The Road to Reality Study Notes: Difference between revisions
→1.2 Mathematical truth: fix typo: "the" → "to"
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There was a need to define a more rigorous method for differentiating truth claims. The Greek philosopher [https://en.wikipedia.org/wiki/Thales_of_Miletus Thales of Miletus] (c. 625-547 BC) and [https://en.wikipedia.org/wiki/Pythagoras Pythagoras of Samos] (c. 572-497 BC) are considered to be the first to introduce the concept of ''mathematical proof''. Developing a rigorous mathematical framework was central to the development of science. Mathematical proof allowed for much stronger statements to be made about relationships between the arithmetic of numbers and the geometry of physical space. | There was a need to define a more rigorous method for differentiating truth claims. The Greek philosopher [https://en.wikipedia.org/wiki/Thales_of_Miletus Thales of Miletus] (c. 625-547 BC) and [https://en.wikipedia.org/wiki/Pythagoras Pythagoras of Samos] (c. 572-497 BC) are considered to be the first to introduce the concept of ''mathematical proof''. Developing a rigorous mathematical framework was central to the development of science. Mathematical proof allowed for much stronger statements to be made about relationships between the arithmetic of numbers and the geometry of physical space. | ||
A mathematical proof is essentially an argument in which one starts from a mathematical statement, which is taken to be true, and using only logical rules arrives at a new mathematical statement. If the mathematician hasn't broken any rules then the new statement is called a ''theorem''. The most fundamental mathematical statements, from which all other proofs are built, are called ''axioms'' and their validity is taken | A mathematical proof is essentially an argument in which one starts from a mathematical statement, which is taken to be true, and using only logical rules arrives at a new mathematical statement. If the mathematician hasn't broken any rules then the new statement is called a ''theorem''. The most fundamental mathematical statements, from which all other proofs are built, are called ''axioms'' and their validity is taken to be self-evident. Mathematicians trust that the axioms, on which their theorems depend, are actually ''true''. The Greek philosopher [https://en.wikipedia.org/wiki/Plato Plato] (c.429-347 BC) believed that mathematical proofs referred not to actual physical objects but to certain idealized entities. Physical manifestations of geometric objects could come close to the Platonic world of mathematical forms, but they were always approximations. To Plato the idealized mathematical world of forms was a place of absolute truth, but inaccessible from the physical world. | ||
=== 1.3 Is Plato's mathematical world "real"? === | === 1.3 Is Plato's mathematical world "real"? === | ||