Why Is Kurt Gödel Significant?

Kurt Gödel (1906-1978)

WHY IS KURT GÖDEL SIGNIFICANT?

Kurt Gödel (1906-1978) is a world renowned logician, mathematician, and analytic philosopher. He set forth mathematical proofs for what are known today as the Incompleteness Theorems. His contribution to intellectual history is revolutionary, pivotal, and enduring.

Popularly expressed, the first theorem declares that any formal axiomatic system in mathematics always contains statements that can be neither proved nor disproved within the system.

The second theorem states that it cannot be proven within the system that the system is consistent, meaning, without logical contradiction.

In other words, any formal axiomatic system in mathematics is always incomplete, that is, it always contains statements that cannot be proved or disproved.

It also cannot be proved within the system that the system itself is free of logical contradictions.

Therefore, if we limit ourselves to the internal logic of the system, we are forced to conclude that the system, potentially—sometimes, actually—harbors logical contradictions.

On the other hand, it is possible to prove that the system is consistent if we go outside the system, e.g. we base our proof on one or more axioms external to the system.

A popular explanation of Gödel’s Incompleteness Theorems is given in this YouTube video by Marcus du Sautoy of Numberphile:

https://www.youtube.com/watch?v=O4ndIDcDSGc 

 

—Numberphile, “Gödel’s Incompleteness Theorem - Numberphile,” YouTube video, 13:52 minutes, May 31, 2017

A more precise explanation, not fully detailed, is given by Britannica:

https://www.britannica.com/topic/history-of-logic/Godels-incompleteness-theorem 

 

—“Gödel’s incompleteness theorems,” Britannica.com 

Gödel’s Incompleteness Theorems brings to a logical conclusion the rationalist enterprise begun by Descartes in the 17^th century.

What is the rationalist enterprise? It is the quest for indubitable knowledge according to the rationalist method. Descartes located indubitable knowledge in formal mathematics, its system of self-evident axioms and logically demonstrated theorems.

“In our search for the direct road towards truth,” Descartes proclaimed in Rules for the Direction of the Mind, “we should busy ourselves with no object about which we cannot attain a certitude equal to that of the demonstrations of arithmetic and geometry.” See:

https://books.google.com.ph/books?id=jjWPe-9NPoEC&pg=PA7&lpg=PA7&dq=in+our+search+for+the+direct+road+towards+truth,+we+should+busy+ourselves+with+no+object+about+which+we+cannot+attain+a+certitude+equal+to+that+of+the+demonstrations+of+arithmetic+and+geometry&source=bl&ots=8zFqHz_vjL&sig=ACfU3U2Jk98kQs_7fLPQ3HU1I49mNPFo7g&hl=en&sa=X&ved=2ahUKEwj0mtC0ovrmAhXPad4KHX6sAlUQ6AEwAXoECAwQAQ#v=onepage&q=in%20our%20search%20for%20the%20direct%20road%20towards%20truth%2C%20we%20should%20busy%20ourselves%20with%20no%20object%20about%20which%20we%20cannot%20attain%20a%20certitude%20equal%20to%20that%20of%20the%20demonstrations%20of%20arithmetic%20and%20geometry&f=false 

 

—René Descartes, “Rules for the Direction of the Mind,” Key Philosophical Writings, transl. by Elizabeth S. Haldane and G. T. S. Ross, ed. and with an Introduction by Enrique Chávez-Arvizo (Hertfordshire, UK: Wordsworth Editions Limited, 1997)

Building on his belief that he could obtain similarly indubitable results by applying the mathematical method of formal reasoning universally, Descartes used his rationalist approach to investigate philosophical questions, thereby deriving his own very personal, somewhat idiosyncratic conclusions.

Cogito, ergo sum, he famously declared. It is the dogma at the foundation of his metaphysics.

Repudiating Descartes, Gödel’s Incompleteness Theorems show that the quest for indubitable, definitive truth, at least if we travel by the road of formal logic and mathematics, is a dead end.

We inevitably conclude that our knowledge thereby of reality—indeed, if we choose to go deeper, all knowledge—is necessarily imperfect construction.

It is a conclusion that has consequences not only logical and epistemological but also metaphysical.

Gödel’s Incompleteness Theorems undermine our belief that any mathematical description, wholly or partially considered, of reality is in fact true, that is, an accurate representation of reality. In some pervasive, subversive, and ineluctable sense, our mathematical understanding of reality is incomplete and possibly contradictory, and in the foregoing two respects incontrovertibly untrue.

We are forever bereft of perfect certainty, therefore, stranded, as it were, inhabiting an island of circumscribed knowledge, surrounded by an unattainable ocean that is at best partially accessible, so that the reality we apprehend is never equal to and always less than what it completely, fully is.

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