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The development of mathematical proof is primarily the product of ancient Greek mathematics, and one of its greatest achievements. [9] Thales (624–546 BCE) and Hippocrates of Chios (c. 470–410 BCE) gave some of the first known proofs of theorems in geometry.
The independence of the mathematical objects is such that they are non physical and do not exist in space or time. Neither does their existence rely on thought or language. For this reason, mathematical proofs are discovered, not invented. The proof existed before its discovery, and merely became known to the one who discovered it. [13]
Thus the only thing we do not have is a formal proof of consistency of whatever version of set theory we may prefer, such as ZF. In practice, most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC, generally their preferred axiomatic system. In most of mathematics as it is practiced ...
Bertrand's postulate and a proof; Estimation of covariance matrices; Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational
A good example of this was the machine-aided proof of the four color theorem, which was very controversial as the first claimed mathematical proof that was essentially impossible to verify by humans due to the enormous size of the program's calculation (such proofs are called non-surveyable proofs).
Turing's proof, although it seems to use the "diagonal process", in fact shows that his machine (called H) cannot calculate its own number, let alone the entire diagonal number (Cantor's diagonal argument): "The fallacy in the argument lies in the assumption that B [the diagonal number] is computable" [3] The proof does not require much ...
In other words, despite their claim of mathematics being human, established mathematical knowledge — which is what we learn in school — is assumed to be and treated as abstract, completely detached from its physical origin. It cannot account for the way learners could access to such knowledge. [7] WMCF is also criticized for its monist ...
God Created the Integers: The Mathematical Breakthroughs That Changed History is a 2005 anthology, edited by Stephen Hawking, of "excerpts from thirty-one of the most important works in the history of mathematics." [1] Each chapter of the work focuses on a different mathematician and begins with a biographical overview. Within each chapter ...