Logical intuition

Logical Intuition, or mathematical intuition or rational intuition, is the ability to perceive logical or mathematical truth.[1] Humans apply logical intuition in proving mathematical theorems,[2] validating logical arguments,[3] developing algorithms and heuristics,[4] and in related contexts.[5] The ability to recognize logical or mathematical truth may vary from person to person and could be subject to cultivation in students.[6] The ability may not be realizable in a computer program by means other than genetic programming or evolutionary programming.[7]

History

Plato and Aristotle 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.[8]

Henri Poincaré distinguished logical intuition from other forms of intuition. In his book The Value of Science he points out:

...[T]here are many kinds of intuition. I have said how much the intuition of pure number, whence comes rigorous mathematical induction, differs from sensible intuition to which the imagination, properly so called, is the principal contributor.[9]

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, and to allow one to comprehend logical developments.[10]

Bertrand Russell, though critical of intuitive mysticism,[11] 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:

When a certain number of logical principles have been admitted, the rest can be deduced from them; but the propositions deduced are often just as self-evident as those that were assumed without proof. All arithmetic, moreover, can be deduced from the general principles of logic, yet the simple propositions of arithmetic, such as 'two and two are four', are just as self-evident as the principles of logic.[12]

Kurt Gödel demonstrated based on his incompleteness theorems that intuition-based propositional calculus cannot be finitely valued.[13] Gödel likened logical intuition to sense perception and considered the mathematical constructs that humans perceive to have an independent existence of their own.[14] The human mind's ability to sense such abstract constructs may not be finitely describable.[15]

Discussion

Dissent regarding the value of intuition in a logical or mathematical context may often hinge on the breadth of the definition of intuition.[16] Dissent regarding the implications of logical intuition in the fields of artificial intelligence and cognitive computing may similarly hinge on definitions.[17] However, similarity between the potentially infinite nature of logical intuition posited by Gödel and the hard problem of consciousness posited by David Chalmers suggest that the realms of intuitive knowledge and experiential consciousness may both have aspects that are not reducible to classical physics concepts.[18]

See also

References

  1. ↑ Parsons, Charles (1980). "X - Mathematical Intuition". 80 (New Series). Proceedings of the Aristotelian Society: 145–168. JSTOR 4544956.
  2. ↑ Lipton, Richard (2010). "Mathematical Intuition—What Is It?".
  3. ↑ Nakamura, Hiroko; Kawaguchi, Jun (2016). "People Like Logical Truth: Testing the Intuitive Detection of Logical Value in Basic Propositions". PLOS ONE.
  4. ↑ "Intuitive way to understand tree recursion". StackOverflow.com. 2014.
  5. ↑ "Godel and the Nature of Mathematical Truth - A Talk with Rebecca Newberger Goldstein". Edge Foundation, Inc. 2005.
  6. ↑ "Developing Your Intuition For Math". BetterExplained.com.
  7. ↑ Rucker, Rudy. Infinity and the Mind. Princeton University Press. , section 330 "Artificial Intelligence via Evolutionary Processes"
  8. ↑ Piętka, Dariusz (2015). "The Concept of Intuition and Its Role in Plato and Aristotle". Organon.
  9. ↑ Poincaré, Henri (1905). "Intuition and Logic in Mathematics, from the book The Value of Science".
  10. ↑ Poincaré, Henri (1905). The Value of Science.
  11. ↑ Popova, Maria (2016). "A Largeness of Contemplation: Bertrand Russell on Intuition, the Intellect, and the Nature of Time". BrainPickings.org.
  12. ↑ Russell, Bertrand (1912). Problems of Philosophy. Chapter XI "On Intuitive Knowledge"
  13. ↑ Kennedy, Juliette (2015). Kurt Gödel. Stanford Encyclopedia of Philosophy.
  14. ↑ Ravitch, Harold (1998). "On Gödel's Philosophy of Mathematics".
  15. ↑ Solomon, Martin (1998). "On Kurt Gödel's Philosophy of Mathematics".
  16. ↑ XiXiDu (2011). "Intuition and Mathematics".
  17. ↑ Szangolies, Jochen (2015). "Does Strong AI conflict with Gödel's incompleteness theorem?".
  18. ↑ Aas, Benjamin (2011). "Body-Gödel-Mind: The unsolvability of the hard problem of consciousness" (PDF).
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