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Certainty

[sur-tn-tee]

A related article is titled uncertainty.

For statistical certainty, see probability.

Certainty can be defined as either (a) perfect knowledge that has total security from error, or (b) 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 historically) seeks this state. It is widely held that certainty is a failed historical enterprise.

Emotion

Strictly speaking, certainty is not a property of statements, but a property of people. 'Certainty' is an emotional state, like anger, jealousy, or embarrassment. When someone says "B is certain" they really mean "I am certain that B". The former is often used in everyday language, as it has a rhetorical advantage. It is also sometimes used to convey that a large number of people are certain about B. However the fact that certainty is an emotional state is not always heeded in the literature. The truth is, certainty is an emotional state that is attained by many people every day. In this sense, certainty is linked to 'faith' as a similar state of consciousness or of emotion.

History

Socrates- ancient Greece

Socrates, often thought to be the first true philosopher, had a higher a criterion for knowledge than others before him. The skeptical problems that he encountered in his philosophy were taken very seriously. As a result, he claimed to know nothing. Socrates often said that his wisdom was limited to an awareness of his own ignorance.

Al-Ghazali- Islamic theologian

Al-Ghazali was a professor of philosophy in the 11th century. His book titled 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 René Descartes, George Berkeley and David Hume. He described the necessity of proving the validity of reason- independently from reason. He attempted this and failed. The doubt that he introduced to his foundation of knowledge could not be reconciled using philosophy. Taking this very seriously, he resigned from his post at the university, and suffered serious psychosomatic illness. It was not until he became a religious sufi that he found a solution to his philosophical problems, which are based on Islamic religion; this encounter with skepticism led Ghazali to embrace a form of theological occasionalism, or the belief that all causal events and interactions are not the product of material conjunctions but rather the immediate and present will of God.

Descartes- 18th Century

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 with Descartes' Meditations on First Philosophy it is actually put forward in his Discourse on Method however, due to the implications of inferring the conclusion within the predicate he changed the argument to "I think, I exist" this then becomes his first certainty.

Ludwig Wittgenstein- 20th Century

On Certainty, is a book by Ludwig Wittgenstein. 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".

Degrees of Certainty

See inductive logic, philosophy of probability, philosophy of statistics.

Rudolph 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.

Foundational crisis of mathematics

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: an undesirable situation in which every mathematical statement that can be formulated in a proposed system (such as 2 + 2 = 5) can also be proved in the system.

Various schools of thought on the right approach to the foundations of mathematics were fiercely 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 thought 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 ZFC, 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.