Thursday, January 20, 2011

How to Easy Understand the Role of Mathematics in Physics, Economics, and Other Fields

Where the ideas in mathematics come from? Where the inventions come from? What is the reason mathematics can be “applied” in so many fields?

Mathematics does not care where the numbers come from. Fundamental axioms of mathematics tell you that number exists a priori, it’s given, and that you don't need to prove the existence of a number in mathematics. Also, roughly, Zermelo-Fraenkel (ZFC) axioms, in essence, are saying that you are allowed to do the basic operations on counts, numbers. The only reason why these statements are called axioms is that the concepts of set and operations on a set can not be defined with anything INSIDE mathematics. All other concepts in mathematics are defined using sets, but the opposite is not true.

You can read more "How math can be applied to so many different fields and how we can use math in real life".

Again, math doesn't care where the numbers come from, while you DO CARE very much, if you are coming from physics, economics, applied statistics, and other fields with math in it! Look at the measurements. You may measure the distance, time, volume, number of cars, apples, atoms, number of CDs in your collection, you can count number of songs in your iTunes collection, number of dollars in your bank account, the price when you purchase something. What do you count, and why, i.e. any motivation why would you be engaged in counting, defines the area, the domain of business, science, that needs mathematical analysis in it. For instance, in physics, you count physical properties and you discover, in addition to physical relations between the objects, the quantitative relations between them. But, all those counts and math relations are abstracted, isolated from their source once they enter mathematical world. You have to keep by yourself, the track what you have been counting and what sequence of operations you have been performing on those counts.

The numbers, as math is concerned, are created from thin air inside mathematics, in other words, in math, you are allowed, without any additional proof, to declare existence of any number. Fundamental axioms of mathematics allow you to do that. Don't forget, when you measure distance and you say that distance between two cities is 12 km, it is not math. Only number 12 is math. All other stuff is YOU keeping track of WHAT and WHY you have been counting. Think about it! Then, let’s say you add a distance to another city, say, 5km. The total distance, if you travel, is 17km. It is 12km + 5km. But in order to obtain 17 you do not need kilometers! You need only existence of 12 and the existence of 5 and the desire or decision to add, sum them. So, 12 plus 5 gives 17. Now, you go back to your reasons why you have been counting, what you have been counting. It was kilometers, km, and you associate that description (which is not math!) to the count of 17. Note how math is isolated, how numbers are independent from what is counted. You can count cars too. First you counted 12 cars then 5 cars. Total cars counted is 17 cars. But look, math again does not care where the numbers 12 and 5 came from. For math it was enough to start with number 12, no matter where it came from, and get number 5, and, again no matter where it came from, to add them together to get count of 17.

Hence, math is only a tool for your physical, economical or other areas for which you may be interested to do quantitative analysis. There is World # 1 where specific, non mathematical, relationships exist between objects and entities you want to count. For instance, in physics, YOU decide you want to measure distance and at the same time you decide to measure time as well. In another example, car speed depends on the driver decision, and the relationship between the driver and the acceleration pedal is non mathematical. You only may supply to mathematics the counts of two objects, namely 15m and, say 3 sec, to calculate speed. YOU also, and not math, decide that you may be interested in their division, and you tell math, I want to divide these two numbers. Math will do that for you. Hence 15/3 = 5. But it is you that keep tracks what has been counted and divided. And YOU can add a label, in small letters, beside the number, to remind you what you have just counted, what you have measured, what you have quantified. It will look like this, 15m divided by 3 sec gives 5 meters for each second. Shorthand notation will be 15m/3sec = 5 m/sec.

Counts need not to be produced by physical process or by physical laws to enter mathematics. They can be produced by any rule. If that rule makes some sense to you, it can be a law. In economics, when you exchange goods, or buy something, it is the market that determines the exchange price. Say one apple for two pears. Or, one DVD for $12. How much five DVDs will cost you? It will be 5 * 12 = $60. But look, in order to perform calculation, math did not care where the numbers 12 and 5 came from. Math only needed 5, 12, and multiplication request to do the job.

Let's call the world of math the World # 2. Other worlds, like, physics, economics, engineering, that can produce or generate counts, we will call them World #1. Rules within World # 1 will generate counts, sequence of required math operations, rules that will label sets, and label counts. Independence of math is quite clear. Looking at 12 + 5 = 17 you can not guess did it come from physics, from measuring distance, or from counting number of cars, or from counting how many DVDs you bought this month (12) and next month (5). Math has no idea, nor you will know if you just look at the numbers, where the 12 + 5 = 17 comes from. That is because in math you are allowed to enter any number, as a starting point, and do any math operations on them. That's math by its definition.

I want to show that there are two distinct set of rules, system of rules when you want to use math. First set of rules are the rules within World # 1, like in physics, economics, statistical biology, engineering, chemistry, trading. These rules in most cases have nothing to do with math. In economics the count of how much items need to be produced come from the consumers subjective choice, or from supply or demand. One feels he will buy today three apples, and tomorrow five, perhaps. So, these rules, i.e. why to buy so many apples, are not mathematical, but once the number is defined it can enter the math world, World # 2 and you can do calculations with it. Also, relationships within physics between say charges, distance and force, like in Coulomb's Law for electrostatic force, are not mathematical. They describe the objects that interacts in certain way between each other, in this case producing force. Now, once you supply counts for charge, once you quantify charge, and distance, you can calculate the force, but note that math does not know what you have measured, it just adds or multiply counts as you requested. You have to keep track what physical law you have been using.

The entries from Worlds # 1 to math world, World # 2 always look like an original invention, from math side, and it is true because math is not aware why you needed to, say, add or multiply two numbers!

There is a set of distinct, non mathematical rules present in each field, outside mathematics. These rules, in their own world can define the initial counts (sometimes called boundary or initial conditions) and define set of required math operations. Math world, in turn, accepts those request, does the calculation, with its own rules (theorems, lemmas etc) and gives back the result to the World # 1. World # 1 keeps track what is counted, measured, quantified, and assigns the math result to the appropriate object. And this is how you can use math in any discipline where you can and want to quantify objects and their relationships.

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