Certainty
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Certainty (also known as epistemic certainty or objective certainty) is the
Importantly, epistemic certainty is not the same thing as psychological certainty (also known as subjective certainty or certitude), which describes the highest degree to which a person could be convinced that something is true. While a person may be completely convinced that a particular belief is true, and might even be psychologically incapable of entertaining its falsity, this does not entail that the belief is itself beyond rational doubt or incapable of being false.[2] While the word "certainty" is sometimes used to refer to a person's subjective certainty about the truth of a belief, philosophers are primarily interested in the question of whether any beliefs ever attain objective certainty.
The
Ludwig Wittgenstein – 20th century
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
Degrees of certainty
Physicist
Alternatively, one might use the
If knowledge requires absolute certainty, then knowledge is most likely impossible, as evidenced by the apparent fallibility of our beliefs.
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
Various schools of thought were opposing each other. The leading school was that of the
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 cannot 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 is not consistent, there exists 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 be 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
- Almost surely
- Fideism
- Gut feeling
- Infallibility
- Justified true belief
- Neuroethological innate behavior, instinct
- Pascal's Wager
- Pragmatism
- Scientific consensus
- Skeptical hypothesis
- As contrary concepts
References
- ^ a b c "Certainty". Stanford Encyclopedia of Philosophy. Retrieved 12 July 2020.
- ^ Reed, Baron. "Certainty". plato.stanford.edu. Retrieved 2022-07-22.
- ^ Wittgenstein, Ludwig. "On Certainty". SparkNotes.
- ^ "question center, SHAs – cognitive tools". edge.com. Archived from the original on 2013-12-05. Retrieved 2011-03-03.
- ISBN 0195079299.
External links
- "Certitude". Catholic Encyclopedia. 1913.
- certainty, The American Heritage Dictionary of the English Language. Bartleby.com
- "certainty vs. doubt". About.com. Archived from the originalon 2009-03-01. Retrieved 2008-02-23.
- Reed, Baron. "Certainty". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
- "Certainty". Internet Encyclopedia of Philosophy.
- The certainty of belief