Group: formalism
Group: mathematics
Group: meaning and truth
Topic: abstraction
Topic: analytic truth
Topic: constructing proof and program together
Topic: logic
Topic: logic programming
Topic: mathematical proof as a social process
Topic: mathematics as a formal system
Topic: mathematics by proofs and refutations
Topic: program proof via assertions
Topic: proofcarrying code
Topic: theorem proving systems
 
Subtopic: indubitable mathematical truth
Quote: a mathematician depends on an innate sense of rightness; seldom attempts a proof if not already convinced of its truth [»browGS_1972]
 Quote: showed that arithmetic laws are analytic judgments (a priori) [»fregG_1884]
 Quote: chain spaces is the proofgenerated concept that yields an indubitable formulation of Euler's theorem [»lakaI_1976]
 Quote: mathematical truth is not something that we ascertain merely by an algorithm; we must 'see' its truth [»penrR_1989]
 Subtopic: fallible mathematical truth
Quote: even though formalism, platonism, and constructivism treat mathematics as indubitable truth, mathematical truth is fallible and corrigible [»daviPJ_1981]
 Quote: in Frege's and Russell's systems can give infallible proofs if the axioms and translation are correct; but both are fallible [»lakaI_1976]
 Quote: mathematical heuristic is like scientific heuristic but with different conjectures, explanations, and counterexamples [»lakaI_1976]
 Subtopic: fruitful mathematical truth
Quote: a real proof is fruitful, its conclusion is in a sense more general than its premises
 Subtopic: proof
Quote: both lemmas and subroutines separate proof/implementation from use
 Quote: both mathematical proof and programming use divideandrule in breaking a proof/program into lemmas/subroutines [»dijkEW_1982]
 Quote: a program proof is for refutation of certain forms of incorrectness; math proofs demonstrate existence [»dobsJ4_1989]
 Quote: Euclid's algorithm for greatest common divisor; with proof [»eucl_300, OK]
 Quote: proof that intersection of planes is a straight line [»eucl_300, OK]
 Subtopic: proof as exploration
Quote: a mathematician depends on an innate sense of rightness; seldom attempts a proof if not already convinced of its truth [»browGS_1972]
 Quote: the process of building a mathematical proof is exploratory, a process of filling in gaps; the end result is highly structured
 Quote: designing a mathematical theory requires experimentation with new notations and concepts [»dijkEW_1982]
 Quote: mathematical invention often occurs after a long period of unconscious work following and followed by conscious work [»poinH_1908, OK]
 Subtopic: proof by induction
Quote: proof by induction rests upon the notion of a chain, i.e., a sequence of numbers
 Quote: proof by induction allows us to generalize, to pass from the finite to the infinite [»poinH_1902, OK]
 Quote: for Russell, mathematical induction is a definition, not a principle as for Poincare [»russB_1919, OK]
 Quote: we define the natural numbers as those to which proofs by mathematical induction can be applied
 Quote: mathematical induction is the essential characteristic that distinguishes the finite from the infinite [»russB_1919, OK]
 Subtopic: proof verification
Quote: proof verification is analytical and leads to nothing but the premises translated into another language [»poinH_1902, OK]

Related Topics
Group: formalism (9 topics, 478 quotes)
Group: mathematics (23 topics, 560 quotes)
Group: meaning and truth (18 topics, 634 quotes)
Topic: abstraction (62 items)
Topic: analytic truth (51 items)
Topic: constructing proof and program together (22 items)
Topic: logic (84 items)
Topic: logic programming (34 items)
Topic: mathematical proof as a social process (14 items)
Topic: mathematics as a formal system (30 items)
Topic: mathematics by proofs and refutations (31 items)
Topic: program proof via assertions (61 items)
Topic: proofcarrying code (7 items)
Topic: theorem proving systems (20 items)
