Can we define equations as a vital fluid to understand reality?
Human beings need equations – moved by the trait to be eager – and able – to add something new to familiar landscapes across acts of thinking and tentative to grasp and understand mathematical ratios.
Therefore, equations appear to be as a vital fluid that is an essential part of our scientific definitions. However, even if they define evergreen lymph and creative flow in mathematics, sometimes they tend to appear as a matter of fear and mockery according to a nice percentage of our society.
It is no coincidence that a nice percentage of the editors promoting popular science texts explicitly ask their authors to leave more room for strings of text rather than the ones dedicated to show articulate calculation. The reason appears to be pretty clear, right? Well, people get annoyed fast from long and detailed explanation of calculus and sales have always the priority over everything else. But we know that already.
The point here is that the so-called “general public”, towards whom one aims to tell something, must be shielded from overly serious, academic, formal and apparently complicated concepts. However, it must not be prevented from a careful and reasoned content elaboration.
I must admit that even if I’m in an intimate affinity with science, there are days and situations when I find difficult to keep the thread of full interest and concentration across series of complex equations that require accurate knowledge – and interconnection – of the rules and theorems outlining their constitution. So, the expectation refer to the brilliance of the author to be able to balance the data ensamble, guiding me in the reasoning process.
Freezing for a moment this very true – personal and general – condition, equations continue to have a fundamental role in shaping the world we live in today. They helped us to create and use the maps of the Global Navigation Satellite System, the music patterns and series we love and need to be creative, the electronics, the digitalization and the signal theory that make our beloved television trustworthy.
Now, as you may already know, there are two types of equations in mathematics that seem very similar at first glance. The first type presents the relationships between different mathematical quantities and their purpose is to show that the relationships they handle are very much true.
The other type gives us information about an unknown quantity (every one of us had to deal with the so called “incognita” for my Italian readers) that mathematicians must know to derive from the equation itself, transforming the unknown quantity into a known one.
I remember that since the period I was in middle school, my mathematics books presented mainly equations replicating the rules of this first typology (equazioni di primo tipo), revealing the existence of fascinating recursive and regular patterns. Patterns made by equations that mirror our fundamental hypotheses about the logical structure of mathematics.
What they express does not present alternatives.
A point that is surprising if we think about the precariousness in which we live, as prey and kinda of masters halfway through situations defined by entropy and emotional randomness of social relationships that regard us with glances that are not always true, exact, clear. Pythagoras’ theorem is an example: if we accept Euclid’s hypotheses for geometry, it turns out to be exact. No margin of error or doubt is admitted when faced with a tautology.
The second type of equations reveals us what is presented in mathematics and mathematical physics. These domains offer us with literal and numerical symbols, data about the real world. They express even properties that concern the relationships in which we live and the laws that govern the Universe whose rules related to phenomena, in principle, may also have been different.
Let’s take Newton’s law of universal gravitation as a direct example: thanks to it we know that the force that attracts two material bodies depends on their masses and on how much distance separates them. Thanks to the equations that define this fundamental law, we can then understand how the planets rotate along their orbits around the Sun. Think about that: our ability to design the orbit that a spacecraft must follow to stay on course comes directly from this law!
Yet, strictly speaking, Newton’s law is not a mathematical theorem since its correctness and usability depends on observations concerning physics. This point can be better understood thinking about how the law of gravity has changed after the general theory of Albert Einstein who has been able to integrate and improve Newton’s law specifying data able to maintain even the correctness of Newton’s laws.
Equations have changed our history many times, being endowed with hidden powers, capable of revealing what nature wants to keep secret and matter of intimate use.
An equation is powerful when its strength comes from an essential ingredient: simplicity.
It guides us in the process of understanding two calculations that are apparently different from a formal point of view but have the same answer. The nature of a stable equation establishes stable relationships between mathematics -the gift of creative human intelligence – and physics, the science that helps us to understand the nature of the outside world.
I’m thinking about how James Clerk Maxwell has managed to transform the experimental observations he made, through empirical laws on magnetism and electricity, only thanks to four equations! The ones that have the merit of having quantitatively described the electromagnetism giving the light to our beloved radio. As well as the radars informing us if it will rain while we plan to go for sports or a picnic outdoors or the television without which we would not know how to plan prevention measures for not becoming the next victims (at least not in terms of phobia) by Coronavirus.
The tendency to be attracted, by nature, by some characteristics that concern what surrounds us, can be seen as a starting point to appreciate the history of the equations that textbooks aiming to present, mostly in the form of symbols and quantities that ask to be automatically remembered.
I’m the idea that we should accept another challenge: look beyond them, questioning symbols to understand the convincing stories they can tell us.
You might ask what is the why behind this extra effort… Well, symbols are quite powerful as – in their turn – they reveal themselves as a safe landing isle to what is guaranteed by language.
In fact, just think on how the latter provides us carte blanche to explore through personal descriptions and interpretations, elements existing around us. The ones that would afterwards define part of our personal domains made of symbols, words and interpretations that science and technology tend, for good reason, to put aside.
Is also true that – sometimes – we even realize that the trend to rely too much on language indicates a price to pay, in terms of uncertainty, inaccuracy and limitation of tracing a path that is completely free from human (pre)conceptions and limited (not only scientific) knowledge maintained throughout and across cultures and civilizations.
Fortunately, equations have managed – choosing to be silently industrious in the history’s backstage – to keep stable their perceptible influence on the fundamental elements defining the mechanisms capable of setting out our current understanding of reality.
I leave you with a reflection, which will most likely lead me to share something more on this subject:
Usually, when you write an equation, wonderfully simple and effective, the world tend to remains in its “comfort zone”, immune from consequences that are the basis of a change…
The latter, in order to have the necessary power and strength, require people capable of imagining and leveraging the content that the equations mediate, driving to be discovered. These effective procedures make the difference, mediated both by knowledge, combination and deep understanding of their current and possible effects on our current representations and areas of research.
Again, is the human creativity factor that, learning to approach and to play with complexity, place intelligence as a constant offer to maintain live the fire of new operative calculus strategies and development, with new data, algorithms and simulations based on dynamic traditional systems.