What is mathematical platonism?


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Mathematical platonism is the idea that physical objects can only approximately embody mathematical structures, and that mathematics exists in an ideal world of forms. This philosophy of mathematics was articulated by Plato during the fourth century B.C. and was likely influenced by Pythagoras' belief that reality is actually created by numbers. In the strictest sense of this idea, the physical phenomena we observe in the world around us are governed by the properties of mathematical truths which exist independently of human thought and physical reality, and are eternal and unchanging. This view dominated the philosophy of mathematics from the time of Plato until the nineteenth century, and many mathematicians and philosophers still hold this view in a weaker form today. More recently, this viewpoint has been adjusted to only assert that mathematics exists as a property of nature, and that our models of the universe must conform to this property. Proponents of mathematical platonism often argue that the great usefulness of mathematics as a tool for modeling physical processes is evidence of its existence as a natural property of the universe. Since the nineteenth century, some developements in mathematics and physics have undermined this view. Before that point, it was believed that Euclidean geometry contained the only possible set of axioms, or rules for defining a gemetric system, that could exist. A set of geometric axioms that violated Euclid's fifth postualte was published by Bolyai in 1830. In the following decades of the nineteenth century, numerous other possibilities for consistent sets of geometric axioms were developed by the mathematical community. In the early twientieth century, the theory of general relativity predicted that the universe exhibited non-Euclidean properties on a cosmological scale. This prediction was later supported by observational evidence. Over the past century, various interpretations of the foundations of mathematics have been developed. Set theory has figured prominently in modern definitions of mathematical structures, which builds nearly all mathematics from a simple set of rules rather than physical motivation. There have been over two hundred logically self-consistent systems of algebra developed, some of which are based on rules that would be mutually exclusive with those of other systems. Perhaps the piece of mathematics that is most damaging to an idealized view of mathematics is Godel's incompleteness theorem, that states that any logical system that can be proven consistent and complete unto itself must be incomplete. It seems that logically consistent mathematics can be invented b the human mind, as well as inspired by physical motivations.