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Indeed, the field of proof theory studies formal proofs and their properties, the most famous and surprising being that almost all axiomatic systems can generate certain undecidable statements not provable within the system. The definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics.
The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory.
In carefully designed scientific experiments, null results can be interpreted as evidence of absence. [7] Whether the scientific community will accept a null result as evidence of absence depends on many factors, including the detection power of the applied methods, the confidence of the inference, as well as confirmation bias within the community.
Mathematical statements are good examples. Like all formal sciences, mathematics is not concerned with the validity of theories based on observations in the empirical world, but rather, mathematics is occupied with the theoretical, abstract study of such topics as quantity, structure, space and change.
…[the] use of computers in mathematics, as in the [Four-Color Theorem], introduces empirical experiments into mathematics. Whether or not we choose to regard the [Four-Color Theorem] as proved, we must admit that the current proof is no traditional proof, no a priori deduction of a statement from premises. It is a traditional proof with a ...
Roberto Poli of McGill University delivered a number of lectures entitled The unreasonable ineffectiveness of mathematics in cognitive sciences in 1999. The abstract is: My argument is that it is possible to gain better understanding of the "unreasonable effectiveness" of mathematics in study of the physical world only when we have understood the equally "unreasonable ineffectiveness" of ...
Social-science methodologist Donald T. Campbell, who emphasized hypothesis testing throughout his career, later increasingly emphasized that the essence of science is "not experimentation per se" but instead the iterative competition of "plausible rival hypotheses", a process that at any given phase may start from evidence or may start from ...
The theory of evolution does not attempt to explain the origin of life [131] or the origin and development of the universe. The theory of evolution deals primarily with changes in successive generations over time after life has already originated. [132]