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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 the axioms and previously proved theorems.
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Philosophiæ Naturalis Principia Mathematica often referred to as simply the Principia, is a book by Isaac Newton that expounds Newton's laws of motion and his law of universal gravitation. The Principia is written in Latin and comprises three volumes, and was authorized, imprimatur, by Samuel Pepys, then-President of the Royal Society on 5 July 1686 and first published in 1687.
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic". An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics.
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Owing to its origin in ancient Greece and Rome, English rhetorical theory frequently employs Greek and Latin words as terms of art. This page explains commonly used rhetorical terms in alphabetical order. The brief definitions here are intended to serve as a quick reference rather than an in-depth discussion. For more information, click the terms.
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
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Mathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics. Mathematicians may express this pleasure by describing mathematics as beautiful or describe mathematics as an art form, or, at a minimum, as a creative activity.
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 due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics.
References
- ↑ Bennett, Bo. "Proof by Intimidation". logicallyfallacious.com. Retrieved 2019-12-01.
- ↑ Michael H. F. Wilkinson. "Cogno-Intellectualism, Rhetorical Logic, and the Craske-Trump Theorem" (PDF). Annals of Improbable Research . 6 (5): 15–16. Retrieved 2008-02-22.
- ↑ Tony Hey (1999). "Richard Feynman and computation" (PDF). Contemporary Physics. 40 (4): 257–265. Bibcode:1999ConPh..40..257H. doi:10.1080/001075199181459 . Retrieved 2008-02-22.
- ↑ Marjorie K. Jeffcoat (July 2003). "Junk science: Appearances can be deceiving". Journal of the American Dental Association . 134 (7): 802–803. doi:10.14219/jada.archive.2003.0268. PMID 12892436.
- ↑ Fisher, Ronald Aylmer (1930). The genetical theory of natural selection. Oxford: The Clarendon Press. p. 27. OCLC 18500548.
- ↑ Caswell, Hal (2001). Matrix Population Models. Sinauer Associates, Incorporated. p. 92. ISBN 0-87893-096-5.
- ↑ Goodman, Leo A. (1968-03-01). "An elementary approach to the population projection-matrix, to the population reproductive value, and to related topics in the mathematical theory of population growth". Demography. 5 (1). Duke University Press: 382–409. doi: 10.1007/bf03208583 . ISSN 0070-3370. S2CID 46970216.