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Posts Tagged ‘**Mathematical concepts**’

**Nature is worth a set of equations; (Nov. 17, 2009)**

I have been reading speeches and comments of Albert Einstein, a great theoretical physicist in the 20^{th} century.

Einstein is persuaded that mathematics, exclusively, can describe and represent nature’s phenomena; that all nature’s complexities can be comprehend and imagined as the simplest system in concepts and principles.

The fundamental creative principle resides in mathematics. And formulas have to be the simplest and most beautifully general. Mathematical concepts can be suggested by experience, the unique criteria of utilization of a mathematical construct.

I got into thinking.

I read this dictum when I was graduating in physics and I have been appreciating this recurring philosophy ever since. The **basic goal in theoretical physics for over a century was to discover the all encompassing field of energy** that can unite the varieties of fields that experiments have been popping up to describing particular phenomena in nature, such as electrical and magnetic fields as well as all these “weak” and “stronger” fields of energy emanating from atoms, protons, and all the varieties of smaller elements.

I got into thinking.

Up until the first quarter of the 20^{th} century most experiments in natural sciences were done by** varying one factor at a time**; experiments never used more than one independent variable and more than one dependent variable (objective measuring variable or the data). Even today, most engineers perform these kinds of totally inefficient and worthless experiments: **no interactions among variables** can be analyzed, the most important and fundamental intelligences in all kinds of sciences. These engineers have simply not been exposed to** experimental designs** in their required curriculum!

Although the theory of probability was very advanced, the field of practical statistical analysis of data was not yet developed; it was real pain and very time consuming doing all the computations by hand for slightly complex experimental designs.

Sophisticated and specialized statistical packages constructs for different fields of research evolved after the mass number crunchers of computers were invented.

Consequently, early theoretical scientists refrained from complicating their constructs simply because they had to solve their exercises and compute them by hand in order to verify their contentious theories.

Thus, theoretical scientists knew that the experimental scientists could not practically deal with complex mathematical constructs and would refrain from undertaking complex experiments in order to confirm or refute any complex construct.

**The trend, paradigm, or philosophy for the theoretical scientists was to promoting the concept that theories should be the simplest with the least numbers of axioms (fundamental principles); they did their best to imagining one general causative factor that affected the behavior of natural phenomena or would be applicable to most natural phenomena.**

When Einstein mentioned that equations should be beautiful in their simplicity he had not in mind graphic design; he meant they should be simple for computations.

This is no longer the case.

Nature is complex; no matter how you control and restrict the scope of an experiment in order to reducing the numbers of manipulated variables to a minimum there are always more than one causative factor that are interrelated and interacting to producing effects.

Currently, physicist and natural scientists can observe many independent variables and several dependent variables and analyze huge number of data points.

Still, nature variables are countable and pretty steady over the experiment. Unlike experiments involving” human subjects” that are in the hundreds and hard and sensitive to control.

**Man is far more complex than nature to study his behavior**.

Psychologists and sociologists have been using complex experimental designs for decades in order to study man’s behavior and his hundreds of physical and mental characteristics and variability.

All kinds of mathematical constructs were developed to aid “human scientists” perform experiments commensurate in complexity with the subject matter.

The dependent variables had no longer to be objectively measurable and many subjective criteria were adopted.

Certainly, “human scientists” did not have to know the mathematical constructs that the statistical packages were using, just the premises that justified their appropriate use for their particular field.

Anyway, these mathematical models were pretty straightforward and no sophisticated mathematical concepts were used: the human scientists should be able to understand the construct if they desired to go deeper into the program without continuing higher mathematical education.

**Nature is complex, though far less complex than human variability**.

Theoretical natural scientists should acknowledge that complexity. And studying nature is worth a set of equations!

Simple and beautiful general equations are out the window. There are no excuses for engineers and natural scientists for not expanding their imagination and focusing their intuition on complex constructs that may account for many causative factors and analyzing simultaneously many variables for their interactions.

** There are no excuses that experimental designs are not set up to handle three independent variables (factors) and two dependent variables; the human brain is capable of visualizing the interactions of 9 combinations of variables two at a time. **

Certainly, scientists can throw in as many variables as they need and the powerful computers will crunch the numbers as easily and as quickly as simple designs; **the problem is the interpretation part of the reams and reams of results.**

Worst, how your audience is to comprehend your study?

A set of coherent series of relatively complex experiments can be designed to answer most complex phenomena and yet be intelligibly interpreted.

**It is time to account for all the possible causatives factors, especially those that are rare in probability of occurrence (at the very end tail of probability graphs) or for their imagined little contributing effects: it is those rare events that have surprised man with catastrophic consequences.**

If complex human was studied with simple sets of equations THEN nature is also worth sets of equations.

Be bold and make these equations as complex as you want; the computer would not care** as long as you understand them for communication sake**.