Certain Equation
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Certainty (also known as epistemic certainty or objective certainty) is the
epistemic Epistemology is the branch of philosophy that examines the nature, origin, and limits of knowledge. Also called "the theory of knowledge", it explores different types of knowledge, such as propositional knowledge about facts, practical knowledg ...
property of
belief A belief is a subjective Attitude (psychology), attitude that something is truth, true or a State of affairs (philosophy), state of affairs is the case. A subjective attitude is a mental state of having some Life stance, stance, take, or opinion ...
s which a person has no rational grounds for doubting. One standard way of defining epistemic certainty is that a belief is certain if and only if the person holding that belief could not be mistaken in holding that belief. Other common definitions of certainty involve the indubitable nature of such beliefs or define certainty as a property of those beliefs with the greatest possible justification. Certainty is closely related to
knowledge Knowledge is an Declarative knowledge, awareness of facts, a Knowledge by acquaintance, familiarity with individuals and situations, or a Procedural knowledge, practical skill. Knowledge of facts, also called propositional knowledge, is oft ...
, although contemporary philosophers tend to treat knowledge as having lower requirements than certainty. 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. 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
philosophical Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
question of whether one can ever be truly certain about anything has been widely debated for centuries. Many proponents of
philosophical skepticism Philosophical skepticism (UK spelling: scepticism; from Ancient Greek, Greek σκέψις ''skepsis'', "inquiry") is a family of philosophical views that question the possibility of knowledge. It differs from other forms of skepticism in that ...
deny that certainty is possible, or claim that it is only possible in ''
a priori ('from the earlier') and ('from the later') are Latin phrases used in philosophy to distinguish types of knowledge, Justification (epistemology), justification, or argument by their reliance on experience. knowledge is independent from any ...
'' domains such as logic or mathematics. Historically, many philosophers have held that knowledge requires epistemic certainty, and therefore that one must have infallible justification in order to count as knowing the truth of a proposition. However, many philosophers such as
René Descartes René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...
were troubled by the resulting skeptical implications, since all of our experiences at least seem to be compatible with various skeptical scenarios. It is generally accepted today that most of our beliefs are compatible with their falsity and are therefore fallible, although the status of being certain is still often ascribed to a limited range of beliefs (such as " I exist"). The apparent fallibility of our beliefs has led many contemporary philosophers to deny that knowledge requires certainty.


Ludwig Wittgenstein – 20th century

''
On Certainty ''On Certainty'' (, original spelling ) is a philosophical book composed from notes written by Ludwig Wittgenstein over four separate periods in the eighteen months before his death on 29 April 1951. He left his initial notes at the home of Eli ...
'' is a series of notes made by
Ludwig Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. From 1929 to 1947, Witt ...
just prior to his death. The main theme of the work is that
context In semiotics, linguistics, sociology and anthropology, context refers to those objects or entities which surround a ''focal event'', in these disciplines typically a communicative event, of some kind. Context is "a frame that surrounds the event ...
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 ropositionsserve in language is to serve as a kind of framework within which empirical propositions can make sense".


Degrees of certainty

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 certaintyand politicians are not always aware of (or do not make it clear) how much certainty we are working with.
Rudolf Carnap Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism. ...
viewed certainty as a matter of degree ("degrees of certainty") which could be objectively measured, with degree one being certainty.
Bayesian analysis Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
derives degrees of certainty which are interpreted as a measure of subjective
psychological Psychology is the scientific study of mind and behavior. Its subject matter includes the behavior of humans and nonhumans, both consciousness, conscious and Unconscious mind, unconscious phenomena, and mental processes such as thoughts, feel ...
belief A belief is a subjective Attitude (psychology), attitude that something is truth, true or a State of affairs (philosophy), state of affairs is the case. A subjective attitude is a mental state of having some Life stance, stance, take, or opinion ...
. Alternatively, one might use the legal degrees of certainty. These standards of
evidence Evidence for a proposition is what supports the proposition. It is usually understood as an indication that the proposition is truth, true. The exact definition and role of evidence vary across different fields. In epistemology, evidence is what J ...
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 meetwhich serves only to terminate the list). 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 Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathem ...
ran into difficulties one after the other in the 20th century, the assumption that mathematics had any foundation that could be stated within
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
itself began to be heavily challenged. One attempt after another to provide unassailable foundations for mathematics was found to suffer from various
paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictor ...
es (such as
Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. Russell's paradox shows that every set theory that contains ...
) and to be
inconsistent In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences o ...
. Various schools of thought were opposing each other. The leading school was that of the formalist approach, of which
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...
was the foremost proponent, culminating in what is known as
Hilbert's program In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early 1920s, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to ...
, which sought to ground mathematics on a small basis of a
formal system A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in ma ...
proved sound by metamathematical finitistic means. The main opponent was the
intuitionist In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of ...
school, led by
L.E.J. Brouwer Luitzen Egbertus Jan "Bertus" Brouwer (27 February 1881 – 2 December 1966) was a Dutch mathematician and philosopher who worked in topology, set theory, measure theory and complex analysis. Regarded as one of the greatest mathematicians of the ...
, 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 ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...
'', the leading mathematical journal of the time.
Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...
, 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 systemsuch as necessary to axiomatize the elementary theory of
arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
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 In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
(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 In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...
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 In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur ha ...
*
Fideism Fideism ( ) is a standpoint or an epistemological theory which maintains that faith is independent of reason, or that reason and faith are hostile to each other and faith is superior at arriving at particular truths (see natural theology). The ...
*
Gut feeling According to the ''American Psychological Association, APA Dictionary of Psychology'', a feeling is "a self-contained phenomenal experience"; feelings are "subjective, evaluative, and independent of the sensations, thoughts, or images evoking th ...
*
Infallibility Infallibility refers to unerring judgment, being absolutely correct in all matters and having an immunity from being wrong in even the smallest matter. It can be applied within a specific domain, or it can be used as a more general adjective. Th ...
* Justified true belief * Neuroethological innate behavior,
instinct Instinct is the inherent inclination of a living organism towards a particular complex behaviour, containing innate (inborn) elements. The simplest example of an instinctive behaviour is a fixed action pattern (FAP), in which a very short to me ...
* Pascal's Wager *
Pragmatism Pragmatism is a philosophical tradition that views language and thought as tools for prediction, problem solving, and action, rather than describing, representing, or mirroring reality. Pragmatists contend that most philosophical topics†...
*
Scientific consensus Scientific consensus is the generally held judgment, position, and opinion of the majority or the supermajority of scientists in a particular field of study at any particular time. Consensus is achieved through scholarly communication at confer ...
* Skeptical hypothesis * As contrary concepts: **
Fallibilism Originally, fallibilism (from Medieval Latin: ''fallibilis'', "liable to error") is the philosophical principle that propositions can be accepted even though they cannot be conclusively proven or justified,Haack, Susan (1979)"Fallibilism and Nece ...
**
Indeterminism Indeterminism is the idea that events (or certain events, or events of certain types) are not caused, or are not caused deterministically. It is the opposite of determinism and related to chance. It is highly relevant to the philosophical pr ...
**
Multiverse The multiverse is the hypothetical set of all universes. Together, these universes are presumed to comprise everything that exists: the entirety of space, time, matter, energy, information, and the physical laws and constants that describ ...


References


External links

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certainty
The American Heritage Dictionary of the English Language ''The American Heritage Dictionary of the English Language'' (''AHD'') is a dictionary of American English published by HarperCollins. It is currently in its fifth edition (since 2011). Before HarperCollins acquired certain business lines from H ...
. Bartleby.com * * *
The certainty of belief
{{Sufism terminology Cognition Concepts in epistemology Concepts in the philosophy of mind