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]

Ancient Greece[edit]

Major elements of philosophical skepticism - the idea that things cannot be known with certainty, which the ancient Greeks expressed by the word acatalepsia - are apparent in the writings of several ancient Greek philosophers, particularly Xenophanes and Democritus. The first Hellenistic school that embraced philosophical skepticism was Pyrrhonism, which was founded by Pyrrho of Elis. Pyrrho's skepticism quickly spread to Plato's Academy under Arcesilaus, who abandoned Platonic dogma and initiated Academic Skepticism, the second skeptical school of Hellenistic philosophy. The major difference between the two skeptical schools was that Pyrrhonism's aims were psychotherapeutic (i.e., to lead practitoners to the state of ataraxia - freedom from anxiety, whereas those of Academic Skepticism were about making judgments under uncertainty (i.e., to identify what arguments were most truth-like).

Descartes – 17th century[edit]

In his Meditations on First Philosophy, Descartes first discards all belief in things which are not absolutely certain, and then tries to establish what can be known for sure.[citation needed] 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.[citation needed] 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.[citation needed]

Ludwig Wittgenstein – 20th century[edit]

If you tried to doubt everything you would not get as far as doubting anything. The game of doubting itself presupposes certainty.

Ludwig Wittgenstein, On Certainty, #115

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".[1]

Degrees of certainty[edit]

Physicist Lawrence M. Krauss suggests that the need for 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.[2]

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 proves that there is no hope to prove the consistency of any system that contains an axiomatization of elementary arithmetic, and, in particular, to prove the consistency of Zermelo–Fraenkel set theory (ZFC), the system which is generally used for building all mathematics.

However, if ZFC would not be consistent, there would exist a proof of both a theorem and its negation, and this would imply a proof of all theorems and all their negations. As, despite the large number of mathematical areas that have been deeply studied, no such contradiction has ever been found, this provides an almost certainty of mathematical results. Moreover, if such a contradiction would eventually be found, most mathematicians are convinced that it will possible to resolve it by a slight modification of the axioms of ZFC.

Moreover, the method of forcing allows proving the consistency of a theory, provided that another theory is consistent. For example, if ZFC is consistent, adding to it the continuum hypothesis or a negation of it defines two theories that are both consistent (in other words, the continuum is independent from the axioms of ZFC). This existence of proofs of relative consistency implies that the consistency of modern mathematics depends weakly on a particular choice on the axioms on which mathematics are built.

In this sense, the crisis has been resolved, as, although consistency of ZFC is not provable, it solves (or avoids) all logical paradoxes at the origin of the crisis, and there are many facts that provide a quasi-certainty of the consistency of modern mathematics.

See also[edit]

References[edit]

  1. ^ Wittgenstein, Ludwig. "On Certainty". SparkNotes.
  2. ^ "question center, SHAs – cognitive tools". edge.com.

External links[edit]