By the later part of the nineteenth-century attempts to develop a logically consistent basis for number and arithmetic not only threatened to undermine the efficacy of the classical view of correspondence debates before the advent of quantum physics. They also occasioned a debate about epistemological foundations of mathematical physics that resulted in an attempt by Edmund Husserl to eliminate or obviate the correspondence problem by grounding this physics in human subjective reality. Since there is a direct line to descent from Husserl to existentialism to structuralism to deconstructionism, the linkage between philosophical postmodernism and the debate over the foundations of scientific epistemology is more direct than we had previously imagined.
A complete history of the debate over the epistemological foundations of mathematical physics should probably begin with the discovery of irrational numbers by the followers of Pythagoras, the paradoxes of Zeno and Gottfried Leibniz. Both since we are more concerned with the epistemological crisis of the later nineteenth-century, beginning with the set theory developed by the German mathematician and logician Georg Cantor. From 1878 to 1897, Cantor created a theory of abstract sets of entities that eventually became a mathematical discipline. A set, as he defined it, is a collection of definite and distinguishable objects in thought or perception conceived as a whole.
Georg Cantor (1845-1918) attempted to prove that the process of counting and the definition of integers could be placed on a solid mathematical foundation. His method was to repeatedly place the element in one set into 'one-to-one' correspondence with those in another. In the case of integers, Canto showed that each integer (1, 2, 3, . . . n) could be paired with an even integer (2, 4, 6, . . . n), and, therefore, that the set of all integers was equal to the set of all even numbers.
Formidably, Cantor discovered that some infinite sets were larger than others and that infinite set formed a hierarchy of ever greater infinities. After this failed attempts to save the classical view of logical foundations and internal consistency of mathematical systems, it soon became obvious that a major crack had appeared in the seemingly solid foundations of number and mathematics. Meanwhile, an impressive number of mathematicians began to see that everything from functional analysis to the theory of real numbers depended on the problematic character of number itself.
In 1886, Nietzsche was delighted to learn the classical view of mathematics as a logically consistent and self-contained system that could prove it might be undermined. And his immediate and unwarranted conclusion was that all of the logic and the whole of mathematics were nothing more than fictions perpetuated by those who exercised their will to power. With his characteristic sense of certainty, Nietzsche derisively proclaimed, 'Without accepting the fictions of logic, without measuring reality against the purely invented world to the unconditional and self-identical, without a constant falsification of the world by means of numbers, man could not live'.
A complete history of the debate over the epistemological foundations of mathematical physics should probably begin with the discovery of irrational numbers by the followers of Pythagoras, the paradoxes of Zeno and Gottfried Leibniz. Both since we are more concerned with the epistemological crisis of the later nineteenth-century, beginning with the set theory developed by the German mathematician and logician Georg Cantor. From 1878 to 1897, Cantor created a theory of abstract sets of entities that eventually became a mathematical discipline. A set, as he defined it, is a collection of definite and distinguishable objects in thought or perception conceived as a whole.
Georg Cantor (1845-1918) attempted to prove that the process of counting and the definition of integers could be placed on a solid mathematical foundation. His method was to repeatedly place the element in one set into 'one-to-one' correspondence with those in another. In the case of integers, Canto showed that each integer (1, 2, 3, . . . n) could be paired with an even integer (2, 4, 6, . . . n), and, therefore, that the set of all integers was equal to the set of all even numbers.
Formidably, Cantor discovered that some infinite sets were larger than others and that infinite set formed a hierarchy of ever greater infinities. After this failed attempts to save the classical view of logical foundations and internal consistency of mathematical systems, it soon became obvious that a major crack had appeared in the seemingly solid foundations of number and mathematics. Meanwhile, an impressive number of mathematicians began to see that everything from functional analysis to the theory of real numbers depended on the problematic character of number itself.
In 1886, Nietzsche was delighted to learn the classical view of mathematics as a logically consistent and self-contained system that could prove it might be undermined. And his immediate and unwarranted conclusion was that all of the logic and the whole of mathematics were nothing more than fictions perpetuated by those who exercised their will to power. With his characteristic sense of certainty, Nietzsche derisively proclaimed, 'Without accepting the fictions of logic, without measuring reality against the purely invented world to the unconditional and self-identical, without a constant falsification of the world by means of numbers, man could not live'.
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