To better understand the unique nature of these disobedient equations, let us first establish a cognitive perspective.
Most, if not all, of the following discussion is based upon Where Mathematics Comes From by Lakoff & Nunez, 2000. This exceptional book is one of the Author’s sacred texts. In this context, a sacred text is written by a certified expert(s) in the field, an Insider. As such, the Author, as an Outsider, takes what is written as gospel truth. In other words, he doesn’t challenge any of the statements. George Lakoff and Rafael Nunez, are highly respected professionals in the field of cognitive science. Having published many articles and books, these scholars are on the cutting edge of their field. More importantly, the ideas that they communicated rang true with the Author. In the terms of the Music Metaphor (to be developed), the concepts resonated with the Author’s personal experience. Hence, he accepted them as true. To establish the cognitive foundations of the current discussion, we will quote extensively from this work, whose subtitle is instructive – How the Embodied Mind Brings Mathematics into Being.
Up to this point in our discussion, we’ve been employing the word ‘metaphor’ casually. It is time for a more technical definition.
“Metaphor, long thought to be a figure of speech, has recently been shown to be a central feature of everyday thought. Metaphor is not a mere embellishment; it is the basic means by which abstract thought is made possible. One of the principle results in cognitive science is that abstract concepts are typically understood, via metaphor, in terms of more concrete concepts. This phenomenon has been studied extensively for more than two decades and is in general as well established as any result in cognitive science … . One of the major results is that the metaphorical mappings are systematic and not arbitrary.” (pp. 39-41)
In other words, metaphor is not just a literary technique that we learn in school. Instead, it is the way in which we organize our mental world. Cognitive science calls this foundational construct of abstract thought, a conceptual metaphor. Conceptual metaphors, with their systematic mappings are qualitatively different from the lyrical mappings of the literary metaphor. The technical definition follows:
“Conceptual metaphor is a cognitive mechanism for allowing us to reason about one kind of thing as if it were another. … ‘Conceptual metaphor’ has a technical meaning. It is a grounded, inference preserving cross-domain mapping – a neural mechanism that allows us to use the inferential structure of one conceptual domain (say geometry) to reason about another (say arithmetic). … Each choice of metaphor provides different inferences and determines a different subject matter.” (pp. 6-7)
The conceptual metaphor does not apply to merely specialized circumstances, such as mathematical thought. Instead, it is employed regularly and relatively continuously to organize and communicate our experience.
“Hundreds of such conceptual metaphors have been studied in detail. … On the whole, they are used unconsciously, effortlessly, and automatically in everyday discourse; that is they are part of the cognitive unconscious. Many arise naturally from correlations in our commonplace experience, especially our experience as children.” (p. 41)
Cognitive science has only come upon the notion of the conceptual metaphor relatively recently. It is one of the big three, in terms of current developements in modern cognitive theory.
“In recent years, there have been revolutionary advances in cognitive science …. Perhaps the most profound of these new insights are the following:
1. The embodiment of mind. The detailed nature of our bodies, our brains, and our everyday functioning in the world structures human concepts and human reason. …
2. The cognitive unconscious. Most thought is unconscious – not repressed in the Freudian sense but simply inaccessible to direct conscious introspection. We cannot look directly at our conceptual systems and at our low-level thought processes. …
3. Metaphorical thought. For the most part, human beings conceptualize abstract concepts in concrete terms, using ideas and modes of reasoning grounded in the sensory-motor system. The mechanism by which the abstract is comprehended in terms of the concrete is called conceptual metaphor.” (pp. 4-5)
Needless to say, the conceptual metaphor is of huge importance in the understanding of human cognition. While some of these advances in cognitive theory are based in linguistics, the study of language usage, it turns out that the conceptual metaphor has a neurological basis as well. This has to do with the notion of conflation.
“Conflation is part of embodied cognition. It is the simultaneous activation of two distinct areas of our brain, each concerned with distinct aspects of experience, like the physical experience of warmth and the emotional experience of affection. In a conflation, the two kinds of experience occur inseparably. The coactivation of two or more parts of the brain generates a single complex experience – an experience of affection-with warmth or difficulty–with–a-physical-burden. It is via such conflations that links across domains are developed – links that often result in conceptual metaphor, in which one domain is conceptualized in terms of the other.” (p. 42)
The process of conflation that results in a conceptual metaphor is not arbitrary, but has distinct rules.
“Each such conceptual metaphor has the same structure. Each is a unidirectional mapping from entities in one conceptual domain to corresponding entities in another conceptual domain. As such conceptual metaphors are part of our system of thought. Their primary function is to allow us to reason about relatively abstract domains using the inferential structure of relatively concrete domains. In metaphor, conceptual cross-domain mapping is primary, metaphorical language is secondary, deriving from conceptual mapping.” (p. 42)
This is probably more than you ever wanted to know about cognitive science in general or conceptual metaphors specifically. However, this basic understanding of current cognitive constructs is essential for the discussion that follows. We are going to be leaning heavily upon these words.
Let us reiterate what we have learned to reinforce understanding.
In their groundbreaking book, Where Mathematics Comes From, Drs. Lakoff and Nunez develop the notion that all abstract thought is based upon a metaphorical relationship with our concrete sensory/motor network. This relationship is not lyrical, but conceptual. The inferential structure of the concrete sensual experience is mapped onto the abstract realm of thought. In other words, we apply the way that we reason about our empirical world of the senses to the abstract world of thought. Abstract reasoning is embodied in our sensory relationship with our environment.
Further, this metaphorical mapping is not arbitrary, approximate or conscious. Instead, the mapping always occurs from the concrete to the abstract realm and follows precise rules that operate on unconscious levels. The application of the inferential structure of our senses to the abstract realm of thought occurs automatically, without effort or consciousness. Conflation is the name that cognitive scientists give to this automatic neurological process, where two experiences are merged into one. “The coactivation of two or more parts of the brain generates a single complex experience.”
Complicating the story, the metaphors are merged in conceptual blends to create even more abstract structures. Multilayered metaphorical blends are the norm rather the exception in our mental landscape. These composite neurological structures are not experienced as blends, but as a single complex experience. Again, this neurological blending occurs automatically, effortlessly and unconsciously, rather than consciously or deliberately.
Although the neurological conflation of experience occurs spontaneously, the human species has consciously followed a similar procedure to develop a consistent system of logic. Mathematicians, logicians and scientists have employed multilayered conceptual blending to create an impressive tightly-defined structure of thought that includes mathematics, science and logic. The Greeks called this structure – Logos.
Unconsciously patterning themselves after our neurological operations, the mathematical community has tended to view Logos, as a single complex experience – a monolithic singularity, rather than as a blend. In similar fashion the West has tended to view our God as a single complex experience, rather than as a conceptual blend. We even tend to view each other and ourselves as a singularity, rather than as a composite structure.
The problems with turning a composite metaphorical blending into a singular experience are based upon a simple indisputable fact. ‘Each choice of metaphor provides different inferences and determines a different subject matter.’ In other words, each metaphor applies to its own realm, but not necessarily straight-across-the-board to other realms.
The logical similarities between the inferential structures allows the effective employment of metaphorical blending. However, with each blending, with each abstraction of the concrete, certain innate difficulties arise. They fall in 2 general categories.
Because the mapping is not exact, but metaphorical, truncations must occur to enable an exact fit. For instance, no one in their right mind would apply the laws of thermodynamics to a warm smile. Nor would anyone attempt to weigh the heavy burden of an individual’s life. Although a mother’s bodily warmth is conflated with her emotional love, each realm has distinct differences that can’t be compared with each other. In other words, the conceptual metaphor inherently truncates or excludes part of the experience so that the inferential structure continues to apply. In similar fashion, the mathematical blending of metaphors automatically excludes certain aspects of existence in order to remain consistent.
The second difficulty arises at the boundaries of the metaphorical blends. The logical fit might be excellent in the broad center, i.e. for most aspects of the metaphorical blend. For instance, geometry and arithmetic are an excellent logical fit, many even thought a perfect fit. However, when the logical structure was extended to its limits in the late 19th century, a split occurred. Based in visual logic, geometry is able to conceive of space and time as continuous. Based in number, arithmetic must digitize the notions of space and time. The analog and digital perception of reality, while similar, have some distinct differences. As an example, note the reemergence of the analog record, despite the many advantages of the digital CD.
However, due to our neurological tendency to turn our metaphorical blends into a single complex experience, the mathematician/logicians of the day, as we shall see, tended to view the arithmetization of geometry as an absolute logical solution to certain paradoxical problems. This monolithic approach of necessity truncated the continuity of experience from their worldview in order to remain logically consistent.
A more appropriate response to metaphorical blending is to consciously differentiate (deconstruct) the inferential structure of each metaphor. Although the blending occurs unconsciously, as it is an innate feature of unconscious cognition, the differentiation is a conscious process. The differentiation of metaphor from reality is a common feature of logical discourse. We apply our reasoning skills to determine the limits of literary metaphor – to which features of reality it applies, and to those which it doesn’t apply.
In similar fashion, we are going to consciously differentiate the conceptual metaphorical blends of mathematics. Via this differentiation process, we hope to elucidate the visual basis of geometry and numerical basis of arithmetic. Both of these logical systems lead to the Container logic of traditional set theory with their obedient equations. This context will enable us to better understand the auditory logic of disobedient equations, such as the Living Algorithm.
Now that we have established a cognitive foundation, we can move on to the history of math from a cognitive perspective. This will enable us to understand the historical context for 1) the glorification of obedient equations, 2) the disdain for, or prejudice against, disobedient equations, and 3) why disobedient equations are required to model the behavior of living matter, i.e. humans – us.