Isaac Newton, 1641-1727 Apple tree outside Newton's birthplace. Newton's vague idea (the proverbial apple on the noggin): There is an unseen force of gravity that affects all objects, whether they are apples a few feet away or planetary satellites many miles away. This force brings about observable movements in the objects. This force acts instantaneously and across any medium for infinite extent. The force is greater for larger objects, and the force diminishes with distance. (Pretty bizarre idea, isn't it? You can read Newton's Principia online.)

Newton's precise formalizations:

Law of Universal Gravitation:    

• applies to particles
• applies independently (additively) to all particles simultaneously
• G is a parameter, estimated from observed data. It turns out that G is essentially the same for all known locations and situations in the universe.

Stroboscopic photography verifies a prediction of Newton's theory.

A simple prediction of the theory:

For a small object of mass m_o near the surface of the earth, the object's acceleration a is given by:

A related prediction:

Because acceleration is constant g (for small distances near the surface of the earth), we can integrate to find the equation for the position of the object as it falls:

(In these integrations I have left out the constants that represent the initial velocity and position.) So, with constant increments in time, the distance fallen should be proportional to time squared, i.e., 0, 1, 4, 9, 16, etc. Galileo had, a few decades before, already observed this with objects rolling down an inclined plane. He noted that the distance travelled, during consecutive time intervals, increased by a constant: 1, 3, 5, 7, etc., just as Newton's laws predict. (Newton was born the year that Galileo died.)

Models, in general, of the physical world:

In general, models of the physical world try to describe changes in observable physical attributes in terms of unseen physical components that cause the changes. Newton's theory of gravity explains changes in position with time (the observables) in terms of gravity and mass (which are not directly observable).

Models in science are replete with constructs that represent unseen physical entities: In physics there are electrical forces, magnetic forces, nuclear forces, etc. In chemistry there are electron orbitals, with shells and subshells. In biology there are germs that carry disease and genes that carry inherited traits (both unobservable when first proposed).

Observable locations of particles at one moment in time input ====> Unobservable physical processes output =====> Observable locations of particles at next moment in time || || || V Selection, Measurement, Summarization ^ || || V ? Selection, Measurement, Summarization || || || V Formal representation of locations at time t input ====> Formal transformation output =====> Formal representation of locations at time t+1 (prediction) ? = Formal representation of locations at time t+1 (empirical data)

Modelers of the physical world want this diagram to commute; i.e., the formal representation of the predicted world should match the formal representation of the actual world. This diagram will be discussed at length in lecture.

Models go beyond vague ideas about hidden constructs. Models formalize the ideas. Formalization has several attractive qualities: First, it makes the idea publically accessible. When an idea is only a vague intuition in the mind of the theorist, then the only way to really know what the theory predicts is to ask the theorist. When the model is formalized, then anyone can determine what the theory predicts. The predictions might be derivable from analytical mathematical methods, or they might only be derivable from computer simulations. Second, formalization makes the theory precise. The predictions are quantitative and specific, so that the theory might actually be tested. And, if the theory holds up, then its predictions are quantitatively useful.

When a theory works extremely accurately in a wide variety of situations, then people start to believe that the formal constructs in the model represent actual hidden entities in the world. Because Newton's theory of gravity has worked so remarkably well, people actually began to think that there was an unseen force that acted instantaneously at all distances through all materials, pulling comets to the sun and knees to the ground. The concept of gravity has become part of our cultural intuition about the physical world. So when Einstein came along and obviated gravity, replacing it with geodesics in space-time, these new formal constructs were a "revolution." No circumspect scientist reifies theoretical constructs, regardless of how well they fit observations. The theoretical constructs are always just a metaphorical description. But this does not mean that they are trivial or frivolous. Good theories are, ultimately, widely useful for applications, and also deeply edifying for the explanatory perspective they yield.

Models, in general, of the cognitive world:

Modeling in cognitive science has essentially the same structure as modeling in physical science. All that differs is the selection of physical attributes and processes that are deemed interesting. Those physical attributes of the world that seem relevant for explaining cognition are called "stimuli," and those physical attributes of the world that seem relevant as indicators of cognition are called "behavior" or "responses."

Observable stimuli input ====> Unobservable cognitive processes output =====> Observable behavior (a.k.a. responses) || || || V Selection, Measurement, Summarization ^ || || V ? Selection, Measurement, Summarization || || || V Formal representation of stimuli input ====> Formal transformation output =====> Formal representation of behavior (prediction) ? = Formal representation of behavior (empirical data)

Modelers of the cognitive world want this diagram to commute; i.e., the formal representation of the predicted behavior should match the formal representation of the actual behavior. This diagram will be discussed at length in lecture.

Models in cognitive science specify formal descriptions of the incoming stimuli, the outgoing behavior, and the intermediate transformed states. These formal descriptions are called representations. A representation explicitly describes particular contents in a particular format. For example, a phone book represents particular contents about people, --their phone number, address, and name-- in a particular format --alphabetical by last name. Models in cognitive science also specify formal descriptions of transformations on these representations. Such formal transformations are called processes.

Any model in cognitive science specifies representation and process. But (and here's the news) so does any model in physical science. For example, Newton's model of gravity specifies certain representations of the world --position, time, mass, force-- and certain processes that transform positions at one time into new positions at a later time --multiplicative combination of masses and distances. Thus, represenation and process are in the model, whether the model refers to the locations of particles or to the behavior of cognizant beings.

Just as physicists can begin to believe that the representations and processes in a successful model actually exist in the world, cognitive scientists can begin to believe that the representations and processes in a successful model actually exist in the mind. I believe that such reification of models is a mistake. There is a lot of philosophical controversy about whether or not the mind actually has representations and processes, and exactly what these terms mean in this context, and whether or not artificial cognition can really be created by copying the representations and processes on other hardware. I believe that such issues can remain unresolved while significant progress is made in models or cognition, just as significant progress has been made in physics despite philosophical uncertainty about whether or not a force of gravity really exists.

Summary of the main points:

• Formal models are specifications of (more vague) explanatory theories.
• Models posit formal constructs that reflect hidden processes and explanatory principles.
• Models in the physical sciences and in the cognitive sciences have essentially the same (meta-)structure. They both specify formal representations and processes.
• Reification of formal constructs can lead to unresolvable philosophical conundrums.