Adonis Diaries

Posts Tagged ‘Nature is worth a set of equations

Einstein speaks on theoretical physics; (Nov. 18, 2009)

The creative character of theoretical physicist is that the products of his imagination are so indispensably and naturally impressed upon him that they are no longer images of the spirit but evident realities. Theoretical physics includes a set of concepts and logical propositions that can be deduced normally. Those deductive propositions are assumed to correspond exactly to our individual experiences.  That is why in theoretical book the deduction exercises represent the entire work.

Newton had no hesitation in believing that his fundamental laws were provided directly from experience.  At that period the notion of space and time presented no difficulties: the concepts of mass, inertia, force, and their direct relationship seemed to be directly delivered by experience.  Newton realized that no experience could correspond to his notion of absolute space which implicates absolute inertia and his reasoning of actions at distance; nevertheless, the success of the theory for over two centuries prevented scientists to realize that the base of this system is absolutely fictive.

Einstein said “the supreme task of a physician is to search for the most general elementary laws and then acquire an image of the world by pure deductive power. The world of perception determines rigorously the theoretical system though no logical route leads from perception to the principles of theory.” Mathematical concepts can be suggested by experience, the unique criteria of utilization of a mathematical construct, but never deducted. The fundamental creative principle resides in mathematics.

Logical deductions from experiments of the validity of the Newtonian system of mechanics were doomed to failures. Research by Faraday and Maxwell on the electro-magnetic fields initiated the rupture with classical mechanics. There was this interrogation “if light is constituted of material particles then where the matters disappear when light is absorbed?” Maxwell thus introduced partial differential equations to account for deformable bodies in the wave theory. Electrical and magnetic fields are considered as dependent variables; thus, physical reality didn’t have to be conceived as material particles but continuous partial differential fields; but Maxwell’s equations are still emulating the concepts of classical mechanics.

Max Plank had to introduce the hypothesis of quanta (for small particles moving at slow speed but with sufficient acceleration), which was later confirmed, in order to compute the results of thermal radiation that were incompatible with classical mechanics (still valid for situations at the limit).  Max Born pronounced “Mathematical functions have to determine by computation the probabilities of discovering the atomic structure in one location or in movement”.

Louis de Broglie and Schrodinger demonstrated the fields’ theory operation with continuous functions. Since in the atomic model there are no ways of locating a particle exactly (Heisenberg) then we may conserve the entire electrical charge at the limit where density of the particle is considered nil. Dirac and Lorentz showed how the field and particles of electrons interact as of same value to reveal reality. Dirac observed that it would be illusory to theoretically describe a photon since we have no means of confirming if a photon passed through a polarizator placed obliquely on its path. 

      Einstein is persuaded that nature represents what we can imagine exclusively in mathematics as the simplest system in concepts and principles to comprehend nature’s phenomena.  For example, if the metric of Riemann is applied to a continuum of four dimensions then the theory of relativity of gravity in a void space is the simplest.  If I select fields of anti-symmetrical tensors that can be derived then the equations of Maxwell are the simplest in void space.

The “spins” that describe the properties of electrons can be related to the mathematical concept of “semi-vectors” in the 4-dimensional space which can describe two kinds of elementary different particles of equal charges but of different signs. Those semi-vectors describe the magnetic field of elements in the simplest way as well as the properties electrical particles.  There is no need to localize rigorously any particle; we can just propose that in a portion of 3-dimensional space where at the limit the electrical density disappears but retains the total electrical charge represented by a whole number. The enigma of quanta can thus be entirely resolved if such a proposition is revealed to be exact.

Critique

            Till the first quarter of the 20th century sciences were driven by shear mathematical constructs.  This was a natural development since most experiments in natural sciences were done by varying one factor at a time; experimenters never used more than one independent variable and more than one dependent variable (objective measuring variable or the data).  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. 

            Thus, early theoretical scientists refrained from complicating their constructs simply because the experimental scientists could not practically deal with complex mathematical constructs. Thus, the theoretical scientists promoted the concept or philosophy that theories should be the simplest with the least numbers of axioms (fundamental principles) and did their best to imagining one general causative factor that affected the behavior of natural phenomena or would be applicable to most natural phenomena.

            This is no longer the case. The good news is that experiments are more complex and showing interactions among the factors. Nature is complex; no matter how you control an experiment 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.

            Consequently, the sophisticated experiments with their corresponding data are making the mathematician job more straightforward when pondering on a particular phenomenon.  It is possible to synthesize two phenomena at a time before generalizing to a third one; mathematicians have no need to jump to general concepts in one step; they can consistently move forward on firm data basis. Mathematics will remain the best synthesis tool for comprehending nature and man behaviors.

            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 the probability graphs) or for their imagined little contributing effects: it is those rare events that have surprised man with catastrophic consequences.

            Theoretical scientists of nature’s variability should acknowledge that nature is complex. Simple and beautiful general equations are out the window. Studying nature is worth a set of equations! (You may read my post “Nature is worth a set of equations”)

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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 20th 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 20th 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.


adonis49

adonis49

adonis49

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