Monday, June 27, 2011

From Real World Math Applications to Pure Math and Back

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 ]

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