Corollary
A corollary ( or UK ) is a statement that follows readily from a previous statement.
In mathematics a corollary typically follows a theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective. Proposition B is a corollary of proposition A if B can readily be deduced from A or is self-evident from its proof, but the meaning of readily or self-evident varies depending upon the author and context. The importance of the corollary is often considered secondary to that of the initial theorem; B is unlikely to be termed a corollary if its mathematical consequences are as significant as those of A. Sometimes a corollary has a proof that explains the derivation; sometimes the derivation is considered self-evident.
Peirce on corollarial and theorematic reasonings
Charles Sanders Peirce held that the most important division of kinds of deductive reasoning is that between corollarial and theorematic. He argued that, while finally all deduction depends in one way or another on mental experimentation on schemata or diagrams,^{[1]} still in corollarial deduction "it is only necessary to imagine any case in which the premisses are true in order to perceive immediately that the conclusion holds in that case", whereas theorematic deduction "is deduction in which it is necessary to experiment in the imagination upon the image of the premiss in order from the result of such experiment to make corollarial deductions to the truth of the conclusion."^{[2]} He held that corollarial deduction matches Aristotle's conception of direct demonstration, which Aristotle regarded as the only thoroughly satisfactory demonstration, while theorematic deduction (A) is the kind more prized by mathematicians, (B) is peculiar to mathematics,^{[1]} and (C) involves in its course the introduction of a lemma or at least a definition uncontemplated in the thesis (the proposition that is to be proved); in remarkable cases that definition is of an abstraction that "ought to be supported by a proper postulate."^{[3]}
See also
References
- Weisstein, Eric W., "Corollary", MathWorld.
- corollary at dictionary.com
- Chambers's Encyclopaedia. Volume 3, Appleton 1864, p. 260 (online copy, p. 260, at Google Books)
- ^ ^{a} ^{b} Peirce, C. S., from section dated 1902 by editors in the "Minute Logic" manuscript, Collected Papers v. 4, paragraph 233, quoted in part in "Corollarial Reasoning" in the Commens Dictionary of Peirce's Terms, 2003–present, Mats Bergman and Sami Paavola, editors, University of Helsinki.
- ^ Peirce, C. S., the 1902 Carnegie Application, published in The New Elements of Mathematics, Carolyn Eisele, editor, also transcribed by Joseph M. Ransdell, see "From Draft A - MS L75.35-39" in Memoir 19 (once there, scroll down).
- ^ Peirce, C. S., 1901 manuscript "On the Logic of Drawing History from Ancient Documents, Especially from Testimonies', The Essential Peirce v. 2, see p. 96. See quote in "Corollarial Reasoning" in the Commens Dictionary of Peirce's Terms.