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

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

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.

You can download this post as an article in a PDF file format by clicking on the picture below or from here.

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

From Real World Math Applications to Pure Math and Back

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

Can you go from pure math to real world math applications? The answer is no. You can not start with math only and then “apply” to real world scenario. It is, however, possible to use math in real world (even in fictional world, like in Harry Potter movies), practical applications, but the path and direction are different and needs to be clarified.

The reason you can not go from pure math to real world math application is in the fact and in the nature, definition (in a sense) of a number. A number is obtained as an abstraction, a common property for many objects. By this very definition, because it is abstracted from counted objects, because it is, now, a separate concept, representing a pure count, without any object associated to it, you can not tell, by looking at the number only, where it came from, what and if anything has been counted to obtain that number. In other words, a number does not carry any information about any object extraneous to mathematics! Hence, you can not say, just looking at the number only, or at the sequence of math operations on numbers, what its or their application, in real world, may be. Newton did not learn calculus first, then applied it to the gravitational problems! Quite the opposite happened. Newton was dealing with non mathematical objects and relationships like apple falling from the tree, Moon orbiting Earth, and other body motions. Unless they are quantified, these are not mathematical objects nor relationships. If they were, then you would see theorems in math books proved by apples, Moon, speed, etc. but it is not so. Math theorems are stipulated and proved using only mathematical objects, like numbers, sets, set of numbers, or other mathematical theorems and axioms. So, let make that clear, Newton first dealt with physical objects and only then he invented calculus. So, when someone tells you you will learn math then apply it, it is not quite true.

When you deal with a number, you deal with an abstracted common property, a separated concept abstracted from all the objects whose count it represents. The very moment you start adding 2 apples and 3 apples you are doing two distinct steps. First one is recognizing that the object to be counted is an apple. That recognition process, a categorization, that you are looking or holding an apple is outside mathematics, since you can be counting apples, pears, cars, books, cups. This recognition is a focus of research of cognitive science, psychology, biology, color research, even socal sciences. The second step (which is, actually, common to counting all those objects) is dealing with numbers 2 and 3. Even if you may not notice, when dealing with 2 and 3, you are dealing with counts that can represent not only apples, but millions of other objects that can be counted, or put in sets, then counted, to obtain 2 and 3. Hence, the result you obtained for 2 and 3 apples, i.e. 2 + 3 = 5, can be used in ANY other situation where you have 2 objects and 3 objects and you want to add their counts. You right away invented and used "pure" mathematics when counted these apples. It is this universality, the common numeric property of counted objects, that gives mathematics ability to be a separate discipline, to deal only with numbers. While to us, and to the field that uses quantification, is very important what, when, why and where something is counted or measured, to investigate only the counts' properties is the task of pure math. Pure math does not care where the numbers or counts are coming from. It is very similar when we create a set of any objects (of interest), but we are only interested how many elements are in the set and not which objects are part of the set (that information, which objects are elements of the set, we keep track of on a separate sheet of paper) ! Math knows and should know only about numbers and sets of numbers. Notice how math may be "motivated" by counting apples, but, the result obtained, i.e. 2 + 3 = 5, can be used when counting any other objects!

Also, math can't tell real world from fictional one! Look! If Harry Potter flies 10 m/s how many meters he will advance after flying 5 seconds?

Mathematics deals with numbers and with numbers only. It does not care where the numbers are coming from (but, in the field of applied math, we do care where the numbers are coming from). Now, you may ask, how we can apply mathematics at all, if the trace what is counted is lost in this abstraction, in this definition of pure number? Well, here is how.

It is true that pure mathematics deals with numbers only and, of course with mathematical operations on them. We have abstracted, separated a concept of number from all possible real world objects that might have been counted. Thus, when you say 5 + 3, you right away know that the answer will be 8. No real world objects are mentioned nor even thought of when we did this addition. We just selected two numbers, and decided to do addition (we could also decide to do subtraction or multiplication). Now, how then we can apply math to real world if we don’t have a trace of what is counted? There is a way! When we want to “apply” or rather, use, math in real world, we will drag the names of objects counted into the math! We will keep track of numbers obtained, to know where they come from, which objects’ counts they represent. How we do that? We will add  a small letter, or abbreviation, or a word, name, just beside the count to tell us what we have counted. For instance, we can write numbers, 3, 5, 7, 10 after we counted (or measured) something. In order to keep track what we have counted we will add small letters right beside the numbers, like 3m, 5m, 7seconds, 10apples. Now, very important thing. These added letters do not represent math. They are for us to keep track what is counted. Unfortunately, frequently, this is all mixed up and students are often told they are doing math even when they describe what are they counted, why (to give 10 apples), where (apples from the basket, he went 3m downhill, then 5 meters uphill,), when (7seconds ago, not after). All this reasoning, descriptions, units, abbreviations, meters, seconds, apples, ago, before, after, downhill, uphill,  DO NOT BELONG TO MATH. Why is that? Because, if you look in any pure mathematical textbook or book, you will clearly see that no theorems are stated or proved by mentioning apples, meters, seconds, pears, etc. All the theorems are proved strictly in terms of mathematical objects, numbers, sets, set of numbers, using other theorems and axioms. No outside objects or descriptions, like apples, cars, downhill, uphill, will ever enter a mathematical theorem or its proof.

Now, when we distinguished what is pure math and what is applied part of it, we can make more interesting and significant conclusions. Pure mathematics deals with numbers and numbers only. Since number 3, say, can represent an abstracted count of so many, many objects, wouldn’t be interesting to have its properties investigated? It looks like there is some value in the fact that one concept, a number, stands for counts for so many objects. We can compare number 3 with other numbers. We can say which numbers are greater than or less than other numbers. We can multiply them and see what numbers we are getting. And all the time we deal only with numbers. The value is, if we get some interesting result for a certain number or numbers, from our “pure” number investigation, we can use that result for all those examples in real world. That’s the value of applied math. But, in order to use it, say, in order to use 5 + 3 = 8, we have to make a match between pure math numbers and real world counts. Hence, IF we count say, CDs and IF we get 5CDs, and, again we count another set of CDs just mentioned. This is a simple example, but there are more complex mathematical results where we don’t need to reinvent the wheel each time we get the real world count, but instead we take advantage of ready to use mathematical result, procedure, theorem, solution.

Mathematics does not see the reasoning from other worlds. Math will see number 6 (given, or picked), math will see number 10 (given, or picked), and math will see the selected, required operation, addition (it could be subtraction or division too). Does it mean that these numbers, 10 and 6 came from thin air to mathematics? They came from counting apples, but, where are the apples then? The point is, remember, when we said that numbers, in pure math, are abstraction for all the objects they can represent count of. They are not from thin air, they exist in our math as our starting point. Our pure math has already numbers available for our use, 1, 2, 3,…10, …, etc.. How? We did not even need to find objects to count to obtain different numbers. We can start with 1, then add 1 to get 2, then add 1 to get 3, etc. That’s why we have numbers already available for us. We only pick them, and do the operations. Math, for that matter, doesn’t need to know if we counted apples, or pears, of chewing gums. It is enough for math to tell there is number 10 and number 6 and that we want to add them. It is us who will keep track why we counted (because Peter was hungry),  what we counted (apples, they are edible), when we counted (in the evening, when was the time for dinner).

How we are allowed, at all, to go from non-axiomatic worlds, physics, economics, finance, to, so strictly defined, axiomatic world of mathematics? Apparently, mathematics does not care whether the problems come from axiomatized system or not! And that tells us that mathematics can not correct logical steps or see the flaws in the system it proudly claims it models. Assumptions coming from non-axiomatized fields, like physics, economics, finance, and into a strict axiomatic system, like mathematics, can produce results that can wreak havoc back in the field where the mathematics is applied.

You may ask, at this point, how we can mix these two logic worlds. One world appear to be very fluid, the real world, with objects selected, of any type, and any kind of relationships. On the other hand we have mathematical logic world, where we only deal with numbers, or sets of numbers only, and with what appears to be quite precise rules, axioms, logic, and well defined sequences of math operations (of addition, subtraction, multiplication, division). Is there a logic that will merge and connect these two worlds? YES! You can mix these two worlds, but you have to be very careful with the World # 1, the real world's objects and scenarios. Your logic there has to be correct. Then, you can use logic used to link these two worlds, and it's  is the logic already familiar to you, but with new domains of application. How it is done? Here is how. Let’s say that logical statement from the first, real world, is labelled as “p” ,  and logical statement from the second, mathematical logic world is labelled as “q”, then we can create a new, logical statement IF “p” AND “q” THEN “s”, or in shorter notation, p ^ q => s. I am sure you are familiar with this logical statement. Here is example. Let p = “Peter is hungry” and let q = “there are 10 apples”. Then we can form a statement “IF Peter is hungry and IF there are 10 apples, THEN he will get 6 apples”.

Mathematics is defined not by objects it counts, nor by reasons or logic why those objects are counted, but with concepts used to define matheatics axioms and to define proofs of mathematical theorems. 

[ applied math, applied mathematics, learn math, math applications, math concepts ]