As someone who regularly rides the donkey backwards to thumb my nose at convention, it pleased me tremendously to discover that the equations that have constituted the focus of my Life’s work are misbehaving equations. In fact, it makes me laugh to think how appropriate it is that Providence would lead me to these unusual equations that continue to perplex mathematicians. Over the decades my Muse inspired me to investigate two distinct types of disobedient equations. The Living Algorithm, the primary mathematical focus in this project, is one of these unruly equations. It is in a class by itself for reasons that we shall explore. The Root Beings are another type of disobedient equation that I have investigated thoroughly.
The Author of these many articles had little, if anything, to do with the choice of areas of investigation. In other words, he did not set out to study non-well-founded equations that don’t obey the axiom of regularity. In fact, he had never heard of any of these concepts until just recently (March 2013), quite late in his game. In 1978, when he discovered the Living Algorithm, fractal equations, the most popular disobedient equation, didn’t even exist as a concept. Further, the computer age, which was to provide an infinity of pragmatic uses for this unique mathematical process was only in its infancy. When he inadvertently came upon his Root Being Equations in 1999, hyper-set theory, which finally incorporates these misbehaving equations into a general mathematical framework, had only recently been proposed (1991) and was just gaining general acceptance.
As someone who resided far from the hallowed halls of academia, the Author was in the dark about these erudite mathematical constructs and their applicability to his investigations. In fact, he imagined that mathematicians had already encompassed his humble studies into a larger framework. But then, he happened to read Where Mathematics Comes From by George Lakoff and Rafael Nunez, 2000. It was only at that point that he realized that he had devoted his life to disobedient equations.
What does it mean for an equation to be disobedient? To understand disobedience, we must first understand what it is to be an obedient, well-behaved, well-founded equation.
A significant branch of Science, some might even claim the main branch, has to do with the application of mathematical symbols to empirical reality. Included in this branch are all the hard sciences, computer science, and most of the soft sciences. This aspect of the scientific endeavor has been based, almost exclusively, upon obedient equations. Only in the last forty years, has the scientific community begun to even consider disobedient equations as a way of describing reality. This shift of focus was due, at least in part, to Mandelbrot’s development of fractal geometry combined with the computer revolution of the late 1970s. Up until that time, any scientific endeavor that had mathematics associated with it was devoted to obedient equations. Even now, obedient equations still have the center stage.
What does it mean for an equation to be obedient?
An obedient equation belongs to the set of well-founded functions in the mathematical lingo. For our purposes, a well-founded function obeys the axiom of regularity. Briefly, this means that the function doesn’t contain itself. In other words, if a function contains itself, it violates the axiom of regularity and is booted from the club of well-behaved functions.
What is it about self-containment that is so repellent to the scientific community that they have ignored this class of equations? Even the labels that they apply to the two classes of functions reveal their collective abhorrence for what we call the disobedient functions. The obedient equations are considered well-founded and regular, while the disobedient equations are considered to be non-well-founded and not regular. It seems that the obedient equation conform to the rules, while the disobedient are non-conforming. This seems to imply that there is something wrong with this branch of the mathematical family.
Even the subtexts of the mathematical labels suggest that this type of function is unstable and irregular. It is no wonder that respectable scientists would shun this unsavory type. “If it is not well-founded, count me out. How can I possibly rely upon non-well-founded functions that don’t obey the axiom of regularity? What kind of scientist do you think I am? I rely on logical precision and accuracy for my results. This ambiguous character does not seem to be my type. Those mathematicians might be interested in this type as an intellectual curiosity, but we hard scientists are only interested in empirical reality. We need a mathematics that can provide us with absolute statements about relationships between variables. Well-founded functions are famous for this ability. This is why we love them so, and are suspicious of the other.”
Why can well-founded functions provide scientists with absolute statements regarding relationships between variables?
To answer this question let us turn to set theory from mathematics. Traditional set theory encompasses well-founded functions. Set theory, in turn, is based in the container metaphor. Something is either in the box or out of the box – and by mathematical extension, either in the set or out of the set. There is no other alternative. The either/or reasoning behind the container metaphor has been the basis of Western logic ever since Aristotle first formulated the laws of this form of logic over 2 millennia ago. Note that disobedient equations are left out of this system.
Based in container logic, as it is, the scientific endeavor has entailed finding a box, read category, for every phenomena in the Universe. Once a category-box is found, it can go through endless refinements – boxes within boxes, forever and ever, Amen. In fact, one scientific sentiment holds that the entire Universal Flux can be put into boxes.
With the urge to box things up, comes the simultaneous urge to identify essence. The identification of essence enables one to place things in their appropriate box. This is true whether you are a scientist, mathematician or are just trying to organize your apartment. Silverware, plates, and well-founded functions, each have a unique essence that determines which box they are to be placed in. Determining essence is a crucial part of the procedure of putting something in a box.
If everything has a box, then everything has a permanent essence, as well. If the essence is not permanent, then it is not a real essence. Due to the importance of essence for container logic, the scientific community has spent an abundance of mental energy upon precisely determining the permanent essence of things. Under this way of thinking, everything has an essence that determines which category box they are to be put in.
Once the scientific community has organized reality as precise essences and placed them in equally precise containers then the mathematician scientists can take over. These modern day wizards employ the symbolic representations of mathematics, the algebra, to precisely define the permanent relationships between these essences. This is true whether the boxes apply to matter, energy and the speed of light; or pressure, volume and heat.
This entire process defines a significant, if not prime, some claim the only, aspect of the scientific endeavor. Identify the permanent features of both Category Boxes and the Essences of Things. Then employ Mathematical Symbols to characterize the permanent Relationships between these Essences. Many in the scientific community, though not all, feel that this process is one of the essences of Science.
If this process is truly an Essence of Science, then any endeavor that doesn’t contain this process is not really science. However many of the so-called Life Sciences, such as Biology, Psychology and Botany, are more about identifying essences and category boxes, and less about identifying permanent mathematical relationship between these boxes. The matter that hard scientists specialize in is particularly susceptible to a permanent mathematical characterization. Living matter is not. However, the Life Sciences do employ mathematics to validate Essence and Category, even though Relationship, in general, is inaccessible. Further, the mathematics of Probability is frequently employed to determine not permanent Essences, Categories and Relationships, but partial determinations – as in, most of the people some of the time.
So as not to exclude the Life Sciences from the Science Box, the notion of Science is expanded to include any discipline that has rigorous standards for determining Essence and Category Boxes. This is where subcategories comes in – the refinement of boxes within boxes. For purposes of illustration, the hard science box could include any science that employs mathematical symbols to characterize permanent relationships between permanent essences. These hard sciences happen to be the material sciences. The soft science box could include any science that includes a rigorous analysis of Essences, Boxes, and Relationships, but without being held to permanent mathematical relationships. The soft sciences happen to be the life sciences. Coincidence or not?
Although there is a significant difference between the soft and hard sciences, they still, in general, participate in the container metaphor of set theory. So at this point we could expand our definition of Science to include any endeavor that participates in the either/or logic of set theory. This definition would include any discipline that attempts to identify Box, Essence and Relationship. Once we’ve expanded our definition in this fashion, the umbrella of Science is able to include many more disciplines.
Remember the well-founded functions, the obedient equations, are the mathematical foundation of traditional set theory. Also set theory, based as it is, in the container metaphor provides the foundation for traditional logic as well. Hence, logic and obedient equations form an inseparable relationship that provides a major essence of Science, as we’ve currently defined it.
To further identify the nature of this traditional logic let us speak briefly about one of its prime components – the Excluded Middle. This is ‘the principle that every proposition is either true or false, so that there is no 3rd truth value and no statements lack truth value.’ In terms of this discussion, this means that everything is either in the box or out of the box. There is no 3rd alternative. Further, there are no other statements besides true and false statements. Due to the absolutism, this is sometimes called either/or logic.
There is a logical fallacy associated with this principle, appropriately called the Fallacy of the Excluded Middle. Here is a typical example: "You are either for us or you are against us." The fallacy of this type of argument is that it tries to eliminate the middle ground. Perhaps the individual is in favor of some of the group’s features and against others. Perhaps the individual values the group, but disagrees with their course of action. The individual may even feel that a group choice may not be in the best interests of the group. However, the general nature of the statement excludes these possibilities as logical alternatives.
In terms of either/or logic, the solution to this fallacy is to break the categories into smaller boxes. The individual refines what he is for and what he is against regarding the group. These categories, of course, create more excluded middles. However as the refinements multiply, the area that the excluded middle encompasses grows smaller and smaller. In fact, a significant section of the scientific community believes that with enough refinements that the excluded middle actually shrinks to zero. Many believe that the absolute precision of mathematics performs the function of eliminating the middle.
Indeed, until the last few decades, theorists, in general, have held the position that either/or logic is an essence of Science. In other words, any phenomenon that couldn’t be characterized by precise true and false statements wasn’t science. Either you’re with us in the Science camp of container logic, or you are against us. The traditional scientific perspective had faith that the logical middle of disobedient functions can be excluded from the scientific endeavor.
This Either/Or Science attitude participates in the Fallacy of the Excluded Middle. In this case, the excluded middle is any type of scientific endeavor that isn’t based in the container metaphor. This volume is devoted to exploring the nature of the Excluded Middle of traditional Science. To this end, we explore the nature of the logic of disobedient equations. Our prime example in this regard is the Living Algorithm.
One reason that obedient equations with their either/or logic are held in such high regard has to do with their amazing ability to characterize the behavior of matter. In contrast, disobedient equations play no role in this regard. Due to this lack of utility, scientists that are devoted to a material explanation reason that this type of equation is just a mathematical curiosity. As such, they feel no qualms about excluding the middle ground of logic from the scientific endeavor. To see why the physical universe is such an ideal subject for obedient equations, check out our next article – Obedient Equations for Obedient Matter.
Before proceeding to new material, we recommend reading a summary of this articles from Life’s metaphorical perspective. To assist assimilation of this unusual material, read Life's affinity with the Living Algorithm based in Disobedience.