Another method of proof of existence, global analysis, uses Sard's lemma and the Baire category theorem; this method was pioneered by G�rard Debreu and Stephen Smale.
22.
He showed that ? ( " G " ) is nilpotent and, in so doing, developed a method of proof known today as Frattini's argument.
23.
Zeno of Elea, a pupil of Parmenides, had the idea of a standard argument pattern found in the method of proof known as " reductio ad absurdum ".
24.
This method of proof is noteworthy in that it is an example of the idea that finitistic algebraic relations in fields of model theoretic principles to prove an elementary statement about polynomials.
25.
Ribenboim says " The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption ."
26.
Zeno's arguments are perhaps the first examples of a method of proof called " reductio ad absurdum ", literally meaning " to reduce to the absurd ."
27.
Prior to then, compurgation was the most usual method of proof, and the ordeal was used in cases where there was some presumption of guilt against the accused or when the accused was bound to fail in compurgation.
28.
Since commutes with the vector field corresponding to the period flow, the inductive method of proof used for the Dirichlet problem works equally well in this case : for the difference quotients preserve the boundary condition when expressed in terms of.
29.
One method of proof, usable when there are only a finite number of cases that could lead to counterexamples, is known as " brute force " : in this approach, all possible cases are considered and shown not to give counterexamples.
30.
The first section, "'Discipline of Pure Reason "', compares mathematical and logical methods of proof, and the second section, "'Canon of Pure Reason "', distinguishes theoretical from practical reason.
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