Applied vs Pure Mathematics - Infinity is Really Big

Calculus 2, Lecture 1 (Videotaped Fall 2016)

Population modeling is an important part of applied mathematics in calculus. It is a good example to start with in looking at the nature of applied vs pure mathematics. — Population modeling is an important part of applied mathematics in calculus.

Calculus is at the heart of much of modern mathematics and its applications. Its discovery by Newton and Leibniz sparked the scientific revolution and enlightenment of the 1600’s and 1700’s and brought about our modern world.

Throughout mathematical history, there have been many interactions between applied vs pure mathematics. The history of calculus is no exception to this general rule.

Generally-speaking, the early pioneers of calculus were applied mathematicians. In fact, people such as Newton and Gauss are more famous as scientists than as mathematicians. They created and used their mathematics in order to model physical reality. This is what applied mathematics is about.

In more recent times, since the mid-1800’s, many mathematicians have focused on doing mathematics for its own sake. This is what pure mathematics is about. Within the subject of calculus, this study goes by the broad name mathematical analysis.

Oftentimes, the interactions between applied and pure mathematics are positive. Applied mathematics provides inspiration for pure mathematicians. And pure mathematics provides justification and unification of what applied mathematicians do.

Sometimes, the interactions between applied vs pure mathematics are negative. Usually, the problems are more with the practitioners of these subjects. Some pure mathematicians may look down on applied mathematics as being dull. On the other hand, some applied mathematicians may look down on pure mathematics as being too esoteric.

My Personal Perspectives on Applied vs Pure Mathematics

Personally, I like and appreciate both aspects of mathematics. Pure mathematics comes more naturally to me, but I very much enjoy it when pure mathematics can be applied to the real world. I like to see what the pure mathematics “is good for”.

As a Christian, I also praise God for both aspects of mathematics. I like pure mathematics for its beauty, a beauty that reflects the creativity of the Creator. And I like applied mathematics because it is useful to help make the world a better place.

In either case, I ask God to help me take the bible verses Colossians 3:23-24 to heart.

²³Whatever you do, work at it with all your heart, as working for the Lord, not for human masters, ²⁴since you know that you will receive an inheritance from the Lord as a reward. It is the Lord Christ you are serving.
Colossians 3:23-24 (New International Version)

Philippians 4:8 is also very near to my heart.

Finally, brothers and sisters, whatever is true, whatever is noble, whatever is right, whatever is pure, whatever is lovely, whatever is admirable—if anything is excellent or praiseworthy—think about such things.
Philippians 4:8 (New International Version)

I want both aspects of mathematics to lead me to my ultimate goal in life. That is, to love and serve God with all my heart, giving praise to Him through Jesus Christ, in whom I believe all things have been created, both through Him and for Him (Colossians 1:16).

Calculus 2, Lecture 1: The Nature of Applied vs Pure Mathematics

Ultimately, this is the first of a series of blog posts on my Calculus 2 Lectures at Bethel University during the Fall of 2016. I have a similar series of blog posts on my Calculus 1 Lectures at Bethel during the Fall of 2019. The first blog post in that series is titled “The Big Ideas of Calculus”.

Here is Lecture 1 for Calculus 2, the main topic of this post.

Calculus 2, Lecture 1: Course Structure, Nature of Applied vs Pure Mathematics, Mathematica Demonstrations

You’ll want to skip ahead to time 21:22 in the video because the first part of the lecture is about the syllabus and other content that is only relevant for the students sitting in the classroom. Also, I’m sorry that the volume gets lower upon zooming. That doesn’t usually happen in later videos.

Applied vs Pure Mathematics in Calculus

How do applied mathematicians, including everyone who uses calculus to model the real world, think about calculus? Here’s a quote from the lecture:

Calculus consists of methods to model quantities that are related to each other and change over time and, perhaps, make predictions about the behavior of systems and the future. Implementation on computers, such as with Mathematica, can allow us to “tweak” these models to see what happens if parameters are changed.
Bill Kinney, Calculus 2, Lecture 1: Course Structure, Nature of Applied vs Pure Mathematics, Mathematica Demonstrations, Fall 2016

The key things that make this applied are that we are using calculus to “model quantities” and “make predictions” about “the behavior of systems and the future”.

For example, we could try to model the temperature, the nature of the wind (speed and direction), and barometric pressure to predict tomorrow’s weather.

Predator-Prey Modeling (Applied Mathematics)

We might also want to model animal populations. This is part of the subject of ecology. Of particular interest are “predator-prey” populations — like rabbits and foxes (but wait a minute, aren’t rabbits and foxes friends now?).

The graphs below show hypothetical rabbit and fox populations over time, based on the model (which is a system of ordinary differential equations).

For both the foxes and the rabbits, the populations oscillate with a period of about 5 units of time (maybe years). The peaks of these oscillations are “out of phase” with each other, however.

Graphs of fox and rabbit populations in a predator-prey model. This is applied mathematics, because it is useful for modeling the real world. — Rabbit and fox populations over time. This behavior is typical of predator-prey mathematical models. The populations oscillate with the same period, though they are ‘out of phase’ with each other.

What does it mean for these peaks to be out of phase? Notice that the rabbit population (in red) reaches a peak (for example, at time $t=4$ ) about 1 unit of time before the fox population reaches a peak (for example, at time $t=5$ ). Something similar happens for the low points, though those are closer to being 2 units of time out of phase.

Does this make intuitive sense? When the rabbit population is highest, the foxes have a lot of available food, so the population of foxes is still growing. On the other hand, when the fox population reaches its peak, there are so many foxes eating rabbits that the rabbit population is going down.

When the rabbit population is lowest, the foxes don’t have much to eat, so they are dying off. On the other hand, when the fox population is lowest, the rabbit population is growing because there are not many foxes to eat them.

You should not expect such a model to be perfect. At best, it is a very rough reflection of reality. For example, it ignores complicating factors such as the abundance or lack of food for the rabbits. However, it is a starting point.

Pure Mathematics

How do pure mathematicians, those who like to do mathematics for its own sake, view calculus? Here is a quote from the lecture:

Calculus gets a new name, ‘Analysis’, and consists of structures and methods to prove mathematical facts in rigorous ways using deductive logic. These methods and facts are useful to others, but are of interest in and of themselves to pure mathematicians. The foundations go all the way back to the nature of numbers and arithmetic.
Bill Kinney, Calculus 2, Lecture 1: Course Structure, Nature of Applied vs Pure Mathematics, Mathematica Demonstrations, Fall 2016

One big topic of concern within pure mathematics is the nature of numbers. At the most foundational level, pure mathematicians ask and try to answer the question: what are numbers?

Does the Square Root of Two Exist?

This is more difficult than you might expect. We might wonder, for example, whether the square root of 2, written $\sqrt{2}$ , actually exists. That is, how do we know there is a number $x>0$ , representing, for example, a length, with the property that $x^{2}=2$ ?

Does $x=1.4$ ? No, because $1.4^{2}=1.96\not=2$ . Does $x=1.41$ ? No, because $1.41^{2}=1.9881\not=2$ . How about $x=1.42$ ? No, because $1.42^{2}=2.0164\not=2$ . How about $x=1.414$ ? No, because $1.414^{2}=1.999396\not=2$ .

In fact, we can continue playing this “game” forever and ever, adding more and more decimal places of accuracy that get us closer and closer to the “true” $\sqrt{2}=1.414213562\ldots$ . But we would never “get there”.

In fact, if $\sqrt{2}$ is assumed to exist, it can be shown to NOT be a rational number (it is NOT a ratio of two whole numbers). So its decimal expansion goes on forever and ever without a repeating pattern. No being but God knows all the infinitely-many digits of $\sqrt{2}$ !

So, I’ll ask it again: how do we know that $\sqrt{2}$ actually does exist? We can’t possibly square its infinite decimal expansion and show it equals 2 exactly!

Here at Infinity is Really Big, I wrote a blog post titled “Does the Square Root of Two Exist” to answer this question. Suffice it to say here that the answer is pretty challenging to understand. I would encourage you to attempt to understand it anyway.

0.99999… = 1

Here’s another interesting thing to ponder about real numbers. Not all real number decimal expansions are unique. For example, point-nine-repeating equals one (0.9999… = 1).

Why? There are numerous arguments for it. One reason is that if they are different, there would be a number between them (their average, for example). However, there is clearly no number between them (and, no, there is no such number as 0.9999…..5, where the “5” occurs at the “end” of infinitely many “9’s” — such a “decimal expansion” is meaningless).

In fact, any real number whose decimal expansion “terminates” has more than one decimal representation. For example, $\frac{1}{4}=0.25=0.249999\ldots$ .

Prime Numbers and the Twin Prime Conjecture

The theory of whole numbers (a.k.a. integers, when negatives are included) is also full of inherent interest. Especially of interest to many people is the theory of prime numbers and prime factorization.

For example, it has been known (and proved) for thousands of years that there are infinitely many prime numbers $2, 3, 5, 7, 11, 13, 17, 19, 23,\ldots$ . The various proofs of this fact are considered to be particularly beautiful, in fact.

But did you know that, to this day, no one knows whether there are infinitely many pairs of “twin primes”? These are pairs of primes that differ by two: 3 and 5, 5 and 7, 11 and 13, 17 and 19, 29 and 31, etc…

The conjecture that there are infinitely many pairs of twin primes seems to be true (it is the content of the “twin prime conjecture”), because we keep finding more and more such pairs. But no one in all history has every found a logical argument for its truth or falsity.

If you do, you’ll be world-famous!!

Conclusion

I hope this post has inspired your interest in both applied and pure mathematics. In the posts and lectures of this series to come, I will get into the details of the subjects in Calculus 2.

I chose to focus on the nature of applied vs pure mathematics in this first post and in lecture 1 because Calculus 2 is a subject in which both parts of mathematics play very important roles.