Why were Equations so Important?

Well, let’s look at the past…
Nearly everything we take for granted today, the technologies of telecommunications, medical science, weather forecasting, music, radar, and many others, rest squarely on some of the most fruitful insight unveiled in another era. Under the most unassuming of circumstances and often with little appreciation of their profound impact on society or human development. Some, at their conception, could have hardly foreseen the enormous consequential impacts these obscure mathematical steps would unlock.

In his Book- 17 Equations that Changed the World, Professor Ian Stewart,
Emeritus Professor of Mathematics at Warwick University outlines in great detail the historical circumstances behind how these came into use. Some by sheer accident, while others over long sustained pioneering efforts and persistence.

What are equations anyway? What’s behind some of the strange symbols they contain? The short answer is, they are descriptions. Powerful ways to condense and compress a description of natural phenomena or processes. Underlying intricate laws encompassing physics and deterministic laws that govern how the natural world operates. Equations, if you will, represent a shorthand way to define the order of things. The Theory of Everything is an elusive concept that suggests that all the natural forces can be unified into one grand description where the ultimate causation of all processes can be condensed into one single equation.

Then, there are the probability functions that operate in a highly probabilistic world that attempt to describe the likelihood of something occurring. The frequency of past events is tabulated into equations to provide a deeper understanding of historical events and use these frequencies to create statistical equations. These equations attempt to depict the probabilities of outcomes. Life tables or actuarial tables are a case in point. Within these equations certain conventions apply, variables and constants are used in the description and influence of factors under study. Coefficients regulate the extent any single variable influences the total outcome. Equations come in many kinds, some are mathematical truths, some, tautologies, think of Napier’s logarithms.

Some Powerful Reminders…

Fourier Transforms –And Digital Photography

Would Fourier ever have thought his work led to such devices that compressed and stored pictures in digital Photography or be developed to clean up digital noise for recorded music?

Stokes-Navier Equation

CFD, Computational Fluid dynamics is the study of fluid and its movements are grounded in the work carried out in the field. It led the way to understand how airflow ————————–moving across solid bodies influenced the dynamic forces. Formula-One racing car design and the study turbulence makes extensive use and application of the work developed by Claude Louis Navier, a French Engineer, and Iris mathematician and physicist George Gabriel Stokes.

Normal Distribution patterns

Only in the 19 th century did mathematicians see the power of understanding statistical patterns and how influential —- these became in understanding randomness and chance events. ———- The above formula defines how probability is measured and defined.

Let’s go back to our original question– why are they important? How do they shape our understanding of things?

Well, the short answer is, they allow us to understand the deeper aspects of natural phenomena, and the more subtle factors at work that underline causation. Understanding causation is the bedrock of being able to explain how events take place within complex systems or the effects we see in everyday life. The difference between describing something –and explaining something- is fundamental to science, and the study of behaviours of natural forces. Describing something is how we observe the natural order of things, explaining them is to see beyond the obvious, as to how these matters come about.

Let’s not forget – we are mere observers to these things, but our insights become clearer as a result of the power of mathematics and equations Extending them into a broader canvas, allows us to model and predict how Independent variables influenced the outcomes of events under study. All implicitly predicated on the notion that most things were deterministic and linear. Then, Chaos Theory came along and brought us to understand a lot more intricate processes were at work. Non-linear patterns emerged where certain natural outcomes were more complex to understand. They did not readily fit into any known formulaic descriptions, instead, they represented far more mysterious and more elusive in trying to describe and compres these phenomena into convenient equations. Suddenly, we discovered small changes in one set of variables could have massive and extensive consequences to the overall picture.

Added to that was the realisation of sensitivity. Initial conditions, aspects that took place at the onset in a sequence became magnified and powerful beyond their starting values. Tiny perturbations in one location lead to enormous consequences elsewhere. A butterfly flaps its wings causing minute movement in the surrounding air in one location, triggers a sequential progressive domino effect that cascades into the physics of a monster storm hundreds of miles away.

More complex models were needed, more powerful and sophisticated equations that could handle and describe these forces with some degree of reliability. This lead to the discovery of Complex Numbers one of the most powerful areas of mathematics leading to the basis of quantum mechanics. Imaginary numbers such as i that were baffling to mathematicians. Henc the square root of minus 1.

These weren’t real numbers, how could they be used to any great value.? Real numbers in arithmetic could be manipulated in traditional computations, but imaginary numbers made no sense! However there was one interesting difference – when you square an imaginary number, the result is negative. That ought not to be possible because squares are always positive. This would prove to be an invaluable feature in later mathematical formalism.

This was all back in 1572 when mathematics was enjoying a great peak in interest, which eventually gave birth to imaginary numbers. Only in the eighteenth century did mathematicians figure out what they were, but it took the nineteenth century before they finally felt comfortable with them. It finally dawned on mathematicians of that era that imaginary numbers represented reality in different ways and in different contexts. There was some sublime underlying duality behind their existence. The imaginary number i united the two most remarkable numbers in mathematics, e and n in a single elegant equation. Imaginary numbers became powerful aids to understanding waves, heat, electricity, and magnetism and to complex numbers, one of the most powerful fields of mathematics.

Is there a deeper construct to all this.?
Perhaps, but, lurking somewhere else. Somewhere, that has nothing to do with numerical sciences or mathematics. Perhaps all these formalistic equations were the tip of something more prosaic, more mysterious. Embedded in everything discussed above regarding these equations, suggested the presence of precision. One cursory inspection of these abstract equations and their make up hinted at an underlying order that followed laws in obedience to something beyond our knowledge of mathematical integrity. This duality was stalking our efforts to understand or de-couple cause and effect systems. This, might at first sight, suggest ambiguity without appreciating some deeper truths. Perhaps all is not as it seems.

Before Einstein’s great breakthrough in understanding the nature of gravity, Isaac Newton’s law of attraction ruled. Simply stated this law postulated that gravity as a force was a function of the Mass of one body exerted an influence on another body, which was proportional to the distance between them. Some force reached out and attracted smaller bodies to the larger.

Einstein’s elegant idea was breathtaking. It suggested that gravity wasn’t so much a force at all but a consequence, of warping Space-time. Think, bowling ball on a trampoline surface. Its mass distorts the immediate area around it, drawing in smaller objects. What looked like an attractive force (gravity) turns out to be something more sublime.

For centuries this commonly held view of gravity failed to appreciate that perhaps all was not as it seemed. It took James Clerk Maxwell using the rigorous tools of his time, to finally realise that magnetism and electricity were close first cousins inextricably linked that lead to the development of power generating turbines. An even more momentous event in the late 1870’s was the realisations of another pairing of duality:- that light itself might be more intriguing than first thought and behaved as a wave.! The use of mathematical equations became more refined with deeper insights growing our understanding of causation.

But as boundaries are pushed beyond the conventions of mathematical logic new paradigms open up. More metaphysical conventions may impose new rules for thinking about these issues.

Consider a more amusing anecdote and, slightly far-fetched illustration. Two friends decide to go on a camping trip into the desert. One carries water but has never seen ice before. He lived a somewhat sheltered life. His companion, on the other hand, brings ice to keep the water chilled. He too has lived a closeted existence and has never seen water before. On their journey, both marvel at each other’s possession. Water and its beautiful liquid appearance and sparkling reflections, while the other is amazed at the magnificent crystalline structure of ice cubes. As the heat of the desert sun exerts its presence, the ice-cubes begin to react and melt in response to temperature. This causes dismay to both friends as each sees the transformation from ice to water. Further dismay follows at nightfall when the chill of the desert night freezes the water into ice. At that point both companions are faced with a new reality. Something they viewed as being different, is in fact connected at a deeper level. Ice and water are essentially the same, but under certain circumstances emerge the appearance of being different seen in a different context. The duality of these two ideas, at first sight, seemed to be contradictory and mutually exclusive. The notion of duality of the ice-water relationship was missed.

This hidden duality exposed secrets that transcended the world-view of our two campers. Perhaps somewhat simplistic, but the story ties to illustrate the possibility that some things lie outside the purview of mathematical treatment or human imagination. Quantum Mechanics ordains there are at least 10 dimensions that underpin our physical reality and yet, we only experience four. Three physical planes, and one of Time. The remainder of these missing dimensions await to be uncovered or… as, Francois Burgoyne our Jesuit bell ringer in Chronicles of Time suggests, “Are not meant to be discovered at all”

The Mysterious Role of Constants of Nature

Earlier we mentioned two powerful symbols in mathematics,  and e as fundamental properties, both representing a mysterious role in the laws that govern how our universe exists. These constants along with the values in the periodic table, or the speed of light, c play a significant role in the making our reality.

They are mysterious because of their precise numerical value they contain. The numerical value,  (3.1416…) and e (2.71828) apparently remain the same in any part of our universe exercising almost some kind existential function in mathematics. Again, the evidence of pristine order, along with a precision that decrees the most fundamental forces of nature operates to some mystical choreography of laws. John Barrow, in his superb book, The Nature Of Constants, describes these values as “a golden thread weaving continuously through all natural laws” If these numerical constants were different to any degree, our world, and our very experiences of reality, would be completely different.

Perhaps it’s no accident that our physical reality is ordered such, that our organic brains are capable of understanding our surroundings. The seen and the unseen are arranged in a manner to allow us to make sense. Our brains are designed to the degree that we can decode and algorithmically compress and absorb just enough information to comprehend and, remain sane at the same time. We’re hard-wired to handle a version of reality that allows us to bypass the underlying complexity of things and happily coexist in this edition or version of the world we inhabit.

Yet, Theosophy Literature (derived from the Greek theos (“god”) and sophia
(“wisdom”), is generally understood to mean “divine wisdom.” states “Behind everything seen or unseen there is an eternal, boundless and immutable reality, which is beyond the range of human thought. Both matter and consciousness (or spirit) are the two polar aspects of this Reality.”

Equally of interest but more direct was H. P. Blavatsky who said,

“Deity is Law. There are no mechanical laws. The universe is pervaded by a non-anthropomorphic intelligence, which is both immanent and transcendent. Therefore, intelligence is at the basis of all laws of nature. At the same time, no super-natural miracles are possible.”

Here was a man who liked to hedge his bets.

Yet, some great thinkers saw more potency beyond the power of equations to explain everything. Men like, Charles Babbage (1897) English Mathematician and Analytical Philosopher who said: –

“The more man inquires into the laws, which regulate the material universe, the more he is convinced that all its varied forms arise from the action of a few simple principles. These principles themselves converge, with accelerating force, towards some still more comprehensive law to which all matter seems to be submitted. Simple as that law may possibly be, it must be remembered that it is only one amongst an infinite number of simple laws: that each of these laws has consequences at least as extensive as the existing one, and therefore that the Creator who selected the present law must have foreseen the consequences of all other laws.”

Theosophy also holds that evolution is purposeful and directed toward greater sensitivity of forms, greater responsiveness of intellect, and greater awareness of spiritual unity. In Christian terms, evolution is God’s method of creating, perfecting, and redeeming the world. Contemporary science, notably quantum physics, has more kinship with biblical Christianity than did the mechanistic physics of Newton, great as he was in his time. The universe as we know it today looks “more like a thought than a machine.”

Perhaps one of the most mysterious places in our known universe is the realm of the quantum world of particle physics. There are no known laws or equations of a quantum mechanical nature that comes close to explaining the workings of this bizarre and surreal existence. The mere act of observance alters the entire vista at the atomic level. Even the scales involved are staggering. Physicists regularly mention in passing the experimentation of particle behavior, at the Planck level. To put this into a context that we can all appreciate –if the entire Milky Way galaxy size equates to a simple atom, then the Planck-level drills down to the level approximate to the height of an average tree! That’s the depth of sub- atomic experimentation. Staggering and mind-boggling as it is, it’s beside the point. Recall we mentioned earlier that certain aspects of the observable world we inhabit, is not always as it seems, then here, at the quantum level nothing makes sense.

Brian Green, in his book, The Elegant Universe, describes this as microscopic weirdness. No one knows or understands quantum mechanics why such strange goings-on takes place. No formulae or equations exist or likely to appear in the foreseeable future, that makes sense of what takes place here. Nothing follows known laws that are comprehensible to mere mortals. Everything behaves in some form of highly probabilistic ritual, where randomness reigns. This suggests a Jekyll and Hyde existence operates at the heart of quantum mechanics and confronts both physicists and mathematicians alike with a staggering conundrum. If these fundamental elements that govern atomic structures operate in such incomprehensible levels of chaotic behavior at that scale: why does this not translate upward to the scale of everyday existence? Where the known laws of physics conform to more deterministic norms. What suspends or disrupts this weirdness that reigns at the subatomic level, from escalating up to our level and turning our world upside down.

This almost becomes an existential problem – and perhaps requires us to seek answers in more metaphysical places.

This, in turn, invites a more introspective understanding between physics and metaphysics. Are we content to settle with the notion that physics as we understand it, defines the workings of laws governing life, while at the same time, the other tries to understand what comes next, in the afterlife?

Life after Death?

According to the Theosophical tradition, when the process of bodily dying is completed, our personality undergoes a transformation. The impulses, tendencies, emotions, habitual thoughts, and automatic reactions, which we ordinarily identify as our “self,” are sorted out and ordered. All those products of our past life are divided into two major parts. One consists of the entirely personal and selfish, the transitory aspect of the past life, which forms a shell or cocoon around our inner core. That core, the other part of us, consists of our generous, unselfish nature, transcending the merely personal. It includes everything from our life that is worth preserving.

It was Plato who said it best, “Numbers are humanity’s greatest shared experience” And where is God in all of this?

Despite the evidence of duality, and its sublime role in nature, the presence of mystical numerical constants, or the inbuilt precision lurking in the fabric of reality, is it possible to conceive the existence of a watch, without the existence of a watchmaker. Does Heaven permit the existence of physics, or does physics permit the existence of Heaven?

Cardinal Joseph Ratzinger before he was elected to become, Benedict XV1, was a highly regarded theologian as well as a keen follower of scientific matters, was asked how he reconciled science and his faith. His answer was interesting when confronting the issues of conflict between dogma and scientific thought, he answered, “Where the heart can go, the mind can follow”.