In their in-depth deconstruction of the conceptual blends of mathematics, Drs. Lakoff and Nunez regularly link the abstract thinking of mathematics with our concrete sensory/motor network. The arrow points one way only – from the concrete to the abstract. By the conclusion of their book, Where Mathematics Comes From, the illustrious authors have deconstructed the complex metaphorical blends that are at the foundation to the entire structure of mathematics. From the basic conceptual metaphors that are grounded in specific senses, the authors proceed step-by-careful step to increasingly complex conceptual blends. Inevitably, the entire discipline of mathematics is encompassed from their cognitive perspective.
This mathematical deconstruction is critical, as well as descriptive. In addition to describing the very solid foundations of mathematics, the authors also expose the inconsistencies at the borders of the increasingly complex conceptual blends. They also illustrate how entire areas of our cognitive existence are excluded from investigation due to the discipline’s demand for rigor. These are the two fallacies of metaphorical thinking, exclusion and paradox (discussed above).
The authors’ intent to deconstruct mathematics has a cognitive basis. As mentioned in a prior article, conflation is the process by which two neurological events are joined as a single complex experience. This process occurs effortlessly, unconsciously, and automatically. One of the results is a conceptual metaphor, which is based upon an experience grounded in the concrete reality of the senses. The innate neurological inferential structure of the senses is applied straight across the board to specific features of all forms of abstract thinking, including words.
However, this linkage is based in metaphor. As such, there are distinct limitations to the association, as there would be with any metaphor, literary or conceptual. First and foremost, metaphorical associations, no matter how seemingly accurate they are, do not constitute reality. Metaphor mirrors reality, but is not reality. Although linked with a color, the word ‘blue’ is not the color. Although Newton’s equation for gravity accurately reflects the relationships between objects, it is not gravity.
Due to the neurological conflation of our innate cognitive processes, we take these conceptual metaphors to be real. In other words, we have an inherent tendency to believe that our conceptual metaphors and then our conceptual blends are real. We take the symbol for reality. This tendency is based in the neurological reality of unconscious conflation.
When a literary metaphor is originally employed, there is distinct separation between the metaphor and the reality. We humans regularly employ metaphor to explain unusual phenomenon. “It tastes like chicken.” However, once an individual or culture employs the metaphor for a sufficient length of time, the metaphorical connection is lost. For instance, when an individual says they have a heavy workload, they do not mean that it weighs a lot. A warm smile does not refer to temperature. The actual definition of the word expands to include new phenomena within its scope. Yet, the metaphorical descriptor, while allowing us to broaden the base of our communication, is no longer connected to the sensual reality.
The differences between literary metaphor and reality are distinct and somewhat obvious. It is qualitatively harder to differentiate the neurological reality of conceptual blends from empirical reality. We attribute reality to our conceptual blends almost immediately. As a rule, the longer we embrace a conceptual blend the more reality we attribute to it. As another rule, the more reality we attribute to any abstract construct, such as patriotism, the greater our emotional investment. In other words, the deconstruction of conceptual metaphors that people take to be real causes an emotional reaction – most often anger. Deconstructing long-term conflations tends to make people mad.
In fact for those that practice self-cultivation, anger is a sign that we are attached to our conflation of reality. Accordingly, we deliberately and consciously attempt to deconstruct these metaphorical blends, as they are at the root of emotional attachment. Breaking these emotional chains sets us free from the anger response that disturbs our peace and the mental conditioning that frequently leads to detrimental behavior. Further, by clearing the path of mental obstructions, we are open to more possibilities. In brief, deconstructing our metaphorical blends enables us to grow as human beings. Similarly deconstructing the metaphorical blends of our culture, while painful, serves the purpose of freeing the community from emotional responses that disturb inner peace and the knee-jerk responses that are so destructive to the common good. Further the deconstruction of illusion opens the culture up to new ways of thinking about existence.
In this regard, Drs. Lakoff and Nunez took it upon themselves to deconstruct the unconscious conflations of the mathematical community. Of course, this strategy evoked the ire of those who had come to believe that their mathematical symbols constituted objective reality. Those with no emotional investment in the supposed reality of their profession reacted with curiosity rather than emotion.
Reiterating for better retention, conflation is a real neurological phenomenon that joins events as a single complex experience. The conceptual metaphor is one result of conflation. By employing the logic of our sensory/motor system, we create metaphorical symbols, both verbal and mathematical, that enable us to better understand and respond to environmental stimuli. However, we have an innate tendency to believe that these conflationary symbols are real. The mental investment in this reality creates an emotional attachment. As such, deconstructing the conflation is frequently resisted with an angry response. This anger is perhaps due to cognitive dissonance. It hurts to discover that our presumed reality is but a metaphor.
To achieve freedom from these destructive emotions, it is essential to deconstruct the metaphorical relationships that are at the basis of thought. With greater emotional freedom, the individual and culture are more able to fulfill potentials. As a community service, Drs. Lakoff and Nunez attempt to deconstruct the metaphorical blends that are the basis of mathematical understanding and misunderstanding.
By exposing the misunderstandings, the authors create some openings and reveal some holes in the supposedly monolithic structure of mathematics. By creating these openings, it is easier to see where disobedient equations, such as the Living Algorithm, fit in. By revealing the holes, the gaps in understanding are also more apparent. Could disobedient equations be required to fill some of these holes? As we shall see, one major gap has to do with dynamic, self-reflective, interactive, process-based systems, such as Life. Is it possible that the dynamic, self-referencing, interactive, process-based Living Algorithm might fill this gap? In that sense, these articles deconstruct the mathematical conceptual blends in order to better understand the context and purpose of the Living Algorithm. We believe and will argue that the Living Algorithm System is the most appropriate mathematical metaphor for Life.
To deconstruct the supposedly monolithic logical structure of mathematics, we begin with the foundation – Number. Most take Number to be a permanent, inviolable essence. It is an entity that we take for granted – thinking instinctively, that the concept is its symbol. Yet this neurological conflation is based in a metaphorical connection, as we shall see. And no metaphor is an exact fit for reality.
The conceptual metaphor blends our innate mathematical abilities with other types of cognition to create an inferential structure that can be employed to better understand our world. To deconstruct this conceptual blend, we must first examine our innate mathematical abilities. To that end, check out the next article in the series – Subitizing, our Innate Mathematical Ability.