A7:  Mathematical Modeling

*

Some readers already know about it--although very many don't--therefore, we will start by explaining what is "mathematical modeling" and, most important, to what it is good for. Now, we study our environment and we permanently discover interesting physical phenomena. Because we have been gifted with intelligence, we want to understand the nature of those physical phenomena as much as it is possible. For that, we use mathematics, and we imagine models that describe mathematically particular physical behavior.

Besides from a deeper scientific understanding of physical phenomena, the advantage of using mathematical models is, if our mathematical models are good, we could continue discovering other true, physical phenomena working this time ONLY with mathematical models! We cannot think of a better example than the discovery of the LASER (Light Amplification by Stimulated Emission Radiation) principle. The LASER discovery has generated a particularly complex psychological and economical impact in our life. The less important aspect for our civilization, the economic one, was translated into thousands of useful applications--we wouldn't have the CD drive today without it--and we have absolutely no doubts that many more exciting LASER applications are going to make our life even more comfortable in the future.

However, what is truly remarkable is the psychological impact the LASER discovery brought into the scientific world. It happened the LASER was the first application that has been first discovered on paper, theoretically, using the available mathematical models, and then implemented into very successful, practical applications. Well, we say it was the first one, although this is not quite true. Anyway, the point to note are psychical implications the LASER discovery brought into the scientific world. The most intelligent and prominent scientific minds at that time said: "That is an example of rigorous scientific development. All future discoveries must have a solid theoretical fundament first, and then they should be tested experimentally." (Of course, they said something similar.)

Consequently, the scientific world rushed on studying and developing the existing mathematical models: there was a gold mine waiting to be discovered in there! Even now, with the latest born physics' child "Cold Fusion", the most intelligent minds of our days struggle to design a mathematical model that will "fit" that anomalous behavior using, of course, the existing mathematical models. Is that good? Is that bad? Fact is, we do need mathematical models no matter what. Now, we have seen "what", and "why", but the real problem is "how"!

The history of our civilization is marked by few major technological levels: fire, wheel, sail, steam power, internal combustion engine, reactor engine, uranium fission, and semiconductor technologies. Overall, they are not very many, but it is certain many more will be added into the future. The interesting thing to note is the efforts, and the time needed to step from one technological level to the other, on one hand; the other aspect is, each new technological level has changed our society, and our life, in a dramatic way.

There was a time when Euclid's geometry was regarded as the most pure, exact, and rigorous mathematical model; later, the sphere geometry came to negate all laws and rules governing Euclid's geometry. Other scientists have been passionate for centuries trying to discover the "golden numbers" which were supposed to explain all secrets of our life. Many have searched for the "philosopher's stone" which was supposed to transform everything it touched into pure gold--incidentally, Cold Fusion seems to be able to do something similar. Over the centuries, the research work was focused on two fronts: mathematics and physics--that was, more or less independently. However, we need them both, if we want any real, valuable, results.

Today, we have at our disposition many complex mathematical models trying to account for various, true, physical phenomena. The point to note, however, is mathematical models are just TOOLS. Trying to discover our reality using the existing tools is not going to help us much, because new, radical discoveries require new, better tools. Suppose you have a big hammer tool and you use it to break little atoms hoping to discover what is inside them. That big hammer represents Quantum Mechanics Theory we use today to find out more about atom's structure--what we try suggesting here is, we need way better tools for that.

It happens that we do not have, and we cannot have physical instruments which could penetrate inside the quarks, for example, to test our theories; therefore, the only true, reliable tool we have to work on those atomic levels is a mathematical model--well, there are other tools but ... we will explain them later ... perhaps. Now, a bad mathematical model could hold back our technological development for hundreds of years, while with a good one ... 

Let's take a true life example. The most known formula in Quantum Mechanics is:
E = m * c²
When Albert Einstein has formulated it the first time, that formula has generated a revolution having many aspects: theoretical, physical, and philosophical. It is still one of the most important formulas we have, because it says mass (matter) is nothing else but pure energy. That is the colossal importance of the formula: atoms, and atoms' components are constructed out of ONLY energy!

The above formula is very simple and it tells us nothing about the NATURE of the energy: it could be heat, potential energy, electromagnetic energy, or something else. Now, try to imagine the squared light-speed constant is replaced by a variable having a certain physical dimension: in that case you could discover that mass (the atom) is in fact equal to the sum of various types of [energies]. We used square brackets intentionally, because in that case the energy would cease to be energy, as we know it: it will have to be something else, something of which the energy itself is built. We will continue developing this topic but, for now, we recommend MERCY for additional details.

One of the most troublesome aspects is the constants we use in our existing mathematical models, and they are very many. A true mathematical model needs to be funded on experimental data, and on some fixed, reference points, in order to test its validity: the constants provide those references we need. For example, Quantum Mechanics is founded on the idea that light-speed is a constant, and the maximum possible speed in the Universe.

If the above hypothesis is true, then we are in big trouble: we will never be able to meet with our dear brothers and sisters living nearby, in our beautiful Milky Way galaxy. We would be left as we were 2000 years ago, when people were watching the immensity of the ocean thinking it was impossible, even dangerous, to reach the other shore--in fact, people thought there was no other shore in the opposite side of the ocean. Now, 2000 years later, anybody can cross the ocean in about 7 hours, and even less; it is just a matter of little more money. The new ocean for us today is the galaxy, and the only tools we have to conquer it are new, better mathematical models.

As mentioned, a mathematical model requires constants, and we have discovered very many. Let's take a look at the most known one:
F = G ((m1 * m2) / d²)
In the above formula G is "Universal gravity constant" equal to 6.67*10^-11 [N * m²/ Kg²]

The above formula is "Newton's Law of Universal Gravity", and it works so well that it was successfully used to discover theoretically new planets in our solar system. The only problem is, at galaxy level the above formula tells us there should be at least 100 times more matter, in order to satisfy the existing galaxy AND the formula. That "extra" nonexistent matter has been poetically named "Dark Matter", because the assumption was, and it still is, the universal gravity constant is a "true" constant. Fact is, there is no more matter in our galaxy, and in the Universe, and the universal gravity constant is not constant: it has other values at galaxy level.

The implications of a wrong constant in our real life are enormous. One constant which is not quite constant means that other tens of constants are not quite constants, and then hundreds of theories and even physical laws are not right. In the end, our entire physical-mathematical apparatus will have to be revised. Scary, isn't it?

Some readers may agree with this article, but we are certain that very many will not. Of course, we will have to detail these brief "assumptions"; we will do that, although we cannot guarantee it is going to be tomorrow. Meanwhile, you could study MERCY and THREE STORIES FROM HILSA'N TASSA GALAXY: these two exceptional books reveal a lot--possibly, even too much ...

***

First published on May 25, 2005 
© Corollary Theorems Ltd. All rights reserved.