describes nature. Aristotle's approach is today referred to as formalism .
In both interpretations the axioms are identified following observations. Aristotle, for instance, carried out extensive dissections of animals to learn about their anatomy. But once the axioms were adopted and mathematical results were derived, neither approach deemed it necessary to carry out experiments to compare mathematical results with what actually occurs in nature. Both of these great scholars and their Greek disciples actually opposed performing such experiments. They justified their opposition to experiments by claiming that appearances are affected by optical illusions and are likely to mislead, whereas logic is irrefutable. (It was not until thousands of years later that mental illusions were recognized and studied.) Thus, as soon as we find the self-evident axioms, the path forward using the power of logic is superior to progress based on appearances. This outlook held sway for thousands of years. It was not until the seventeenth century that the currently prevalent scientific practice, which demands experiments that corroborate the theory, became established. Onepossible reason for the fact that both Plato and Aristotle and most of the leading philosophers who succeeded them held extreme anti-experiment views is that they came from wealthy aristocratic families who considered physical work to be menial, inferior, and even contemptible.
The debate initiated by Plato and Aristotle on the essence of mathematics has continued for two thousand four hundred years and continues to this day. Scholarly articles supporting one view or the other, or proposing improvements to them, continue to appear in professional literature. The debate has no direct effect on mathematicians carrying out research or extending the boundaries of mathematics. Thus, so the story goes, it is quite normal for a mathematician, when asked if he or she is a Platonist or a formalist, to give one answer on weekdays and the opposite answer on the weekend. The reason is that if in the course of your work you discover a startling mathematical theorem or formula, you think of it as being an independent and significant entity, as Plato tells us. During the weekend, however, when you are asked to explain the essence of the entity you discovered, it is more convenient to avoid the discussion and to hide under the cover of formalism.
12. MODELS OF THE HEAVENLY BODIES
The Greeks inherited a wealth of astronomic measurements from the Babylonians and the Egyptians. These included information on the movement of the planets, the lengths of a year and a month, the cyclicality of the relation between the solar year and the lunar year, the effects of all these on agriculture, and so on. The Greeks made great efforts over many years to improve the measurements and to add new ones, and they developed the ability to forecast celestial events with the greatest precision. For example, whereas the Egyptians gave the length of the year as 365 days, the Greeks of the fifth century BCE determined it as 365 days, 6 hours, 18 minutes and 56 seconds, which is a deviation of just half an hour from the correct figure. In 130 BCE the Greeks achieved greater precision and set the length of the year at 365 days, 5 hours, 55 minutes and 12 seconds, a deviation of only six minutes and twenty-six seconds from the true figure. Thesemeasurements were achieved following the development of advanced mathematical methods of calculation and measurement. We will not expand on those methods here, as our focus is on how mathematics explains nature.
We will now review the conceptual and mathematical contributions made by the Greeks to the development of a model of the heavenly bodies, both in the classical period and thereafter. This development reached its peak in Ptolemy's (Claudius Ptolemaeus's) model. It should be mentioned again that there is no direct written evidence from the classical period, and the
In both interpretations the axioms are identified following observations. Aristotle, for instance, carried out extensive dissections of animals to learn about their anatomy. But once the axioms were adopted and mathematical results were derived, neither approach deemed it necessary to carry out experiments to compare mathematical results with what actually occurs in nature. Both of these great scholars and their Greek disciples actually opposed performing such experiments. They justified their opposition to experiments by claiming that appearances are affected by optical illusions and are likely to mislead, whereas logic is irrefutable. (It was not until thousands of years later that mental illusions were recognized and studied.) Thus, as soon as we find the self-evident axioms, the path forward using the power of logic is superior to progress based on appearances. This outlook held sway for thousands of years. It was not until the seventeenth century that the currently prevalent scientific practice, which demands experiments that corroborate the theory, became established. Onepossible reason for the fact that both Plato and Aristotle and most of the leading philosophers who succeeded them held extreme anti-experiment views is that they came from wealthy aristocratic families who considered physical work to be menial, inferior, and even contemptible.
The debate initiated by Plato and Aristotle on the essence of mathematics has continued for two thousand four hundred years and continues to this day. Scholarly articles supporting one view or the other, or proposing improvements to them, continue to appear in professional literature. The debate has no direct effect on mathematicians carrying out research or extending the boundaries of mathematics. Thus, so the story goes, it is quite normal for a mathematician, when asked if he or she is a Platonist or a formalist, to give one answer on weekdays and the opposite answer on the weekend. The reason is that if in the course of your work you discover a startling mathematical theorem or formula, you think of it as being an independent and significant entity, as Plato tells us. During the weekend, however, when you are asked to explain the essence of the entity you discovered, it is more convenient to avoid the discussion and to hide under the cover of formalism.
12. MODELS OF THE HEAVENLY BODIES
The Greeks inherited a wealth of astronomic measurements from the Babylonians and the Egyptians. These included information on the movement of the planets, the lengths of a year and a month, the cyclicality of the relation between the solar year and the lunar year, the effects of all these on agriculture, and so on. The Greeks made great efforts over many years to improve the measurements and to add new ones, and they developed the ability to forecast celestial events with the greatest precision. For example, whereas the Egyptians gave the length of the year as 365 days, the Greeks of the fifth century BCE determined it as 365 days, 6 hours, 18 minutes and 56 seconds, which is a deviation of just half an hour from the correct figure. In 130 BCE the Greeks achieved greater precision and set the length of the year at 365 days, 5 hours, 55 minutes and 12 seconds, a deviation of only six minutes and twenty-six seconds from the true figure. Thesemeasurements were achieved following the development of advanced mathematical methods of calculation and measurement. We will not expand on those methods here, as our focus is on how mathematics explains nature.
We will now review the conceptual and mathematical contributions made by the Greeks to the development of a model of the heavenly bodies, both in the classical period and thereafter. This development reached its peak in Ptolemy's (Claudius Ptolemaeus's) model. It should be mentioned again that there is no direct written evidence from the classical period, and the
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