Report on Mathematical Proofs
Added on 2020-04-15
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Running head: MATHEMATICAL PROOFS1Mathematical ProofsNameInstitution
MATHEMATICAL PROOFS2Mathematical Proofs“What are the possible benefits of a computer program that can check proofs?”One of the highlights of mathematics in the 20th century was the realization that proofs can be made completely precise and proofs can be checked precisely. These can be carried out tothe level of putting a significant amount of mathematics on a computer and having the computer check the validity of the proofs. A Mathematical ProofIn mathematics, a proof refers to “an inferential argument for some mathematical statement whereby previously established theorems are applied” (Buch Berger & Paule, 2013). Proofs trace back to axioms as well as inference rules, which are assumed statements preferably referred to “self-evident.” Buch Berger and Paule define axioms as conditions that a statement has to meet before it applies. Additionally, proofs are adapted from inductive reasoning and differ from empirical arguments in that a demonstration must accompany a proof that a statementis true. Therefore, a proof should list all possible outcomes and deduce, from the cases, that a given reasoning holds. Proofs, just like conjecture, employ logic but usually contain some elements of natural language. In fact, the majority of proofs in mathematics are considered as applications of informal logic and not discussed in proof theory. Notably, the difference between formal and informal proofs has led to more emphasis on mathematical practice and mathematics philosophy as a whole. In mathematics, indicating that a proof is wrong does not necessarily mean that the conclusion based on the claim is not right. However, in the article on mathematics and skepticism, Kasman (2014) states that:
MATHEMATICAL PROOFS3“If you cannot completely prove your claims in mathematics, the new results will not be accepted by the mathematical community, and they will not be published in a journal, and -- to be blunt -- you won’t be a mathematician for long...”Similarly, there are instances where mathematical proofs with errors have been publishedand remained unnoticed for quite a long time. These have been the cases for the inconspicuous journals that few people read. Such proofs had received public focus prior to the error discovery. For instance, the faulty “Four-color map Theorem” proof by Alfred Kembe that was published in1879. The error was discovered 11 years later (Michael, 2015). Therefore, mathematical proofs are similar to scientific theories in the sense that: If an error is found, it immediately invalidates the proposition. With the advancement in technology that has seen revolutionization of the computer science world, some proofs have been verified by the computer.Computer Checked Mathematical ProofsThese are mathematical proofs which have been partially or fully generated by the computer. According to Harris (2015), lengthy computations necessitated the use of computer programs. As a result, most of the computer-based proofs nowadays have been executed of “proofs by exhaustion of a mathematical theorem.” Computers programs are used to perform the calculations and then provide a proof based on the results obtained from a given theorem. In the field of artificial intelligence, attempts have been made to create new proofs that are smaller and explicit by use of machine reasoning techniques. Such as the heuristic search. In the recent past, heated debates have risen. Many ask “can we trust a mathematical proof simply because it is accompanied by a claim of ‘checked by computer’?” Some claim that an error in Godel’s original “Proof of Incompleteness” is unlikely since the proofs have been
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