Certainty
Encyclopedia
Certainty can be defined as either:
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.
It is widely held that certainty about the real world is a failed historical enterprise (that is, beyond deductive truths, tautology
, etc.). This is in large part due to the power of David Hume
's problem of induction
. Physicist Carlo Rovelli
adds that certainty, in real life, is useless or often damaging (the idea is that "total security from error" is impossible in practice, and a complete "lack of doubt" is undesirable).
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.
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
.
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.
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".
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.
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
.
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).
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 paradox
es (such as Russell's paradox
) and to be inconsistent
.
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.
- perfect knowledge that has total security from error, or
- the mental stateMental state* In psychology, mental state is an indication of a person's mental health**Mental status examination, a structured way of observing and describing a patient's current state of mind...
of being without doubtDoubtDoubt, a status between belief and disbelief, involves uncertainty or distrust or lack of sureness of an alleged fact, an action, a motive, or a decision. Doubt brings into question some notion of a perceived "reality", and may involve delaying or rejecting relevant action out of concerns for...
Objectively defined, certainty is total continuity and validity of all foundational
Foundationalism
Foundationalism is any theory in epistemology that holds that beliefs are justified based on what are called basic beliefs . This position is intended to resolve the infinite regress problem in epistemology...
inquiry, to the highest degree of precision. Something is certain only if no skepticism
Skepticism
Skepticism has many definitions, but generally refers to any questioning attitude towards knowledge, facts, or opinions/beliefs stated as facts, or doubt regarding claims that are taken for granted elsewhere...
can occur. Philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
(at least historical Cartesian philosophy) seeks this state.
It is widely held that certainty about the real world is a failed historical enterprise (that is, beyond deductive truths, tautology
Tautology
Tautology may refer to:*Tautology , using different words to say the same thing even if the repetition does not provide clarity. Tautology also means a series of self-reinforcing statements that cannot be disproved because the statements depend on the assumption that they are already...
, etc.). This is in large part due to the power of David Hume
David Hume
David Hume was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and skepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment...
's problem of induction
Problem of induction
The problem of induction is the philosophical question of whether inductive reasoning leads to knowledge. That is, what is the justification for either:...
. Physicist Carlo Rovelli
Carlo Rovelli
Carlo Rovelli is an Italian physicist who has worked in Italy, the USA, and France. His work is mainly in the field of quantum gravity. He is among the founders of the Loop Quantum Gravity theory.-Life and career:...
adds that certainty, in real life, is useless or often damaging (the idea is that "total security from error" is impossible in practice, and a complete "lack of doubt" is undesirable).
Pyrrho - ancient Greece
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.
Al-Ghazali – Islamic theologian
Al-Ghazali was a professor of philosophy in the 11th century. His book titled The Incoherence of the PhilosophersThe Incoherence of the Philosophers
The Incoherence of the Philosophers is the title of a landmark 11th century polemic by the Sufi sympathetic Imam Al-Ghazali of the Asharite school of Islamic theology criticizing the Avicennian school of early Islamic philosophy...
marks a major turn in Islamic epistemology, as Ghazali effectively discovered philosophical skepticism
Philosophical skepticism
Philosophical skepticism is both a philosophical school of thought and a method that crosses disciplines and cultures. Many skeptics critically examine the meaning systems of their times, and this examination often results in a position of ambiguity or doubt...
that would not be commonly seen in the West until René Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...
, George Berkeley
George Berkeley
George Berkeley , also known as Bishop Berkeley , was an Irish philosopher whose primary achievement was the advancement of a theory he called "immaterialism"...
and David Hume
David Hume
David Hume was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and skepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment...
. 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
Occasionalism
Occasionalism is a philosophical theory about causation which says that created substances cannot be efficient causes of events. Instead, all events are taken to be caused directly by God...
, 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
God in Islam
In Islamic theology, God is the all-powerful and all-knowing creator, sustainer, ordainer, and judge of the universe. Islam puts a heavy emphasis on the conceptualization of God as strictly singular . God is unique and inherently One , all-merciful and omnipotent. According to the Islamic...
.
Descartes – 17th century
Descartes' Meditations on First PhilosophyMeditations on First Philosophy
Meditations on First Philosophy is a philosophical treatise written by René Descartes and first published in 1641 . The French translation was published in 1647 as Méditations Metaphysiques...
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
On CertaintyOn Certainty
On Certainty is a philosophical book composed from the notes written by Ludwig Wittgenstein just prior to his death. Some of the notes were left at the home of G. E. M...
is a series of notes made by Ludwig Wittgenstein
Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He was professor in philosophy at the University of Cambridge from 1939 until 1947...
just prior to his death. The main theme of the work is that context
Context
Context may refer to:* Context , the relevant constraints of the communicative situation that influence language use, language variation, and discourse summary...
plays a role in epistemology. Wittgenstein asserts an anti-foundationalist
Anti-foundationalism
Anti-foundationalism as the name implies, is a term applied to any philosophy which rejects a foundationalist approach, i.e. an anti-foundationalist is one who does not believe that there is some fundamental belief or principle which is the basic ground or foundation of inquiry and knowledge...
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
Physicist Lawrence M. KraussLawrence M. Krauss
Lawrence Maxwell Krauss is an American theoretical physicist who is professor of physics, Foundation Professor of the School of Earth and Space Exploration, and director of the Origins Project at the Arizona State University. He is the author of several bestselling books, including The Physics of...
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.
Rudolph Carnap viewed certainty as a matter of degree (degrees of certainty
Degrees of Certainty
Degrees of Certainty may refer to:*Rudolf Carnap*Bayesian analysis*Confidence Interval...
) which could be objectively
Objectivity (philosophy)
Objectivity is a central philosophical concept which has been variously defined by sources. A proposition is generally considered to be objectively true when its truth conditions are met and are "mind-independent"—that is, not met by the judgment of a conscious entity or subject.- Objectivism...
measured, with degree one being certainty. Bayesian analysis derives degrees of certainty
Degrees of Certainty
Degrees of Certainty may refer to:*Rudolf Carnap*Bayesian analysis*Confidence Interval...
which are interpreted as a measure of subjective
Subjectivity
Subjectivity refers to the subject and his or her perspective, feelings, beliefs, and desires. In philosophy, the term is usually contrasted with objectivity.-Qualia:...
psychological belief
Belief
Belief is the psychological state in which an individual holds a proposition or premise to be true.-Belief, knowledge and epistemology:The terms belief and knowledge are used differently in philosophy....
.
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
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
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
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
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
itself began to be heavily challenged.
One attempt after another to provide unassailable foundations for mathematics was found to suffer from various paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
es (such as Russell's paradox
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
) and to be inconsistent
Consistency proof
In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e. there exists an interpretation under which all...
.
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
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
was the foremost proponent, culminating in what is known as Hilbert's program
Hilbert's program
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies...
, which thought to ground mathematics on a small basis of a formal system
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
proved sound by metamathematical
Metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories...
finitistic
Finitism
In the philosophy of mathematics, one of the varieties of finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps...
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
Mathematische Annalen
Mathematische Annalen is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann...
, the leading mathematical journal of the time.
Gödel's incompleteness theorems
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
, proved in 1931, showed that essential aspects of Hilbert's program could not be attained. In Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
'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
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
—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
Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its...
.
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
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
and category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
), they may be avoided.
Quotes
See also
- Almost surelyAlmost surelyIn probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...
- FideismFideismFideism is 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...
- Gut feelingGut Feeling"Gut Feeling" is a song by Devo. It appears on their 1978 album Q: Are We Not Men? A: We Are Devo!. The track was used on the soundtrack of the Wes Anderson film The Life Aquatic with Steve Zissou, for which Devo member Mark Mothersbaugh composed new music, and also used in the Stacy Peralta...
- InfallibilityInfallibilityInfallibility, from Latin origin , is a term with a variety of meanings related to knowing truth with certainty.-In common speech:...
- Justified true beliefJustified true beliefJustified true belief is one definition of knowledge that states in order to know that a given proposition is true, one must not only believe the relevant true proposition, but one must also have justification for doing so. In more formal terms, a subject S knows that a proposition P is true if,...
- NeuroethologicalNeuroethologyNeuroethology is the evolutionary and comparative approach to the study of animal behavior and its underlying mechanistic control by the nervous system...
innate behavior, instinctInstinctInstinct or innate behavior is the inherent inclination of a living organism toward a particular behavior.The simplest example of an instinctive behavior is a fixed action pattern, in which a very short to medium length sequence of actions, without variation, are carried out in response to a... - Pascal's WagerPascal's WagerPascal's Wager, also known as Pascal's Gambit, is a suggestion posed by the French philosopher, mathematician, and physicist Blaise Pascal that even if the existence of God could not be determined through reason, a rational person should wager as though God exists, because one living life...
- PragmatismPragmatismPragmatism is a philosophical tradition centered on the linking of practice and theory. It describes a process where theory is extracted from practice, and applied back to practice to form what is called intelligent practice...
- Skeptical hypothesis
- As contrary concepts
- FallibilismFallibilismFallibilism is the philosophical principle that human beings could be wrong about their beliefs, expectations, or their understanding of the world...
- IndeterminismIndeterminismIndeterminism is the concept that events are not caused, or not caused deterministically by prior events. It is the opposite of determinism and related to chance...
- MultiverseMultiverseThe multiverse is the hypothetical set of multiple possible universes that together comprise all of reality.Multiverse may also refer to:-In fiction:* Multiverse , the fictional multiverse used by DC Comics...
- Fallibilism
External links
- certainty, The American Heritage Dictionary of the English LanguageThe American Heritage Dictionary of the English LanguageThe American Heritage Dictionary of the English Language is an American dictionary of the English language published by Boston publisher Houghton Mifflin, the first edition of which appeared in 1969...
. Bartleby.comBartleby.comBartleby.com is an electronic text archive, headquartered in New York and named after Herman Melville's story "Bartleby the Scrivener". It was founded under the name "Project Bartleby" in January 1993 by Steven H. van Leeuwen as a personal, non-profit collection of classic literature on the website... - certainty.co.uk - The UK's National Will Register