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,  ]

The Definition of Number

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After a number of years dealing with mathematics in your primary, secondary school, there still may be a question what the number is. Moreover, unless you are a professional mathematician, with PhD in your resume, I can safely assume that your frustration and fear of mathematics is still present.

What is a number? Seemingly popular approach I am using here does not reduce the strength and significant clarity of the definition. Bear with me, and listen carefully :-) You may find out many interesting things!

Here is the clearest approach to defining number.   

Number is a count.

I will repeat again, number is a count. The purity and significance of this definition can not be emphasized more. While it is simple, it conveys many more important messages than other definitions and approaches you may have read about before. One of the most important message, in my view, of this definition is that it implicitly specifies what you can do with counts. Knowing what you can do with counts, you actually filter out all non mathematical concepts that may be mixed during "bad" mathematical lectures over the years. Also, thinking of numbers as counts, you define what pure mathematics is about! And that can help you answering the questions how math can be applied (about what "applied" means we will see later) in so many different fields, and what differentiate pure and applied math.

What you can do with counts is what mathematics is all about! So, what can you do with counts? You can add them, subtract them, divide, multiply. You can, then, do any number of these operations in any sequence you want. No apples, pears needed to do that! Count 5 is a universal count. It can come from counting apples, pears, cars, atoms, money, steps, seconds. That number 5, count 5 is a universal thing for all of them. While you can eat 5 apples, drive 5 cars, wait 5 seconds, with count 5 you can not do that. But you can add another number 5 to it! or deduct count of 3 from it. Or do any other "counting" operation! Note very important thing -> how you call these counts, i.e. are they integers, positive, negative, odd, even, rational, are just labels we attach and associate to the concept of a count! There is only count we are dealing it all the time.

Looking at count 5 only, i.e. number 5 only, you can not tell whether it came from apples, pears, or counting seconds. So, how you will differentiate count 5 of apples and count 5 of seconds if count, number is actually so universal concept? There is no other way than to keep track by yourself what you have counted. Technically, you will write a small letter beside the number, beside the count to remind you what it is a count of! Or you can remember that in your mind. Whatever works for you.

So, whenever you read about those exotic mathematical concepts, like matrices, determinants, integrals, equations, algebra, arithmetic, you will know one thing - it is all and only about pure counts we have just talked about. There is nothing else there. For instance, a matrix is a set of counts arranged in rectangular fashion on page. But, you do not need even that rectangle. You can just imagine in your mind the same set of counts and differentiate between them in any way you want. It just happened that it was convenient to write those numbers in a rectangular grid on paper! It was just convenience.

The logic, the reasoning in the world of counted objects is separate from the logic that deals only with counts. You can investigate properties of counts only, completely independent from the real world objects they might represent count of. This is the topic of mathematics. To find what is true about counts i.e. numbers. Hence the proof. But note here, we are really interested in counts' characteristics, no matter which objects have been counted! What these characteristics can be? We can have odd numbers, or even! We can have prime counts. Some counts can be divided by others, while some not. But, the numbers are numbers, it is us who give them names to keep track of some of their properties or just we want to deal with some numbers while leaving other numbers alone!. Naming numbers is not a mathematical operation. It just help us describe, label numbers, counts, we want to deal with.

We can compare the way we can deal with numbers to the way a sculptor deals with clay. It is only the clay that he works with and nothing else. Clay! But, what clay represents when it is shaped, what sculpture represents is not about clay! The motivation how the sculptor will twist, press, mold, shape clay is outside clay's world. Same in math! The reason why we add, subtract, divide, or even select numbers to deal with, frequently are outside mathematics! The motivation can come from us buying CDs or from an economist measuring supply and demand, or police measuring speed of the car.

Now, back to sculpture again. The sculpture can represent anything. The similar thing is with numbers. In mathematics we are dealing with numbers only, the same way sculptor deals only with clay! But, if we want to interpret math results and use math in some other fields then we will have to keep track what we have counted, measured, keep track which objects are numbers, counts associated with! That would be called applied math. And, again, the numbers can represent count of many, many different things.

In developing and understanding a subject, axioms come late. Then in the formal presentations, they come early. - Rueben Hersh. #math

The view that mathematics is in essence derivations from axioms is backward. In fact, it's wrong. - Rueben Hersh http://books.google.ca/books?id=R-qgdx2A5b0C&printsec=frontcover&dq=what+is+mathematics+really&hl=en&sa=X&ei=6O9BT5RIqdfRAcyqteMH&ved=0CDsQ6AEwAA#v=onepage&q=what%20is%20mathematics%20really&f=false

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Concept of a Set and of a Number

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For instance, let's take a look at the cars on a highway, apples on a table, coffee cups in a coffee shop, pears in a basket. Without our initiative, our thought action, will, our specific direction of thinking, objects will sit on table or in their space undisturbed and unanalyzed. They are apples, cars, coffee cups, pears. But then, on the other hand, we can think of them in any way we wish. We can think how we feel about them, are they edible, we can think about theory of color, their social value, utility value, psychological impressions they make. We can think of them in any way we want or find interesting or useful, or we can think of them for amusement too. They are objects in the way they are and they need not to be members of any set, i.e. we don’t need to count them.

Now, imagine that our discourse of thought is to start thinking of them in terms of groups or collections, what whatever reason. Remember, it's just came to our mind that we can think of objects in that way. The fact that the apples are on the table and it looks like they are in a group is just a coincidence. We want to form a collection of objects in our mind. Hence, apples on a table are not in a group, in a set yet. They are just spatially close to each other. Objects are still objects, with infinite number of conceptual contexts we can put them in.
Again, one of the ways to think about them is to put them in a group, for whatever reason we find! We do not need to collect into group only similar objects, like, only apples or only cars. Set membership is not always dictated by common properties of objects. Set membership is defined in the way we want to define it! For example, we can form set of all objects that has no common property! We can form a group of any kind of objects, if our criterion says so. We can even be just amused to group objects together in our mind. Hence, the set can be specified as “all objects we are amused to put together”. Like, one group of a few apples, a car, and several coffee cups. Or, a collection of apples only. Or, another collection of cars and coffee cups only. All in our mind, because, from many directions of thinking we have chosen the one in which we put objects together into a collection.

Without our initiative, our thought action, objects will float around by themselves, classified or not, and without being member of any set! Objects are only objects. It is us who grouped them into sets, in our minds. In reality, they are still objects, sitting on the table, driven around on highways, doing other function that are intrinsic to them or they are designed for, or they are analyzed in any other way or within another scientific field.

Since, as we have seen, we invented, discovered a direction of thinking which did not exist just a minute ago, to think of objects in a group, we may want to proceed further with our analysis.
Roughly speaking, with the group, collection of objects we have introduced a concept of a set. Note how arbitrary we even gave name to our new thought that resulted in grouping objects into collections. We had to label it somehow. Let's use the word set!

Now, if we give a bit more thought into set, we can see that set can have properties even independent of objects that make it. Of course, for us, in real world scenarios, and set applications, it is of high importance whether we counted apples or cars. We have to keep tracks what we have counted. However, there are properties of sets that can be used for any kind of counted objects. Number of elements in a set is such one property. If we play more with counts and number of elements in a set we can discover quite interesting things. Three objects plus six objects is always nine objects, no matter what we have counted!  The result 3 + 6 = 9 we can use in any set of objects imaginable, and it will always be true. Now, we can see that we can deal with numbers only, discover rules about them, in this case related to addition that can be used for any objects we may count.

Every real world example for mathematics can generate mathematical concepts, mainly sets, numbers, sets of numbers, pair of numbers. Once obtained, all these pure math concepts can be, and are, analyzed independently from real world and situations. They can be analyzed in their own world, without referencing any real world object or scenario they have been motivated with or that might have generate them, or any real world example they are abstracted from. How, then, conception of the math problems come into realization, if the real world scenarios are eliminated, filtered out? Roughly speaking, you will use word “IF” to construct starting points. Note that this word “IF” replaces real world scenarios by stipulating what count or math concept is “given” as the starting point.
But, it is to expect. Since a number 5 is an abstracted count that represents a number of any objects as long as there are 5 of them, we can not, by looking at number 5, tell which objects they represent. And we do not need to that since we investigate properties of sets and numbers between themselves, like their divisibility, which number is bigger, etc. All these pure number properties are valid for any objects we count and obtain that number! Quite amazing!

Moreover, even while you read a book in pure math like "Topology Fundamentals" or "Real Variable Analysis" or "Linear Algebra" you can be sure that every set, every number, every set of numbers mentioned in their axioms and theorems can represent abstracted quantity, common count, and abstracted number of millions different objects that can be counted, measured, quantified, and that have the same count denoted by the number you are dealing with. Hence you can learn math in the way of thinking only of pure numbers or sets, as a separate concepts from real world objects, knowing they are abstraction of so many different real world, countable objects or quantifiable processes (with the same, common count), or, you can use, reference, some real world examples as helper framework, so to speak, to illustrate some of pure mathematical relationships, numbers, and sets, while you will still be dealing, really, with pure numbers and sets.

There may be, also, a question, why it is important to discover properties of complements, unions, intersections, of sets, at all? These concepts look so simple, so obvious, how such a simple concepts can be applied to so many complex fields?

Let’s find out! Looking at sets, there is really only a few things you can do with them. You can create their unions, intersections, complements, and then find out their cardinalities, i.e. sizes of sets, how many elements are there in a set. There is nothing else there. Note how, in math, it is sufficient to declare sets that are different from each other, separate from each other. You don’t have to elaborate what are the sets of, in mathematics. You do not even need to use labels for sets, A, B, C,… It’s sufficient to imagine two (or more) different sets. In mathematics, there are no apples, meters, pears, cars, seconds, kilograms, etc. So, if we remove all the properties of these objects, what properties are left to work with sets then? Now, note one essential thing here! By working with sets only, by creating unions, complements, intersections of sets, you obtain their different cardinalities. And, in most cases, we are after these cardinalities in set theory, as one of the major properties of sets, and hence in mathematics. Roughly speaking, cardinality is the size of a set, but also, after some definition polishing, it represents a definition of a number too. Hence, if we get a good hold on union, complement, intersection constructions and identity when working with sets, we have a good hold on their cardinalities and hence counts and numbers. And, again, that's what we are after, in general, in mathematics!

As for real world examples, you may ask, how distant is set theory or pure mathematical, number theory from real world applications? Not distant at all. Remember the fact how we obtained a number? A number is an abstraction of all counted objects with the same count, of all sets of objects with the same number of elements (apples, cars, rockets, tables, coffee cups, etc). Hence, the result we have obtained by dealing with each pure, abstracted number can be immediately applied to real world by deciding what that count represents or what objects we will count that many times. Or, the other way is, even if we dealt with pure math, pure numbers all the time, we would've kept track what is counted, with which objects we have started with. There is only one number 5 in mathematics, but in real world applications we can assign number 5 to as many objects as we want. Hence, 5 apples, 5 cars, 5 rockets, 5 thoughts, 5 pencils, 5 engines. In real world math applications scenarios it matter what you have counted. But that fact and information, what you have counted (cars, rockets, engines, ..) is not part of math, as we have just seen. Math needs to know only about a specific number obtained. Number 5 obtain as a number of cars is the same as number 5 obtained from counting apples, from the mathematical point of view. But, it can and does represent sizes of two sets, cars and apples. For math, it is sufficient to write 5, 5 to tell there are two counts, but for us, it is practical to drag a description from the real world, cars, apples, to keep track what number 5 represents.

[ concept of a set, math, math tutoring, mathematics, number concept, number definition, numbers, set, set concept, sets and numbers, tutoring ]