Time for a little more cognitive science.
Reiterating for assimilation: Cognitive science has established beyond a reasonable doubt that conceptual metaphor is the basis of most abstract thought. In this context, abstract thought is unconscious cognition. This unconscious cognition is the basis of conscious thought, as well as words and language. Conceptual metaphor, as opposed to literary metaphor, is the precise mapping of the inferential structure of a concrete experience onto an abstract experience. Once a conceptual metaphor is formed, it is then blended with other conceptual metaphors with similar cognitive structures to create increasingly complex cognitive structures.
“Conceptual metaphor and conceptual blending are among the most basic cognitive mechanisms that take us beyond minimal innate arithmetic and simple counting to the elementary arithmetic of natural numbers.” (p. 52)
It takes 2 types of metaphors to construct the exceedingly complex metaphorical blends of advanced mathematics – grounding and linking metaphors. Grounding metaphors are based in concrete sensory-motor experience. Linking metaphors join other branches of mathematics with arithmetic to construct conceptual blends. In the current discussion we are primarily interested in grounding metaphors, the basic conceptual metaphor.
It takes four of these grounded conceptual metaphors to develop the notion of the real number line. Along with the whole counting numbers, the real number line includes whole numbers, as well as rational, irrational, transcendental and negative numbers. On the surface, it appears as if all numbers are merged in this seemingly continuous line. Due to the importance of the real number line in the development of increasingly complex conceptual blends, Drs. Lakoff and Nunez referred to the 4 grounding metaphors collectively as the 4Gs.
Three of these 4 grounding metaphors lead to a unique conception what a number is. These 3 unique perspectives on Number enable the construction of 3 unique forms of mathematics. The logical structure of each form of mathematics is exceedingly similar. In fact the logic behind each mathematical form is so similar that many assume it to be the same logic.
Although the inferential structure behind each of these mathematical disciplines is similar, it is not exact. This is true of any metaphorical correspondence, literary or cognitive. In fact, this lack of exact equivalence is an innate feature of metaphor. Hence, when we say that it takes 4 metaphorical blends to generate the idea of the real number line, this means that the blends are not exact. Of course if the blends don’t have an exact fit, this means that the idea of the real number line has some logical holes as well.
Due to foundational nature of the real number line in advanced mathematics, the mathematical-scientific community has assumed that the real number line is in fact real, hence the name. The underlying assumption of this exceedingly logical community is that the logic behind the real number line is unassailable. In other words, the logic of the real number line is unitary/singular, not a metaphorical blend.
Due to this implicit, hence unexamined, belief, the mere claim that mathematics is based upon metaphorical blends raises the ire of a significant portion of this esteemed community. Anger is frequently, if not always, the result when our conceptions of reality are challenged. Hence, the mere claim that the presumed reality of the real number line, and by extension all of advanced mathematics, is based upon metaphorical blends ignites anger in those attached to the reality of science.
This unconscious attribution of reality to the real number line is based upon a natural cognitive process. As we’ve seen, we have an innate unconscious tendency to conflate events with a similar inferential, hence neurological, structure into a single complex experience. Due to the unitary nature of this cognitive blend, we take it to be real.
Conflation applies to these mathematical structures as well. The mathematical-scientific community has conflated these unique blends into a monolithic notion of number and mathematics. In other words, most of us tend to think that there is only one kind of number. This tendency to conflate experience, further leads to the belief that one logically consistent structure creates a monolithic mathematics consisting of a variety of branches all devoted to the study of the unitary Number.
There is yet another significant offshoot of our unconscious tendency to conflate experience. Many scientists in a multitude of disciplines, including members in both the hard and soft sciences, employ mathematics as a metaphor for experience. The inferential structure of the mathematics and the experience are similar. Due to unconscious neurological conflation, scientists have a tendency to believe that the connection between mathematics and empirical reality is real, not metaphorical.
Drs. Lakoff and Nunez refer to this attitude as part of the complex of beliefs that constitute the ‘romance of mathematics’.
“Ideas are necessary in mathematics and … one cannot, within mathematics, rigorously put ideas into symbols. The reason is that ideas are in our minds; even mathematical ideas are not entities within formal mathematics, and there is no branch of mathematics that concerns ideas. The link between mathematical formalisms using symbols and the ideas they are to represent is part of the study of the mind – part of cognitive science, not part of mathematics. Formalisms using symbols have to be understood, and the study of that understanding is outside mathematics per se.” (Lakoff & Nunez, Where Mathematics Comes From, 2000, p. 321)
Drs. Lakoff and Nunez devote their book, in part, to dispelling the notion that number is of one essence and that mathematics consists of an internally consistent monolithic structure. They effectively develop the notion that number and mathematics are not uni-centric, but are omni-centric instead. In other words, there are many mathematical perspectives, not just one. Further, it takes the unique talents of each of these perspectives to construct an effective mathematical metaphor that encompasses a wider range of human experience.
This series of articles is an attempt to establish that the Living Algorithm, although disobedient, provides a unique mathematical perspective that encompasses a unique range of human behavior. We will further see the current mathematical perspectives have an innate blindness to this feature of existence. In other words, disobedient equations fulfill a unique mathematical function that can’t be fulfilled by obedient equations. In brief, disobedient equations can be employed to mimic the behavior of disobedient living systems, Life. In contrast, obedient equations more appropriately model the behavior of obedient material systems, Matter. Although they are both mathematics, they are, as we shall see, based in a different class of conceptual metaphors. The obedient equations are based in the image schema of sight. In contrast, the Living Algorithm mathematical metaphor is based in our auditory-motor network, as we shall see.
Let us summarize our findings to aid assimilation and memory. Due to neurological conflation, humans tend to believe that metaphorical blends are a single complex experience. Deconstructing metaphorical blends is a psychologically healthy method for separating metaphor from reality. Separating metaphor from reality serves at least 3 positive functions. This differentiation 1) mitigates destructive emotional responses such as anger, 2) reduces negative behavioral momentum and 3) opens us up to new possibilities. For these 3 reasons, it important to spend time deconstructing metaphorical blends. The time invested in deconstruction leads to emotional growth, whether personal or cultural.
It takes 4 grounding metaphors combined with our innate mathematical abilities to create the idea of the real number line. In other words, the real number line is based upon a complex metaphorical blend. The theoretical structure of higher mathematics is founded in part upon the real number line. Due to conflation, the scientific community tends to believe that the real number is real. To open up Science to new possibilities, it is necessary to first deconstruct the preconception that the real number line is real. This unconscious preconception blocks growth, as it is based upon the misconception that mathematics has a singular logic. This monolithic perception excludes fresh perspectives. To include the disobedient equations, such as the Living Algorithm, in the complex metaphorical blend that is mathematics, it is essential to deconstruct the metaphorical blend that constitutes the real number line.