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The concept of proof is formalized in the field of mathematical logic. [ 12] A formal proof is written in a formal language instead of natural language. A formal proof is a sequence of formulas in a formal language, starting with an assumption, and with each subsequent formula a logical consequence of the preceding ones.
Proof theory is a major branch [1] of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed ...
Nevertheless, the subsequent achievements of proof theory at the very least clarified consistency as it relates to theories of central concern to mathematicians. Hilbert's work had started logic on this course of clarification; the need to understand Gödel's work then led to the development of recursion theory and then mathematical logic as an ...
Solomon Feferman (December 13, 1928 – July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic.In addition to his prolific technical work in proof theory, computability theory, and set theory, he was known for his contributions to the history of logic (for instance, via biographical writings on figures such as Kurt Gödel, Alfred Tarski, and Jean van ...
In physics and cosmology, the mathematical universe hypothesis ( MUH ), also known as the ultimate ensemble theory, is a speculative "theory of everything" (TOE) proposed by cosmologist Max Tegmark. [ 1][ 2] According to the hypothesis, the universe is a mathematical object in and of itself. Tegmark extends this idea to hypothesize that all ...
Sir Andrew John Wiles. Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were believed to be impossible to ...
This is a list of mathematical theories. Almgren–Pitts min-max theory; Approximation theory; Arakelov theory; ... Proof theory; Quantum theory; Queue theory; Ramsey ...
This reflective critique in which the theory under review "becomes itself the object of a mathematical study" led Hilbert to call such study metamathematics or proof theory. [ 29 ] At the middle of the century, a new mathematical theory was created by Samuel Eilenberg and Saunders Mac Lane , known as category theory , and it became a new ...