… to continue our exploration into the productive nature of disobedient equations. To provide context for the discussion, we have primarily examined the nature of the obedient equations. Seen from the traditional scientific perspective, the disobedient equations are the shadow of their well-behaved cousin. By revealing what they are not, we come to a better understanding of what disobedient equations are.
Disobedient equations do not adhere to the either-or logic of traditional set theory. The reason is straightforward. Disobedient equations are self-referential. In other words, as functions, they contain themselves. They are continually referring back to themselves to obtain the next result. Fractal geometry is a popular example of the employment of self-referential equations to produce some amazing visual results.
Strangely enough, living systems are also self-referential. In order to survive, they must constantly adjust to external circumstances by monitoring internal responses. For this simple reason, we made the claim that disobedient equations are more appropriate than obedient equations to describe the behavior of disobedient Life. Although this statement may or may not be true, we have not yet addressed what this means.
In other words, how could disobedient equations model living systems if they don’t obey either-or logic? If they don’t obey either-or logic, what type of logic do they obey, if any? If another type of logic is employed, what type of definitive statements can be made? What are the practical applications of a logic that is not based in the either-or logic of traditional set theory?
To understand the logic of disobedient equations, we must first understand the logic of obedient equations.
The dichotomy of obedient and disobedient equations makes it appear as if the mathematical world of equations is divided into two opposing camps. In a broad sense, this polarity is true. However, the two mathematical camps are divided internally. There are many types of disobedient equations, each with their own brand of logic. This is as we would expect from this unruly brand of mathematics.
However, we don’t expect this kind of mathematical and logical diversity from obedient equations. In fact, popular thinking, even in scientific circles, tends to hold that the mathematical logic of obedient equations is unitary. In other words, an implicit belief of the scientific community is that the mathematical logic of well-behaved equations is monolithic – unicentric rather than polycentric. There is only one logic rather than many. Close in thinking is the perception that there is only one science and one math.
This unitary perspective is inherited from the Greeks. Plato et al developed the notion that there was only one Logos, one logic. While the ancient Greek culture had many gods, the Greek philosophers had only one Logos. The Christians adopted this same perspective in their attempt to blend the one Logos with their one God. The Renaissance thinkers continued this legacy. Spinoza, following Descartes, even developed the notion that God was only Logos, nothing else. The scientific community up to the present day has adhered to this belief, except that they have dropped God from the equation. Just as the Biblical religions believe in one God, the scientific community tends to believe in one Logic.
Drs Lakoff and Nunez have successfully dispelled the notion of a singular mathematical logic in their book – Where Mathematics Comes From. Taking a cognitive perspective, they illustrate that advanced mathematics consists of a complex metaphorical blend. In fact, one intent to their book is to deconstruct the complex of ideas that they call the romance of mathematics. Included in this complex is the belief that mathematics is real and singular.
Employing this book as the starting point and foundation of our current discussion of disobedient equations, we have begun the deconstruction of mathematical logic. We start by examining our innate mathematical abilities – our ability to subitize our world. This innate ability has so do with our ability to count. The mathematics of discrete counting numbers eventually led to the development of computers with their binary either-or language of 1s and 0s. The digital world is a testament to the power of counting numbers.
To better understand the unique nature of these disobedient equations, we must first examine their historical context from a cognitive perspective. To accomplish this aim, we must first understand the cognitive perspective. That is the function of the next article in the stream – The Conceptual Metaphor of Cognitive Science.
•To truly understand and appreciate the logic behind disobedient equations, such as the Living Algorithm, it is important to first establish a cognitive perspective. To that end read the next article in the stream – The Conceptual Metaphor of Cognitive Science.