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| 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:
—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:
—“Gödel’s
incompleteness theorems,” Britannica.com
Gödel’s
Incompleteness Theorems brings to a logical conclusion the rationalist enterprise
begun by Descartes in the 17th 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:
—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.
Wir werden nie wissen.
Public domain photo
ReplyDeletePhoto link: https://www.flickr.com/photos/levanrami/24246848265
Gonzalinho
From a philosophical point of view the implications of the Incompleteness Theorems is radical. But from another standpoint, e.g. the mathematical or the practical, the ramifications are not so consequential and possibly even trivial.
ReplyDeleteGonzalinho
THE END OF CERTAINTY
ReplyDeleteThe ancient Greek philosophers, including Socrates, Plato and Aristotle, asked “How do we know truth?” From this question came the idea of setting out some postulates or axioms that everyone could agree were true and then use logic to deduce other truths. This was the basis of Euclidean geometry that was developed to determine all truths in plane geometry from 5 basic postulates. When you studied geometry in high school, you learned a variety of theorems that were true because they had been deduced from these basic axioms. The ancient mathematicians sought a set of axioms from which all mathematical truths could be deduced, and this quest continued after the Renaissance.
To summarize: Mathematics is a collection of theorems derived by deduction from a set of basic a priori assumptions, called axioms. The sequence of deductions that lead from the axioms to the statement of a theorem is known as a proof of that theorem. So, mathematics may be regarded as a collection of proofs–links from a priori “assumptions” to inevitable conclusions.
When mathematical exploration was resuscitated in the Renaissance, the axiomatic structure of mathematics introduced by Euclid continued to be the basis of certainty and “truth” in geometry. However, the intuitive nature of the axioms that caused problems in geometry were also causing difficulties in the other branches of mathematics. Algebra and analysis, like geometry had also relied on intuitive notions that were not well defined and proofs were not rigorous in the modern sense. Newton’s development of calculus using “fluxions” had been based on intuitive notions of motion and change rather than precisely defined concepts. As mathematics began to venture into the realm of infinite quantities, convergence, and limits, the formal manipulation of symbols often led to contradictions. Mathematicians became increasingly aware of the importance of examining all axioms for hidden assumptions that might later yield contradictions. Even number theory, known as “the higher arithmetic,” came under scrutiny. In 1889, Giuseppe Peano published a set of nine axioms, precisely formulated in the language of set theory. These axioms were designed to put algebra on a firm footing by replacing all intuitive notions of whole numbers with unambiguously stated properties.
In 1899, David Hilbert revised Euclid’s axioms in a similar way, replacing intuitive notions with precisely-stated properties relating points, planes, and lines. His revision of Euclid’s axioms was part of what became known as the Hilbert program. In 1920, Hilbert proposed that a new research project be launched with a two-fold purpose:
1. To underpin all of mathematics with a finite set of axioms.
2. To develop a “metalanguage” that could be used to prove those axioms consistent.
The mathematicians who followed the Hilbert approach, subscribed to the idea that mathematics can be reduced to rules for manipulating formulas without any reference to their meaning. Members of this so-called formalist school believed that the mathematical symbols, and the inferential rules that govern their relationships constitute the totality of mathematical thought.
To be continued
Gonzalinho
THE END OF CERTAINTY
DeleteContinued
While Hilbert was attempting to achieve rigor by showing that all of mathematics could be deduced from a set of basic axioms and simple rules of inference, without using the concept of number or set, Gottlob Frege and Bertrand Russell were attempting to use set theoretic language and formal symbolic logic to achieve absolute rigor. Followers of this latter philosophy were said to be members of the logicist school.
When Gödel published his incompleteness theorems in 1931, its implications were unnoticed by the mathematics community at large. His theorem stated:
In any mathematical system complex enough to contain simple arithmetic, there exists an undecidable proposition–that is, a proposition that is not provable and whose negation is not provable.
Those who were working in the foundations of mathematics soon recognized the far-reaching implications of his powerful theorem and corollary. The brilliant wunderkind, John von Neumann, who had published The Axiomatization of Set Theory in 1928, was one of the first to perceive the “truth and importance of Gödel’s work.” It became apparent that the goals of the Hilbert formalist school and Russell’s logicist school were unattainable. This was, indeed, dream-shattering for Hilbert and Russell who had dedicated a significant chunk of their lives striving for an unattainable goal.
In 1978, an article in The New York Times described Gödel’s Theorem as “the most significant mathematical truth of this century, incomprehensible to laymen, revolutionary for philosophers and logicians.” Today, many researchers in AI invoke Gödel’s theorem to suggest that AI is based on internal computer logic circuits and may therefore have innate limitations. The debate continues.
https://www.quora.com/qemail/track_click?al_imp=eyJ0eXBlIjogMzMsICJoYXNoIjogIjB8MXwxMHwyMjgyMDc5NjUifQ%3D%3D&al_pri=0&aoid=9nnFpK82Uua&aoty=4&aty=4&cp=1&ct=1753064950949303&et=153&id=4d9fb5d7346a44d0a783d5e1e128e5f7¬if_type=508&request_id=508&snid=88603612959&src=1&st=1753064950951668&stories=%5B(%3Cstory_types.tribe_post%3A+10%3E%2C+201983462)%2C+(%3Cstory_types.tribe_post%3A+10%3E%2C+202060486)%2C+(%3Cstory_types.tribe_post%3A+10%3E%2C+201708166)%5D&tribe_item_ids=oiDaux72KmA%7CLAnpMtJMQfY%7Cb5SjVXQzN6u&uid=YFlmIIlXQC6&v=0
—Brendan Kelly, “Why Was Gödel’s Discovery A “Dream-Shattering Revelation” For Mathematicians Like Hilbert And Russell?” Quora, July 17, 2025
The quest for certainty in mathematical formalism is over. Today we are ready to toll the bell and write elegies. The greatest and finest minds in the history of the world have pursued this quest and concluded it. Do we consider it a failure? No, it remains a continuing challenge to investigate the variously novel questions it raises in what is more likely than not going to be an interminable thread.
Gonzalinho