"When we are engaged in investigating the foundations of a science, we must set up a system of s which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas; and no statement within the realm of the science... is held to be correct unless it can be derived from axioms by means of a finite number of logical steps. Upon closer consideration the question arises: Whether, in any way, certain statements of single axioms depend upon one another, and whether the axioms may not therefore contain certain parts in common, which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another. But above all, I wish to designate the following as the most important among numerous questions which can be asked with respect to the axioms: To prove that they are not contradictory, that is, that a finite number of logical steps based upon them can never lead to contradictory results."
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are ex
"Paul Erdős, although an atheist, spoke of an imaginary book, in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would exclaim "This ones from the Book!". This viewpoint expresses... that mathematics, as the... foundation on which the laws of our universe are built, is a natural candidate for what has been personified as God..."
Mathematical proof"Erdős was a genius at finding brilliantly simple proofs of deep results, but, until recently, very much of his work was ignored..."
Mathematical proof"Proofs are for the mathematician what experimental procedures are for the experimental scientist: in studying them one learns of new ideas, new concepts, new strategies—devices which can be assimilated for ones own research and be further developed."
Mathematical proof"Under the same assumptions made in the Chord papers, the [SIGCOMM] version of the protocol is not correct, and not one of the properties claimed invariant in [PODC] is actually invariantly true of it."
Mathematical proof"Perhaps we should discard the myth that mathematics is a rigorously deductive enterprise... hand-waving is intrinsic. We try to minimize it and we can sometimes escape it, but not always, if we want to discover new theorems."
Mathematical proof"are expressed using no numbers or other symbolic formalisms. Though the nitty-gritty details of the proof are formidably technical, the proofs overall strategy, delightfully, is not. ...They belong to a branch of mathematics known as formal logic or mathematical logic, a field which was viewed, prior to Gödels achievement, as mathematically suspect."
Mathematical proof