How Free or Constrained We Are in Applying Mathematics to Real and Fictional Worlds

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There are several ways to “apply” mathematics, or more importantly, to obtain numbers and work with them. Here they are:

1. If you have 3 apples and you say that each one costs $2, how much money you will earn by selling all of them?
2. Measure the distance.
3. Physics laws, initial conditions, results of formula calculations.
4. Harry Potter or Hunger Games story.
5. Mathematical axioms.

The first example arbitrary associates a number with an apple. No measurements or physical law is required. Economical exchange and the quantity to exchange are solely based on human values. The selection of the price is usually how trader perceives the value, and it can be subjective, yet that subjectivity is the only way to go when agreeing on an exchange price of goods.

The second example is selective counting. The same way we define apples and want to count apples (and no other things), we decide we want to count how many of some unit length are in the given distance. We have in advance a unit length, say inches, or meters, and then a distance we want to measure. Note here that measurement is not a part of mathematics. Precision of a measurement is also outside mathematics. It is a method in the realm of physical world, how to count something, in this case length or distance. Measurement implies only that we agreed what and how to count, how to obtain numbers that will enter the numerical world of mathematics, often as pure starting points. Let’s say, we have 1m as a unit, and the length between two tables in a coffee shop. After the measurements we found that the distance between the tables is 2.3m

Let’s compare first and second example. First one has arbitrary numbers put together and multiplication selected as math operation due to need to sell the apples. Hence, math will see: 3, 2, multiply. 3 x 2 = 6. In the second example math will see: 1, and the count 2.3. That’s it. The difference in these two examples is that in the second one you are constrained by the physical distance you want to measure. You also specified the unit of length, 1m. Once these two things are specified, the measurement is not arbitrary. But, note, technically, it was arbitrary which units of length you have selected, and, in a sense, it is arbitrary which distance you want to measure. However, once this is established, selecting numbers is not arbitrary any more, it actually depends on the length and measurement unit.

The third example is a firm physics law. A physics law specifies what needs to be counted and then, very important, the relations between these counts. Are they are to be added, divided, multiplied, etc.. Note how you, in a physics formula, you still deal with counts, but you keep track aside what are those counts of. Now, in physics law, we have even less arbitrary things. It is not arbitrary anymore what needs to be counted (time, force, mass, energy, distance) but also the mathematical relations are firmly established (addition, division, multiplication etc). Interesting things is, mathematics, again, will see these quantities as given as starting point only. Specifying formula is extraneous to math.

Let’s look at  Newton formula F = ma. Virtually, no numbers are given compared to apples and price. What is given then? You are told that if you count mass and count acceleration of a body, then multiply these counts, you will get the quantity of force that is acting on the body. So, where is the freedom here, and where is the law, or constrain? You are completely free to select, arbitrary if you wish, completely up to you, a mass of a body, and acceleration. Example is, you arbitrary chose a car to drive from a dealer’s parking lot, and arbitrary accelerate when on the road, to test it. Of course, when you see other drivers driving their cars, you will have to measure their mass and measure acceleration, i.e. not arbitrary any more, it’s given by other’s driver’s arbitrary selection to you. The formula now tells you that it is the multiplication you have to perform on these two numbers to obtain the force on the car. That’s the value of the formula. A genius is required to select what to count and then to establish, discover, the relationships between these counts. Of course, the very first thing is to want to count something, as oppose to look for some other things in order to explain certain behaviour.

The fourth example, a Harry Potter story, signifies the fact that mathematics can not distinguish real from fictional world. Yes, math can be applied to real life and quantitative relations within physical world are important. But, math deals with numbers you supply to it, and with numbers only. It can not distinguish where these numbers are coming from. It is you who use the math and keep track where the numbers are coming from. Have you really counted, measured something, or just say you think that the number should be like that, math doesn’t care. If Harry Potter flies on his broom with the speed of 5 m/s, what is the distance he will advance after 7 seconds? The result is 5 x 7 = 35. He will fly over the distance of 35m. Note how math did not really care how you specified the numbers. Harry Potter’s broom or a rocket, or from the fictional world of Hunger Games, math does not know where the starting numbers and operations are coming from.

The fifth example tells you that, for math, it is sufficient, just to say, hey, here is the number 5, here is the number 7, do the multiplication and give me the result back. This is axiomatic approach and it is called pure math. Axioms of mathematics, more or less, tell you that the counts and operations are already available, you can pick them and define any sequence of operations on them. This is the fifth way you can obtain and play with numbers. No rockets, no apples, no currency, no physics laws, no length measurements are required to deal with numbers and hence to develop mathematics. Counts are there and you deal with them. One of the values of pure mathematics is that counts, numbers themselves and relations between numbers and sets of numbers, have some interesting properties, and results of that investigation can be used when you obtain numbers by any of the previous four ways, because the results will be applicable in each of them. Like, even if you don’t know what is counted, you will know that 3 + 5 = 8, in pure counts, pure numbers. It is a generally applicable result. For math, only the numbers you provide to it exists. You say here is the number 3, here is the number 5, add them. If this comes from any of the previous four examples, it is you, and not math, who will have to keep track what is counted and why you have chosen addition and not, say, division.

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[ Harry Potter, Hunger Games, applied math, applied mathematics, math and real life, real world math, examples of natural numbers,  counting, number concept,  ]

How to Teach Your Kids and Yourself to Think More Freely About Math and Real World Math Applications

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When I was a kid, I remember, I was often asked, during a math chat, “How many apples are there?” or “If you subtract 3 apples from 10 apples how many apples are left?”. Or, "Peter has 5 apples. How many apples he will have if he gives Susan 3 apples"? Then, there were many more examples of counting objects.

I am sure that many of us have had this kind of first encounters with mathematics.

When I was 4th grade, an interesting idea came to my mind. Why do we need to know concept of an apple in order to subtract 3 from 10? How and when we become aware that an object is an apple, and not, say pear? And what if Peter doesn't own an apple? Isn't the ownership a legal concept? Does Peter really own an apple? Then, if Susan is really hungry and Stephen is not, then giving Susan 3 apples is way more important than actually counting apples. Also, turning left and drive 2km is not the same as turning right and drive 2km. Yet, left and right were not part of math. They are not quantities, yet they are so important. Those were my thoughts when I was in 4th grade..

After a while, and after many examples involving apples, cherries, cars, marbles I have been shown, it became clear to me that 3 and 10 are separate, abstracted concepts of numbers, and hence they can be dealt with independently of any apples, oranges, pears, and other objects used as counting example. Teachers are constantly talking about universal nature of math and even about pure math. Yet, somehow, teachers were always sticking to real world examples when explaining math concepts. At the same time they were talking where the math can be applied, but miss to inform us what is it we are actually applying, i.e. pure math. Fortunately, I have realized that math can be independent discipline and that the math tasks and problems can be tied to the numbers only, without any real world examples. But, that was a trouble, when I looked how the math is taught. I  did not talk to teachers about that, they were very strict, and I was afraid to raise any issue. Yet, I was the best student in primary school, in my class, and one of the best in school, in math. And the issue was why would you expect from kids to know non mathematical concepts in order to add two numbers? Why kids should know social structures and social relationships, legal concepts and then learn math from those examples? Math can be taught without them. The best approach is to say that math can be motivated by real world affairs and also it can be developed from math axioms.

Then, there was another puzzle. The opposite (of knowing non mathematical concepts) was also true. As a kid, I realized that I didn't know how the automobile engine works, yet I can count engines. Then, in 7th grade when physics kicked in, we have been counting objects that we only vaguely knew what they are, like, electrons, atoms, waves, particles. To me, if math says that definition what actually belongs to a set, in order to be counted, must be clear and non ambiguous, it should be followed. Yet, we do not know what is electron, but it was apparently sufficient to stick to "it looks like it's a particle, but, it can be a wave too, and it may appear and disappear time after time". So, apparently we can define set "of objects we don't know exactly what they are" and have that count too.

Throughout “schooling” teachers (not all), most likely, failed to educate us what math really is. What is left from so called math education, for many of us, is a bitter, frustrating, scary feeling, tight test schedules, unresolved problems, and many confusion about math concepts. Moreover, teachers were running through material without giving any deeper insights.

And that view I questioned through primary school. Later, it became clear that math can be developed as a discipline without waiting for real world examples. However, it is true that certain directions in math were motivated by physics problems, like calculus, but still, calculus can be developed from scratch without physics input. You can not expect from kids to fully know concept of force, energy, electrical field, in order to introduce calculus. Calculus can be introduced way earlier, right after irrational numbers.

Well, I did have my personal “fight” with math, and I think I won! And this happened relatively early, around 4^th grade, when I realized that you can work with numbers independently from any real world examples, but I did question how we define sets, and can we really always define clearly what belongs to a set. Later, that proved to be related to Russell's paradox. That was, at least for me, key point when I started to like math, to appreciate its elegance, its independence as a discipline. After primary school I enrolled in Mathematical Gymnasium, then later, finished my Bachelor's Degree in Electrical Engineering (later redefined as Masters Degree, Bologna, around 2000), studied Astronomy along the way, and have exciting career in these fields.

Here, I would try to keep examples simple, intuitive, and I would always emphasize that particular mathematical concept can be defined and derived without real world examples, but can be motivated by them.

In this article I want to explain a method how to teach kids, or primary school students, or anyone who is interested to understand math better, to think more freely about math, and to see that math is a game, puzzle game sometimes, whose variations can be created by you as well. You do not need to wait for someone else to give you mathematical problems and tasks. I hope that, after you read this post, you will have an unstoppable desire to revisit other parts of math, and to go on your own journey which will also include YOU creating new mathematical problems to solve.

Let’s start. Counting objects is the core of mathematics. Let’s say we have two kids, Peter and Stephen, who wants to learn math. Usually the learning starts with showing some objects, say apples, and asking kids to count them. Then, it comes addition of apples, subtraction, and perhaps later multiplication and division. Now, after, say one hour or so, or after even a few days, Peter and Stephen will always expect to see objects then to count them. That’s where the word “given” in math comes from, or even word “suppose”. Peter or Stephen will always expect to have a problem in front of him, to be “given” that he needs to solve. Needless to say, kids may be, at this point, scared to asked where these problems come from? Who “gives” them? Here is the answer.

It would be nice to show to kids that 2 + 3 = 5 is independent of any objects counted. This can be done by going through a number of examples with apples, oranges, cars, CDs, pencils, even time, like hours and seconds, and then show separately that 2 + 3 equals 5 no matter what are you counting. You can reuse this numerical answer for any objects you may be counting in future. At this point kids should be aware of numbers as separate concepts. They have to be aware that numbers can exist without any real world examples. They have to be aware that they can do two things with numbers. Deal with them as pure numbers, like 2 + 3 = 5, and, obtain these numbers by counting objects. It is very important to explain this fact to kids.

And, now, here is the main step!

Get a bunch of objects together, say 30 apples, in a basket. Ask Stephen to go outside of room. Then, tell Peter to specify how many apples he wants to be taken of the basket and put on table. Suggest to Peter that the number he specified is completely arbitrary, it’s up to him to tell the number. Or, he may have his own logic to do that. Let’s say Peter chose number 5, for whatever reason. Then 5 apples will be taken from the basket and put on table. Ask Peter again to chose another number. Peter says 3 and he takes 3 apples from the basket to table, in another group.

Now, explain to Peter how his arbitrary, starting selection, a starting number of apples dictated how many pencils will be counted. It also dictated the two numbers Stephen will be dealing with. Also, explain to Peter that it is him who specified how many apples were to be counted. There was no pressure from anyone else, no “given” number of apples.  It was Peter who “gave” the numbers.

So, in mathematics, not only that you can count some objects and tell how many of those objects are there, but also you can use a pure number to start with, and tell how many of things is to be counted. This count, specified by you, can be a “given” starting point for someone else’s calculation! Please feel free to select any “starting” number again and play with mathematical operations, addition, subtraction etc, using other numbers by your choice. The numerical results you get is important knowledge, and can be reused in real world applications.

Let’s say, after a while, you have a few pages with numbers chosen by you, and a number of results obtained by multiplication, addition, etc. You may want to sort them, as you wish, maybe by magnitude, smaller numbers first, bigger numbers later. You may ask yourself, how this can be applied in real life? But you can not go from math back to real objects because, as we have seen, when you write 2 + 3 you do not know what is counted. Can be anything. However, there is a way for real world application! If you start with “IF I counted CARS, and get 2, and then IF 3 more cars came by” what is the result? Hence, once you start keeping track WHAT you have counted, and WHY, you can use your mathematical results to match the numbers. Note here that math does not see where the numbers 3 and two came from or why. You can count cars passing by, you can count cars on parking lot, or these can be numbers you tell car dealer to move cars in the garage. Math can’t see that. Math only see you provided to it number 2 and 3 to add. It is you that you have to carefully keep track what did you count and why and where the result will go.

Back to Peter and Stephen. Now, it is time to ask Stephen to specify how many apples will be taken out of the basket while Peter is out of the room. Now Peter will have “given” numbers to add, and Stephen will become aware that it is him who can specify these initial conditions for counting.

These exercises should be continued with measuring and specifying length. First Peter will be the one who will specify the distance between two objects in the room, by moving those objects apart. Then, Stephen will come in the room and measure the distance. Second exercise will be for Stephen and Peter, to specify the distance and then measure and move two objects to match that distance. It is of fundamental importance here to make Peter and Stephen aware that it’s them who specify the initial numbers, counts, and then the change is done in real world by their own specification! During these exercises kids will start thinking more freely about mathematics, and starting creating their own mathematical problems, instead of only dealing with what is “given” to them.

In next post I will write how we separate and connect two logical worlds. First one is how we create rules and keep track what and why is counted and the second world is the world of mathematics that will deal with these numbers. With the second world we are already familiar at this post.

Real World Math Applications

Peter may want to ask “Ok, I have done all this math calculations, and what I get is a number as the result. Where is the real world application?”

Here is the answer. Let’s say Peter obtained number 5 as counting pencils. What now? What possible application can be created using number 5 and pencils?

Here is how the real world examples can be created. Make this sentence:

“If I have 5 pencils AND IF 'something else A' THEN 'something else B .”

Note first “AND IF”! Following that it can be a true statement about anything you can imagine that can depend on the number of pencils counted. You have a true statement that you counted and obtained 5 pencils. Then, you may say, IF I have 5 pencils AND IF it’s sunny outside THEN I will decide to go out and draw landscapes. Note how the number of pencils influenced your decision to go outside and draw landscapes. Please note how open ended logical connective “AND IF ” can be related to any specified fact YOU have in mind! The very true and accurate calculation of 5 pencils can lead to an action that you can attach to this number! It is really up to you or up to your analysis what will depend on number five. It can be your creative new statement, new initiative or new decision that will start once you have 5 (and not 3 or 6) pencils in. That connective on the other side of "AND IF" can be another count, obtained from counting other objects too.

Now, teachers will rarely ask you to do this! Teachers will just stop after you calculated 5 (pencils or other objects) and leave you hanging there asking “what for?” and “what now?” I want to change that. I want to show you how the math calculations are actually applied, used, and utilized in the real world.

[ applied math, applied mathematics, learn math, math, math and real life, math concepts, math education, math tutoring, mathematics ]

How Mathematics Can Be Used in Real Life

You can download all the important posts as  PDF book "Unlocking the Secrets of Quantitative Thinking".

How to distinguish between pure math and applied math? How math can be applied to many different fields? How we can use math in real life? Why math looks so complicated? Where all those volumes of math come from? Can I revisit my math from primary and secondary school and finally understand what is it about? Can I learn math now? How I can teach my kids or students effectively primary and secondary school math?

I will be trying to answer these questions in the next several posts.

Here are more links you might like as well:

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• Math and its relationship with real world
• How math can be applied to so many different fields?
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[ applied math, applied mathematics, axioms, math and real life, math education, math, mathematics, physics concepts, tutoring, calculus, school, education ]