A proof is a demonstration of the truth of a mathematical statement.
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"What I said was just a mathematical statement," Mnuchin told CNBC on Wednesday.
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"This paper makes a mathematical statement around retrocausality," said Renato Renner from ETH Zurich in Switzerland.
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It was just a mathematical statement to say if half these people were to lose their jobs, this is what it would be.
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What I said is just a mathematical statement, which is 40% of the people employed in the private workforce are employed by companies, 500 people and less.
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In theoretical computer science, Arden's rule, also known as Arden's lemma, is a mathematical statement about a certain form of language equations.
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The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. In Brouwer's original intuitionism, the truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can assert the truth of a statement only by verifying the validity of that construction by intuition. The vagueness of the intuitionistic notion of truth often leads to misinterpretations about its meaning. Kleene formally defined intuitionistic truth from a realist position, yet Brouwer would likely reject this formalization as meaningless, given his rejection of the realist/Platonist position.
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The term scientific theory is reserved for concepts that are widely accepted. A scientific law often refers to regularities that can be expressed by a mathematical statement. However, there is no consensus about the distinction between these terms.Scientific Laws And Theories.
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Subsequent work to resolve these problems shaped the direction of mathematical logic, as did the effort to resolve Hilbert's Entscheidungsproblem, posed in 1928. This problem asked for a procedure that would decide, given a formalized mathematical statement, whether the statement is true or false.
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These physical facts justify a mathematical statement that hotness exists on an ordered one-dimensional manifold. This is a fundamental character of temperature and thermometers for bodies in their own thermodynamic equilibrium.Maxwell, J.C. (1872). Theory of Heat, third edition, Longman's, Green & Co, London, p. 45.
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P. Oxy. 29, one of the oldest surviving fragments of Euclid's Elements, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5. A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.
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In toxicology, Haber's rule or Haber's law is a mathematical statement of the relationship between the concentration of a poisonous gas and how long the gas must be breathed to produce death, or other toxic effect. The rule was formulated by German chemist Fritz Haber in the early 1900s.
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"realism/anti-realism," The Oxford Dictionary of Philosophy, 2nd ed. revised, pp. 308–9. Oxford. Because it encompasses statements containing abstract ideal objects (i.e. mathematical objects), anti-realism may apply to a wide range of philosophic topics, from material objects to the theoretical entities of science, mathematical statement, mental states, events and processes, the past and the future.
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The Church–Turing thesis is a mathematical statement of this versatility: any computer with a certain minimum capability is, in principle, capable of performing the same tasks that any other computer can perform. Therefore, computers ranging from a netbook to a supercomputer are all able to perform the same computational tasks, given enough time and storage capacity.
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For example, mathematical physics is the application of mathematics in physics. Its methods are mathematical, but its subject is physical. The problems in this field start with a "mathematical model of a physical situation" (system) and a "mathematical description of a physical law" that will be applied to that system. Every mathematical statement used for solving has a hard-to-find physical meaning.
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Feferman 1999, p. 1 Alonzo Church and Alan Turing proposed formal models of computability, giving independent negative solutions to Hilbert's Entscheidungsproblem in 1936 and 1937, respectively. The Entscheidungsproblem asked for a procedure that, given any formal mathematical statement, would algorithmically determine whether the statement is true. Church and Turing proved there is no such procedure; Turing's paper introduced the halting problem as a key example of a mathematical problem without an algorithmic solution.
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Tamm, Journal of Physics (USSR) 9, 449 (1945) and the method is now named after both. In the late 1940s, Dancoff began a collaboration with the Viennese-refugee physician and radiologist Henry Quastler in the new field of cybernetics and information theory. Their work led to the publication of what is now commonly called Dancoff's Law. A non-mathematical statement of this law is, "the greatest growth occurs when the greatest number of mistakes are made consistent with survival".
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Tyndall scattering, i.e. colloidal particle scattering, is much more intense than Rayleigh scattering due to the bigger particle sizes involved. The importance of the particle size factor for intensity can be seen in the large exponent it has in the mathematical statement of the intensity of Rayleigh scattering. If the colloid particles are spheroid, Tyndall scattering can be mathematically analyzed in terms of Mie theory, which admits particle sizes in the rough vicinity of the wavelength of light.
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Several independent efforts to give a formal characterization of effective calculability led to a variety of proposed definitions (general recursion, Turing machines, λ-calculus) that later were shown to be equivalent. The notion captured by these definitions is known as recursive or effective computability. The Church–Turing thesis states that the two notions coincide: any number- theoretic function that is effectively calculable is recursively computable. As this is not a mathematical statement, it cannot be proven by a mathematical proof.
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To establish a mathematical statement as a theorem, a proof is required. That is, a valid line of reasoning from the axioms and other already-established theorems to the given statement must be demonstrated. In general, the proof is considered to be separate from the theorem statement itself. This is in part because while more than one proof may be known for a single theorem, only one proof is required to establish the status of a statement as a theorem.
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The initial proposal of the SYZ conjecture by Strominger, Yau, and Zaslow, was not given as a precise mathematical statement. One part of the mathematical resolution of the SYZ conjecture is to, in some sense, correctly formulate the statement of the conjecture itself. There is no agreed upon precise statement of the conjecture within the mathematical literature, but there is a general statement that is expected to be close to the correct formulation of the conjecture, which is presented here.Gross, M., Huybrechts, D. and Joyce, D., 2012.
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See e.g. The Internet Encyclopedia of Philosophy, article "Frege" Ernst Zermelo This period overlaps with the work of what is known as the "mathematical school", which included Dedekind, Pasch, Peano, Hilbert, Zermelo, Huntington, Veblen and Heyting. Their objective was the axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory. Most notable was Hilbert's Program, which sought to ground all of mathematics to a finite set of axioms, proving its consistency by "finitistic" means and providing a procedure which would decide the truth or falsity of any mathematical statement.
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Written in Latin, the title of the book is a word Lambert devised from the Greek: φῶς, φωτος (transliterated phôs, photos) = light and μετρια (transliterated metria) = measure. Lambert’s word has found its way into European languages as photometry, photometrie, fotometria. Photometria was the first work to accurately identify most fundamental photometric concepts, to assemble them into a coherent system of photometric quantities, to define these quantities with a precision sufficient for mathematical statement, and to build from them a system of photometric principles. These concepts, quantities, and principles are still in use today.
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The Kantorovich theorem, or Newton–Kantorovich theorem, is a mathematical statement on the semi-local convergence of Newton's method. It was first stated by Leonid Kantorovich in 1948. It is similar to the form of the Banach fixed-point theorem, although it states existence and uniqueness of a zero rather than a fixed point. Newton's method constructs a sequence of points that under certain conditions will converge to a solution x of an equation f(x)=0 or a vector solution of a system of equation F(x)=0.
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Another difficulty is finding the proper analogue for the uncertainty principle, an idea frequently attributed to Heisenberg, who introduced the concept in analyzing a thought experiment involving an electron and a high-energy photon. However, Heisenberg did not give precise mathematical definitions of what the "uncertainty" in these measurements meant. The precise mathematical statement of the position–momentum uncertainty principle is due to Kennard, Pauli, and Weyl. The uncertainty principle applies to situations where an experimenter has a choice of measuring either one of two "canonically conjugate" quantities, like the position and the momentum of a particle.
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Physically, hydrostatic equilibrium connects the gradient of the potential, the density, and the gradient of the pressure, whereas Poisson's equation connects the potential with the density. Thus, if we have a further equation that dictates how the pressure and density vary with respect to one another, we can reach a solution. The particular choice of a polytropic gas as given above makes the mathematical statement of the problem particularly succinct and leads to the Lane–Emden equation. The equation is a useful approximation for self-gravitating spheres of plasma such as stars, but typically it is a rather limiting assumption.
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The term "anti-realism" was introduced by Michael Dummett in his 1982 paper "Realism" in order to re-examine a number of classical philosophical disputes, involving such doctrines as nominalism, Platonic realism, idealism and phenomenalism. The novelty of Dummett's approach consisted in portraying these disputes as analogous to the dispute between intuitionism and Platonism in the philosophy of mathematics. According to intuitionists (anti-realists with respect to mathematical objects), the truth of a mathematical statement consists in our ability to prove it. According to Platonic realists, the truth of a statement is proven in its correspondence to objective reality.
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Physicists use the scientific method to test the validity of a physical theory. By using a methodical approach to compare the implications of a theory with the conclusions drawn from its related experiments and observations, physicists are better able to test the validity of a theory in a logical, unbiased, and repeatable way. To that end, experiments are performed and observations are made in order to determine the validity or invalidity of the theory. A scientific law is a concise verbal or mathematical statement of a relation that expresses a fundamental principle of some theory, such as Newton's law of universal gravitation.
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As a result, some mathematicians have worked on a related conjecture known as the geometric Langlands correspondence. This is a geometric reformulation of the classical Langlands correspondence which is obtained by replacing the number fields appearing in the original version by function fields and applying techniques from algebraic geometry. In a paper from 2007, Anton Kapustin and Edward Witten suggested that the geometric Langlands correspondence can be viewed as a mathematical statement of Montonen–Olive duality.Kapustin and Witten 2007 Starting with two Yang–Mills theories related by S-duality, Kapustin and Witten showed that one can construct a pair of quantum field theories in two-dimensional spacetime.
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More limited versions of constructivism limit themselves to natural numbers, number-theoretic functions, and sets of natural numbers (which can be used to represent real numbers, facilitating the study of mathematical analysis). A common idea is that a concrete means of computing the values of the function must be known before the function itself can be said to exist. In the early 20th century, Luitzen Egbertus Jan Brouwer founded intuitionism as a part of philosophy of mathematics . This philosophy, poorly understood at first, stated that in order for a mathematical statement to be true to a mathematician, that person must be able to intuit the statement, to not only believe its truth but understand the reason for its truth.
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The truth of a mathematical statement, in this view, is represented by the fact that the statement can be derived from the axioms of set theory using the rules of formal logic. Merely the use of formalism alone does not explain several issues: why we should use the axioms we do and not some others, why we should employ the logical rules we do and not some others, why do "true" mathematical statements (e.g., the laws of arithmetic) appear to be true, and so on. Hermann Weyl would ask these very questions of Hilbert: In some cases these questions may be sufficiently answered through the study of formal theories, in disciplines such as reverse mathematics and computational complexity theory.
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Many early scientists attributed the magnetism of lodestones to two different "magnetic fluids" ("effluvia"), a north-pole fluid at one end and a south-pole fluid at the other, which attracted and repelled each other in analogy to positive and negative electric charge. However, an improved understanding of electromagnetism in the nineteenth century showed that the magnetism of lodestones was properly explained not by magnetic monopole fluids, but rather by a combination of electric currents, the electron magnetic moment, and the magnetic moments of other particles. Gauss's law for magnetism, one of Maxwell's equations, is the mathematical statement that magnetic monopoles do not exist. Nevertheless, Pierre Curie pointed out in 1894 that magnetic monopoles could conceivably exist, despite not having been seen so far.
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If a statement P is provable, then it is certainly impossible to prove that there is no proof of P. But even if it can be shown that no disproof of P is possible, we cannot conclude from this absence that there is a proof of P. Thus P is a stronger statement than not-not-P. Similarly, to assert that A or B holds, to an intuitionist, is to claim that either A or B can be proved. In particular, the law of excluded middle, "A or not A", is not accepted as a valid principle. For example, if A is some mathematical statement that an intuitionist has not yet proved or disproved, then that intuitionist will not assert the truth of "A or not A". However, the intuitionist will accept that "A and not A" cannot be true.
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Within human–computer interaction, the law was rediscovered by Johnny Accot and Shumin Zhai, who mathematically derived it in a novel way from Fitts's law using integral calculus, experimentally verified it for a class of tasks, and developed the most general mathematical statement of it. Some researchers within this community have sometimes referred to the law as the Accot–Zhai steering law or Accot's law (Accot is pronounced ah-cot in English and ah-koh in French). In this context, the steering law is a predictive model of human movement, concerning the speed and total time with which a user may steer a pointing device (such as a mouse or stylus) through a 2D tunnel presented on a screen (i.e. with a bird's eye view of the tunnel), where the user must travel from one end of the path to the other as quickly as possible, while staying within the confines of the path.
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