WHAT IS MATHEMATICAL MODELING AND WHY IS IT DIFFICULT?
What is mathematical modeling? Why is it difficult? How to do it right? What are the benefits of doing it right? Let us investigate all these questions here.
What is a model? No , I do not mean the skinny type 🙂 , I mean the ones used in science and engineering. Here is how one dictionary I have handy defines it:
model (n): a miniature representation of something; a pattern of something to be made; an example for imitation or emulation.
I would like to mention an interesting quotation I read regarding models:
“All models are wrong but some models are useful”
The quotation above emphasizes ( rather dramatically so ) the fact that a model is not the reality , not even close to it – sometimes it is indeed necessary to remind oneself of it especially when the models are good and useful. Now what is a mathematical model? I would say it is a model as per the definition above, in which the “miniature representation” is done in terms of the language of mathematics. This brings us to the question : “What is mathematics?” Well that is a complex non-trivial question and I would postpone trying to answer it for maybe another article.
Why Do mathematical modeling?
The motivation to do mathematical modeling lies in the applied side of mathematics. Mathematical modeling is always done to solve a practical problem. This is in contrast with the so called study of pure mathematics . Hence a mathematical model , and the entire mathematical modeling exercise is useful only to the degree that it succeeds in solving the real world practical problem for which it is employed.
And now lets get to the point …. Why is mathematical modeling difficult?
Mathematical modeling is not trivial and easy , if you think otherwise I hope to convince you to my point of view by the time you are done reading this article 🙂
Let us try to explore the difficulty of mathematical modeling by means of an example. Let us come up with a simple example – a very (seemingly) simple example indeed.
EXAMPLE –SIMPLE ADDITION
Alright , here it is , answer this question : What is 2+2? I bet most of you ( I was hoping all of you 🙂 ) would answer : 4. Okay… this is the correct answer , so congratulations you passed the test ! Now let us delve deeper . Is the answer really 4 ? Sure , you would say. But just think : What makes you so sure? Well maybe it is past experience. Let us say , that in your experience you have found that when you put 2 items of anything – books let us say – at a place. And then you add 2 more books to the same collection , and count the total you find that you now have 4 books . So your experience dictates that the statement 2 + 2 = 4 is indeed correct.
Well we seem to be getting somewhere, but there is a problem with the line of reasoning given above. The problem is that we have prior knowledge of the numbers 2 and 4 in this case, even before we supposedly get positive reinforcement about the statement being correct from experience . So this is the question we come to next : just what exactly are the numbers 2 and 4 ? More generally , what exactly is a number? A number can be very casually defined to be a mathematical abstraction signifying quantity . This is not a rigorous definition but let us just live with it for the time being.
Now comes the next question : ” is 2 a number?” You would most probably say that indeed it is , but you would be very wrong! . 2 is just a convenient symbol we are using to represent a number , to get a “handle” on it for using it in mathematics and in daily life. The number it represents is a concept , an abstraction , an idea. The symbol 2 is just its symbolic representation , not the real thing itself.
Now think about this , when did you first “know” that 2+2=4 , and similar (seemingly) trivial things. I am sure most of you can point to some point in elementary school or even before that. Can you remember what was the method through which it was “taught” to you? I am almost sure it will be some variant of the “learn by heart” method . You were just asked to accept it as the truth and to remember by heart the basic additions, and use some simple rules built on top of those basic addition statements. Those basic rules were also “taught” by the “learning by heart” method and by just accepting them as being correct. All of that was of course not useless ( lest you think I am undermining it all ) , and indeed has tremendous pedagogic value .All the more so because the intellectual capabilities of a small child in pre elementary school are not enough to fully comprehend the real nature of deceptively simple concepts like number , addition and arithmetic .
Anyways , let us move on. Then as you grew up , your experience positively reinforced the “truth” of these statements that you learnt by heart earlier . But the real issue to think about here is that in this particular case , why does addition seem to work at all ? Why do abstract concepts like numbers , and operations like addition built for these abstractions seem to work perfectly in the real world? Is it because the design of these abstractions by humans was after all motivated by their real world experience and needs ? Or is it that these abstractions like numbers turn out to have a life and properties of their own as well , independent of the real physical world ? The answers to these questions are not trivial , and we will put them on hold till maybe another article on these questions.
I would like to end the mention of this example of simple addition by quoting a beautiful passage from Chapter 1 titled “The World and Theories” from the book “An Introduction to Information Theory : Symbols , Signals and Noise” by late Professor John R Pierce
This ( the arithmetic properties of numbers representing the quantity of real life objects ) , seems not to depend at all on the nature of the objects. Insofar as we can assign numbers to the members of any collection of objects , the results we get by adding , subtracting, multiplying, and dividing numbers or by arranging the numbers in sequence hold true.The connection between numbers and collection of objects seems so natural to us that we may overlook the fact that arithmetic itself is a mathematical theory which can be applied to nature only to the degree that the properties of numbers correspond to the properties of the physical world
Alright , some of you might be thinking by now , why all this fuss over simple addition , numbers and all? And how is this related to the main point of this article? Well , here is why and how:
WHY THE FUSS?
We can acquire the following lessons from our above discussion of the simple addition example:
1) Most of the basic fundamental mathematical concepts that most of us rely on in daily life and in science and engineering are built on varying degrees of intuition , reinforced by the following methods:
a) Learning by heart
b) Practical experience.
They are not explored fully in the depth and with the rigor that is required as a tool for good mathematical modeling . Paradoxically and ironically , the methodologies of a) and b) above serve their purpose well starting from the pre-elementary school days and later. They build our intuition and trust regarding fundamental mathematics, and we just “use” these concepts throughout life without giving much thought to the reality of these concepts. So far so good , but if and when we want to be good mathematical modelers , the same intuition and trust turns out to be the biggest hindrance in our way. To get to that level of proficiency of mathematical modeling , one must see beyond the surface and really gain insight into the following:
c) The real in depth meaning of the mathematical abstractions and concepts involved in the task at hand.
d) How these abstractions in c) above are “glued” to the real physical world and why exactly do these abstractions work for practical problems in which we are interested?
e) The capabilities and the limitations of the mathematical tools being used in the current problem solving exercise.
2) The point 1 above is true regarding fundamental mathematics but also regarding most of the higher mathematical concepts, abstractions and tools like calculus , fourier transforms , etc. The issue becomes more complicated for higher mathematics tools because they build on top of the pre-requisite fundamental and other higher mathematics . So to truly understand a complex mathematical tool like the fourier transform and analysis , one must understand all its pre-requisites as well as the mathematics of fourier itself as per the details and depth discussed in point 1 above. Only then can one be able to really apply the higher mathematics tool effectively for mathematical modeling of the physical world problem in whose solution we are interested. There is just no other way of being a good mathematical modeler.
The above reasons are precisely why mathematical modeling is non trivial , and difficult. Finally , I would like to end with the concluding thought that in addition to the reasons above , mathematical modeling is also difficult because we lump together a whole range of abstractions , concepts , and tools under the single term of “mathematical modeling” . Pretty much the entire domain of knowledge that we identify with the term “mathematics” , comes under this definition.
So I hope by now you agree with me , that mathematical modeling is difficult 🙂 . But that is not a reason to despair , because when done correctly and in due depth , it is a very powerful tool solve the real physical world problems . We just need to put in the due thought and effort in doing good mathematical modeling. But most importantly , we need to know exactly what kind of effort is required in doing so. That is the key take-away point of the current article.
I would like to know what the readers think of this area , and would love to discuss this topic with them through this blog entries. So any comments , insights , agreements , disagreements are all more than welcome!
I intend to keep writing a series on this topic of mathematical modeling where every subsequent post will discuss successively advance math modeling concepts , so keep checking this blog post for updates .