Over the years, I have heard this question or criticism many times:

Why is so much math required for a computer science degree?

I never questioned the amount of math that was required to earn my degree. I enjoy learning, especially math and science. Although, a few of the classes felt like punishment. I remember the latter part of the semester in *Probability* was especially difficult at the time. Possibly because I was challenged with a new way of thinking that is required for these problems, which can be counter-intuitive.

## What is Computer Science?

Let's start with computer science, and work our way towards math and how the two are interconnected. A definition of what is considered computer science may be helpful before we discuss its foundation in math. Here is an abridged definition from Wikipedia[^]

Computer science is the scientific and practical approach to computation and its applications. It is the systematic study of the feasibility, structure, expression, and mechanization of the methodical procedures...

A computer scientist specializes in the theory of computation and the design of computational systems.

I have had many discussions over the years with students and professionals educated from all different realms. One common sentiment that I hear from all of these groups is this:

Theory is nice, but I need to have relevant skills to get a job

This is a very valid point. It is also true that the majority of college graduates do not continue their career focused in the world of academics. Therefore, we can assume that most people that attend college do so with the intent to join the general workforce, i.e., to get a job. Let's evaluate.

### The Role of Academia

Much of what I have learned about the academic realm is that it has a fundamental focus on expanding the limits of our collective knowledge. Here is a great explanation of the purpose of a Ph.D. by Professor Matt Might at the University of Utah, The illustrated guide to a Ph.D.[^] As these new discoveries are better understood, they get further developed. Some discoveries may continue to live and exist solely in the academic realm. While other theories are applied and practical uses are developed from these discoveries.

### Application of Theory

The concepts taught in math, physics and engineering are based on theory. A theory itself is a model or a collection of ideas that is intended to explain something. The theories taught in these disciplines have been developed and refined over time. It is relatively simple to observe the application the theories that are required for the engineering disciplines based upon physics.

Some of our engineering disciplines start to push the boundaries of what is possible for us to observe, such as electrical engineering and nuclear sciences. However, we are still able to verify these hypothesis through repeatable experiments and observations, even if we can only observe the indirect results of these experiments.

Math and computer science are purely theoretical. They are abstract mental models that describe the relationships and behaviors of the interactions between these models. The application of these theories is a result of mapping these conceptual models into a context that can be represented and observed in physical form.

#### Yeah, yeah that's great (in theory), but...

## I Need Skills to Get a Job

Trade schools are designed to help students learn the skills of a trade needed to get a job in that profession. The curriculum of trade schools is focused on teaching students the programming languages that are in highest demand, used to solve the most common problems that occur *today*.

This typically means that you would study a semester learning and solving problems with JAVA. Then move on and use a scripting language such as Python for a semester. Interfacing with relational databases is also likely to be in the curriculum as it is a very marketable skill.

As for college degrees in computer science; yes, you will learn a variety of computer languages that may help you get a job. However, these languages are used as the basis to teach the principles and give you the opportunity to apply these theories. The end goal is to teach critical thinking, to help students of computer science learn to reason, experiment, observe and understand the theories.

I believe that what you get out of your education, depends on what type of effort that you put into your education. This is true for either path. The difference between the two is typically where the instructors focus your attention. Neither path guarantees a job at the end. However, if you do not have what it takes at a university computer science program, they will flunk you out of the program. The for profit trade-schools are bit more forgiving.

### You Can Program Computers with Minimal Math Skills

This is a true statement.

In many cases, all that is needed to be able to program a computer is a computer, and the programming tools to interface with the computer. With basic arithmetic skills and knowledge of a computer language, you will be able to command that computer.

The same thing goes for many other skills, amateur auto mechanic, home plumbing, renovations, do-it-yourself haircuts. Some of the world's most successful people dropped out of school and started a tech company. For example, John Carmack, who dropped out of college, co-founded Id Software and went on to pioneer some of the game industries ground-breaking graphics technologies; when he needed to know the math, he taught to himself.

I don't have any numbers, but I would imagine that most computer science graduates go on to become software developers of some sort. Many of the software engineers that I work with graduated college with an Electrical Engineering or Mechanical Engineering degree. Two guys I know graduated with law degrees, not a degrees in engineering.

## Where Does Math Factor In?

When math is mentioned, most people generally think of topics that require numbers and are focused on calculating values of some sort. The one notable exception is Geometry. Here is a small sampling of those types of topics:

- Arithmetic
- Geometry
- Algebra
- Calculus
- Linear Algebra
- Fourier Transforms
- Ordinary Differential Equations
- Partial Differential Equations
- etc.

However, it is important to recognize that mathematics deals with more than *just* numbers. Mathematics spans a wide range of topics:

- Logic
- Algorithms
- Fundamental theorems
- Mathematic proofs and models
- Discrete math
- Theory of computation
- Information theory
- Combinatorics
- Set theory
- Graph theory
- Abstract algebra
- etc.

### Math is a Foundation, Which is Built Upon

“Mathematics is the gate and key to the sciences.”

– Roger Bacon

For those of you that have taken Analytic Geometry, Calculus or even Trigonometry, consider how the material is taught. They start with a mathematical foundation and proofs are demonstrated that expand on what you should know going into the class. Each step is then used to derive the next step in the curriculum.

At the end of the course, many of the interim steps are forgotten or rarely used. However, they were a very necessary stepping stone to make a leap to the remainder of the material.

An example that comes to mind appears in Calculus, and is the concept of a *limit* of an equation. The concept of finding the limits of equations is then used to create *derivatives* and *integration*.

You could simply memorize the formulas to plug in your own numbers, and you would be able to *use* the equations. However, when you run into a problem that doesn't fit neatly within the formulas that you have learned, you would be stuck. The critical thinking skills that developed as these math courses progress give you the tools and ability to transform problems into a simpler form that *does* fit the mold of solvable equations.

This is the same structure that is used in computer science related courses. Data structures and algorithms that have been studied for decades have been optimized and we are taught how we reached this point. Again, it is possible to use an algorithm called `sort`

, to organize your data and you do not need to understand how it works.

However, if you find that your `sort`

routine runs much slower as the data-sets that you work with become much larger, you will need to know how to analyze your algorithm to be able to build something better. With a solid foundation, you will understand how to compare the trade-offs and build something based on previous proven work.

Without this foundation, you may still be able to improve upon your original algorithm. However, you will only be guided by previous experience. Making that inspirational leap to a radically different, yet far more efficient design is not always easy.

### Critical Thinking

Critical thinking is essential in professional fields as well as academia. Well, essential to maximize success. Critical thinking is more than analyzing and reasoning well thought-out decisions. It involves evaluating, conceptualizing, applying, observing, and communicating your conclusions.

Along the way, you also need to become self-critical looking for any biases that could add prejudice to your conclusions. Reflection and scientific skepticism help guide you towards accurate observations and well-justified conclusions.

It may seem as though we have meandered away from computer science into philosophy or scientific theory. We may now be discussing concepts that are central to these other topics. However, we have not wandered away from computer science. These skills and concepts are just as important to computer science as they are with every other engineering discipline.

## Programming != Computer Science

It is confirmed then, you do not need advanced math to be able to program a computer, nor do you need a degree in computer science.

Hopefully, it is also obvious that there is a limit to what you will be able to program without knowledge of certain topics, especially math.

IT departments need a lot of programmers to create applications that automate or simplify business logic. These are line-of-business applications that give the companies users the information they need to do their jobs more efficiently. Much of this work centers around large databases and reducing the data set to the pertinent information required by the user.

Software engineering for machines and cutting-edge technologies often does require knowledge of advanced math. All of the wireless technologies that have been developed and become mainstream over the last 20 years are heavily dependent upon signal processing algorithms. Signal processing in itself could cover waveform synthesis and analysis, compression, encryption, forward-error-correction, and modulation/demodulation.

Now consider artificial intelligence, computer vision, image processing, natural-language processing, simulations, robotics, general-purpose GPU programming, distributed computing, the continues to grow each year. With the exception of robotics, I believe that today's smartphones are in some form dependent upon all of those technologies that I listed, all of which require knowledge of advanced math to do well.

Most web-sites do not receive an amount of traffic that cannot be solved by throwing more hardware at the problem. However, even with the commodity prices of hardware these days, more hardware is not always the best or even an easy solution. Experienced developers with strong critical-thinking and reasoning skills are a much better investment.

For some companies like Facebook, more hardware simply is not an acceptable solution. With hundreds of millions of interactions constantly occurring through-out the world, highly specialized algorithms are required to operate their site as efficiently as possible.

## Summary

Math is good. For some people, math is hard. Math is also more than just numbers. Mathematics is a logical foundation that most of our sciences are built upon. Analysis, critical-thinking, and reasoning are just a few abstract skills that are developed by studying and applying mathematic concepts. These are also very important skills to develop in order to become a good software developer. This is true whether you work in a corporate IT department, or are developing the next-generation wireless communications network.

I was required to learn one-year of numerical-based math passed Calculus. The type most people think of when considering math. However, the majority of my math classes were about more abstract topics like Discrete Mathematics, and the Theory of Computation. These are the building blocks that set the foundation for the more advanced topics taught the senior year of my degree. These are also the building blocks that are required for those students that choose to move on to graduate school.

Math is not necessary to be able to program a computer. However, math is very important to engineering and science, and there are just some places that you cannot access with a computer unless you have a solid foundation in logic, critical-thinking and abstract mathematics.