decahedron, water.
This matching today seems extremely naive, but it was in accordance with the mathematical and physical knowledge of that time. These perfect solids would play a role fifteen hundred years later, when Kepler, who knew that the world consisted of more elements than those Plato had enumerated, tried to find a function for the perfect solids in his description of the structure of the world.
11. PLATONISM VERSUS FORMALISM
The ever-closer link between mathematical findings and the depiction of nature required a philosophical study of the question “What is mathematics?” Similar questions about the essence of mathematics continue to be asked even today: Does mathematics stand on its own, or is it perforce bound up with the things it describes? What is the connection between mathematics and nature? Is mathematics created or discovered? Two of Greece's major classical philosophers, Plato and his student Aristotle, addressed these questions.
Their answers reveal differences between their views on the essence of mathematics, but they hardly differ in their ideas on how mathematics should be studied and used. Both start with axioms that are derived from nature but that are “self-evident,” and for both, the next steps rely on the correct use of syllogisms and other logical tools. The differences between them are in their interpretation of the results and the role of those results in describing the world.
Plato's interpretation was firm: mathematics stands on its own. It exists in an abstract world, a world of ideas, irrespective of the people who discover it and of the phenomena it describes. The way to discover it is by using the power of logic and logical arguments. Axioms show the way to the starting point on the path of discovery (etymologically, the Greek word for axiom comes from the word meaning to think correctly). The axioms must be simple and obvious. We derive these simple axioms from the world around us, but care must be taken to choose only axioms that are irrefutably correct. When the axioms have been agreed upon, progress is made only through deduction, according to the rules of inference and logic, and this leads to the discovery of correct mathematics. Hence, according to Plato, mathematics is discovered, not created. The phenomena we observe in nature contain defects and are a contamination of the ideal mathematics. Nature is apt to mislead us, so that we must not rely on appearances but must derive the principles from nature and search for mathematics using the power of logic. In that way we will understand the principles underlying the natural phenomena. This approach is known as Platonism .
Aristotle's interpretation was different. He claimed that mathematics did not exist as an independent entity. Mathematics is simply the outcome of logical operations based on syllogisms, and the search starts with axioms. Mathematics is created, not discovered. The axioms and the mathematical results they yield have no intrinsic significance; they are formal results that have no purpose and no independent existence. Aristotle used the word techne for the creation of mathematics, the origin of the word technology . According to him, the importance of the formal operations comes to light immediately when one discovers “correct” axioms, that is, those that nature satisfies. Aristotle used the word episteme to describe the finding of the correct axioms, which later gave rise to the word epistemology , meaning the study and theory of knowledge, which provides the basis for the common sense that describes how the world operates. When the axioms are correct, the resulting mathematics derived from them constitutes a precise record of what is found in nature. Hence, that is the way to study nature, avoiding the pitfall that could be caused by false appearances. Mathematics does not exist in its own right but only as a formal procedure, the significance of which depends on its interpretation and how it
This matching today seems extremely naive, but it was in accordance with the mathematical and physical knowledge of that time. These perfect solids would play a role fifteen hundred years later, when Kepler, who knew that the world consisted of more elements than those Plato had enumerated, tried to find a function for the perfect solids in his description of the structure of the world.
11. PLATONISM VERSUS FORMALISM
The ever-closer link between mathematical findings and the depiction of nature required a philosophical study of the question “What is mathematics?” Similar questions about the essence of mathematics continue to be asked even today: Does mathematics stand on its own, or is it perforce bound up with the things it describes? What is the connection between mathematics and nature? Is mathematics created or discovered? Two of Greece's major classical philosophers, Plato and his student Aristotle, addressed these questions.
Their answers reveal differences between their views on the essence of mathematics, but they hardly differ in their ideas on how mathematics should be studied and used. Both start with axioms that are derived from nature but that are “self-evident,” and for both, the next steps rely on the correct use of syllogisms and other logical tools. The differences between them are in their interpretation of the results and the role of those results in describing the world.
Plato's interpretation was firm: mathematics stands on its own. It exists in an abstract world, a world of ideas, irrespective of the people who discover it and of the phenomena it describes. The way to discover it is by using the power of logic and logical arguments. Axioms show the way to the starting point on the path of discovery (etymologically, the Greek word for axiom comes from the word meaning to think correctly). The axioms must be simple and obvious. We derive these simple axioms from the world around us, but care must be taken to choose only axioms that are irrefutably correct. When the axioms have been agreed upon, progress is made only through deduction, according to the rules of inference and logic, and this leads to the discovery of correct mathematics. Hence, according to Plato, mathematics is discovered, not created. The phenomena we observe in nature contain defects and are a contamination of the ideal mathematics. Nature is apt to mislead us, so that we must not rely on appearances but must derive the principles from nature and search for mathematics using the power of logic. In that way we will understand the principles underlying the natural phenomena. This approach is known as Platonism .
Aristotle's interpretation was different. He claimed that mathematics did not exist as an independent entity. Mathematics is simply the outcome of logical operations based on syllogisms, and the search starts with axioms. Mathematics is created, not discovered. The axioms and the mathematical results they yield have no intrinsic significance; they are formal results that have no purpose and no independent existence. Aristotle used the word techne for the creation of mathematics, the origin of the word technology . According to him, the importance of the formal operations comes to light immediately when one discovers “correct” axioms, that is, those that nature satisfies. Aristotle used the word episteme to describe the finding of the correct axioms, which later gave rise to the word epistemology , meaning the study and theory of knowledge, which provides the basis for the common sense that describes how the world operates. When the axioms are correct, the resulting mathematics derived from them constitutes a precise record of what is found in nature. Hence, that is the way to study nature, avoiding the pitfall that could be caused by false appearances. Mathematics does not exist in its own right but only as a formal procedure, the significance of which depends on its interpretation and how it
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