Certainty

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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. Philosophy (at least, historical Cartesian philosophy) seeks this state.[citation needed]

History[edit]

Pyrrho – ancient Greece[edit]

Pyrrho is credited as being the first Skeptic philosopher. The main principle of Pyrrho's thought is expressed by the word acatalepsia, which denotes the ability to withhold assent from doctrines regarding the truth of things in their own nature; against every statement its contradiction may be advanced with equal justification. Secondly, it is necessary in view of this fact to preserve an attitude of intellectual suspense, or, as Timon expressed it, no assertion can be known to be better than another.

Ibn-Rushd – Averroes[edit]

Averroes was a purveyor of certain parts of Aristotelian philosophy. His philosophy was considered controversial in Muslim circles.[1] as well as in West with thinkers like St. Thomas Aquinas who said of Averroes, "the Arabian commentator as one who had, indeed, perverted the Peripatetic tradition, but whose words, nevertheless, should be treated with respect and consideration." Averroes' contribution to epistemology is only noted for the fact that he was one of the first to write on the topic and acted as a comparison for the traditional and definitive works of the Western tradition.Turner, William. "Averroes". Catholic Answers. Catholic Answers. Retrieved February 28, 2018. 

Descartes – 17th century[edit]

Descartes' Meditations on First Philosophy is a book in which Descartes first discards all belief in things which are not absolutely certain, and then tries to establish what can be known for sure. Although the phrase "Cogito, ergo sum" is often attributed to Descartes' Meditations on First Philosophy, it is actually put forward in his Discourse on Method. Due to the implications of inferring the conclusion within the predicate, however, he changed the argument to "I think, I exist"; this then became his first certainty.

Ludwig Wittgenstein – 20th century[edit]

On Certainty is a series of notes made by Ludwig Wittgenstein just prior to his death. The main theme of the work is that context plays a role in epistemology. Wittgenstein asserts an anti-foundationalist message throughout the work: that every claim can be doubted but certainty is possible in a framework. "The function [propositions] serve in language is to serve as a kind of framework within which empirical propositions can make sense".[2]

Degrees of certainty[edit]

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.[3]

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[edit]

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.[citation needed] 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.

Quotes[edit]

Doubt is not a pleasant condition, but certainty is absurd.

— Voltaire

In this world nothing can be said to be certain, except death and taxes.

There is no such thing as absolute certainty, but there is assurance sufficient for the purposes of human life.

I don't know what to say

See also[edit]

References[edit]

External links[edit]