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Logical Intuition, or mathematical intuition or rational intuition, is a series of instinctive foresight, know-how, and savviness often associated with the ability to perceive
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
al or
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
truth—and the ability to solve mathematical challenges efficiently. Humans apply logical intuition in proving mathematical
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
s, validating logical arguments, developing
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
s and
heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...
s, and in related contexts where mathematical challenges are involved. The ability to recognize logical or mathematical truth and identify viable methods may vary from person to person, and may even be a result of knowledge and experience, which are subject to cultivation. The ability may not be realizable in a computer program by means other than
genetic programming In artificial intelligence, genetic programming (GP) is a technique of evolving programs, starting from a population of unfit (usually random) programs, fit for a particular task by applying operations analogous to natural genetic processes to t ...
or
evolutionary programming Evolutionary programming is one of the four major evolutionary algorithm paradigms. It is similar to genetic programming, but the structure of the program to be optimized is fixed, while its numerical parameters are allowed to evolve. It was first ...
.


History

Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
and
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
considered intuition a means for perceiving ideas, significant enough that for Aristotle, intuition comprised the only means of knowing principles that are not subject to argument.
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
distinguished logical intuition from other forms of intuition. In his book ''
The Value of Science ''The Value of Science'' (french: La Valeur de la Science) is a book by the French mathematician, physicist, and philosopher Henri Poincaré. It was published in 1905. The book deals with questions in the philosophy of science and adds detail to th ...
'', he points out that: The passage goes on to assign two roles to logical intuition: to permit one to choose which
route Route or routes may refer to: * Route (gridiron football), a path run by a wide receiver * route (command), a program used to configure the routing table * Route, County Antrim, an area in Northern Ireland * ''The Route'', a 2013 Ugandan film * Ro ...
to follow in search of scientific
truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs ...
, and to allow one to comprehend logical developments.
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
, though critical of intuitive
mysticism Mysticism is popularly known as becoming one with God or the Absolute, but may refer to any kind of ecstasy or altered state of consciousness which is given a religious or spiritual meaning. It may also refer to the attainment of insight in u ...
, pointed out that the degree to which a truth is
self-evident In epistemology (theory of knowledge), a self-evident proposition is a proposition that is known to be true by understanding its meaning without proof, and/or by ordinary human reason. Some epistemologists deny that any proposition can be self- ...
according to logical intuition can vary, from one situation to another, and stated that some self-evident truths are practically
infallible Infallibility refers to an inability to be wrong. It can be applied within a specific domain, or it can be used as a more general adjective. The term has significance in both epistemology and theology, and its meaning and significance in both fi ...
:
Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imme ...
demonstrated based on his
incompleteness theorems Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
that intuition-based
propositional calculus Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
cannot be finitely valued. Gödel also likened logical intuition to sense perception, and considered the mathematical constructs that humans perceive to have an independent
existence Existence is the ability of an entity to interact with reality. In philosophy, it refers to the ontology, ontological Property (philosophy), property of being. Etymology The term ''existence'' comes from Old French ''existence'', from Medieval ...
of their own. Under this line of reasoning, the human mind's ability to sense such abstract constructs may not be finitely implementable.


Discussion

Dissent regarding the value of intuition in a logical or mathematical context may often hinge on the breadth of the definition of intuition and the psychological underpinning of the word. Dissent regarding the implications of logical intuition in the fields of
artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech re ...
and
cognitive computing Cognitive computing (CC) refers to technology platforms that, broadly speaking, are based on the scientific disciplines of artificial intelligence and signal processing. These platforms encompass machine learning, reasoning, natural languag ...
may similarly hinge on definitions. However, similarity between the potentially infinite nature of logical intuition posited by Gödel and the
hard problem of consciousness The hard problem of consciousness is the problem of explaining why and how humans have qualia or phenomenal experiences. This is in contrast to the "easy problems" of explaining the physical systems that give us and other animals the ability to d ...
posited by
David Chalmers David John Chalmers (; born 20 April 1966) is an Australian philosopher and cognitive scientist specializing in the areas of philosophy of mind and philosophy of language. He is a professor of philosophy and neural science at New York Universi ...
suggest that the realms of intuitive knowledge and experiential consciousness may both have aspects that are not reducible to classical physics concepts.


See also

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Intuition Intuition is the ability to acquire knowledge without recourse to conscious reasoning. Different fields use the word "intuition" in very different ways, including but not limited to: direct access to unconscious knowledge; unconscious cognition; ...
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Epistemology Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Episte ...
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Philosophy of mind Philosophy of mind is a branch of philosophy that studies the ontology and nature of the mind and its relationship with the body. The mind–body problem is a paradigmatic issue in philosophy of mind, although a number of other issues are addre ...
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Philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's ...
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Cognition Cognition refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses all aspects of intellectual functions and processes such as: perception, attention, thought, ...
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Numerical cognition Numerical cognition is a subdiscipline of cognitive science that studies the cognitive, developmental and neural bases of numbers and mathematics. As with many cognitive science endeavors, this is a highly interdisciplinary topic, and includes ...
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Consciousness Consciousness, at its simplest, is sentience and awareness of internal and external existence. However, the lack of definitions has led to millennia of analyses, explanations and debates by philosophers, theologians, linguisticians, and scien ...
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Hard problem of consciousness The hard problem of consciousness is the problem of explaining why and how humans have qualia or phenomenal experiences. This is in contrast to the "easy problems" of explaining the physical systems that give us and other animals the ability to d ...
*
Panpsychism In the philosophy of mind, panpsychism () is the view that the mind or a mindlike aspect is a fundamental and ubiquitous feature of reality. It is also described as a theory that "the mind is a fundamental feature of the world which exists thro ...
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Transcendental idealism Transcendental idealism is a philosophical system founded by German philosopher Immanuel Kant in the 18th century. Kant's epistemological program is found throughout his '' Critique of Pure Reason'' (1781). By ''transcendental'' (a term that des ...
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Intuitionism 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 fu ...
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Intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
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Continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
*
Logical truth Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement whic ...


References

{{reflist Logic