Proof sheets were the means by which syndicates provided newspapers with black-and-white line art for the reproduction of strips ( which they arranged to have colored in the case of Sunday strips ).

* Gödel's Proof ( 2002 revised edition ) by Ernest Nagel and James R. Newman, edited by Hofstadter ( ISBN 0-8147-5816-9 ).

He famously put the point into dramatic relief with his 1939 essay " Proof of an External World ", in which he gave a common sense argument against scepticism by raising his right hand and saying " Here is one hand ," and then raising his left and saying " And here is another ," then concluding that there are at least two external objects in the world, and therefore that he knows ( by this argument ) that an external world exists.

Proof: If there is no possible move, then the lemma is vacuously true ( and the first player loses the normal play game by definition ).

* Elusive Proof, Elusive Prover: A New Mathematical Mystery, by Dennis Overbye, New York Times, Science, August 15, 2006.

Proof by contradiction is also known as indirect proof, apagogical argument, proof by assuming the opposite, and reductio ad impossibilem.

* Proof by contrapositive

meaning " quod erat demonstrandum " or by one of the tombstone marks "" or "" meaning " End of Proof ", introduced by Paul Halmos following their usage in magazine articles.

* Proof of Fermat's Last Theorem is discovered by Andrew Wiles.

* Proof by verbosity, sometimes colloquially referred to as argumentum verbosum-a rhetorical technique that tries to persuade by overwhelming those considering an argument with such a volume of material that the argument sounds plausible, superficially appears to be well-researched, and it is so laborious to untangle and check supporting facts that the argument might be allowed to slide by unchallenged.

* Proof, 1980 B / W student film by Black-and-white, expanded to Fandango

* Proof ( 1991 film ), an Australian film by Jocelyn Moorhouse

* Proof ( play ), a play by David Auburn

** Proof ( 2005 film ), a film by John Madden, based on the play

* " Proof ", a 1942 hard SF short story by Hal Clement

* " Proof ", a song by Coldplay from " Speed of Sound "

* " Proof " ( I Am Kloot song ), a song by I Am Kloot from their eponymous second album

* " Proof ", a song by Paul Simon from The Rhythm of the Saints

* " Proof ", a song by Happy Rhodes from Many Worlds Are Born Tonight

* Proof by exhaustion, proof by examining all individual cases

Proof by exhaustion, also known as proof by cases, perfect induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases and each case is checked to see if the proposition in question holds.

* Proof by exhaustion

Proof of the existence of traveling wave solutions and analysis of their properties is often done by the phase space method.

* Proof theory, a branch of mathematical logic that represents proofs as formal mathematical objects

Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques.

* Metamath Proof explorer a repository of mathematical theorems along with their proofs using the Metamath proof assistant.

Proof that QCD confines at low energy is a mathematical problem of great relevance, and an award has been proposed by the Clay Mathematics Institute for whoever is also able to show that the Yang – Mills theory has a mass gap and its existence.

: Proof theory is the mathematical study of formalized arguments.

Newman also wrote Gödel's Proof ( 1958 ) with Ernest Nagel, presenting the main results of Gödel's incompleteness theorem and the mathematical work and philosophies leading up to its discovery in a more accessible manner.

This book inspired Douglas Hofstadter to take up the study of mathematical logic, write his famous book Gödel, Escher, Bach, and prepare a second edition of Gödel's Proof, published in 2002.

: Proof: By mathematical induction.

: Proof: We use mathematical induction.

Lessing outlined the concept of the religious " Proof of Power ": How can miracles continue to be used as a base for Christianity when we have no proof of miracles?

Proof examples from 1936 to 1942 and 1950 to 1967 were struck at the Philadelphia Mint ; in 1968 proof production was shifted to the San Francisco Mint.

 He made another major contribution to the field in his 1979 paper, joint with his student Robert A. Reckhow, " The Relative Efficiency of Propositional Proof Systems ", in which they formalized the notions of p-simulation and efficient propositional proof system, which started an area now called propositional proof complexity.

Proof: We only give the proof in the simplified case ; the general case is similar.

This informal analysis can be formalized to make a rigorous proof of the incompleteness theorem, as described in the section " Proof sketch for the first theorem " below.

Proof beyond a reasonable doubt, therefore, is proof of such a convincing character that you would be willing to rely and act upon it without hesitation in the most important of your own affairs.

Proof theory is important in philosophical logic, where the primary interest is in the idea of a proof-theoretic semantics, an idea which depends upon technical ideas in structural proof theory to be feasible.

Proof methods that are not direct include proof by contradiction, including proof by infinite descent.

Discharging these proof obligations is performed using the SPADE Simplifier ( an automated theorem prover ) and the SPADE Proof Checker ( a manual theorem prover, used for those VCs too thorny for the Simplifier to automatically discharge ).

:: Sketch of proof: Proof by contradiction.

Any proof by contrapositive can also be trivially formulated in terms of a Proof by contradiction: To prove the proposition, we consider the opposite,.

A popular front-end for proof assistants is the Emacs-based Proof General, developed at the University of Edinburgh.

Some issues sell for significant sums, for example the 1995-W proof ($ 3, 500 as of 2011 ) and the 2006 20th anniversary set containing a special " Reverse Proof " coin along with a regular proof coin and the new " Burnished Uncirculated " coin ( over $ 300 as of 2011 ).

The LeMats which found their way through the Union blockade were stamped with British proof marks from the Birmingham Proof House, leading to the misapprehension that the pistols were actually manufactured in the UK.

Becker's Postulate later played a role in the formalization given, by Charles Hartshorne, the American process theologian, of the Ontological Proof of God's existence, stimulated by conversations with the logical positivist and opponent of the alleged proof, Rudolf Carnap.