vol I chap 3 sect 3
Previous: 3.2. Generation of a mental Global Positioning System (GPS).
3.3. Cognitive procedures for creating conceptual models.¶
Creating scientific knowledge implies the development of learning activities for asking and answering pertinent questions, as well as the gradual integration and evaluation of practical visions and procedures for better understanding. All together might allow us to obtain good explanations of different phenomena by building better and better models. The word modelx comes from Latin modulus, a measure. In particular, conceptual models are simplified representations of the main characteristics of objects, persons, products, systems or circumstances.
X: https://en.wikipedia.org/wiki/Model
In this section we present four cognitive procedures for creating scientific models with the following purposes: to introduce scientific concepts (inquiring procedures), to propose and test conceptual relationships (training procedures) and to describe and apply successful cognitive structures (comprehension procedures). Then, the critical revision of the previous three procedures will correspond to a metacognitive procedure (to think about was taught). The aim is to have a guide for cognitive navigation for building, understanding, communicating and applying conceptual models.
To accomplish previous purposes, it is helpful to navigate cognitive maps representing models and containing concepts, conceptual relationships and structures. One possibility is to use a cognitive GPS for guiding us to find positions and orientations in a cognitive space of ideas, questions and solutions. The working steps of such cognitive GPS would satisfy similar conditions as those of a typical GPS, like those used in common cellular phones. These cognitive GPS might follow the four cognitive procedures for creating scientific models: Inquiring, Training, Comprehension and Metacognition.
As we have seen in the first part of previous section 3.2, the following technological conditions are required to function in a communication GPS used for localization and orientation: a system of satellites surrounding Earth generates signals indicating their positions as well as the time measured in their clocks (this might correspond to an inquiring procedure); the users´ cellular phones capture the signals from the satellites looking for visualizations of places and trajectories (this might correspond to a training procedure), and a system of radars on earth provides information for the localization and orientation of the satellites as well as for the registration of their orbits around earth (this might correspond to a comprehension procedure). A fourth metacognition procedure might imply in this case to take into account two relativistic corrections to the oscillatory frequency of the atomic clocks located inside the satellites: one is because the intensity of the gravitational field varies with altitude and the other is because the clock moves at high velocities with respect the observer on Earth.
Now we propose to connect each one of the functions of the four neural cells (place, grid, direction and border) that integrates a mental GPS, to the three segments composing a communication GPS for localization and orientation (the space, the control, and the user) and the relativistic corrections made to the oscillatory frequency of the atomic clocks located inside the satellites. Then. we associate these four elements (three segments and a correction) to the cognitive procedures for creating conceptual models where a cognitive GPS functions in terms of procedures for Inquiring, Training, Comprehension, and Metacognition.
Each one of these three GPS is a useful tool for map navigation in a different environment: the mental GPS corresponds to neural networks working in the brain, the communication GPS is for localization and orientation used by humans in technological contexts, and the cognitive GPS is for the development of conceptual models. Next Table summarizes all these connections; afterwards we will present an application of the four cognitive procedures for creating conceptual models to the description of those physical models proposed by Kepler, Newton and Einstein to explain the orbit of the planet Mercury.
Description of the perihelion precession of Mercury.¶
What follows is an example of application of the previously described Cognitive procedures for creating conceptual models. We consider the evolution in the geometrical and physical explanations concerning the planetary orbits of Mercury. We focus on the learning outcomes derived from different levels of explanations of how, when and why the first planet in our planetary system moves. We briefly describe the main characteristics of the physical conceptual models presented by Kepler, Newton and Einstein. We propose to associate Kepler´s model to the Inquiring procedure, Newton´s model to the Training procedure, and Einstein´s model to the Comprehension procedure. The Metacognition procedure is referred to a future theory of quantum gravity.
Inquiring procedure: to identify what is understood according to the three laws of motion formulated by Johannes Kepler(1571-1630). The first two laws were published in 1609 in Astronomia Nova and the third law in 1619 in Harmonice Mundi.
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First Kepler´s law (the law of the orbits): The planets move around the Sun in orbits which are ellipses, with the Sun at one focus.
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Second Kepler´s law (the law of the areas): A line joining a planet to the Sun sweeps over equal areas in equal times.
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Third Kepler’s law (the law of the harmonies): The squares of the periods of revolution of the planets around the Sun are proportional to the cubes of the (mean) radii of their respective orbits.
The formulation of these laws introduced new concepts regarding planetary movements: the motion trajectories were described in terms of elliptic orbits instead of the Ptolemaic notions of epicycles and deferents, and the speed of motion did not correspond to uniform circular orbits but was assumed to vary inversely as the planet distance to the Sun.
Training procedure: to make clear Newton´s scientific reasonings for explaining what previously has been observed, registered and interpreted.
Isaac Newton (1642-1727) published in 1687 three books called Philosophiae Naturalis Principia Mathematica; the first two dealt with general theorems concerning the motions of bodies and the third one contained applications to the solar system. Newton proposed that a gravitational attractive force \(F_G = G [\frac{(m_S m_P)}{r^2}]\) interacted instantly and at distance when the two interacting bodies, the sun of mass \(m_S\) and a planet of mass \(m_P\), were separated by a distance \(r\); \(G\) was a constant. Also, he defined force as “the rate of change of momentum” \(F=\frac{d_p}{d_t}\), where the linear momentum was defined as \(p = mv\), \(m\) representing the mass and \(v\) the velocity.
By replacing the expression of \(F_G\) in the general equation \(F=\frac{d_p}{d_t}\), Kepler´s laws can be explained in terms of the following mathematical considerations:
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The first law is a consequence of the fact that the force \(F_G\) is central, it is oriented in the direction of the sun to the planets.
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The second law is a consequence of the conservation of angular momentum \(l = r \times p\). In the case of planar elliptic orbits this implies that the quantity mvr is constant. This means that the planet moves faster at short distances in the perihelion (the distance sun-planet is the shortest) and it moves slower at long distances in the aphelion (the distance sun-planet is the largest).
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The third law is a consequence of the assumption that gravitational force is inversely proportional to the square of the distance (\(F \thicksim 1/r^2\)).
Newton could explain what Kepler predicted but without describing how the gravitational force was produced. He was able to change the magic approach due to Kepler into a mysterious explanation that promoted understanding but left open questions about the origin of the force that produced the observed planetary movements.
Comprehension procedure: to describe the theories of special and general relativity created by Albert Einstein where he explains the physical properties of the universe in terms of geometrical properties.
Careful measurements have shown that the position of the perihelion of Mercury appeared to be delayed after completing one full year of revolution around the Sun. During a century the principal axe of the elliptic orbit of Mercury had a very small deviation of 43 seconds of arc (Figure 3.5). According to the theory of general relativity this physical effect was produced by the geometrical circumstance that a curvature of space was produced by the enormous concentration of matter existing in the Sun. This explanation indicates that the gravitational force has a geometrodynamic nature.
Figure 3.6. Perihelion precession of planet Mercury showing the change of orientation of the orbital ellipse within its orbital plane as an effect of the curvature of spacetime.
Metacognition procedure: to define the frontiers of ignorance after reviewing assumptions, procedures and results as well as contrasting them with the experimental tests of calculated predictions. Concerning the real structure of physical spaces, according to the available results of observations, experiments and calculations, it is now clear that the mathematical structure of such spaces is not Euclidean. Nevertheless, up to now a quantum gravity theory is still under development.
The explanations given by Kepler, Newton and Einstein are elements in a chain of concepts, models and theories that solve some problems but leave open other questions. All kinds of explanations have their own region of validity. More about gravitation will be discussed in next Chapter 4. Geometrization of the description of motion.