Sunday, April 22, 2012

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

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,  ]

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