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A the core of any philosophical research various academic concepts from practically every scientific field, including mathematical, work together in order to explain hypotheses, especially those that appear controversial at first. Mathematical concepts are popular in this regard, because some subjects are easily defined and connected by formulating the problems in mathematical “language”. The primary objective is to establish a viable connection by utilizing figures and formulas. For example, Galileo’s mathematical-experimental method has been used to explain various geographical and astronomical phenomena, especially those relating to the movement of earth and other objects in space. Specifically, the experimental method regards testing the time (T) and the velocity (V) of falling objects in space. Owing to the difficulty that he had encountered in testing the speed, V, of a falling object in his research regarding the science of motion, he established some experimental methods, which could progressively test it in functionality with distance, D. The progress of his work’s development entailed a synergetic use of mathematical methodization and experimental approaches. These later gave birth to a whole new understanding of physics, taken together with Aristotelian physics. This method is a subject of philosophical postulations, as researchers endors or critique a given subject.

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As mentioned above, the fact that Galileo could not direcxly measure V, through deduction he used a comparative functionality based on reasoning that if V?T, then D? T2.This relation, on the other hand, is experimentally testable. The result was the validation of V?T. The idea of a mathematical-experimental method was born out of the fact that D? T2 was derived from V?T mathematically and then qualified experimentally. The whole methodological concept is referred to as a hypothetico-deductive method, because the hypothesis tested in the analysis cannot be tested directly through an experiment. In this case, a researcher tests the subject through a proxy. This forms the core of Galileo’s mathematical-experimental methodological approach. Essentially, he was not able to directly test the connection between V and T. However, he knew that if V.T was indeed true, then D.T2 has to be true as well, and hence tested the later claim in order to validate the former one through a mathematical experiment in a process of hypothetico-deduction.

The new theory of terrestrial motion

The resolution of the abovementioned supposition was used to formulate physics laws of terrestrial motion. As explained by Bernard Cohen, Galileo’s experiment birthed a new principle that had a lot of ramifications in the science discourse. New approach to physics used principles of hypothesis testing, which engages a deductive process in a gradual manner. Deduction graduates in a progressive design, entailing utilization of a known parameter in order to predict or find out the attributes of the unknown metric of interest. In this case, the prediction is intended to be as precise as possible by observing the salient features of the known objects and by testing them. As Cohen puts it, there is no doubt that instead of testing an object, say ‘I,' Galileo used it to deduce ‘II’, and then experimentally tested the newly formed object. After this, Galileo concluded that ‘II’ would hold. Indeed, in his new analysis of physics, Cohen holds that because ‘II’ is deduced mathematically and because it was tested through an experiment, the process is called the mathematico-deductive method (Cohen, 67).

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The mathematico-deductive method was instrumental for Galileo in his studies of terrestrial objects’ motion. Just as in the already discussed case of measuring velocity, it is hard to directly assign a value to important parameters in the study of movement to. Therefore, Galileo used the same method that was used for testing the distance and time repeatedly in order to determine the velocity of objects in free fall. In his experiments, Galileo used objects that he dropped from an elevated position, for example from a tower. The objects were of varying weights and sizes, and this allowed him to test and formulate his theories of velocity. In this physics experiment, both the mathematico-deductive and hypothetico-deductive methods were combined and such a combination proved to be quite helpful. These methods formed the backbone of Galileo’s work in this regard and resulted in various important findings. In studying various objects, he importantly noted that even though the objects have different weights, they fall in the same way. This enabled him to refute the argument that the speed of falling of objects differed depending on the weight of an object. Galileo reached the conclusion due to the discovery of the fact that every object experienced influence of the same gravitational force, which is practically constant on earth. Thus, the speed of various falling objects, despite their variable weights, should be the same in no other forces are present.

The utilization of methods in this regard was purposely aimed at generating or discovering mathematical laws that can describe the motion of terrestrial bodies. His experiments demonstrated that free objects tend to move horizontally with a constant speed, but their fall is vertical due to a constant acceleration. Therefore, in this regard Galileo came to a conclusion that the resulting vertical and horizontal motions need to be linked together but act independently at the same time, so that parabolic trajectories can be sustained. The mathematical manipulation used in the experiments aimed at generating a new theory of terrestrial motion so that the movements are correctly and logically related to the forces that produce them. Indeed, the main methods explained above gave a clear way of preposition of the position direction and velocity of the terrestrial objects until their settling. These new ideas and scientific approaches led to development of a whole new field of science, which explains movement and settling of the said objects.

Contribution of the theory to the approval of Copernican heliocentric theory

Proposed and developed by Nicolaus Copernicus, Copernican heliocentrism refers to an astronomical model used for estimating the motion of objects surrounding the sun in outer space. The sun is positioned at the center of the earth and other planets move around it on circular orbits. The sun is thought to be motionless, while the planets and other objects have a uniform speed on their orbital paths. The theory takes roots in the Ptolemaic system, and paints a more realistic picture of objects’ movement in the universe. It firmed up the understanding of a scientific revolution and astronomy. The Ptolemaic explanation held that even the spheres on which the planets were embedded in the cosmos would be rotating with the planets in their motion. Galileo, on the other hand, used his hypothetico-deductive theory for accepting and furthering the Copernican theory after he succeeded in understanding the behavior of object on earth. His approach enabled him to understand and appreciate that earth could circularly move around the sun without necessarily losing its moon. Despite the ensuing controversy between the Copernican theory and the popular at the time alternative explanation, Galileo stuck with the theory, owing to his experiments that painted a probable picture of objects’ movement in space.

The hypothetico-deductive theory disapproved the assertion that experiments on the earth were not sufficient for demonstrating the movement patterns of earth itself. Indeed, Copernican held that it was possible for the earth to shuttle circularly in space around the sun, while the sun still has the moon. Galileo believed that there was no way of separating the solar system, in which the satellites move along their orbits, and the universe (Galilei, Andrea, and Marenzana, 90). This presented an opportunity for Galileo to identify the salient features of the universe with regards to the attributes he studied in his experiments. The hypothetico-deductive approach became very instrumental in sorting the velocity, density and time attributes of the motion of the concerned objects. By utilizing this method, the attributes and the places of objects in the universe which were not measurable due to the unavoidable circumstances, became relatively easy to study and determinable by way of deduction and logical reasoning. Therefore, Galileo’s theory made a very significant contribution to understanding the application of the heliocentric nature of terrestrial objects and their motion and subsequent acceptance of the Copernican theory. Grasping the motion aspects regarding the speed and the variations impacted on them by the difference in their weights when falling is clearly put by Galileo in his theories.

Disapproval of the Aristotelian objections

Aristotelian objections to the hypothesis of earth in motion due to observations of free-falling bodies are contained in Aristotelian approach to physics. For Aristotle, physics is a wholesome body of natural knowledge containing principles that govern every aspect of both living and non-living things’ existence. Physics covers aspects such as change in places, all of the motions and changes due to size, number, and weight. It also considers the qualitative changes of any kind. Understanding the Aristotelian concept of motion and the universe helps in appreciating his arguments. In his physics, he held that everything below the sphere of the moon was in four main forms; the earth air, water, and fire.

Aristotle regarded motion as a process of change, especially from one place to another. He then regarded change as being manifested as an alteration in quality, corruption, and generation. Further, he saw corruption and generation as forms of change that are biological in nature, while alteration in quality is organic, such as a a falling leaf which changes its color from green to orange. Change can still be inorganic, giving an example of a served beverage changing from hot temperature to cold. Moreover, his understanding of motion and change regards it as not mathematical, an argument that directly contradicts any modern application and understanding of physics, which is completely mathematically-based. Aristotelian approach to physics does not attempt to link the concept of motion any object, terestiral or not, with a mathematical explanation or a theory, an idea completely contrary to Galileo’s approach. Finally, he believes that an object can only have one type of motion at a time, mostly natural.

The elements are in motion, which can be either radial or rectilinear. The heavier ones move downwards towards the center of earth, while the light ones remain higher. In other words, the density and weight of an object have direct influence on the velocity of a falling object, and, therefore, its effect when it falls (Galilei, Andrea, and Marenzana, 56). Accordingly, the falling objects have the original falling speed that is pre-defined, mainly depending on its speed. Moreover, the four elements have natural places they occupy in the universe. For the element earth, its place is in the center of the cosmos while that of water is just above the earth. The place of the element fire is at the periphery of the terrestrial demesne while air is just below it. The motion of the items is occasioned by the effort of the elements trying to reach their natural position. This motion is continuing because they have not been able to arrive at the said destination. He also points out that the moving elements can be forced to move in an unnatural direction such as tossing a piece of rock or stone to move upwards.

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The essence of Aristotle’s explanation of objects’ movement was that an object is falling at a speed proportional to the weight of the object. Galileo refuted such a claim by conducting his experiments to test the arguments. He demonstrated that an object’s speed does increase as they fall (Galilei, Andrea, and Marenzana, 34). However, the increase in speed has no relationship with the weight of the falling object, because the friction of the air is negligible. In fact, Aristotelian argument displaces the fundamental role played by the force of gravity, which usually acts consistently on any object. His disapproving arguments were somewhat accurate and logical given that he conducted and observed them. His discussions of the findings were also apt given that the experimentations established mathematical reasoning to establish the logical nexus.

Another one of refuted Aristotle’s arguments was that celestial bodies, such as the moon and the sun, had smooth surfaces (Galilei, Andrea, and Marenzana, 78). Aristotle also held in the same argument that the said objects belong to a different composition, not the universe and that have a spherical shape. Galileo disputes these assertions by discovering through his experiments that the surface of the moon had some resemblance with the bodies found in some places within the universe (Galilei, Andrea, and Marenzana, 78). Furthermore, Galileo’s positions and arguments were convincing and believable because, unlike Aristotle, whose claims were based on assumptions and rather simple philosophical reasoning, Galileo’s ideas and approaches took ground in experimentation. He did not conduct practical experiments in order to prove his theories.

Conclusion

Galileo’s mathematical and experiment-based approaches have made an immense contribution to explaining the motion of various objects in the universe. His experiments focused on objects of varying masses and they took in account the roles played by various aspects as gravity, wind and the atmosphere. Therefore, the results are believable, forming the basis for the acceptance of Copernican heliocentric theory and rejection of Aristotelian physics. Galileo’s synergic application of mathematico-deductive and hypothetico-deductive methods paints a practical picture of the happenings in the universe. For Aristotle on the other hand, his explanations are easily developed on simple assumptions that are not investigated. Galileo’s reasoning, therefore, comes out a strong contradiction to Aristotle’s assertions.

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