He lived 1601 to 1665, invented Fermat's minor theorem and Fermat's last theorem [1640], and studied differential calculus, maxima/minima in war and astronomy [1629], curve and surface tangents and normals in optics and motion, and infinite-descent method.
He lived 1654 to 1705 and studied calculus of variations, variable separation, lemniscate curve, differential equations, and inflection points. He invented Bernoulli's theorem and Bernoulli equation. If event can have two outcomes, each with probability, after many independent events, relative frequency approaches probability {weak law of large numbers, Bernoulli}. Perhaps, probabilities are inferable from frequencies.
He lived 1667 to 1748 and studied series, arithmetic series, geometric series, differential equations [1691], astroid [1691], curvature radius, Bernoulli numbers, and curve rectification. Wire shape allows bead to slide from one end to the other in shortest possible time {brachistochrone, Bernoulli} [1696].
He lived 1707 to 1783 and invented Euler's formula, Euler number, and Euler constant. He studied incompressible non-rotating non-viscous fluid flows {potential flow}. He studied non-homogeneous nth order differential equations, partial-fractions method, explicit and implicit functions, networks, harmonic and divergent series, hypergeometric functions, natural logarithms, partial derivatives, multiple integrals, calculus of variations, finite-differences method, and gamma and beta functions.
First-order equations can be exact differentials. Newton's laws can depend on a maximizing-minimizing principle {principle of stationary action} {stationary action principle} {Euler-Lagrange equations}.
European mathematicians used induction, generalization, agreement method, difference method, analogy, and causal generalization.
He lived 1752 to 1833, studied number theory and elliptic integrals, and invented Legendre function and Legendre differential equation.
European mathematicians studied populations, statistics, distributions, sampling, correlation, and correlation coefficient.
He lived 1777 to 1855 and studied Earth magnetic field. In statistics, he developed Gaussian distribution {normal distribution, Gauss}, variance, standard deviation, mean standard error, least-squares method, and regression. In number theory, he worked on analytic number theory, algebraic numbers, complex numbers, hypercomplex numbers, Diophantine analysis, and theory of forms. In geometry, he invented seventeen-sided regular polygons, used substitute parallel axiom for non-Euclidean geometry, and studied curvature, congruence theory, and Gaussian coordinates. In algebra, he invented fundamental theorem of algebra and studied elliptic functions, Gauss characteristic equation, and central limit theorem. In vector theory, he worked with dot product and cross product. In physics, he developed dynamic equations that minimized quantity and Principle of Least Constraint.
He lived 1802 to 1829 and invented elliptic-function addition theorems and integrals. He studied quintic polynomials [1824], elliptic functions, series, fields, and rings. He invented Abelian integrals [1826], Abel's theorem, Abel summability, and Abelian group or commutative group.
He lived 1805 to 1859 and invented Dirichlet series, Dirichlet conditions, and Dirichlet principle or Thomson principle. He studied analytic number theory [1825].
European mathematicians studied transitivity, reflexivity, symmetry, equivalence, axial symmetry, radial symmetry, rotation, reflection, translation, and inversion.
He lived 1793 to 1856 and invented Lobachevsky's rule. He used substitute parallel axiom, applied to two boundary lines with angle to the perpendicular, to make non-Euclidean hyperbolic geometry {Lobachevskian geometry, Lobachevski}. People do not know Euclid's axioms with certainty, and they are not true a priori.
He lived 1823 to 1891 and helped develop intuitionism. He invented Kronecker delta function and studied fields [1881].
He lived 1815 to 1897, used arithmetic concepts for mathematical analysis, and studied real number theory, analytic and elliptic functions, and uniform convergence. He invented Weierstrass-Bolzano theorem [1854]. To remove contradictions introduced by infinitesimals, he reformulated calculus using limits and exhaustion method. Elliptic complex functions are sums of convergent power functions. Irrational numbers are rational-number-series convergences.
He lived 1849 to 1925 and set forth Erlangen program [1872]. He invented Klein's bottle and metric. In three dimensions, all metric geometries are projective geometry augmented by a quadric {absolute, geometry} or a curve related to absolute.
He lived 1849 to 1917 and studied linear algebra [1878], series, and groups.
He lived 1843 to 1921 and invented Schwarz statistics criterion, Schwarz's inequality [1885], and Schwarz's paradox.
He lived 1871 to 1956 and studied functions using series and measure theory [1895], invented Heine-Borel theorem, and helped develop intuitionism.
He lived 1862 to 1943 and studied formal systems, proof theory, metamathematics, and Erlanger Program. He studied real numbers using connection, calculation, order, and continuity axioms. He invented Hilbert space and Hilbert-Schmidt theorem. He posed problems {Hilbert program} for 20th century mathematicians to solve [1900]. His tenth problem {Entscheidungsproblem} asked if theorem-proving algorithms are possible. Integral equations and complete orthogonal-system theories relate.
Epistemology
Mathematics can depend on proofs using symbol language {formalism, Hilbert}. Mathematics branches can be formal and studied at higher level {metamathematics, Hilbert}, but do not need infinitely high level. Meaningful mathematics is about finite objects and relations. The infinite hotel {Hilbert hotel} has an infinite number of rooms, so it has infinitely many vacancies, no matter how many people.
European mathematicians studied topology or analysis situs, as in neighborhood, completeness, compactness, connectedness, winding number, homeomorphy, Königsberg bridges problem, four-color theorem or map problem, manifold, simplex, and tesselation. They studied impredicative definition, as in Burali-Forti paradox, barber paradox, Richard's paradox, heterological paradox, Russell's paradox, hangman's paradox, and Newcomb's problem.
He lived 1875 to 1941 and invented Lebesgue integral and Riemann-Lebesgue lemma, studied measure theory [1901], and helped develop intuitionism.
He lived 1865 to 1963 and studied functionals [1903], characteristic equations and helped develop intuitionism.
He lived 1869 to 1951 and studied hypercomplex numbers, Lie group theory, differential geometry [1904], and exterior derivatives.
He lived 1854 to 1912, helped develop intuitionism, and studied function theory, differential equations, orbits, and combinatorial topology. He found special-relativity equations [1905]. He showed how to keep distances constant as observed from different constant motions in flat space-time {Poincaré motion} {inhomogeneous Lorentz motion}, by lengthening light-cone along space dimensions and shrinking light-cone along time dimension. After systems reach largest phase-space region, they can return to all smaller regions over times much longer than universe age {Poincaré recurrence}.
Epistemology
Mathematical thinking is purely mental and so can reveal what is essential in mind. Unconscious thinking has preceded insight. Mind unconsciously selects possible solutions using innate or consciously formulated rules. Thinking appears to move in one direction and has purpose. Aesthetic value is an important creativity component.
Thinking converges on truth, but absolute truth is unattainable. Statement is possibly true if it is not necessary that it is not true. Contradictions are necessarily not true. Statements that do not involve contradiction state logical possibility. Not all contradictions are apparent. Nature contains contradictions, so contradictions can state possibilities.
Science decides what is naturally possible and naturally impossible. Epistemic possibility is what is consistent with human knowledge states. Possible truth is true in at least one possible world. Necessary truth is true in all possible worlds. Possibility and necessity are arbitrary rules about word use. Concept meaning depends on possible and impossible.
Definition can quantify over all class objects {vicious-circle principle, Poincaré} {impredicative definition, Poincaré} or not include them {predicative definition, Poincaré}.
He lived 1884 to 1944, invented Birkhoff's theorem [1909], proved Poincaré's Last Geometric Theorem [1913], discovered ergodic theorem [1931 to 1932], studied asymptotic series, and helped develop quantum logic.
He lived 1885 to 1955, studied integral equations, helped develop intuitionism, and studied universe symmetries. Abstract objects exist only if they have predicative definitions {predicative theory, Weyl}. Predicative definition must be countable.
European mathematicians invented Students' t distribution, F distribution, chi-square distribution, variation coefficient, factor analysis, sequential analysis, estimation, unbiased estimate, and confidence interval.
Mathematicians, including Claude Chevalley, André Weil, Henri Cartan, and Jean Dieudonné, studied modern-mathematics foundations.
European mathematicians studied operations research, linear programming, simplex method, marginal value principle, formal languages, algorithms, and decision problem.
European mathematicians studied coding, check bits, parity checking, logical sum checking, weighted check sums, modulo 37 with progressive digitizing, Hamming code, geometric code, variable length code, instantaneous code, Huffman code, compression, hashing, and Gray code [Hamming, 1960].
He lived 1922 to 1974. He founded empirical mathematics philosophy, in which people can know truth by Methodology of Scientific Research Programmes. He opposed the philosopher of science Paul Feyerabend.
Kolmogorov lived 1903 to 1987 and developed measure theory [1965].
He wrote popular science.
He wrote popular science.
Middle Eastern mathematicians solved general linear and quadratic equations using variables.
He lived 770 to 840, used Hindu numbers and fractions, and studied algebra. Adelard of Bath and Gerard of Cremona [1100 to 1150] translated his works and so transferred Indian and Islamic philosophy to Europe.
He lived 953 to 1029 and invented completing the square.
He lived 1501 to 1576 and found general-cubic-equation and general-quartic-equation solutions. He studied negative-number square roots and essentially discovered complex numbers, finding complex-number roots of x + y = 10 and x*y = 40. Cubic equation can be x^3 = p*x + q, where p and q can be zero, positive, or negative. Cubic equation can have up to three roots, all real numbers. If there are three roots {casus irreducibilis}, intermediate steps to solution can require complex numbers. He also studied game probability and began probability theory.
Politics
Culture and politics relate, as actual states and history show.
He lived 1499 to 1557 and found general solution to cubic equation.
He lived 1522 to 1565 and found general solution to quartic equation [1542].
He lived 1560 to 1621, used modern algebra notation, and studied trajectories. He invented refraction law.
He lived 1704 to 1752 and invented Cramer's rule.
He lived 1735 to 1796 and invented determinant minor.
He lived 1791 to 1858 and studied algebra systems and permanence of form.
He lived 1806 to 1871 and studied divergent series. He invented De Morgan's laws [1849] of algebra of classes: commutation, association, inverse, identity, distribution, and null.
He lived 1871 to 1928 and studied algebraic field theory.
He lived 1887 to 1985 and studied problem solving and problem-solving heuristics and invented counting formula. Plausibility depends on authority or reliability of information source used to justify proposition, not on probabilities of alternatives. Reasoning-chain plausibility is least-plausible-proposition plausibility.
He lived 1928 to ? and studied hyperfunctions [1958].
He lived 1350 to 1425, founded Kerala School of mathematics, developed infinite series, and started mathematical analysis.
He lived 1789 to 1857, used arithmetic concepts for mathematical analysis, and began complex-variable function theory [1814]. He invented Cauchy's principle, Cauchy convergence criterion, and Cauchy integral theorem. He studied method of characteristics, theory of content, and spaces. Separating first-order partial-differential-equation variables can make ordinary-differential-equation systems. First-order partial differential equation systems can describe elastic-media properties.
He lived 1784 to 1846 and invented Bessel equation [1817 to 1824] and Bessel's inequality.
He lived 1804 to 1851 and studied elliptic integrals, function theory, and inverse elliptic functions {theta function, Jacobi}. He invented Jacobian.
He lived 1798 to 1852 and worked with elliptic functions [1840 to 1841]. Elliptic functions are sums of converging power terms.
He lived 1822 to 1901 and invented Hermitean operators and Hermite functions [1858 to 1864].
He lived 1833 to 1872 and studied genus of curves {Clebsch-Gordan coefficients}.
He lived 1837 to 1912.
He lived 1845 to 1879 and invented geometric product and Clifford algebras [1878]. He studied complex analysis. Addition does not necessarily combine two units of same kind but instead defines relations, as in complex numbers or hypernumbers. People have innate learning, which developed through evolution {evolutionary epistemology, Clifford}. Mind grows by evolution {creative evolution}.
She lived 1850 to 1891 and studied elliptic functions and power-series sums.
He lived 1892 to 1945 and studied functional analysis, projection theorem, triangle inequality, and adjoints. He invented Banach spaces, Hahn-Banach theorem [1922], and Banach algebra.
He lived 1879 to 1934 and invented Hahn-Banach theorem [1922].
He lived 1920 to 1990 and developed generalized-function spaces [1932].
He lived 1883 to 1953 and developed measure theory [1941].
He lived 1915 to 2002, developed distribution theory {theory of distributions}, and developed generalized-function theory, allowing discontinuous-function derivatives [1945 to 1950].
It had 12 months and predicted yearly flooding.
Year was 365 days long, with 12 months of 30 days. Calendar predicted Nile flooding.
Babylonian mathematicians predicted Sun and Moon positions.
He lived -1680 to -1620, solved practical architecture problems, calculated astronomical events, and used simple interest, compound interest, principal, and rate. Multiplication is repeated doubling, and division is repeated halving.
He lived -580 or -569 to -500 and invented gnomon and Pythagorean theorem. He used similar figures, proportions, Pythagorean triples, Golden Ratio, Golden Section, and Golden Rectangle, and triangular, square, perfect, amicable, and prime numbers.
Babylonian mathematicians used number system based on 60 {sexagesimal number system}. They predicted Sun, Moon, and planet positions based on previous positions that they had recorded.
Platonists used inference, proof, deduction, and induction. They studied regular polyhedra, conic sections, prism, pyramid, cone, cylinder, perimeter, area, volume, and surfaces. Regular polyhedra are tetrahedron, icosahedron, and dodecahedron.
Platonists studied prisms, pyramids, cylinders, cones, conic sections, and the five regular polyhedra. They gave circle 360 degrees. They assumed proposition is true and then deduced consequences, until statement is clearly true or false {method of analysis}. They assumed theorem and showed that theorem leads to contradiction, so theorem is false {reductio ad absurdum, Platonists}. They used deductive proof {deduction, Platonists}.
It depends on Persian books.
He lived 836 to 901.
He lived 868 to 929 and found ecliptic angle and solar-year length.
He lived 965 to 1039 and studied perspective, projection, vanishing points, and cubic equations.
He lived 1114 to 1185, followed Brahmagupta, and used combinations and permutations.
He lived 1390 to 1450 and used base-ten number system, decimals, and negative powers.
He lived 1843 to 1930 and studied geometry foundations [1882], especially line and point interchangeability.
He lived 1858 to 1932. He invented logical notation, which Russell used. He studied axiomatic number systems. He invented Peano's postulates about rational numbers, based on Dedekind's work. He used reflexive, symmetric, and transitive axioms to derive rational numbers from natural numbers.
He lived 1880 to 1960 and axiomatized geometry using ideas of point and order.
He lived 1871 to 1952 and invented line and space axiomatic systems, building from points to lines to space. The three complete-quadrilateral diagonal points are never collinear {Fano's axiom}.
He lived 1906 to 1978. First-order predicate calculus and first-order logic are complete [1930]. All formal arithmetic systems must be incomplete [1931]. For all formal and consistent arithmetic systems, at least one true arithmetic proposition cannot be formally decidable. Neither proposition nor negation has proof, so arithmetic system is incomplete {Gödel's first incompleteness theorem}. Propositions are statements about numbers. Propositions have Gödel-number codes. Systems have propositions about propositions, and at least one such statement is not provable, because proofs use self-referential number statements. Therefore, it is impossible to prove system consistency using arithmetic.
Formal or logical systems are logically equivalent to recursively definable functions and arithmetic systems. Computing machines embody such functions. Therefore, machines can never prove their consistency or completeness.
The continuum hypothesis is consistent with basic set-theory axioms [1938 to 1939].
Epistemology
Definitions can specify class elements and their relations, and relations can make new elements {recursive definition}.
Mathematical objects and concepts are real and separate from mind. People know fundamental mathematical truths by intuition.
He lived 1900 to 1982. Mathematics branches become more formal over time, until they are deductive systems. Mathematics is about deductive systems.
For axiomatic theories, subsets can use only some original terms but contain same theorems {Craig's theorem}.
He lived 1642 to 1708, invented calculus, and used determinants [1683]. Japanese temple geometry flourished at this time.
He lived 1652 to 1719 and invented Rolle's theorem [1691].
He lived 1700 to 1782. He solved differential equations by isolating variables. He developed cylindrical and spherical wave equations to represent organ-pipe sounds. He invented vibrating string equation. He studied hydrodynamics and invented Bernoulli's law [1734].
He lived 1698 to 1759 and developed dynamics maximizing-minimizing principle (principle of least action or least-action principle or principle of stationary action or stationary-action principle).
He lived 1736 to 1813 and studied calculus of variations, mean-value theorem, spherical coordinates, solution envelopes, adjoint equations, finite-differences method, and perturbation methods. He solved differential-equation systems using conic-section deviations. Newton's laws can depend on principle of stationary action in Euler-Lagrange equations. Natural numbers are sums of four natural-number squares.
He lived 1717 to 1783 and studied differential equations and multiple integrals and invented d'Alembert's test. He found that Newton's 3rd law applies to free bodies {d'Alembert's principle}.
He lived 1749 to 1827 and studied partial differential equations, Laplace transforms and operators, perturbations method, spherical coordinates, finite-differences method, and divergence theorem.
After proving that planetary elliptical orbits can be stable, he said, "Je n'avais pas besoin de cette hypothèse-là" or "I had no need of that hypothesis" when asked by Napoleon why he did not invoke God to explain solar-system stability, as Newton had thought necessary because of chaotic conditions (which are there but just small enough).
Epistemology
Given physical laws and particle motions and positions, people can predict everything in the future.
Metaphysics
Solar system formed from spinning gas cloud {nebular hypothesis}. Gravity and motion correct planetary-orbit perturbations, rather than causing chaos.
He lived 1793 to 1841, invented Green's theorem, and studied double integrals, line integrals, and curvilinear integrals.
He lived 1809 to 1882 and invented Sturm-Liouville theory [1829 and 1837] and transcendental numbers [1851]. Phase-space region volume is constant for Hamiltonian equation {Liouville's theorem, Liouville}, but volumes spread into larger space, leaving empty spaces.
He lived 1795 to 1870 and studied curvilinear coordinates [1840] and invented Lamé's differential equation.
He lived 1819 to 1903 and invented Stokes theorem [1845], fluid-dynamics Navier-Stokes equations, and Stokes lines. Navier-Stokes equations extend Newton's second dynamics law and linear constitutive stress relation.
He lived 1831 to 1889 and classified partial differential equations.
He lived 1856 to 1894 and invented Stieltjes integral.
He lived 1888 to 1970 and solved differential equations by substituting algebraic equations {relaxation method, Southwell}.
He lived 1918 to 1974 and developed the idea of infinitesimals as greater than zero but smaller than all positive numbers {nonstandard analysis, Robinson}. He described infinitesimal neighborhoods of points infinitely close to a point {compactness theorem, Robinson}.
He lived 1903 to 1987, invented Kolmogorov probability, and developed measure theory [1965]. System-complexity measures {algorithmic complexity, Kolmogorov} {Kolmogorov complexity, Kolmogorov} {algorithmic information content, Kolmogorov} can be number of bits for smallest program that can run on universal Turing machines and produce same output. In turbulence, low frequencies transfer energy to higher frequencies throughout fluid.
She lived 1900 to 1998. For non-linear radio amplifiers, equations {Van der Pol equation} can calculate output for sine-wave input. At higher amplifier gains, output period doubles input period and then becomes non-periodic. Van-der-Pol-equation solutions were early chaos-theory ideas.
He lived 1908 to 1968. He proposed neutron stars [1932], and J. Robert Oppenheimer and G. M. Volkov found mass limit {Landau-Oppenheimer-Volkov limit, Landau} for making black holes instead of neutron stars, 2.5 times Sun mass.
Turbulence begins when new frequencies appear in fluid at overlapping velocities and masses. Turbulent motions include oscillatory, skewed varicose, cross-roll, knot, and zigzag. Turbulence is like white noise, with all frequencies.
He lived 1917 to ? and studied complex systems. He invented non-periodic weather-system computer models that were sensitive to initial conditions {butterfly effect, Lorenz}.
He studied fluid convections with circular motions {Rayleigh-Bénard convection, Lorenz}. Equations are dx / dt = 10 * (y - x), dy / dt = x * z + 28 * x - y, and dz / dt = x * y - (8/3) * z.
Paths through phase space never cross. Attractor can move to another surface when it moves to another phase-space region, so surfaces do not intersect.
Complex non-linear systems can have different final states that are not interchangeable {intransitive system}. Systems can be almost intransitive and can flip spontaneously from one state to another.
One-dimensional objects with cycle of period three have all periods.
He lived 1930 to ? and studied non-linear oscillators that had stable, non-repeating, periodic patterns. He studied topology in five or higher dimensions and Poincaré conjecture. He invented topological phase-space transformations {Smale's horseshoe}, in which space stretches, shrinks, and folds multiple times in any dimension. Transformations are sensitive to initial conditions.
He lived 1923 to 2002 and studied catastrophe theory.
He used three independent motions to describe turbulence. However, this was wrong. Phase-space centers can be not equilibria or periodic loops but infinitely long lines in confined space {strange attractor, Ruelle}. Strange attractors are stable, can have few dimensions, and are periodic but not exactly periodic.
He studied feedback systems and devised how to calculate order in one-dimensional-system chaos [1973], using quantum-field-theory renormalization group, stochastic processes, and fractals to remove infinities. Using y = r * (x - x^2) and x(t) = r * sin(pi * x(t - 1)), doubling oscillation period converges geometrically and so scales with constant ratio = 4.6692016090, to predict all doubling values. Functions are recursive {self-referential} and so introduce higher frequencies that indicate turbulence.
He studied conductivity [1973], with Jerry Gollub. He studied phase transitions. Rotating one cylinder inside another causes intervening liquid to flow {Couette-Taylor flow, Swinney}. First, flow streamlines. At faster speed, fluid cylinder separates into layers along cylinder axis, so fluid goes up and down cylinder. At higher frequency, flow is chaotic, with no defined frequencies. Vapor at critical point gives off white glow {opalescence, vapor}.
He analyzed work of Robert May. In one-dimensional systems, regular cycle of period three implies regular cycles of other periods, as well as chaotic behavior.
Assign initial number to logistic difference equation. Low rate values make number go to zero. Medium values make number go higher steady-state numbers. After high initial value, system oscillates between two values. After even higher initial value, system oscillates among four values. After even higher initial values, system oscillates among 8, 16, 32, and so on, values, with smaller differences between rates, until chaos starts {point of accumulation} {accumulation point, complexity}. After that point, oscillations are among all values. However, at higher points, oscillations are among 3 or 7 values, then oscillations are among 6, 9, 12, 14, 21, 28, and so on, values, then chaos returns again.
He lived 1909 to 1984 and studied chaos in vibrating strings {Fermi-Pasta-Ulam theorem}.
He studied stretching, compressing, and folding phase space to get self-similarity {Hénon attractor} [1976]: x(t) = y(t - 1) + 1 - 1.4 * (x(t - 1))^2 and y(t) = 0.3 * x(t - 1). He predicted that globular clusters have center that experienced gravitational collapse {gravothermal collapse} [2002].
They studied catastrophe theory.
He studied random graphs and Boolean networks to try to find complex-system, chaos, and self-organization laws. Most algorithms are their shortest descriptions {incompressibility}. Element physical interactions can order systems {self-organization, Kauffman} [Kauffman, 1995].
Self-organizing systems follow physical laws and describe living-system energy flows.
Brain has hierarchical structure and new properties can arise at highest levels.
He used a liquid-helium box to study turbulence onset and found that it had period doubling, as in other complex non-linear systems [1996]. First, system reaches steady state as cylinders roll, then convection rolls become toruses, then those bifurcate, making 1, 2, 4, 8, 16, and so on, rolls as convection coil goes faster, and turbulence increases.
Phase transitions and critical points can be hierarchies of phase regions that affect neighbors {phase scaling} [1999].
She lived 1815 to 1852. Calculating machines cannot be creative, but only do what program indicates {Lady Lovelace's objection}.
He lived 1912 to 1954 and developed Turing test for intelligence. He developed a code-breaking machine {electronic cryptanalytic machine}, which was the first programmed computer {Colossus computer}. Fixed definite processes {algorithm, Turing} {recursive procedure, Turing} or trial-and-error procedure {heuristic procedure} can solve mathematical problems. Turing machines programmed to perform procedures can solve problems. Universal Turing machines can define all possible operations and solve general problems. Algorithms and heuristics cannot solve some mathematical problems, so machines cannot solve them [Turing, 1950].
He lived 1902 to 1950, studied statistical decision problem, and used minimax [1939].
He lived 1894 to 1964 and studied non-linear problems. He developed animal and machine control and communication theory {cybernetics, Wiener} and feedback-using self-regulating system theory. He helped develop artificial limbs. He studied automata {logical net}, information theory, and principles involved in communication between sources and sinks.
Epistemology
Information has encoding, transmission, and decoding. A possibly noisy channel transmits information. Channel has information capacity. The same information can use different codes, one of which can be optimum.
He lived 1903 to 1957.
In logic, he studied empirical logic, logic with uncertainty, error, and logical-net errors and helped develop quantum logic, with Birkhoff and Mackey.
In computing, he studied linear programming and electronic digital-computer theory and developed first digital computer [1946], called ENIAC. Multiple connections between elements allow system to operate, even if some units fail {multiplexing, Neumann}. Multiple lines can provide multiplexing.
In biology, he studied finite automata as central-nervous-system models.
In geometry, he showed how to use general eigenvalue theory for axiomatic Hilbert spaces and operators.
In game theory, he studied zero-sum games, strategy, Colonel Blotto game, minimax theorem, utility function, prisoner's dilemma, competition, and cooperation. Game theory involves decision-making when conditions are uncertain.
Set theory does not allow sets {paradoxical set, Neumann} to be their own elements {Foundation axiom, Neumann}.
He lived 1915 to 1998 and invented Hamming code [1950] to detect computer-coding errors.
He lived 1934 to ? and invented reversible-computing gates {Fredkin gate} {Conservative Logic Gate}. Mathematical models can reversibly transform into computational models {Fredkin transforms}. Universe computes using discrete and finite quantities {digital mechanics}. The more two alternatives are similar, the harder it is to choose and the less the choice matters {Fredkin paradox, Fredkin}.
He lived 1921 to ?, invented computer programs to prove first-order theorems [1959], and invented infinite series of types. Mathematics is intuitive.
He lived 1928 to ?, studied learning theories {constructionism}, and invented the Logo computer language.
He lived 1923 to ? and wrote ELIZA program to imitate psychologist querying patient.
He studied neural networks.
He studied AI.
They studied AI.
He invented backpropagation learning algorithms.
He invented robots.
He studied neural networks.
He studied quantum computation [1985].
They studied neural networks, with Gregory E. Hinton and R. J. Williams.
He studied computer memory.
He studied computer memory.
To hidden layer, he added units {context layer, Elman} that received a hidden-layer copy and then added back to hidden layer {simple recurrent network, Elman}.
He studied shape from shading in neural networks [1992], with Sidney Lehky.
He invented Mathematica software.
Science does not need laws expressed as mathematical equations. Simple non-linear rules operating on simple units can generate all pattern types and describe all phenomena. Because they can be equivalent to any algorithm, cellular automata can describe all complex processes. Physical systems satisfying differential equations can be cellular automatons, by substituting finite differences and discrete variables for differential equations. His Rule 30 seems to create unpredictable pattern, rather than expected recursiveness. www.stephenwolfram.com/publications/articles/date.html.
Information is only in physical media, which store bits or qubits {information science, Schumacher}. Physical medium can transform and/or transfer information to process information. Output from processing must be verifiable or complete task.
Salesmen want to travel shortest distance among cities, with no path duplication. What is the shortest path {traveling-salesman problem, Kirkpatrick} [2001]? Traveling-salesman problems are NP-complete. Number of possible paths is factorial of number of cities, divided by two, because trips can be in either direction. Tours are vertexes of N-dimensional polygons. Tours that differ by one city are near each other in N-dimensional space. Simulated annealing can find shorter paths but allow longer paths, to avoid local minima. Techniques can find good paths but not necessarily the best.
He lived 1902 to 1977 and studied game theory, competition, and cooperation.
He lived 1928 to ? and invented Nash equilibrium [1950], Nash bargaining solution [1950], and Nash programme [1951].
Cavemen painted designs on cave walls.
Papyrus describes Egyptian geometry.
Babylonian mathematicians used Pythagorean theorem to find distances.
He lived -750 to -690 and constructed circles from rectangles and squares from circles.
Sophists invented geometric proofs and studied circles as many-sided regular polygons. They tried to square circle, trisect angle, and double cube using only straightedge and compass.
He lived -470 to -410 and wrote first geometry text, first calculated curved area using rectilinear figures {quadrature, Hippocrates}, and first proved theorems using earlier theorems {pyramiding theorems}. He invented method of proving something by disproving its opposite {indirect proof, Hippocrates}.
He lived -408 to -355. He studied limits, used infinite polygons to find curved-figure areas and volumes {exhaustion method, Eudoxus}, and developed explicit axioms.
Proportion is magnitude or length. He showed how to prove that two different integer ratios, which make real numbers, are equal or not equal. Proportions are magnitude or length ratios. To compare ratios, find integer pairs such that product of first integer and numerators and product of second integer and denominators makes numerators greater than denominators. If successful, first ratio is greater than second, because new ratio, first/second, is less than first ratio and greater than second ratio. If unsuccessful, find integer pairs such that product of first integer and numerators and product of second integer and denominators makes numerators less than denominators. If successful, first ratio is less than second, because new ratio, first/second, is greater than first ratio and less than second ratio. If not successful, ratios are equal. You can thus approach any real number and so can work with irrational-number square roots of positive integers.
Planetary orbits are nested spheres. He measured year length.
He lived -325 to -265 developed Euclid's theorem and Euclid's algorithm. He studied perpendicular, parallel, superposition, arc, and prime numbers. He used exhaustion method, rather than infinitesimals, to study curves. He systematized plane geometry, number proportions and ratios, prime numbers, and solid geometry.
Book 1 is about congruence, parallel lines, Pythagorean theorem, simple constructions, constructions with equal areas, and parallelograms {rectilinear figure, Euclid}. Sum of two triangle-side lengths is greater than or equal to third-side length. Book 2 is about geometric algebra, using areas and volumes to find products and quadratic equations, and adding line segments to add. Book 3 is about circles, chords, tangents, secants, central angles, and inscribed angles. Book 4 is about figures inscribed in, or circumscribed around, circles. Book 5 is about proportion by magnitudes, commensurable magnitudes, and incommensurable magnitudes. Book 6 is about similar figures, using proportions. Book 7 is about number theory, Euclidean algorithm, and numbers as line segments. Book 8 is about geometric progressions. Book 9 is about square and cubic numbers, plane and solid numbers, geometric progressions, and the theorem that number of primes is infinite. Book 10 classifies incommensurable magnitudes. Book 11 is about convex solids and generation of solids. Book 12 is about curved-surface areas and volumes, using exhaustion method and indirect proof. Book 13 is about regular polyhedrons in spheres and regular polygons in circles.
He lived -276 to -194 and found circumference of Earth.
He lived -262 to -185 and was Neo-Pythagorean and mystic. He invented a systematic theory of parabolas, ellipses, and hyperbolas, based on eccentricity, directrix, and focus. He studied right circular cones, oblique circular cones, hyperbolas, parabolas, ellipses, conjugate diameters, tangents, asymptotes, foci, conic intersections, maximum and minimum conic lengths, conic normals, similar and congruent conics, and conic segments. Two conic tangents meet at poles, and sides are polars. Given three points, lines, or circles, construct a circle tangent to or including the points, lines, or circles {Apollonian problem}.
Ethics
Simple life is best.
Mind
Mind and body are separate realities.
He lived 87 to 150, invented maps with longitude and latitude, discovered Ptolemy's theorem, and invented epicycles to describe planetary motions.
He lived 260 to 350 and proved Pappus' theorem {Guldinus theorem}.
He lived 940 to 997, used secant and cosecant, and constructed using straightedges and circles.
He lived 1201 to 1274 and used tangent and secant. He invented devices to resolve linear motion into sum of two circular motions {Tusi-couple, Nasir-Eddin}.
He lived 1540 to 1603 and invented sine law, cosine law, and Napier's rule.
He lived 1548 to 1620 and used decimal fractions and force parallelograms.
He lived 1598 to 1647, invented Cavalieri's theorem, and studied indivisibles method [1629].
He lived 1591 to 1661, invented Desargue's theorem, and studied projective geometry, involution, harmonic point sets, and poles and polar theory.
He lived 1713 to 1765, studied space curves [1731], invented Clairaut's equation, and determined Earth's shape.
She lived 1718 to 1799 and published discussion of cubic witch of Agnesi curve [1948].
He lived 1746 to 1818, studied developable surfaces, rediscovered projective geometry [1768], and was the "father of descriptive geometry".
He lived 1788 to 1867, rediscovered projective geometry, and studied affine geometry, differential geometry, and harmonic point sets.
He lived 1802 to 1860 and used substitute parallel axiom [1823], applied to intersecting and non-intersecting lines, to make non-Euclidean geometry [1833].
He lived 1801 to 1868 and studied trilinear coordinates and line coordinates.
He lived 1801 to 1883 and invented Plateau's problem.
He lived 1798 to 1867 and analyzed projective geometry without metric and without congruence.
He lived 1826 to 1866. He studied non-Euclidean geometry, differential geometry, complex functions, multiple-valued functions, mapping, prime-number theorems, analytic number theory, and singularities. He invented Riemann surfaces, Riemann-Darboux integral, Riemann zeta function, Riemann mapping theorem, and Riemann hypothesis. Riemann integrals are sums over infinity of step functions. All closed line segments have the same number of points. All points, in plane touching Riemann sphere at South Pole, map to sphere points, with points at infinity mapping to North Pole. Compact-plane points can thus map to limited, closed, and bounded surfaces.
He lived 1838 to 1926.
He lived 1924 to ? and ascribed fluctuations to discontinuous effects and to trends. He studied fractals, self-symmetry, 1/f noise, and 1/f squared noise. Fractal curves have non-integral dimensions. 1/f noise is like Cantor sets. In continuous intervals, continually removing inner third of each remaining continuous interval still leaves infinitely many points, and total empty distance is interval length {Cantor set, Mandelbrot}. Cantor sets are the same at all scales.
He lived 1945 to ?.
He lived 1811 to 1832 and studied group, field, solvability, and factoring representation theory [1830 to 1832].
He lived 1842 to 1899 and studied transformation groups and finite continuous groups {Lie group, Lie}.
He lived 1856 to 1934 and combined equation group theory, group number theory, and infinite transformation groups {abstract group theory} [1901].
She lived 1882 to 1935, studied invariance, and showed that symmetries relate to conservation laws [1915] {Noether's theorem, Noether}. She also studied rings [1921].
He lived 1916 to 2001 and founded information theory and studied transition probabilities.
He lived -287 to -212 and invented Archimedes' theorem, Archimedes spiral, Archimedes axiom, and Archimedes real-number property. He used exhaustion method to find pi and sphere and conic areas and volumes. He used completeness axiom. He found Archimedes buoyancy law {Archimedes' principle, Archimedes} {Archimedes principle, Archimedes}. He put a screw {Archimedes' screw} {Archimedes screw} inside a cylinder, to lift water.
He lived 10 to 70, invented Hero's formula, and studied geodesy, mechanics, and pneumatics. He maintained constant water-clock water supply, using float and needle valve, as in carburetors. He invented steam engine {aeolipile} [62].
He lived 1623 to 1662, was Cartesian and Jansenist, and invented first metal-tooth wheeled calculating machine [1642]. He invented hydraulic press to multiply force, syringe, Pascal's principle, Pascal's theorem, Pascal's triangle, mathematical induction, fundamental enumeration principle, binomial theorem, large-numbers law, and conditional-probability law.
At mechanical equilibrium, with only gravity acting, liquid has hydrostatic pressure {Pascal's law}.
Epistemology
People can neither reject reasoning nor say there is only reasoning. Reason cannot deal with ultimate metaphysical problems. Faith is necessary complement to reason. Expected value of believing in God is more than value of non-belief {Pascal's wager}.
Metaphysics
God exists because man is helpless without God.
He lived 1926 to 1992 and developed a FORTRAN-like programming language {BASIC programming language}.
He lived 1815 to 1864 and studied symbolic logic and logic of classes or extensional logic. Arithmetic and algebras have axioms and theorems allowing independent term or variable meanings. Axioms and theorems can be statements, sets, classes, events, or durations. Syllogisms can use arithmetic notation, and algorithm can prove them {Boolean algebra, Boole}. Boolean algebra has sets, union operation, intersection operation, complement operation, zero element, and unit element. Arithmetic axioms hold for elements and operations.
Epistemology
Mind has ability to conceive class, designate individual class members by common name, perform other logical tasks, and think logically {laws of thought, Boole}. Thought laws are innate and inherited.
He lived 1822 to 1900. Because circle chords can have varying angles to tangents, for example perpendicular to radius and parallel to tangent, different ways of selecting chords lead to different probabilities that chord is less than inscribed-equilateral-triangle side {Bertrand's paradox}.
He lived 1832 to 1898 and studied symbolic logic. Assuming inference rule is not the same as assuming conditional statement.
He lived 1862 to 1956. Integers are describable in words with a finite number of letters. An integer exists that is the least integer not describable in 100 or less letters. However, that phrase has less than 100 letters {Richard's paradox} [1905].
He lived 1881 to 1966, tried to define numbers, and helped develop quantum logic. He helped develop the idea that mathematics requires mental constructions for truth {intuitionism, Brouwer} [1924]. Unconstructed and non-existent things cannot be the basis for truth. Infinities cause excluded-middle-law contradiction, so mathematics cannot use this law.
He lived 1861 to 1947 and was idealist. He studied logical analysis, axiomatized logic, and developed logicism. Events can relate {process, Whitehead}. Relations and events transform object properties. Objects are always changing properties or property values. Reality is about such changes {process philosophy, Whitehead}. Since no properties exist for significant times, processes and relations are more important than matter, time, and position. All things interconnect and continually adjust to environment {philosophy of organism, Whitehead}. Higher properties emerge from lower systems. God is always becoming, and this unifies universe. Qualities are not substances but are mind-activity results.
He lived 1880 to 1940 and helped develop three-valued logic [1910 to 1913].
He lived 1883 to 1964. Elements {Sheffer stroke element} can equal "Not AND" and fire if either, but not both, of two input elements fire. Sheffer-stroke-element combinations can make OR element, AND element, and NOT element. Using many Sheffer stroke elements creates devices whose output fires if and only if most inputs fire.
Skolem lived 1887 to 1963. Lowenheim lived 1878 to 1957. If countable sets have formal models, domain is countable {Löwenheim-Skolem theorem}, as proved by Löwenheim [1915] and Skolem [1920]. However, real numbers are not countable {Skolem paradox}. Models {nonstandard model} can have elements that are not countable.
He lived 1886 to 1939 and invented definition theory. He helped develop quantum logic, based on equivalence {protothetic logic, Lesniewski}, abstract quantifiers {ontology logic, Lesniewski}, and part and whole relations {mereology, Lesniewski}. Logic is not about real world, only about statements. Wholes are not just sets or sums of parts, because parts relate. Because living things can replace parts, modal or temporal logic can maintain integrated wholes by maintaining relations among replaced parts.
He lived 1897 to 1954. Symbol strings can substitute other symbol strings {Post grammar, Post} {Post machine} [1936], to make formal systems. Start with long symbol string and substitute, using symbol-string precedence rules.
Logic can be three-valued {many-valued logic}. Many-valued logic can use cyclic negation, so next truth-value negates previous one. Such systems include all finite-valued logics. Such logics can represent switching circuits with many inputs and/or outputs.
He lived 1891 to 1953, studied analytic philosophy, and helped develop quantum logic. Spaces and times are relative. Probability depends on frequency. Induction depends on frequency. The geometry people use for universe is just conventional, not real, because instruments can systematically alter from expectations.
He lived 1883 to 1964, helped develop modal or relevance logic, developed implication requiring necessity {strict implication, Lewis}, and studied phenomenalism.
He lived 1898 to 1980 and helped develop quantum logic [1930].
He lived 1906 to 1965 and invented natural deduction and worked with infinite-valued logic [1934 to 1936].
He lived 1909 to 1945. He developed formal first-order logic {natural deduction, Gentzen} [1935], which only assumes inference laws. One rule uses premises and operator to make compound statement {introduction rule, Gentzen}. Another rule uses compound statement and statement to make statement. Statements depend on simple and compound sequent statements. Sequent-calculus proofs can be truth-trees or truth-tables {cut elimination theorem, Gentzen}, which eliminate formulas. Natural deduction led to proof theory.
He lived 1902 to 1983, founded modern logical theory, studied part and whole relations {mereology, Tarski}, helped develop quantum logic, and invented Banach-Tarski theorem.
Convention establishes basic-linguistic-element use and meaning {basic vocabulary}, which can construct complex term and sentence meanings {compositional semantics} {recursive semantics}.
Formal languages have consistent syntax, in which sentences form correctly or not. Formal language uses objects to replace language variables and predicates to replace language functions {interpretation, Tarski}. Truth is about interpretation {semantic theory of truth}. Determining truth requires defining what constitutes satisfying interpretation {satisfaction, Tarski}, which requires metalanguage {Tarski's theorem}. Formal languages have true interpretations {model, Tarski}. Premise sets can be models. If premise model is sentence model, sentences are premise-set consequences {theory of logical consequence} {logical consequence theory}.
For two sentence systems, sentences in one system can derive from sentences in other system {equipollence, Tarski}.
He lived 1886 to 1941. Self-applicable can mean thing expresses property that it has. Self-applicable can mean expression applies to itself. If heterological means not-self-applicable, then heterological is both self-applicable and not-self-applicable {Grelling's paradox} {Weyl's paradox}.
He lived 1903 to 1995, studied denotation, and helped develop quantum logic. Symbol strings can represent numbers and functions. Using functions on input function and data strings makes output function and data strings {lambda calculus, Church}. Lambda acts on variable or function, or variable and function combination, which is second-function dummy variable: lambda(x(f(x))) = f, lambda(x(f(x)))(a) = f(a), lambda(f(f(f(x)))) = lambda(f(lambda(x)(f(f(x))))) = lambda(f(x)(f(f(x)))). This expression is a function and precedes a value, which substitutes into function. In particular, after lambda, expressions can have variable zero times, function of variable one time, function of function of variable two times, and so on: 0 = lambda(f(x)(x)), 1 = lambda(f(x)(f(x))), 2 = lambda(f(x)(f(f(x)))). Function of function equals lambda and function of function {abstraction, lambda calculus}: f(f(x)) = lambda(x)(f(f(x))). Really, symbols are functions. Lambda calculus represents recursion, iteration, and algorithm loops. Recursive functions can be equation sets. Recursive functions are computable {Church's theorem}. Functions are computable if they are recursive {Church's thesis, recursion}. Recursive functions can be lambda calculus. Lambda calculus is equivalent to Post grammar and Turing machine and so can express all algorithms. LISP computer language depends on lambda calculus.
Epistemology
Formal systems can prove most theorems {effectively calculable} {computability}. Lambda calculus shows that it is impossible to prove some valid theorems in most formal systems, including arithmetic.
He lived 1909 to 1994, studied recursion theory and formal logic, and added subtraction to lambda calculus. At least one mathematical truth is true intuitionistically but not Platonically [Kleene, 1952].
She lived 1921 to ? and studied modal logic. The possibility that something has an attribute implies that something exists that possibly has the attribute {Barcan formula}, assuming that possible worlds overlap.
He lived 1921 to ? and invented fuzzy-set theory or fuzzy logic.
He lived 1923 to ? and developed laws of form {calculus of indications, Spencer-Brown}, based on differences. Autopoietic theory references his work.
They helped develop relevance logic, modal logic, deontic logic, and logical connectives.
Granting permission for two things can sound like permitting first or second, and so like exclusive OR, but is actually conjunction {confectionary fallacy, Jennings}. It is a deduction fallacy.
He lived 1821 to 1895 and studied matrix theory and invariant theory.
Cavemen carved numbers on bones.
Sumerian sexagesimal number system was for finances, with positional notation but with no zero. They had ordinal numbers, cardinal numbers, odd and even numbers, addition, subtraction, and simple fractions. They also used lines, circles, and angles.
Egyptian mathematicians used hieroglyphs to represent 1, 10, 100, 1000, 10000, and 100000.
Harappans decimal system was for weights and lengths.
Babylonian mathematicians calculated multiplication tables for number squares, cubes, and square roots, using sexagesimal number system with positional notation.
It had nine symbols for numerals 1 through 9, but zeroes were empty spaces.
Manipulating objects can add and subtract.
The word abacus comes from Indo-European root for sand.
Manipulating red and black counting rods can add and subtract.
It has decimal logarithms.
He lived 200 to 284 and studied number theory, algebra symbols, and determinate and indeterminate equations.
He lived 476 to 550, used positional notation, found circle chord lengths, and calculated sine tables.
He lived 505 to 587 and used positional notation.
Manuscript is about arithmetic and algebra.
He lived 598 to 668 and used decimal number system, negative numbers, and zero. He invented Brahmagupta's theorem.
He lived 800 to 870 and used zero, positional notation, decimal system, and negative, irrational, and rational numbers.
He lived 1170 to 1250 and invented Fibonacci numbers and studied number theory.
He lived 1548 to 1620. He invented decimal numbers and fractions, and so real numbers [1585].
He lived 1550 to 1617, used decimal point [1594], and studied logarithms.
He lived 1616 to 1703, studied cryptography, and invented expressions for pi [1655].
He lived 1638 to 1675 and invented expressions for pi [1671].
He lived 1667 to 1754 and invented DeMoivre's theorem [1722].
He lived 1690 to 1764. All even integers greater than 2 are sums of two primes {Goldbach's hypothesis, Goldbach} [1742].
He lived 1745 to 1818 and placed complex numbers on a plane with two perpendicular coordinates.
He lived 1768 to 1822 and placed complex numbers on a plane with two perpendicular coordinates.
She lived 1776 to 1831 and studied number theory and elasticity [1816]. For integers x, y, and z, if x^5 + y^5 = z^5, then x, y, or z must be divisible by 5 [1820] {Germain's theorem}.
He lived 1809 to 1877 and invented hypercomplex numbers {Grassmann variable}. Hypernumbers can represent tensors, quaternions, matrices, determinants, and all number types. Grassmann variables anti-commute: m . n = - n . m. He studied calculus of extension. Perhaps, space-time has extra, Grassmann dimensions to allow supersymmetry and supergravity.
He lived 1788 to 1856 and belonged to school of intuition. He invented quaternions and Cayley-Hamilton theorem. He studied non-commutative algebras. His student was H.L. Mansel. People can know the finite, but people know the rest by faith, based on Kant.
He lived 1831 to 1916 and studied fields, algebraic numbers, and irrational numbers.
He lived 1884 to 1937 and defined number.
He lived 1937 to ? and invented an axiomatic number system, constructing counting numbers, and so all numbers, using rules for right and left. He invented Game of Life [1970], based on cellular automata.
He lived 1947 to ? and proved that no general algorithm can decide if Diophantine-equation systems have integer solutions, based on work by Julia Robertson, Martin Davis, and Hilary Putnam.
Aleph is symbol for infinity levels.
He lived 1661 to 1704 and studied series and invented L'Hospital's rules.
He lived 1685 to 1731 and invented Taylor series and Taylor's theorem.
He lived 1698 to 1746, invented Maclaurin series, and used determinants method to solve linear equations [1726 to 1729].
He lived 1710 to 1761 and invented Simpson's rule.
He lived 1768 to 1830, invented heat equation, and invented Fourier series and Fourier transform: over intervals, any function can be trigonometric series.
He lived 1781 to 1848 and studied continuity and set theory. Real numbers in closed intervals can be in one-to-one correspondence with real numbers in other closed intervals. Infinite sequences in closed intervals have limits {Bolzano-Weierstrass theorem}. Truths can be a priori. Logic is about ideals, not about time or space.
He lived 1834 to 1923 and invented Venn diagrams.
He lived 1845 to 1918 and studied set theory, infinity, continuity, transfinite numbers, union, intersection, conjunction, disjunction, bound, extension principle, abstraction principle, and one-to-one correspondence.
He invented continuum hypothesis. Cardinal-number series and ordinal-number series are infinite. Irrational numbers in closed intervals are rational-number-series limits. Sets of limits can have sets of limits, and so on, to infinity.
Geometrical-figure or space topologies are points related by distance functions or limits. For any real number n, 2^n > n.
He lived 1861 to 1931. Ordinal numbers are well-ordered by definition. Ordinal-number sets must then have a greatest ordinal number. However, the set can be infinite and not have greatest ordinal number. Therefore, infinite ordinal-number sets cannot exist {Burali-Forti paradox}. Ordinal-number sets are higher-ordinal-number-set subsets.
He lived 1871 to 1956 and invented Zermelo-Fraenkel set theory [1904 to 1908]. Infinite sets can contain sets with no elements in common. Methods to choose one element from each set must exist {axiom of choice, Zermelo}. If sets have no defined choice function, sets must use axiom of choice.
He lived 1868 to 1942, invented generalized continuum hypothesis [1907], and invented Hausdorff space.
He lived 1891 to 1965 and improved Zermelo-Fraenkel set theory [1922].
He lived 1934 to ? and proved that continuum hypothesis was indeterminable under set theory [1963].
He lived 1702 to 1761. Expected outcome is worth or gain multiplied by probability. Risk is expected-outcome divided by outcome value {Bayesian theory} [1761].
Epistemology
Census, experimental, or statistical data can determine expected outcomes and find hypothesis probability {Bayesian confirmation theory}. Before evaluating new data, people already have beliefs about hypothesis risk and expected outcome. They know what they expect data to be if hypothesis is correct and what data happen no matter whether hypothesis is true or false.
He lived 1781 to 1840 and invented Poisson distribution.
He lived 1856 to 1922, invented probability theory using Chebyshev continued fraction [1900], and invented Markov process [1913].
He lived 1894 to 1981 and invented Neyman-Pearson hypothesis-testing theory [1933].
He lived 1857 to 1936 and invented Neyman-Pearson hypothesis-testing theory [1933].
Chernoff lived 1923 to ?. Mises lived 1921 to ?.
He lived 1906 to 1985. Probability is graded belief or judgments. If judgments are coherent and consistent, judgments converge to consensus with more data, and probability is relative frequencies observed in nature.
He lived 1853 to 1925 and studied absolute differential calculus [1894]. He started tensors, spinors, invariance, covariant, contravariant, version orientation-entanglement relation, and bivector wedge product.
He lived 1873 to 1941 and studied tensors.
He lived 1790 to 1868, invented Möbius strip, and studied homogeneous coordinates.
He lived 1878 to 1973 and studied function spaces and topology [1906], introducing metric spaces [1906].
He applied Yang-Mills gauge theory to four-dimensional manifolds [1983].
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Date Modified: 2022.0225