Discovery of the structure of the universe, the atom, or the DNA started by developing a model that describes the behavior of these structures. The modeling procedure is the same for different scientific disciplines, although the details of the modeling vary between different disciplines. Modeling of the same problem also changes with time as more knowledge is acquired.
In the above description of the relationship of the Earth to the sky, three different models were introduced, namely:
1. A model of the Earth at the center of the universe with the sky as a dome above Earth.
2. A model of the Earth at the center of the universe with the sky as a sphere above Earth.
3. A model of the Earth rotating around the sun.
Development of physical models will be illustrated by presenting a simple model of a vibrating pendulum or yo-yo. A basic property of the pendulum is frequency. There is nothing complex in this idea; if one watches a pendulum oscillating up and down, the number of oscillations in a second is called the frequency of vibration. This pendulum consists of a spring and a block attached to it. If the spring is held fixed, the block will be stationary at vertical equilibrium position. The spring provides the restoring force against gravity. If the block is to move from equilibrium position, the spring is stretched and it stores potential energy and a force is developed in the spring. When the block is released, the spring force pulls the block towards the equilibrium as the potential energy is converted to kinetic (motion) energy. This process of energy transfer between the potential and kinetic energy continues and causes the block to oscillate about the equilibrium position.
The frequency of the pendulum can be calculated by equating the potential and kinetic energy. This solution is subject to certain assumptions, which should only be made if the solution is easier to solve and the results are accurate enough for whatever use they are intended. Some of the logical assumptions for that spring-block system include ignoring the effects of earthquake and wind. One should not assume that the results would be correct if the pendulum is moving in a windy place or during an earthquake. If wind and earthquake are included in the solution of this system, the resulting equations are usually very complex.
Similarly, when dealing with the dynamics of the universe, one cannot include the effects of every phenomenon in the equations describing the motion of the universe. There are two reasons for that: first there are still many unknowns in the universe; second, including some known phenomena could make these equations impossible to solve. Theories dealing with the dynamics of the universe have much observational studies to support them. Mathematical reasoning is the only way to understand the fundamentals that lie behind observations. This is so because mathematics is a language in which ideas can be formulated and followed in logical steps. Many times mathematics has provided insights available in no other way. Unfortunately, some people suffer mental blocks when they see a simple equation. However, our civilization would scarcely exist without the physical laws and intellectual techniques developed by mathematical research. No one could balance his checkbook without applying mathematical rules. Many of the world’s great thinkers have decided that mathematics represents the absolute truth. "God ever geometrizes," Plato said. "God ever arithmetizes," echoed the 19th Century Prussian scientist Jacobi. In our time the British physicist Sir James Jeans declared, "The Great Architect of the Universe now begins to appear as a pure mathematician."
Today, although mathematicians affirm the universality of their subject, some deny that it possesses any absolute qualities of truth. Bertrand Russell defines mathematics as "the subject in which we never know what we are talking about nor whether what we are saying is true." He may be talking about some mathematicians who expect the universe to follow their theories, instead of searching of the actual laws that describe the behavior of the universe.
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