I must confess that I prefer the Liberal who is personally affected, who is willing to send his own children to a mixed school as proof of his faith.

In any event, the critical productivity of that time is abundant proof that if he was taking laudanum, it was never in command of him to the extent that it had been during his vagrant years.

The appointment of U Thant of Burma as the U.N.'s Acting Secretary General -- at this writing, the choice appears to be certain -- offers further proof that in politics it is more important to have no influential enemies than to have influential friends.

The first is the strictly scientific, which demands concrete proof and therefore may err on the conservative side by waiting for evidence in the flesh.

The idea of the proof is this.

In the notation of the proof of Theorem 12, let us take a look at the special case in which the minimal polynomial for T is a product of first-degree polynomials, i.e., the case in which each Af is of the form Af.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

so that the absence of the hymen is by no means positive proof that a girl has had sex relations.

Most of them, the world over, operate on the same principle by which justice is administered in France and some other Latin countries: the customer is to be considered guilty of abysmal ignorance until proven otherwise, with the burden of proof on the customer himself.

Schwab also declared there is no proof of Weinstein's entering a conspiracy to use the U.S. mails to defraud, to which federal prosecutor A. Lawrence Burbank replied:

level ( when the standard of proof is high, the chances of overlooking

Chaitin prefaces his definition with: " I'll show you can't prove that a program is ' elegant '"— such a proof would solve the Halting problem ( ibid ).

Euclid stipulated this so that he could construct a reductio ad absurdum proof that the two numbers ' common measure is in fact the greatest.

Similarly, all the statements listed below which require choice or some weaker version thereof for their proof are unprovable in ZF, but since each is provable in ZF plus the axiom of choice, there are models of ZF in which each statement is true.

When one attempts to solve problems in this class, it makes no difference whether ZF or ZFC is employed if the only question is the existence of a proof.

It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF.

There is some documentary proof that the Romans named the hot sulfur springs of Aachen Aquis-Granum, and indeed to this day the city is known in Italian as Aquisgrana, in Spanish as Aquisgrán and in Polish as Akwizgran.

The above proof is an example of a non-constructive proof disallowed by intuitionists:

By non-constructive Davis means that " a proof that there actually are mathematic entities satisfying certain conditions would have to provide a method to exhibit explicitly the entities in question.

In addition, some adherents of these schools reject non-constructive proofs, such as a proof by contradiction.

This can be a non-constructive proof ( possibly involving a strategy stealing argument ) that need not actually determine any moves of the perfect play.

The proof is non-constructive ( a pure existence proof ): while it guarantees that if a sentence α follows ( semantically ) from a set of sentences Σ, then there is a proof of α from Σ, it gives no indication of the nature of that proof.

This is in contrast to a non-constructive proof ( also known as an existence proof or pure existence theorem ) which proves the validity of a proposition without considering an example.

Some non-constructive proofs show that if a certain proposition is false, a contradiction ensues ; consequently the proposition must be true ( proof by contradiction ).

Nearly every proof which explicitly relies on the axiom of choice is non-constructive in nature because this axiom is fundamentally non-constructive.

Typically, supporters of this view deny that pure existence can be usefully characterized as " existence " at all: accordingly, a non-constructive proof is instead seen as " refuting the impossibility " of a mathematical object's existence, a strictly weaker statement.

" This theorem can be proven via a constructive proof, or via a non-constructive proof.

Jarden's non-constructive proof proceeds as follows:

The non-constructive proof does not construct an example a and b ; it merely gives a number of possibilities ( in this case, two mutually exclusive possibilities ) and shows that one of them — but does not show which one — must yield the desired example.

( It turns out that is irrational because of the Gelfond – Schneider theorem, but this fact is irrelevant to the correctness of the non-constructive proof.

For finite games, and games where the appropriate instance of Markov's rule can be constructively established by means of bar induction, then the non-constructive proof of a winning strategy for the first player can be converted into a winning strategy.

The use of the infinity lemma in this proof of the Garden of Eden theorem makes it non-constructive, but this is unavoidable, because there cannot exist an algorithm that always terminates and that correctly tests whether a given automaton has a Garden of Eden.

* In some contexts, it means a method of non-constructive existence proof in mathematics called the probabilistic method

That was so because it not only was proof of excessive pride, but also resulted in violent acts by or to those involved.

* Metamath-a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems ; while the Metamath language is not accompanied with an automated theorem prover, it can be regarded as important because the formal language behind it allows development of such a software ; as of March, 2012, there is no " widely " known such software, so it is not a subject of " automated theorem proving " ( it can become such a subject ), but it is a proof assistant.

The second proof does not need this because showing that the error term vanishes proves the existence of the limit directly.

( For groups of low 2-rank the proof of this breaks down, because theorems such as the signalizer functor theorem only work for groups with elementary abelian subgroups of rank at least 3.

Various other speculative proposals have been made in the 19th and 20th centuries, all of them failing to gain significant acceptance because no significant proof of this connection was found.

") have little value as proof because of the placebo effect, the biases inherent in observation and reporting of cases, difficulties in ascertaining who is an expert and more.

Initially, their proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand.

This new proof is similar to Appel and Haken's but more efficient because it reduced the complexity of the problem and required checking only 633 reducible configurations.

In August 1970, Gödel told Oskar Morgenstern that he was " satisfied " with the proof, but Morgenstern recorded in his diary entry for 29 August 1970, that Gödel would not publish because he was afraid that others might think " that he actually believes in God, whereas he is only engaged in a logical investigation ( that is, in showing that such a proof with classical assumptions ( completeness, etc.

Undaunted, Luthor gathers together a new Revenge Squad to fight against invading Kryptonians led by General Zod, leaving Superman alive to provide assistance simply because he believed that this ' invasion ' was proof that he had been right about Superman all along and he wanted Superman to live with that knowledge.

Precise credit is at times difficult or contentious because scientific language and standards of proof changed, so whether medieval statements are equivalent to modern statements or sufficient proof, or instead similar to modern statements and hypotheses is often debatable.

Refraining from these restrictions when there is unequivocal proof that this will lead to the murder of innocents is worse, because movement restricted can later be granted, while dead will never live again.

The validity of sense perception, therefore, is not susceptible to proof ( because it is presupposed by all proof as proof is only a matter of adducing sensory evidence ) nor should its validity be denied ( since the conceptual tools one would have to use to do this are derived from sensory data ).

For the first three years of clad production, in lieu of proof sets, specimen sets were specially sold as " Special Mint Sets " minted at the San Francisco mint in 1965, 1966, and 1967 ( Deep Cameo versions of these spectacular coins are highly valued because of their rarity ).

This result is quite interesting, because the statement is purely algebraic yet the simplest proof is topological.

Also, using proof by contradiction is problematical because the axiomatizations of all practical domains of knowledge are inconsistent in practice.

Many different types of glue held these sections together most commonly Irish glue and bone glue until hilton glue or cement glue was introduced because of its water proof qualities.

In the social proof model, people imitate those who seem similar, despite or even because of societal disapproval.

However, this comes at a cost: because the keystream is now pseudorandom, and not truly random, the proof of security associated with the one-time pad no longer holds: it is quite possible for a stream cipher to be completely insecure.