Related concepts and fundamentals:
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CERTAINTY is perfect knowledge that has total security from error, or the mental state of being without doubt .
Objectively defined, certainty is total continuity and validity of
all foundational inquiry, to the highest degree of precision.
Something is certain only if no skepticism can occur.
It is widely held that certainty about the real world is a failed
historical enterprise (that is, beyond deductive truths, tautology ,
etc.). This is in large part due to the power of
* 1 History
* 2 Degrees of certainty * 3 Foundational crisis of mathematics * 4 Quotes * 5 See also * 6 References * 7 External links
PYRRHO – ANCIENT GREECE
AL-GHAZALI – ISLAMIC THEOLOGIAN
Main article: Al-Ghazali
Al-Ghazali was a professor of philosophy in the 11th century. His
The Incoherence of the Philosophers marks a major turn in
Islamic epistemology , as Ghazali effectively discovered philosophical
skepticism that would not be commonly seen in the West until Averroes
IBN-RUSHD - AVERROES
Main article: Ibn Rushd
DESCARTES – 17TH CENTURY
LUDWIG WITTGENSTEIN – 20TH CENTURY
On Certainty is a series of notes made by
DEGREES OF CERTAINTY
Physicist Lawrence M. Krauss suggests that identifying degrees of certainty is under-appreciated in various domains, including policy making and the understanding of science. This is because different goals require different degrees of certainty—and politicians are not always aware of (or do not make it clear) how much certainty we are working with.
Rudolf Carnap viewed certainty as a matter of degree ("degrees of certainty") which could be objectively measured, with degree one being certainty. Bayesian analysis derives degrees of certainty which are interpreted as a measure of subjective psychological belief .
Alternatively, one might use the legal degrees of certainty . These standards of evidence ascend as follows: no credible evidence, some credible evidence, a preponderance of evidence, clear and convincing evidence, beyond reasonable doubt, and beyond any shadow of a doubt (i.e. undoubtable—recognized as an impossible standard to meet—which serves only to terminate the list).
FOUNDATIONAL CRISIS OF MATHEMATICS
Main article: Foundations of mathematics § Foundational crisis
The foundational crisis of mathematics was the early 20th century's term for the search for proper foundations of mathematics.
After several schools of the philosophy of mathematics ran into difficulties one after the other in the 20th century, the assumption that mathematics had any foundation that could be stated within mathematics itself began to be heavily challenged.
One attempt after another to provide unassailable foundations for mathematics was found to suffer from various paradoxes (such as Russell\'s paradox ) and to be inconsistent .
Various schools of thought were opposing each other. The leading school was that of the formalist approach, of which David Hilbert was the foremost proponent, culminating in what is known as Hilbert\'s program , which sought to ground mathematics on a small basis of a formal system proved sound by metamathematical finitistic means. The main opponent was the intuitionist school, led by L.E.J. Brouwer , which resolutely discarded formalism as a meaningless game with symbols. The fight was acrimonious. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen , the leading mathematical journal of the time.
Gödel\'s incompleteness theorems , proved in 1931, showed that essential aspects of Hilbert's program could not be attained. In Gödel 's first result he showed how to construct, for any sufficiently powerful and consistent finitely axiomatizable system—such as necessary to axiomatize the elementary theory of arithmetic —a statement that can be shown to be true, but that does not follow from the rules of the system. It thus became clear that the notion of mathematical truth can not be reduced to a purely formal system as envisaged in Hilbert's program. In a next result Gödel showed that such a system was not powerful enough for proving its own consistency, let alone that a simpler system could do the job. This dealt a final blow to the heart of Hilbert's program, the hope that consistency could be established by finitistic means (it was never made clear exactly what axioms were the "finitistic" ones, but whatever axiomatic system was being referred to, it was a weaker system than the system whose consistency it was supposed to prove). Meanwhile, the intuitionistic school had failed to attract adherents among working mathematicians, and floundered due to the difficulties of doing mathematics under the constraint of constructivism .
In a sense, the crisis has not been resolved, but faded away: most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of Zermelo–Fraenkel set theory , generally their preferred axiomatic system. In most of mathematics as it is practiced, the various logical paradoxes never played a role anyway, and in those branches in which they do (such as logic and category theory ), they may be avoided.
In this world nothing can be said to be certain, except death and
There is no such thing as absolute certainty, but there is assurance sufficient for the purposes of human life. — John Stuart Mill
* As contrary concepts