers. While some disciplines (like accounting) still train the old-fashioned way (which takes more work and therefore creates more understanding from the effort), computers in general cheapen the required effort to build the math work we do.
  2. Math is all cumulative, and it takes patience to work through it. Unlike other language (like prose), it requires revisiting old concepts repeatedly until you understand them entirely. If you only memorized the information to repeat it back on a test, it will not be useful later.
  3. Some of the higher-level math concepts are filled with dense and sometimes confounding jargon. Integers and integrals, for example, have nothing to do with one another. The ideas are often simpler than the jargon makes them sound.
  4. Higher-education math is composed of many math researchers, but many high-end university cultures revolve around the professors’ actual educating of students being side work. Their ineptitude at the craft of teaching therefore contribute to students’ stereotypes that math is a densely complicated subject.

Math itself isn’t really “easy”, but it is thorough, and it can often be tedious, but there are several techniques to make it easier:

  1. Use little numbers. If the problem has big, gigantic numbers, then swap out that problem for little numbers and try to solve that one instead.
  2. Separate out the concepts. If you see a long formula, break apart the pieces and solve those individual pieces as entirely separate concepts. If there are big numbers, apply #1 to get a firmer grasp of it.
  3. Look beyond the book you’re reading. Sometimes, the textbook author didn’t word it correctly or is a technical idiot, and they made it too difficult for you to understand. For any legitimately useful math, there are many videos, books, and tutorials on the subject (e.g., BetterExplained, StackOverflow.
  4. If you simply want the clear answer, use a calculator to find the answer, then work your way through it (e.g., Wolfram Alpha).
  5. Use website tools as well for step-by-step walkthroughs of your exact problem (e.g., Mathway, Symbolab). If you understand it, that’s much more important than solving that particular problem, since you’ll be able to solve 100 just like it later.
  6. Once you do understand it, rework the problem yourself, without looking at a reference. Take your time, and do not rush the learning process.


Math isn’t 1 discipline, but technically hundreds.

As late as the Renaissance, there were simply 2 branches of mathematics:

  • Arithmetic, which is manipulating numbers
  • Geometry, which is the study of shapes
    • Euclidean geometry uses Euclidean planes (i.e., plane geometry) and the three-dimensional Euclidean space we all inhabit.

During the Renaissance, two more areas appeared:

  • Algebra, which started as mathematical notation, that worked with precise symbolic descriptions of mathematical ideas.
  • Calculus, which studies nonlinear relationships between different quantities (i.e., patterns for how things are related).
    • Differential calculus studies the rates that quantities change (e.g., the average wavelength of light).
    • Integral calculus, or simply integrals, studies the accumulations or volumes of those relationships (e.g., the total sum of all light rays hitting a moving object.

For a while, the 4 domains existed independently, and most of its framework was grounded in very old works (e.g., his book “Elements”).

There was quite a bit of work most mathematicians performed in proving inescapably that the principles they explored were true:

  • Number Theory clarified and qualified precise definitions of what numbers even are.
    • Analytic number theory uses logic to solve problems with numbers.
    • Algebraic number theory uses abstract algebra to study numbers.
    • Geometry of numbers uses geometry to study numbers.
    • Diophantine equations use 2+ unknown numbers that are integers

However, at the end of the 19th century, there was an issue about Euclid’s fifth postulate (or parallel postulate):

  1. Given a point and a line…
  2. When that point is not on a line…
  3. Only 1 other line can pass through that point and be parallel with the existing line.

The problem was that nobody could prove the fifth postulate, and that unleashed a post-modern deconstruction movement called the foundational crisis.

From there, there were a lot of new branches of mathematics, and most of the previous math disciplines became labeled as the “classical” discipline:

  • Non-euclidean geometry, which measures things that do not follow the parallel postulate.