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 study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

al or

mathematical 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 ...

truth—and the ability to solve mathematical challenges efficiently. Humans apply logical intuition in proving mathematical

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

s, validating logical arguments, developing

algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

s and

heuristic A heuristic or heuristic technique (''problem solving'', '' mental shortcut'', ''rule of thumb'') is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless ...

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 Genetic programming (GP) is an evolutionary algorithm, an artificial intelligence technique mimicking natural evolution, which operates on a population of programs. It applies the genetic operators selection (evolutionary algorithm), selection a ...

evolutionary programming Evolutionary programming is an evolutionary algorithm, where a share of new population is created by mutation of previous population without crossover. Evolutionary programming differs from evolution strategy ES(\mu+\lambda) in one detail. All in ...

History

Plato Plato ( ; Greek language, Greek: , ; born BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...

and

Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...

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 (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...

distinguished logical intuition from other forms of intuition. In his book ''

The Value of Science ''The'' is a grammatical article in English, denoting nouns that are already or about to be mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The ...

'', he points out that: The passage goes on to assign two roles to logical intuition: to permit one to choose which route to follow in search of scientific

truth Truth or verity is the Property (philosophy), property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth, 2005 In everyday language, it is typically ascribed to things that aim to represent reality or otherwise cor ...

, 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 philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...

, though critical of intuitive

mysticism Mysticism is popularly known as becoming one with God or the Absolute (philosophy), Absolute, but may refer to any kind of Religious ecstasy, ecstasy or altered state of consciousness which is given a religious or Spirituality, spiritual meani ...

, pointed out that the degree to which a truth is self-evident according to logical intuition can vary, from one situation to another, and stated that some self-evident truths are practically infallible:

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 profoundly ...

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 ...

that intuition-based

propositional calculus The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...

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 state of having being or reality in contrast to nonexistence and nonbeing. Existence is often contrasted with essence: the essence of an entity is its essential features or qualities, which can be understood even if one does ...

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 the capability of computer, computational systems to perform tasks typically associated with human intelligence, such as learning, reasoning, problem-solving, perception, and decision-making. It is a field of re ...

and cognitive computing may similarly hinge on definitions. However, similarity between the potentially infinite nature of logical intuition posited by Gödel and the

hard problem of consciousness In the philosophy of mind, the hard problem of consciousness is to explain why and how humans and other organisms have qualia, phenomenal consciousness, or subjective experience. It is contrasted with the "easy problems" of explaining why and how ...

posited by

David Chalmers David John Chalmers (; born 20 April 1966) is an Australian philosopher and cognitive scientist, specializing in philosophy of mind and philosophy of language. He is a professor of philosophy and neural science at New York University, as well ...

suggest that the realms of intuitive knowledge and experiential consciousness may both have aspects that are not reducible to classical physics concepts.

References

{{reflist Logic

History

Discussion

See also

References