Mathematics & Cognitive Science

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Mathematics & Cognitive Science

In an attempt to make the new and revolutionary theories of cognitive science, as defined in “Philosophy in the Flesh” by Lakoff and Johnson, clear I will attempt to illustrate how this human characteristic, claimed by the theory, is used in developing arithmetic. I am not saying that metaphor theory was available when arithmetic was first ‘invented’ I am saying that the book “Where Mathematics Comes From” written by Lakoff and Nunez illustrates how mathematics can be explained using metaphor theory. In other words because we are metaphorizing creatures we are able to create mathematics.

At birth an infant has a minimal innate arithmetic ability. This ability to add and subtract small numbers is called subitizing. (I am speaking of a cardinal number—a number that specifies how many objects there are in a collection, don’t confuse this with numeral—a symbol). Many animals display this subitizing ability.

In addition to subitizing the child, while playing with objects, develops other cognitive capacities such as grouping, ordering, pairing, memory, exhaustion-detection, cardinal-number assignment, and independent order.

Subitizing ability is limited to quantities 1 to 4. As a child grows s/he learns to count beyond 4 objects. This capacity is dependent upon 1) Combinatorial-grouping—a cognitive mechanism that allows you to put together perceived or imagined groups to form larger groups. 2) Symbolizing capacity—capacity to associate physical symbols or words with numbers (quantities).

“Metaphorizing capacity: You need to be able to conceptualize cardinal numbers and arithmetic operations in terms of your experience of various kinds—experiences with groups of objects, with the part-whole structure of objects, with distances, with movement and location, and so on.”

“Conceptual-blending capacity. You need to be able to form correspondences across conceptual domains (e.g., combining subitizing with counting) and put together different conceptual metaphors to form complex metaphors.”

We commonly think of metaphor as something like analogy. We are trying to explain something to someone and we say this something new is very much like this other something you are familiar with. This is one form of metaphor but there is another metaphor that is automatic and unconscious. The child playing with objects has an experience of collecting objects in a pile. This experience results in a neurological network that we might identify as grouping. This neurological structure that contains some sort of logic related to this activity serves as a primary metaphor.

The child has various experiences resulting from playing with objects. These experiences result in mental spaces with neural structures that contain the logic resulting from the experience. When the child then begins to count perhaps on her fingers these mental spaces containing the experiences automatically map to a new mental space and become the logic and inference patterns to make it possible for the child to count because counting contains similar operations.

Primary metaphors are the contents of mental spaces developed in experience and the contents then pass to another mental space to become the bases for a new concept. The contents of space A is mapped to space B to then be the foundation for the new concept at space B. This mapping is automatic and unconscious.

Quotes from “Where Mathematics Comes From”

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Neural Modeling

Cognitive science has radically attacked the traditional Western philosophical position that there is a dichotomy between perception and conception. This traditional view that perception is strictly a faculty of body and conception (the formation and use of concepts) is purely mental and wholly separate from and independent of our ability to perceive and move.

Cognitive science has introduced revolutionary theories that, if true, will change dramatically the views of Western philosophy. Advocates of the traditional view will, of course, “say that conceptual structure must have a neural realization in the brain, which just happens to reside in a body. But they deny that anything about the body is essential for characterizing what concepts are.”

The cognitive science claim is that “the very properties of concepts are created as a result of the way the brain and body are structured and the way they function in interpersonal relations and in the physical world.”

The embodied-mind hypothesis therefore radically undercuts the perception/conception distinction. In an embodied mind, it is conceivable that the same neural system engaged in perception (or in bodily movements) plays a central role in conception. Indeed, in recent neural modeling research, models of perceptual mechanisms and motor schemas can actually do conception work in language learning and in reasoning.

A standard technique for checking out new ideas is to create computer models of the idea and subject that model to simulated conditions to determine if the model behaves as does the reality. Such modeling techniques are used constantly in projecting behavior of meteorological parameters.

Neural computer models have shown that the types of operations required to perceive and move in space require the very same type of capability associated with reasoning. That is, neural models capable of doing all of the things that a body must be able to do when perceiving and moving can also perform the same kinds of actions associated with reasoning, i.e. inferring, categorizing, and conceiving.

Our understanding of biology indicates that the body has a marvelous ability to do as any handyman does, i.e. make do with what is at hand. The body would, it seems logical to assume, take these abilities that exist in all creatures that move and survive in space and with such fundamental capabilities reshape it through evolution to become what we now know as our ability to reason. The first budding of the reasoning ability exists in all creatures that function as perceiving, moving, surviving, creatures.

Cognitive science has, it seems to me, connected our ability to reason with our bodies in such away as to make sense out of connecting reason with our biological evolution in ways that Western philosophy has not done, as far as I know.

It seems to me that Western philosophical tradition as always tried to separate mind from body and in so doing has never been able to show how mind, as was conceived by this tradition, could be part of Darwin’s theory of natural selection. Cognitive science now provides us with a comprehensible model for grounding all that we are both bodily and mentally into a unified whole that makes sense without all of the attempts to make mind as some kind of transcendent, mystical, reality unassociated with biology.

Quotes from “Philosophy in the Flesh”

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Metaphors

Throughout our life we constantly make judgments about such abstract matters as difference, importance, difficulty, and morality, and we have subjective experiences such as affection, desire, love, intimacy and achievement. Cognitive science claims that the manner in which we conceptualize and reason about these matters are determined, to one extinct or another, by sensorimotor domains of experience. CS claims that, in many cases, early experiences of normal mundane manipulations of objects become the prototypes from which these later concrete and abstract judgments are made.

“When we conceptualize understanding an idea (subjective experience) in terms of grasping an object (sensorimotor experience) and failing to understand an idea as having it go right by us or over our heads” we are using a sensorimotor experience as the metaphor for the subjective experience. The metaphor ‘understand is grasp’ results from our conflating a sensorimotor happening with a later subjective experience.

Metaphor is a standard means we have of understanding an unknown by association with a known. When we analyze the metaphor ‘bad is stinky’ we will find: we are making a subjective judgment wherein the olfactory sensation becomes the source of the judgment. ‘This movie stinks’ is a subjective judgment and it is made in this manner because a sensorimotor experience is the structure for making this judgment.

Why is the premise “A straight line is the shortest distance between two points” self-evident. It is because this is one of the first things an infant learns and it is verified and reinforced constantly throughout life by our sensorimotor experiences. The metaphor ‘more is up’ is not so pervasive in our experience but its rationale is similar.

If we recognize metaphor as a means to associate something new with something old, something known with something unknown, we can begin to understand what CS is proposing in this revolutionary theory. CS is presenting a theory based upon empirical evidence gathered by the combined effort of linguists, philosophers, and neural physicists that metaphor is a very necessary element of our ability to reason as we do.

We normally think of metaphor as a tool of language whereby one can enlighten another by making an association of an unknown with a known. CS is making a much more radical use of metaphor.

CS is claiming that the neural structure of sensorimotor experience is mapped onto the mental space for another experience that is not sensorimotor but subjective and that this neural mapping, which is unconscious and automatic, serves as part of the “DNA” of the subjective experience. The sensorimotor experience serves the role of an axiom for the subjective experience.

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Complex Metaphor Becomes Mathematics

Primary metaphors function somewhat like atoms that can be joined into molecules and these into a compound neural network. On the back cover of “Where Mathematics Comes From” is written “In this acclaimed study of cognitive science of mathematical ideas, renowned linguist George Lakoff pairs with psychologist Rafael Nunez to offer a new understanding of how we conceive and understand mathematical concepts.”

“Abstract ideas, for the most part, arise via conceptual metaphor—a cognitive mechanism that derives abstract thinking from the way we function in the everyday physical world. Conceptual metaphor plays a central and defining role in the formation of mathematical ideas within the cognitive unconscious—from arithmetic and algebra to sets and logic to infinity in all of its forms. The brains mathematics is mathematics, the only mathematics we know or can know.”

We are acculturated to recognize that a useful life is a life with purpose. The complex metaphor ‘A Purposeful Life Is a Journey’ is constructed from primary metaphors: ‘purpose is destination’ and ‘action is motion’; and a cultural belief that ‘people should have a purpose’.

A Purposeful Life Is A Journey Metaphor
A purposeful life is a journey.
A person living a life is a traveler.
Life goals are destinations
A life plan is an itinerary.

This metaphor has strong influence on how we conduct our lives. This influence arises from the complex metaphor’s entailments: A journey, with its accompanying complications, requires planning, and the necessary means.

Primary metaphors ‘ground’ concepts to sensorimotor experience. Is this grounding lost in a complex metaphor? “Not at all.” Complex metaphors are composed of primary metaphors and the whole is grounded by its parts. “The grounding of A Purposeful Life Is A Journey is given by individual groundings of each component primary metaphor.”

The ideas for this post come from “Philosophy in the Flesh”. The quotes are from “Where Mathematics Comes From”