Mathematics is seemingly one of the most feared subjects in all curricula. At least that appears to be the case here in the United States. While indisputably powerful and useful (look no further than the latest machine learning and artificial intelligence hype), many of us “mere mortals” (i.e. non-mathematicians) feel it’s something far beyond our comprehension. Moreover, after we’ve finished up our course work many of us often stop learning advanced math altogether. This suggests a common sentiment that there is no further use of higher-order mathematics in our everyday lives. I want to reiterate that this is certainly not true. If nothing else, it will improve your reasoning skills if not also your problem abstraction skills.
Mathematics is Not Computation
Much work as an engineer is done using logic, physics, differential equations, and various other forms of esoteric computation. However, memorizing common formulas and plugging/chugging through them is insufficient for being a top-tier engineer. Similarly, performing this sort of rote computation side-steps the importance of learning math in the first place.
The difference between computation and math is like cooking a hot pocket vs. preparing a 4 course meal. Namely, when cooking a hot pocket, everything is prepared for you; all you have to understand is how to operate microwave or oven. On the other hand, a 4 course meal requires the same knowledge of using those tools, however, it requires you to also understand the synergy between flavors and how to transform your raw ingredients into special dishes.
In summary, believing that math is all about “crunching numbers” will only hinder your ability to study it.
Mathematics is Not Easy
No matter what anyone tells you, math is not easy. While learning is highly non-linear and learning rates are highly specific from person to person, I assert that– in general– hard work and persistence are required for actually advancing your knowledge. Many of math’s groundbreaking theorems and insights come from years of devoted work by some of the world’s most intelligent and persistent human beings. Truly understanding all of that insight in a short time span is simply unreasonable.
The point is, do not underestimate yourself and do not overestimate the intelligence of others. Once someone has grasped a concept, it should seem “easy” to them. As a result, if you are speaking with a colleague and they mention something is “easy” that you find complex, just realize that they’ve somehow captured their idea. Eventually, the same ideas will become natural to you as soon as you wrap your head around them.
Mathematics is Approachable
I’ve been asked a few times, “where do you study math?” Well, the internet is a good place to start though I personally prefer structured texts for in-depth learning on a specific topic. Assuming you have the background, a well-structured math text should build a strong foundation for introducing key concepts in any particular field. On the other hand, I often like to delve into areas of math which I know nothing about. At the start of this process, everything is a bit more exploratory and reading a text on the topic is often beyond my of comprehension. In such a situation, I find myself jumping down the rabbit hole trying to gather a strong enough foundation to grapple with the ideas I wanted to learn in the first place.
For instance, suppose you know nothing– but want to learn– about affine spaces. If you’re unfamiliar with linear algebra or Euclidean spaces, you may want to start learning about those first. Similarly, if you delve into those subjects and don’t understand their foundations, recursively repeat the process (a sort of depth-first traversal on the subject-matter). While this may be time-consuming the first few times you do it, eventually you’ll start to notice significant overlap in the foundational maths across concepts and this sort of research will become easier over time.
Ultimately, if you take the time to learn the base concepts of a subject, math continues to build in a logical way. As a result, seeing that “natural” progression of building enables higher-level mathematics to be approachable to anyone patient enough to explore fundamental concepts.
Mathematics is Reasoning
Above all, mathematics is the skill of reasoning. Since we have computers in our modern age, it’s less useful to optimize our thought processes for crunching numbers. Instead, we should focus on understanding high-level concepts and algorithms. More concretely, understanding underlying proofs of key theorems often points to useful features of the underlying field of math.
Even if your day-to-day work doesn’t directly require mathematics, it certainly demands that you reason about situations. By studying more advanced maths, you will find yourself more capable of effectively handling a variety of situations and thinking abstractly. Similarly, studying math will help you to question the “obvious” facts of any situation. All of this will make everyone a better problem solver.