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Using only the very elementary framework of finite probability spaces, this book treats a number of topics in the modern theory of stochastic processes. This is made possible by using a small amount of Abraham Robinson's nonstandard analysis and not attempting to convert the results into conventional form.

Martingales (Mathematics) --- Stochastic processes. --- Probabilities. --- Martingales (Mathematics). --- Stochastic processes --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Random processes --- Probabilities --- Abraham Robinson. --- Absolute value. --- Addition. --- Algebra of random variables. --- Almost surely. --- Axiom. --- Axiomatic system. --- Borel set. --- Bounded function. --- Cantor's diagonal argument. --- Cardinality. --- Cartesian product. --- Central limit theorem. --- Chebyshev's inequality. --- Compact space. --- Contradiction. --- Convergence of random variables. --- Corollary. --- Correlation coefficient. --- Counterexample. --- Dimension (vector space). --- Dimension. --- Division by zero. --- Elementary function. --- Estimation. --- Existential quantification. --- Family of sets. --- Finite set. --- Hyperplane. --- Idealization. --- Independence (probability theory). --- Indicator function. --- Infinitesimal. --- Internal set theory. --- Joint probability distribution. --- Law of large numbers. --- Linear function. --- Martingale (probability theory). --- Mathematical induction. --- Mathematician. --- Mathematics. --- Measure (mathematics). --- N0. --- Natural number. --- Non-standard analysis. --- Norm (mathematics). --- Orthogonal complement. --- Parameter. --- Path space. --- Predictable process. --- Probability distribution. --- Probability measure. --- Probability space. --- Probability theory. --- Probability. --- Product topology. --- Projection (linear algebra). --- Quadratic variation. --- Random variable. --- Real number. --- Requirement. --- Scientific notation. --- Sequence. --- Set (mathematics). --- Significant figures. --- Special case. --- Standard deviation. --- Statistical mechanics. --- Stochastic process. --- Subalgebra. --- Subset. --- Summation. --- Theorem. --- Theory. --- Total variation. --- Transfer principle. --- Transfinite number. --- Trigonometric functions. --- Upper and lower bounds. --- Variable (mathematics). --- Variance. --- Vector space. --- W0. --- Wiener process. --- Without loss of generality.

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In today's unpredictable and chaotic world, we look to science to provide certainty and answers--and often blame it when things go wrong. The Blind Spot reveals why our faith in scientific certainty is a dangerous illusion, and how only by embracing science's inherent ambiguities and paradoxes can we truly appreciate its beauty and harness its potential. Crackling with insights into our most perplexing contemporary dilemmas, from climate change to the global financial meltdown, this book challenges our most sacredly held beliefs about science, technology, and progress. At the same time, it shows how the secret to better science can be found where we least expect it--in the uncertain, the ambiguous, and the inevitably unpredictable. William Byers explains why the subjective element in scientific inquiry is in fact what makes it so dynamic, and deftly balances the need for certainty and rigor in science with the equally important need for creativity, freedom, and downright wonder. Drawing on an array of fascinating examples--from Wall Street's overreliance on algorithms to provide certainty in uncertain markets, to undecidable problems in mathematics and computer science, to Georg Cantor's paradoxical but true assertion about infinity--Byers demonstrates how we can and must learn from the existence of blind spots in our scientific and mathematical understanding. The Blind Spot offers an entirely new way of thinking about science, one that highlights its strengths and limitations, its unrealized promise, and, above all, its unavoidable ambiguity. It also points to a more sophisticated approach to the most intractable problems of our time.

Uncertainty (Information theory) --- Science --- Measure of uncertainty (Information theory) --- Shannon's measure of uncertainty --- System uncertainty --- Information measurement --- Probabilities --- Questions and answers --- Science and society --- Sociology of science --- Social aspects. --- Acknowledgment (creative arts and sciences). --- Algorithm. --- Ambiguity. --- Analogy. --- Approximation. --- Axiom. --- Axiomatic system. --- Basic research. --- Big O notation. --- Calculation. --- Certainty. --- Chaos theory. --- Circumference. --- Computation. --- Concept. --- Conjecture. --- Consciousness. --- Consistency. --- Contingency (philosophy). --- Continuous function. --- Continuum hypothesis. --- Contradiction. --- Counting. --- David Bohm. --- Dynamism (metaphysics). --- Emergence. --- Euclidean geometry. --- Explanation. --- Feeling. --- Fermat's Last Theorem. --- Geometry. --- Gestalt psychology. --- Gregory Chaitin. --- Gödel's incompleteness theorems. --- Human behavior. --- Human intelligence. --- Hypothesis. --- Ideology. --- Inference. --- Integer. --- Irrational number. --- Learning. --- Logic. --- Logical reasoning. --- Mathematician. --- Mathematics. --- Measurement. --- Methodology. --- Modernity. --- Molecule. --- Natural number. --- Nature. --- Paradigm shift. --- Paradox. --- Participant. --- Phenomenon. --- Philosopher. --- Philosophy of mathematics. --- Philosophy of science. --- Philosophy. --- Platonism. --- Prediction. --- Principle. --- Probability. --- Pythagoreanism. --- Qualitative property. --- Quantification (science). --- Quantity. --- Quantum mechanics. --- Randomness. --- Rational number. --- Rationality. --- Real number. --- Reality. --- Reason. --- Reductionism. --- Relationship between religion and science. --- Result. --- Science. --- Scientific method. --- Scientific progress. --- Scientific theory. --- Scientist. --- Self-reference. --- Set theory. --- Special case. --- Subatomic particle. --- Subjectivity. --- Suggestion. --- Technology. --- The Philosopher. --- Theorem. --- Theoretical physics. --- Theory of everything. --- Theory. --- Thomas Kuhn. --- Thought. --- Uncertainty. --- Universality (philosophy). --- Writing. --- Sociology of knowledge

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The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic.

Research & information: general --- Mathematics & science --- quine --- logic --- ontology --- multiple conclusion rule --- disjunction property --- metadisjunction --- axiomatizations of arithmetic of natural and integers numbers --- second-order theories --- Peano's axioms --- Wilkosz's axioms --- axioms of integer arithmetic modeled on Peano and Wilkosz axioms --- equivalent axiomatizations --- metalogic --- categoricity --- independence --- consistency --- logic of typical and atypical instances (LTA) --- logic of determination of objects (LDO) --- quasi topology structure (QTS) --- concept --- object --- typical object --- atypical object --- lattice --- filter --- ideal --- discussive logics --- the smallest discussive logic --- discussive operators --- seriality --- accessibility relation --- Kotas' method --- modal logic --- deontic logic --- ontology of situations --- semantics of law --- formal theory of law --- Wittgenstein --- Wolniewicz --- non-Fregean logic --- identity connective --- sentential calculus with identity --- situational semantics --- deduction --- (dual) tableau --- Gentzen system --- deductive refutability --- refutation systems --- hybrid deduction-refutation rules --- derivative hybrid rules --- soundness --- completeness --- natural deduction --- meta-proof theory --- synthetic tableaux --- principle of bivalence --- cut --- first-order theory --- universal axiom --- Peano's axiomatics of natural numbers --- Leśniewski's elementary ontology --- Frege's predication scheme --- Frege's Zahl-Anzahl distinction --- term logic --- Franz Brentano --- Lewis Carroll --- logic trees --- logic diagrams --- paraconsistent logic --- paraconsistency --- Sette's calculus --- the law of explosion --- the principle of ex contradictione sequitur quodlibet --- semantic tree --- distribution --- Aristotle's logic --- syllogistic --- Jan Łukasiewicz --- axiomatic system --- axiomatic refutation --- temporal logic --- intuitionistic logic --- minimal system --- knowledge --- sequent-type calculi --- nonmonotonic logics --- default logic --- rejection systems --- Kripke models --- logics of evidence and truth

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The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic.

quine --- logic --- ontology --- multiple conclusion rule --- disjunction property --- metadisjunction --- axiomatizations of arithmetic of natural and integers numbers --- second-order theories --- Peano’s axioms --- Wilkosz’s axioms --- axioms of integer arithmetic modeled on Peano and Wilkosz axioms --- equivalent axiomatizations --- metalogic --- categoricity --- independence --- consistency --- logic of typical and atypical instances (LTA) --- logic of determination of objects (LDO) --- quasi topology structure (QTS) --- concept --- object --- typical object --- atypical object --- lattice --- filter --- ideal --- discussive logics --- the smallest discussive logic --- discussive operators --- seriality --- accessibility relation --- Kotas’ method --- modal logic --- deontic logic --- ontology of situations --- semantics of law --- formal theory of law --- Wittgenstein --- Wolniewicz --- non-Fregean logic --- identity connective --- sentential calculus with identity --- situational semantics --- deduction --- (dual) tableau --- Gentzen system --- deductive refutability --- refutation systems --- hybrid deduction–refutation rules --- derivative hybrid rules --- soundness --- completeness --- natural deduction --- meta-proof theory --- synthetic tableaux --- principle of bivalence --- cut --- first-order theory --- universal axiom --- Peano’s axiomatics of natural numbers --- Leśniewski’s elementary ontology --- Frege’s predication scheme --- Frege’s Zahl-Anzahl distinction --- term logic --- Franz Brentano --- Lewis Carroll --- logic trees --- logic diagrams --- paraconsistent logic --- paraconsistency --- Sette’s calculus --- the law of explosion --- the principle of ex contradictione sequitur quodlibet --- semantic tree --- distribution --- Aristotle’s logic --- syllogistic --- Jan Łukasiewicz --- axiomatic system --- axiomatic refutation --- temporal logic --- intuitionistic logic --- minimal system --- knowledge --- sequent-type calculi --- nonmonotonic logics --- default logic --- rejection systems --- Kripke models --- logics of evidence and truth --- n/a --- Peano's axioms --- Wilkosz's axioms --- Kotas' method --- hybrid deduction-refutation rules --- Peano's axiomatics of natural numbers --- Leśniewski's elementary ontology --- Frege's predication scheme --- Frege's Zahl-Anzahl distinction --- Sette's calculus --- Aristotle's logic