![]() ![]() What Gödel did to the work of Russell and Whitehead, Heisenberg did to Laplace’s concept of causality. ![]() Laplace envisioned a world of ever more precise prediction, where the laws of physical mechanics would be able to forecast nature in increasing detail and ever further into the future, a world where “the phenomena of nature can be reduced in the last analysis to actions at a distance between molecule and molecule.” In theory, if we knew the position and velocity of each molecule, we could trace its path as it interacted with other molecules, and trace the course of the physical universe at the most fundamental level. In the Laplacean view, molecules are just as subject to the laws of physical mechanics as the planets are. He later extended this theory to the interaction of molecules. In the early 1800s Laplace had worked extensively to demonstrate the purely mechanical and predictable nature of planetary motion. Like granite outcroppings piercing through a bed of moss, these apparently trivial contradictions were rooted in the core of mathematics and logic, and were only the most readily manifest examples of a limit to our ability to structure formal mathematical systems. Just four years before Gödel had defined the limits of our ability to conquer the intellectual world of mathematics and logic with the publication of his Undecidability Theorem, the German physicist Werner Heisenberg’s celebrated Uncertainty Principle had delineated the limits of inquiry into the physical world, thereby undoing the efforts of another celebrated intellect, the great mathematician Pierre-Simon Laplace. Yet Russell and Whitehead had, after all that effort, missed the central point. trivial or not, the matter was a challenge.” Attempts to address the challenge extended the development of Principia Mathematica by nearly a decade. He wrote that “it seemed unworthy of a grown man to spend his time on such trivialities, but. ![]() By the end of 1901, Russell had completed the first round of writing Principia Mathematica and thought he was in the homestretch, but was increasingly beset by these sorts of apparently simple-minded contradictions falling in the path of his goal. This problem did not totally escape Russell and Whitehead. This same sort of self-referentiality is the keystone of Gödel’s proof, where he uses statements that imbed other statements within them. The key point of contradiction for these two examples is that they are self-referential. If you prove the statement is false, then that means its converse is true – it is provable – which again is a contradiction. Or, even more trivially, a statement like “This statement is unprovable.” You cannot prove the statement is true, because doing so would contradict it. A sheet of paper has the words “The statement on the other side of this paper is true” written on one side and “The statement on the other side of this paper is false” on the reverse. The flavor of Gödel’s basic argument can be captured in the contradictions contained in a schoolboy’s brainteaser. In 1931, he wrote a treatise entitled On Formally Undecidable Propositions of Principia Mathematica and Related Systems, which demonstrated that the goal Russell and Whitehead had so single-mindedly pursued was unattainable. Although Gödel did face the risk of being liquidated by Hitler (therefore fleeing to the Institute of Advanced Studies at Princeton), he was neither a Pole nor a Texan. Momentous though it was, the greatest achievement of Principia Mathematica was realized two decades after its completion when it provided the fodder for the metamathematical enterprises of an Austrian, Kurt Gödel. Russell recounted that “every time that I went out for a walk I used to be afraid that the house would catch fire and the manuscript get burnt up.” The other three were Texans, subsequently successfully assimilated.” The complex mathematical symbols of the manuscript required it to be written by hand, and its sheer size – when it was finally ready for the publisher, Russell had to hire a panel truck to send it off – made it impossible to copy. Three of those were Poles, subsequently (I believe) liquidated by Hitler. Russell observed of the dense and demanding work, “I used to know of only six people who had read the later parts of the book. Bertrand Russell with Alfred North Whitehead, in the Principia Mathematica aimed to demonstrate that “all pure mathematics follows from purely logical premises and uses only concepts defined in logical terms.” Its goal was to provide a formalized logic for all mathematics, to develop the full structure of mathematics where every premise could be proved from a clear set of initial axioms.
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