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Non-surveyable Proofs
In the philosophy of mathematics, a non-surveyable proof is a mathematical proof that is considered infeasible for a human mathematician to verify and so of controversial validity. The term was coined by Thomas Tymoczko in 1979 in criticism of Kenneth Appel and Wolfgang Haken's computer-assisted proof of the four color theorem, and has since been applied to other arguments, mainly those with excessive case splitting and/or with portions dispatched by a difficult-to-verify computer program. Surveyability remains an important consideration in computational mathematics. Tymoczko's argument Tymoczko argued that three criteria determine whether an argument is a mathematical proof: * ''Convincingness'', which refers to the proof's ability to persuade a rational prover of its conclusion; * ''Surveyability'', which refers to the proof's accessibility for verification by members of the human mathematical community; and * ''Formalizability'', which refers to the proof's appealing to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophy Of Mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's lives. The logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts. The philosophy of mathematics has two major themes: mathematical realism and mathematical anti-realism. History The origin of mathematics is subject to arguments and disagreements. Whether the birth of mathematics was a random happening or induced by necessity during the development of other subjects, like physics, is still a matter of prolific debates. Many thinkers have contributed their ideas concerning the nature of mathematics. Today, some philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deductive Reasoning
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be false. For example, the inference from the premises "all men are mortal" and "Socrates is a man" to the conclusion "Socrates is mortal" is deductively valid. An argument is ''sound'' if it is ''valid'' and all its premises are true. Some theorists define deduction in terms of the intentions of the author: they have to intend for the premises to offer deductive support to the conclusion. With the help of this modification, it is possible to distinguish valid from invalid deductive reasoning: it is invalid if the author's belief about the deductive support is false, but even invalid deductive reasoning is a form of deductive reasoning. Psychology is interested in deductive reasoning as a psychological process, i.e. how people ''actually'' draw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proofs
A mathematical proof is an inferential 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 examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols, along ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automated Proof Checking
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer. System comparison * ACL2 – a programming language, a first-order logical theory, and a theorem prover (with both interactive and automatic modes) in the Boyer–Moore tradition. * Coq – Allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, and extracts a certified program from the constructive proof of its formal specification. * HOL theorem provers – A family of tools ultimately derived from the LCF theorem prover. In these systems the logical core is a library of their programming language. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kepler Conjecture
The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is around 74.05%. In 1998, Thomas Hales, following an approach suggested by , announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof, and the Kepler conjecture was accepted as a theorem. In 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Teller (mathematician)
Paul Scott Teller (born February 21, 1971) is the executive director of Advancing American Freedom in Washington DC, Mike Pence's advocacy organization. Teller previously served under President Donald Trump as Special Assistant to the President for Legislative Affairs, as well Director of Strategic Affairs for Vice President Mike Pence. Earlier, Teller had been chief of staff for Texas Senator Ted Cruz and executive director of the United States House of Representatives Republican Study Committee. In a profile published shortly after Cruz appointed Teller his chief of staff, The Hill described him as "Cruz's agitator in chief." In January 2017, Trump and Pence appointed him Special Assistant to the President for Legislative Affairs, with a focus on Senate and House conservatives. In February 2020, Pence appointed Teller as Deputy Assistant to the President and Director of Strategic Initiatives for the Vice President. Early life Raised on Long Island, Teller graduated from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book, ''The principles of arithmetic presented by a new method'' ( la, Arithmet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Software Bug
A software bug is an error, flaw or fault in the design, development, or operation of computer software that causes it to produce an incorrect or unexpected result, or to behave in unintended ways. The process of finding and correcting bugs is termed " debugging" and often uses formal techniques or tools to pinpoint bugs. Since the 1950s, some computer systems have been designed to deter, detect or auto-correct various computer bugs during operations. Bugs in software can arise from mistakes and errors made in interpreting and extracting users' requirements, planning a program's design, writing its source code, and from interaction with humans, hardware and programs, such as operating systems or libraries. A program with many, or serious, bugs is often described as ''buggy''. Bugs can trigger errors that may have ripple effects. The effects of bugs may be subtle, such as unintended text formatting, through to more obvious effects such as causing a program to crash, freezing th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bonnie Gold
Bonnie Gold (born 1948) is an American mathematician, mathematical logician, philosopher of mathematics, and mathematics educator. She is a professor emerita of mathematics at Monmouth University. Education and career Gold completed her Ph.D. in 1976 at Cornell University, under the supervision of Michael D. Morley. She was the chair of the mathematics department at Wabash College before moving to Monmouth, where she also became department chair. Contributions The research from Gold's dissertation, ''Compact and \omega-compact formulas in L_'', was later published in the journal ''Archiv für Mathematische Logik und Grundlagenforschung'', and concerned infinitary logic. With Sandra Z. Keith and William A. Marion she co-edited ''Assessment Practices in Undergraduate Mathematics'', published by the Mathematical Association of America (MAA) in 1999. With Roger A. Simons, Gold is also the editor of another book, ''Proof and Other Dilemmas: Mathematics and Philosophy'' (MAA, 2008) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Priori And A Posteriori
("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current experience (e.g., as part of a new study). Examples include mathematics,Some associationist philosophers have contended that mathematics comes from experience and is not a form of any a priori knowledge () tautologies, and deduction from pure reason.Galen Strawson has stated that an argument is one in which "you can see that it is true just lying on your couch. You don't have to get up off your couch and go outside and examine the way things are in the physical world. You don't have to do any science." () knowledge depends on empirical evidence. Examples include most fields of science and aspects of personal knowledge. The terms originate from the analytic methods found in ''Organon'', a collection of works by Aristotle. Prior analytic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Experiment
An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome occurs when a particular factor is manipulated. Experiments vary greatly in goal and scale but always rely on repeatable procedure and logical analysis of the results. There also exist natural experiment, natural experimental studies. A child may carry out basic experiments to understand how things fall to the ground, while teams of scientists may take years of systematic investigation to advance their understanding of a phenomenon. Experiments and other types of hands-on activities are very important to student learning in the science classroom. Experiments can raise test scores and help a student become more engaged and interested in the material they are learning, especially when used over time. Experiments can vary from personal and in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is an inferential 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 examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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