This document discusses the relationships between logic, mathematics, and science. It provides examples of how philosophers have used logic to explore truths, such as William Paley's argument for the existence of a creator based on biological complexity. The development of symbolic logic allowed mathematics to be treated as a logical system, as in Bertrand Russell's Principia Mathematica. However, Kurt Gödel's incompleteness theorems proved that a fully consistent and complete logical system is impossible. While logic has limitations, it remains an important tool for understanding and acquiring knowledge through the scientific method.
This document summarizes the key ideas from Gödel's incompleteness theorems and their implications. It discusses how Gödel proved that within any consistent axiomatic system, there will exist true propositions that cannot be proven within that system. It also discusses how Turing built on this work to show that there is no algorithm to determine whether a program running on a Turing machine will halt. Finally, it discusses how Penrose argued that human intuition allows for insights like grasping mathematical truths that artificial systems cannot match. The document evaluates how these mathematical findings suggest reality cannot be fully described by physical laws alone and have implications for various metaphysical views.
The document discusses the scientific method and its application to process improvement. It begins by discussing key thinkers who helped establish the scientific method, such as Einstein, Pearson, Broad, Popper, Dewey, Simon, and Ackoff. It then covers concepts like the scientific method, theory development and testing, bounded rationality in decision making, and systems thinking. The document concludes by discussing statistical process control pioneers like Shewhart, Juran, and their contributions to using statistics and understanding process dominance to analyze and improve processes, setting up the DMAIC method as a strategic approach.
The History and Evolution of the Concept of InfinityJohn Batchelor
This document provides a detailed overview of the history and evolution of the concept of infinity in mathematics. It discusses how early Greek mathematicians like Euclid, Archimedes, and Eudoxus began the formal study of infinity and developed key concepts like π and methods like exhaustion. Later mathematicians like Euler, Gauss, and Cantor made major contributions, with Cantor developing set theory and the distinction between different types of infinities. The document also examines how infinity relates to fields like astronomy, art, and philosophy. It discusses open problems and paradoxes involving infinity that remain unsolved.
1) The theory of everything has been elusive for over 100 years due to faulty postulates underlying general relativity and quantum mechanics. New primordial postulates are needed to derive corrected postulates for these theories.
2) Physicists' specialization and the selection process to become a physicist weeds out those who may think differently and have the gift to solve the theory of everything.
3) Space-time is a medium, not a void, which has been an incorrect assumption that has guided physics research down the wrong path for over 100 years.
This document provides an overview of the development of logic and physics that motivates the need for a new approach called transdisciplinarity. It discusses how non-Euclidean geometries and Gödel's incompleteness theorems challenged classical logic. It also explains how quantum mechanics experiments revealed phenomena like wave-particle duality and nonlocality that are contradictory to classical physics. This introduced issues of incompleteness and plurality into physical theories. The document argues that transdisciplinary logic is needed to provide a formal characterization of theories that accounts for these developments across disciplines.
I. This document discusses over two dozen theistic arguments, including ontological and metaphysical arguments. Some key arguments mentioned are:
1. The argument from intentionality, which claims that propositions represent reality and intentionality requires a mind, suggesting propositions are divine thoughts.
2. The argument from collections, which claims that sets are collections requiring an infinite mind like God's given their vast number and complexity.
3. The argument from numbers, which claims that numbers seem dependent on intellectual activity but there are too many for human minds, suggesting they are God's ideas.
"God’s dice" is a qubit: They need an infinite set of different symbols for all sides of them
INTRODUCTION
I A SKETCH OF THE PROOF OF THE THESIS
II GLEASON’S THEOREM (1957) AND THE THESIS
III GOD’S DIE, GLEASON’S THEOREM AND AN IDEA FOR A SHORT PROOF OF FERMAT’S LAST THEOREM
IV INTERPRETATION OF THE THESIS
V GOD’S DICE (A QUBIT) AS A LAW OF CONSERVATION AND TRANSFORMATION
VI CONCLUSION
This document summarizes the key ideas from Gödel's incompleteness theorems and their implications. It discusses how Gödel proved that within any consistent axiomatic system, there will exist true propositions that cannot be proven within that system. It also discusses how Turing built on this work to show that there is no algorithm to determine whether a program running on a Turing machine will halt. Finally, it discusses how Penrose argued that human intuition allows for insights like grasping mathematical truths that artificial systems cannot match. The document evaluates how these mathematical findings suggest reality cannot be fully described by physical laws alone and have implications for various metaphysical views.
The document discusses the scientific method and its application to process improvement. It begins by discussing key thinkers who helped establish the scientific method, such as Einstein, Pearson, Broad, Popper, Dewey, Simon, and Ackoff. It then covers concepts like the scientific method, theory development and testing, bounded rationality in decision making, and systems thinking. The document concludes by discussing statistical process control pioneers like Shewhart, Juran, and their contributions to using statistics and understanding process dominance to analyze and improve processes, setting up the DMAIC method as a strategic approach.
The History and Evolution of the Concept of InfinityJohn Batchelor
This document provides a detailed overview of the history and evolution of the concept of infinity in mathematics. It discusses how early Greek mathematicians like Euclid, Archimedes, and Eudoxus began the formal study of infinity and developed key concepts like π and methods like exhaustion. Later mathematicians like Euler, Gauss, and Cantor made major contributions, with Cantor developing set theory and the distinction between different types of infinities. The document also examines how infinity relates to fields like astronomy, art, and philosophy. It discusses open problems and paradoxes involving infinity that remain unsolved.
1) The theory of everything has been elusive for over 100 years due to faulty postulates underlying general relativity and quantum mechanics. New primordial postulates are needed to derive corrected postulates for these theories.
2) Physicists' specialization and the selection process to become a physicist weeds out those who may think differently and have the gift to solve the theory of everything.
3) Space-time is a medium, not a void, which has been an incorrect assumption that has guided physics research down the wrong path for over 100 years.
This document provides an overview of the development of logic and physics that motivates the need for a new approach called transdisciplinarity. It discusses how non-Euclidean geometries and Gödel's incompleteness theorems challenged classical logic. It also explains how quantum mechanics experiments revealed phenomena like wave-particle duality and nonlocality that are contradictory to classical physics. This introduced issues of incompleteness and plurality into physical theories. The document argues that transdisciplinary logic is needed to provide a formal characterization of theories that accounts for these developments across disciplines.
I. This document discusses over two dozen theistic arguments, including ontological and metaphysical arguments. Some key arguments mentioned are:
1. The argument from intentionality, which claims that propositions represent reality and intentionality requires a mind, suggesting propositions are divine thoughts.
2. The argument from collections, which claims that sets are collections requiring an infinite mind like God's given their vast number and complexity.
3. The argument from numbers, which claims that numbers seem dependent on intellectual activity but there are too many for human minds, suggesting they are God's ideas.
"God’s dice" is a qubit: They need an infinite set of different symbols for all sides of them
INTRODUCTION
I A SKETCH OF THE PROOF OF THE THESIS
II GLEASON’S THEOREM (1957) AND THE THESIS
III GOD’S DIE, GLEASON’S THEOREM AND AN IDEA FOR A SHORT PROOF OF FERMAT’S LAST THEOREM
IV INTERPRETATION OF THE THESIS
V GOD’S DICE (A QUBIT) AS A LAW OF CONSERVATION AND TRANSFORMATION
VI CONCLUSION
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
RSS stands for Really Simple Syndication or Rich Site Summary, depending on who you ask. Many websites publish a rich set of RSS feeds, which can be processed by other websites as a form of syndicated content. But the regular structure of RSS as an XML application means that feeds can be easily edited (“munged”) and combined (“mashed up”). Programming libraries exist for processing feeds, but Yahoo! Pipes makes this easy with a graphical user interface and no coding. We will discuss methods and applications of RSS feeds which might be suitable for a course website—for instance, combining feeds from SlideShare and scribd and publishing them to Facebook, or publishing your office hours on your blog automatically. (Received September 21, 2010)
Limits are used to find the value of a function as the input gets infinitely close to a certain value. They allow us to calculate things like derivatives, which describe the rate of change of a function. Limits are an essential building block of calculus and are used throughout its applications.
This document provides instructions for solving linear inequalities in one variable. It explains that to solve an inequality, the same operations can be performed on both sides as with an equation, except that the inequality symbol must be reversed if multiplying or dividing by a negative number. It also describes how to graph the solutions of inequalities on a number line, including whether endpoints are included or excluded. Examples are given of solving and graphing different inequalities on number lines. The homework assignment is to complete problems 1 through 10 on page 97.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
Lesson 27: Integration by Substitution (Section 041 slides)Matthew Leingang
The document contains notes from a Calculus I class at New York University on December 13, 2010. It discusses using the substitution method for indefinite and definite integrals. Examples are provided to demonstrate how to use substitutions to evaluate integrals involving trigonometric, exponential, and polynomial functions. The key steps are to make a substitution for the variable in terms of a new variable, determine the differential of the substitution, and substitute into the integral to transform it into an integral involving only the new variable.
The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It also proves the limit laws, such as the fact that limits can be combined using arithmetic operations and the rules for limits of quotients and roots.
Calculus is the mathematical study of change. It has two main areas: differential calculus concerns slopes and rates of change, while integral calculus concerns area and volume. The foundations of calculus were established in the late 17th century by Newton and Leibniz, who recognized that differentiation and integration are inverse processes. Calculus is based on the concept of a limit, which allows approximating values that cannot be calculated directly. The document then provides an example of using limits to calculate the area under a curve by dividing it into rectangular elements and taking the sum as the number of elements approaches infinity.
Lesson 24: Areas, Distances, the Integral (Section 041 slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
This document is a class supplement for Calculus I at New York University that covers optimization techniques. It provides objectives, outlines the topics to be covered which include recalling previous concepts and working through examples. Examples covered include finding two positive numbers with a product constraint that minimize their sum, finding the closest point on a parabola to a given point, and using derivatives to solve optimization problems with constraints. The document reviews methods like the closed interval method, first derivative test, and second derivative test to find maxima and minima.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
Lesson 13: Exponential and Logarithmic Functions (slides)Matthew Leingang
The document provides an outline and definitions for sections 3.1 and 3.2 of a calculus class, which cover exponential and logarithmic functions. It defines exponential functions, establishes conventions for exponents of all types, and graphs exponential functions. Key points covered include the properties of exponential functions and defining exponents for non-whole number bases.
Listen in for a quick "Tips in 20" webinar to learn how to build a strategic framework that will allow your online community to evolve and achieve ongoing success.
The document discusses an introductory calculus class and provides announcements about homework due dates and a student survey. It also outlines guidelines for written homework assignments, the grading rubric, and examples of what to include and avoid in written work. The document aims to provide students information about course policies and expectations for written assignments.
This document provides information about a quantitative reasoning course. The course aims to help students gain a comprehensive understanding of mathematics and the ability to think critically and logically. It will cover topics like logical and quantitative thinking, arguments and reasoning, and the relationship between logic, science and mathematics. The course goals are to understand mathematics as a body of knowledge and a way of thinking, and to reason quantitatively on issues relevant to students and society. On completing the course, students should be able to analyze and evaluate arguments, understand mathematical concepts, and apply problem-solving skills to quantitative problems.
Social constructivism as a philosophy of mathematicsPaul Ernest
- Social constructivism views mathematical knowledge as a social and historical construct. It rejects the notion that mathematical knowledge is absolutely valid or certain.
- Key aspects of social constructivism include viewing mathematical concepts and proofs as evolving through a conversational process of proposing ideas and subjecting them to criticism and refinement. Mathematical knowledge is seen as intersubjective rather than purely objective.
- On this view, mathematical texts and concepts can be understood as participating in an ongoing conversation, with proponents putting forth ideas and critics examining them for weaknesses. The acceptance of mathematical ideas and proofs occurs through this social and dialogical process rather than being intrinsically certain.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
RSS stands for Really Simple Syndication or Rich Site Summary, depending on who you ask. Many websites publish a rich set of RSS feeds, which can be processed by other websites as a form of syndicated content. But the regular structure of RSS as an XML application means that feeds can be easily edited (“munged”) and combined (“mashed up”). Programming libraries exist for processing feeds, but Yahoo! Pipes makes this easy with a graphical user interface and no coding. We will discuss methods and applications of RSS feeds which might be suitable for a course website—for instance, combining feeds from SlideShare and scribd and publishing them to Facebook, or publishing your office hours on your blog automatically. (Received September 21, 2010)
Limits are used to find the value of a function as the input gets infinitely close to a certain value. They allow us to calculate things like derivatives, which describe the rate of change of a function. Limits are an essential building block of calculus and are used throughout its applications.
This document provides instructions for solving linear inequalities in one variable. It explains that to solve an inequality, the same operations can be performed on both sides as with an equation, except that the inequality symbol must be reversed if multiplying or dividing by a negative number. It also describes how to graph the solutions of inequalities on a number line, including whether endpoints are included or excluded. Examples are given of solving and graphing different inequalities on number lines. The homework assignment is to complete problems 1 through 10 on page 97.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
Lesson 27: Integration by Substitution (Section 041 slides)Matthew Leingang
The document contains notes from a Calculus I class at New York University on December 13, 2010. It discusses using the substitution method for indefinite and definite integrals. Examples are provided to demonstrate how to use substitutions to evaluate integrals involving trigonometric, exponential, and polynomial functions. The key steps are to make a substitution for the variable in terms of a new variable, determine the differential of the substitution, and substitute into the integral to transform it into an integral involving only the new variable.
The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It also proves the limit laws, such as the fact that limits can be combined using arithmetic operations and the rules for limits of quotients and roots.
Calculus is the mathematical study of change. It has two main areas: differential calculus concerns slopes and rates of change, while integral calculus concerns area and volume. The foundations of calculus were established in the late 17th century by Newton and Leibniz, who recognized that differentiation and integration are inverse processes. Calculus is based on the concept of a limit, which allows approximating values that cannot be calculated directly. The document then provides an example of using limits to calculate the area under a curve by dividing it into rectangular elements and taking the sum as the number of elements approaches infinity.
Lesson 24: Areas, Distances, the Integral (Section 041 slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
This document is a class supplement for Calculus I at New York University that covers optimization techniques. It provides objectives, outlines the topics to be covered which include recalling previous concepts and working through examples. Examples covered include finding two positive numbers with a product constraint that minimize their sum, finding the closest point on a parabola to a given point, and using derivatives to solve optimization problems with constraints. The document reviews methods like the closed interval method, first derivative test, and second derivative test to find maxima and minima.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
Lesson 13: Exponential and Logarithmic Functions (slides)Matthew Leingang
The document provides an outline and definitions for sections 3.1 and 3.2 of a calculus class, which cover exponential and logarithmic functions. It defines exponential functions, establishes conventions for exponents of all types, and graphs exponential functions. Key points covered include the properties of exponential functions and defining exponents for non-whole number bases.
Listen in for a quick "Tips in 20" webinar to learn how to build a strategic framework that will allow your online community to evolve and achieve ongoing success.
The document discusses an introductory calculus class and provides announcements about homework due dates and a student survey. It also outlines guidelines for written homework assignments, the grading rubric, and examples of what to include and avoid in written work. The document aims to provide students information about course policies and expectations for written assignments.
This document provides information about a quantitative reasoning course. The course aims to help students gain a comprehensive understanding of mathematics and the ability to think critically and logically. It will cover topics like logical and quantitative thinking, arguments and reasoning, and the relationship between logic, science and mathematics. The course goals are to understand mathematics as a body of knowledge and a way of thinking, and to reason quantitatively on issues relevant to students and society. On completing the course, students should be able to analyze and evaluate arguments, understand mathematical concepts, and apply problem-solving skills to quantitative problems.
Social constructivism as a philosophy of mathematicsPaul Ernest
- Social constructivism views mathematical knowledge as a social and historical construct. It rejects the notion that mathematical knowledge is absolutely valid or certain.
- Key aspects of social constructivism include viewing mathematical concepts and proofs as evolving through a conversational process of proposing ideas and subjecting them to criticism and refinement. Mathematical knowledge is seen as intersubjective rather than purely objective.
- On this view, mathematical texts and concepts can be understood as participating in an ongoing conversation, with proponents putting forth ideas and critics examining them for weaknesses. The acceptance of mathematical ideas and proofs occurs through this social and dialogical process rather than being intrinsically certain.
This document provides an overview of a course on the philosophy of science. It discusses the interaction between philosophy and science, key concepts in the philosophy of science like scientific realism and falsificationism, and views of the scientific method from thinkers like Popper, Duhem and Kuhn. The course will examine general questions about science as well as specific issues in cosmology and cognitive sciences.
The laboratoryandthemarketinee bookchapter10pdf_mergedJeenaDC
This document discusses two modes of scientific knowledge production:
(1) Modus 1 knowledge is produced within established disciplines through deductive reasoning and empirical testing. It aims to build consistent theoretical systems.
(2) Modus 2 knowledge is produced transdisciplinarily to solve practical problems. It is shaped by social and economic factors beyond any single discipline. Knowledge takes the form of narratives that guide practices rather than deductive systems.
The key difference is that Modus 1 seeks fundamental theoretical understanding while Modus 2 focuses on knowledge applicable to real-world problems across disciplinary boundaries. Both make use of models, analogies, and metaphors to give meaning and applicability to formal theories and findings.
1. This document discusses different perspectives on mathematics, including its history and nature. It explores whether math is discovered or invented, and how culture and different fields influence mathematical thought.
2. Key aspects of math discussed are its basis in axioms, deductive reasoning, and theorems. Different views are presented on whether mathematical truths are empirical, true by definition, or insights into reality.
3. The document prompts reflection on relationships between math and other domains like logic, religion, intuition, science, language, and beauty. It encourages examining how perception of math may differ depending on factors like one's background or profession.
Logic is the study of reasoning and correct thinking. It involves analyzing concepts, establishing general laws of
truth, and determining valid forms of argument. Logic is applicable to all fields as it provides standards for
consistent and evidence-based reasoning. It has wide scope and helps with social studies, engineering, mathematics,
science, and computer programming through modeling reality, simplifying complex problems, and representing
information processing in a logical way. Studying logic is important as it helps develop critical thinking skills
needed to make rational decisions, adapt to new situations, and form justifiable beliefs.
This document discusses the concept of revolution in various contexts such as astronomy, politics, and science. It provides definitions and examples of different types of revolutions including political revolutions, social revolutions, and scientific revolutions. Key scientific revolutions discussed include Copernicus' theory that the Earth revolves around the sun, Darwin's theory of evolution by natural selection, and Einstein's theory of relativity. The document also examines concepts in the philosophy of science such as theories, hypotheses, laws, paradigms, and different views on scientific change and progress put forth by thinkers like Kuhn, Popper, Lakatos, and Feyerabend.
Mathematics Provides More Information Than You ThinkMattHill96
This document discusses the relationship between mathematics and reality. It provides perspectives from scientists and mathematicians on how mathematics effectively describes the physical world. Some key points made include:
- Mathematics appears to describe the universe in an uncanny way, as if it was designed for that purpose. This is a mystery that thinkers have struggled with for centuries.
- Mathematical concepts and ideas can be discovered and exist objectively outside of human thought and the physical world. Discoveries in mathematics are often made within the logical framework of mathematics itself rather than being inferred from physical reality.
- The universe appears to have a beginning at the Big Bang, but current physics theories break down at this point. While we can't recreate the
1) Pythagoras proclaimed that "All Things are Number," inspiring centuries of attempts to understand the physical world in mathematical terms.
2) Kepler formulated his "zeroth law" relating planetary orbits to Platonic solids, beautifully realizing this Pythagorean ideal, but it was proven wrong.
3) Quantum mechanics revived this Pythagorean vision by defining atomic structure in terms of whole numbers, as in Bohr's model of the hydrogen atom. This marked the triumphant return of the ancient idea that number underlies physical reality.
Explanation in science (philosophy of science)Anuj Bhatia
This document discusses scientific explanation and the challenges involved. It summarizes Carl Hempel's covering law model of explanation and identifies some of its limitations, such as not respecting asymmetry and allowing irrelevant explanations. The document also discusses how causality may provide a better approach than covering laws. Additionally, it notes that while science can explain many phenomena, some things like the origin of life may remain unexplained or be fundamentally unexplainable by science. The document also discusses reductionism and how higher-level sciences are autonomous due to multiple realization.
Dr. Aldemaro Romero Jr. talks about the fundamentals of the History and Philosophy of Science in this documentary. You can also watch it for free and in full at: https://www.academia.edu/courses/plqxp1?tab=0&v=Ee0Anb
This is a presentation about the nature of science of my source "History and Philosophy of Science". You can watch the video version at: https://www.academia.edu/courses/plqxp1?tab=0&v=DPrRKE
Research Methods in Architecture - Theory and Method - طرق البحث المعمارى - ا...Galala University
This document discusses different types of theories and how they relate to research methodologies in architecture. It begins by defining theory and explaining how theories emerge from systematic explanations. It then discusses the key components of theories, including propositions, logical connections, conclusions, empirical links, assumptions, and testability. Different types of theories are described, such as positive vs normative, big/medium/small, and polemical theories. Scientific theories are contrasted with design theories, with the latter focused more on persuasion than prediction. Finally, seven types of architectural research methods are outlined: interpretive-historical, qualitative, correlational, experimental, simulation, logical argumentation, and case study.
The document provides an overview of a lecture on research methods in science. It discusses key concepts like the scientific method, hypothesis generation and testing, deductive and inductive logic, parsimony, and the presuppositions, domains, and limits of science. It also briefly covers the Bayesian and frequentist approaches, the humanistic side of science, and ethics in scientific research. The overall lecture aims to outline general principles that pervade scientific inquiry across different disciplines.
Fue theory 4 lecture 3 - theory in relation to methodGalala University
This document provides an overview of theory and its relationship to research methodology in architecture. It discusses several key points:
1. There are different types of theories, including scientific/positive theories that can be tested, and normative/design theories that provide guidance but cannot be proven.
2. Theories generally have six components: observations, logical connections, conclusions, links to empirical reality, assumptions, and testability.
3. Theories can be categorized as big/grand, medium-range, or small based on their scope. Polemical theories take opposing positions on design concepts.
4. Scientific theories aim to predict through identifying causal links, while design theories focus more on persuasion given their
This slides explain about the philosophy of science. Philosophy and natural science.
logical positivism and logical empiricicism.
epistemology. Empiricism. induction.
This paper presents the evolution of the scientific method that has been instrumental in promoting the advancement of science and also technology throughout history. It is important to note that the scientific method refers to a cluster of basic rules of how to be the procedure in order to produce scientific knowledge, either new knowledge, either a correction or an increase of previously existing knowledge. The scientific method, therefore, is nothing more than the logic applied to science. The search for a suitable scientific method guided the action of most thinkers of the sixteenth and seventeenth highlighting among them Galileo Galilei, Francis Bacon, René Descartes and Isaac Newton, who with their contributions were crucial to the structure of what we call today of modern science. In addition to these thinkers, it was also important later contributions of Hegel, Marx, Engels, Popper, Russell, Duhem, Poincaré, Morin, etc.
While medical research requires significant funding, it also leads to important advances that benefit public health and reduce healthcare costs over time. Preventative vaccines and treatments for diseases like polio and dengue fever show that medical research does not always result in expensive treatments. As the medical sector progresses, costs for common illnesses and medicines tend to decrease due to new discoveries about cheaper substances that can be used. Therefore, continuing medical research is worthwhile even if not all can initially afford its benefits.
This article aims to present the concept of scientific truth, the methods adopted for the search for scientific truth, the questions about the scientific method and how to prove the scientific truth.
The document discusses common fallacies of reasoning. It aims to help readers recognize and discard errors in reasoning by describing fallacies of relevance, including subjectivism, appeal to ignorance, limited choice, appeal to emotion, appeal to force, inappropriate appeal to authority, personal attack, begging the question, and non sequitur. It also discusses fallacies involving numbers and statistics such as appeal to popularity, appeal to numbers, hasty generalization, availability error, false cause, and issues with percentages. The overall goal is to help evaluate information critically and carefully exercise reasoning.
The document discusses deductive and inductive arguments. It provides examples of valid and invalid deductive arguments using categorical propositions and conditional premises. It also discusses inductive arguments, noting that inductive conclusions generalize from specific premises rather than necessarily following from them. The document then compares deductive and inductive arguments and discusses their uses in everyday life and mathematics. It concludes by introducing some common rules of inference for deductive arguments.
3.2 geometry the language of size and shapeRaechel Lim
Geometry began with ancient Egyptians and Babylonians using practical measurements and Pythagorean relationships in construction. Greeks like Euclid later formalized geometry, establishing five postulates including the parallel postulate. Many unsuccessfully tried to prove this postulate, leading to non-Euclidean geometries developed by Bolyai, Lobachevsky, and others. These geometries have different properties than Euclidean geometry and opened new areas of mathematical exploration. Fractal geometry, developed by Mandelbrot, describes naturally occurring structures through fractional dimensions and infinite complexity across all scales.
3.1 algebra the language of mathematicsRaechel Lim
Algebra began with ancient Egyptians and Babylonians solving linear and quadratic equations. Over time, mathematicians generalized arithmetic operations and developed symbolic notation. Modern algebra arose from studying abstract structures like groups. A group is a set with operations where elements combine associatively, every element has an inverse, and a unique identity element exists. Groups include addition modulo m and permutations. Abstract algebra studies algebraic properties of different mathematical systems.
The document discusses strategies and techniques for solving quantitative problems. It emphasizes that problem solving requires creativity, organization, and experience. Some key points made include: keeping track of units is an important problem solving tool; no single strategy always works so flexibility is important; understanding the context and restating the problem can help clarify the solution; and the most effective problem solvers view challenges as opportunities to improve their skills through practice.
2.2 measurements, estimations and errors(part 2)Raechel Lim
Measurements and estimations involve uncertainty that arises from imprecision, random errors, and systematic errors. Numbers can be categorized as exact or approximate, with approximations involving uncertainty. Uncertainty must be expressed either implicitly by careful rounding or explicitly using ranges. Significant digits indicate the precision of measurements and estimations, and implied uncertainty ranges can be determined from them. When combining approximate values, answers must be rounded or expressed as ranges consistent with the least precise input to properly account for accumulated uncertainties.
2.1 numbers and their practical applications(part 2)Raechel Lim
This document discusses the development of the modern number system, including natural numbers, integers, rational numbers, real numbers, imaginary and complex numbers. It provides examples and definitions of these different types of numbers. The document also focuses on prime numbers, describing properties like twin primes and Mersenne primes. Methods for finding primes like the Sieve of Eratosthenes are explained. Applications of prime numbers in cryptography and factoring are also mentioned.
2.1 lbd numbers and their practical applicationsRaechel Lim
The document discusses the history and development of numbers and numerical systems. It begins with early counting methods using tallies and evolved to include the Egyptian, Babylonian, Roman, and Hindu-Arabic systems. The modern number system is then built up from the natural numbers to integers to rational numbers to real numbers, which include irrational numbers. Imaginary and complex numbers were later introduced to solve problems involving square roots of negative numbers. Place value systems and the ability to represent zero were important developments.
This document contains mathematics exercises involving trigonometric functions and identities. It asks the student to 1) find the values of trig functions given angle measures, 2) prove various trigonometric identities, and 3) use half-angle identities to find trig functions of divided angles given the functions of the original angles.
This document contains instructions for a take-home math exercise assigned to be completed and turned in on Wednesday November 23rd. The exercise includes 4 problems: 1) finding the fundamental solution set of 4 equations, 2) finding the solution set of 2 other equations and checking the solutions, 3) solving a triangle with given side lengths, and 4) calculating the distance to a boat given the angle of depression and height of a lighthouse.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
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9
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Liberal Approach to the Study of Indian Politics.pdf
Chapter 1 (part 4)
1.
2. We describe the inter-relationships among logic,
mathematics and science, which open the way to
understanding the scientific method, the
principal means by which knowledge is
acquired today.
Hopefully with this,
you will be skeptical
about the untested claims of
pseudo-science.
2
1.3 The Search For Truth And Knowledge
3. William Paley (Natural Theology, 1802)
“Suppose I pitch my foot against a stone, and were asked
how this stone came to be there; I might possibly answer,
that for anything I knew to the contrary, it had lain there
forever… But suppose I had found a watch upon the
ground, and it should be inquired how the watch happened
to be in the place; I should hardly think of the answer
which I had given before, that, for anything I knew, the
watch might have always been there… The inference, we
think, is inevitable; that the watch must have had a maker.”
3
1.3.1 The Search for Truth
Can the question of the existence of God, or Creator,
be settled through logic?
4. Paley’s argument:
Premise: The existence of something as
complex and functional as a
watch implies the existence of a
watchmaker.
Premise: Biological systems are more
complex and functional as a
watch.
Conclusion:
Biological systems imply the
existence of a Creator.
4
5. • This provides us an example of how
philosophers have tried to use logic to
ascertain ultimate truths.
• It also demonstrates the inter-relationships
among logic, mathematics and science.
5
6. Classical Logic and Mathematics
Mathematics and logic always have been
intertwined.
â The study of logic is considered to be a
branch of mathematics.
â Just as logic is used to test the validity of
arguments, mathematics is used to
establish the truth of mathematical
propositions.
6
7. âMathematics establishes the truth of a
theorem by constructing a proof.
âConsider, for example, the famous
Theorem of Pythagoras (580-500 BC)
7
Pythagorean Theorem. For any right
triangle whose legs measure a and b
units and whose diagonal measures c
units, a2 + b2 = c2.
8. Proof of Bhaskara (12th century Hindu
mathematician)
8
a
b
c
Given any right triangle
with bases a and b and
hypotenuse c, construct
a square whose length of
a side measures c units.
c
c
c
c
9. 9
c
c
c
c
a
a - b
The area of the outer
square region is c2.
This is the same as the
sum of the areas of the 4
triangular regions and
the inner square region.
The area of a triangular
region is .ab
2
1
The area of the inner
square region is . 2
ba
b
b
11. • The similarity between logic and mathematics explains
why many philosophers were also considered
mathematicians.
• Aristotle is a well-known Greek philosopher, tutor of
Alexander the Great
- probably the first person who attempted to
give logic a rigorous foundation.
- believed that truth could be established from
three basic laws:
11
12. Three Basic Aristotelian Laws
• The law of identity
A thing is itself.
• The law of excluded middle
A statement is either true or false.
• The law of non-contradiction
No statement is both true and false.
12
13. • Aristotle’s laws were the basis of the logic
used by the Greek mathematician Euclid to
establish the foundations of geometry (in his
famous treatise The Elements (300 BC).
• Euclid began with only 5 postulates or
premises from which he derived all of
classical geometry, also known as Euclidean
geometry.
13
14. • There is great trust in the validity of classical
or Aristotelian logic.
• This led directly to the development of the
modern scientific method and the
accompanying advances in human
knowledge and technology.
• This interplay of logic and mathematics may
have been the single greatest factor in the
rise of the Western world, beginning in the
Renaissance, as the center of scientific,
industrial, and technological development.
14
15. Leibniz’s Dream
• Recognizing that logic could be
used to establish mathematical
truths, could logic also be used to
establish other truths? Could it be
used to determine “universal
truths”?
15
Gottfried Leibniz
Leibniz (1646-1716) attempted to establish a calculus of
reasoning which can be used to decide all arguments;
suggested that an international symbolic language for
logic be developed with which equations of logic could
be written and used to calculate a “solution” to any
argument.
16. What happened with Leibniz’ dream?
• Leibniz had little progress. Real work on
creating a symbolic logic had to wait
nearly 200 years until George Boole
published “The Laws of Thought” in 1854.
• Boole tried to treat logic as a mechanical
process akin to algebra and developed the
fundamental ideas for using mathematical
symbols and operations to represent
statements and to solve problems in logic.
16
18. • Bertrand Russell and Alfred North Whitehead in their work
“Principia Mathematica” (published 1910-1913) sought to
put all of mathematics into a standard logical form by
attempting to derive all known mathematics from symbolic
laws of thought.
• Russell hoped that this would lead to Leibniz’s dream of
creating a system of logic in which all truths could be
derived from a few basic principles.
18
19. So what really became of Leibniz’s dream?
• Kurt Gödel in 1931 proved that
the dream could never be
achieved.
• Leibniz’s dream was shattered!
• But this ushered in a new period
in the relationship between logic
and mathematics, often termed
the period of modern logic.
19
Kurt Gödel
20. The History of Logic
20
Classical Logic
(300 BC to mid 1800’s)
Symbolic Logic
(mid 1800’s to 1931)
Modern Logic
(since 1931)
Aristotelian
logic
Euclidean
Geometry
Algebra
of Sets
Godel’s
Theorem
21. 1.3.2 The Limitations of Logic
Gödel’s Theorem
Mathematicians believed that for an
ultimate system of logic to be realized, a
first step is to show that mathematics could
be wholly understood as a system of logic.
Only then could mathematical logic be
developed into Leibniz’s dream of a
calculus of reasoning.
21
22. Mathematics as an Axiom System
David Hilbert sought to formalize
mathematics as a system in which all
mathematical truths, or theorems, could be
derived from a few basic assumptions called
axioms, by applying rules of logic.
22
23. Required Properties of an Axiom System
• It must be finitely describable, that is, the number of basic
axioms should be limited.
• It must be consistent, that is, it should have no internal
contradictions (statements that are both true and false).
• It must be complete, that is, the basic axioms should allow
analysis of every possible situation.
23
24. In 1931, Kurt Gödel, an Austrian
mathematician, proved that no formal
system of logic can possess all three
required properties. He proved that no
system can be simultaneously complete,
consistent and finitely describable.
24
25. Implications of Gödel’s Theorem
Gödel’s theorem spawned entirely new
branches of mathematics and philosophy.
Some of its consequences are:
• Some true mathematical theorems can never
be proven.
• Some mathematical problems can never be
solved.
• No systematic approach to mathematics can
answer all mathematical questions.
25
26. In other words, no absolute way exists to
define the concept of truth.
Gödel’s theorem virtually hit the nail on
the coffin of Leibniz’s dream. A
calculus of reasoning that resolves all
kinds of arguments can never be found.
Gödel’s theorem may well be one of the
most important discoveries of human
history.
26
27. The Value of Logic
If no system of logic can be perfect, what
good is logic then?
• Logic allows the discovery of new
knowledge and the development of new
technology.
• Logic provides ways to address disputes,
even if it cannot always ensure their
resolution.
27
28. 28
• Through logic, you can study
your personal beliefs and societal
issues.
• Logic can help you study the
nature of truth, though logic
cannot ultimately answer all
questions.
29. Though logic alone may fail under
some circumstances, logical reasoning
is an excellent tool for understanding
and acquiring knowledge.
Finding the proper balance
between logic and other processes
of decision making is one of the
greatest challenges of being human.
29
30. 1.3.3 Logic and Science
What is science?
Lat. scientia which means “having
knowledge” or “to know”.
Science is knowledge acquired through
careful observation and study;
knowledge as opposed to ignorance or
misunderstanding.
30
31. What is the scientific method?
• It is a set of principles and procedures, based on
logic, for the systematic pursuit of knowledge.
• It depends on logical analysis both in
determining how to pursue knowledge and in
testing and analyzing proposed theories.
• It depends on mathematics, not only in the close
historical ties between mathematics and science,
but also in the demand for quantitative
measures.
31
32. Fact, law, hypothesis and theory
Fact - a simple statement that is indisputably or
objectively true, or close as possible to being so.
Law - a statement of a particular pattern or order in
nature
Hypothesis - a tentative explanation for some set of
natural phenomena, sometimes called “an educated
guess”
Scientific theory - an accepted (that is, extensively
tested and verified) model that explains a broad
range of phenomena
32
33. The Scientific Method
(an idealization of the process used to discover or
construct new knowledge)
1. Recognition and formulation of a problem
2. Construction of a hypothesis
3. New predictions
4. Unbiased and reproducible tests of new predictions
5. Modification of hypothesis
6. Hypothesis passes many tests and becomes a theory
7. Theory continually challenged and re-tested for
refinement, expansion, and/or replacement
33
34. Science, Nonscience and Pseudoscience
Many people still seek knowledge through ways that
do not follow the basic tenets of the scientific method.
Nonscience - any attempt to search for knowledge
that knowingly does not allow the scientific method
Pseudoscience - that which purports to be science
but, under careful examination, fails tests conducted
by the scientific method
34
35. What distinguishes science from these?
• The central claims of non-science and pseudo-
science are not borne out in scientific tests; tests
are either unimportant (for non-science) or
biased (for pseudo-science).
• The distinguishing quality of the scientific
method is its unbiased and reproducible testing.
• As the scientific method is an idealization,
boundaries between them are not always clear.
35
36. Is science objective?
• By their very definition, scientific theories must be
objective, not subject to individual interpretation or
biases.
• However, individual scientists are always biased.
• Biases can show up in the following ways:
1. Opinion may matter in the choice of the
hypothesis.
2. Commission of the so-called “scientific fraud”
in hypothesis testing.
• The scientific method allows continued testing by
many people, thus mistakes based on personal
biases will eventually be discovered.
36
37. 1.3.4 Paradoxes
• A paradox is a situation or statement that seems to violate
common sense or to contradict itself.
• Paradoxes allow for the recognition of problems which may
lead to new principles, to new facts, or to a new scientific
theory.
• Paradoxes may or may not be resolved.
37
41. If the barber doesn’t shave
himself, then the barber
shaves him, a contradiction.
If the barber shaves himself,
then the barber does not
shave him, a contradiction.
42. 42
The Paradox of Light
Is light a particle or wave?
In the 20th century, physicists have
shown that light is both particle and
wave.
This discovery led to the a new area
of physics known as quantum
mechanics.
44. Imagine a race between the
warrior Achilles and a tortoise.
The tortoise is given a small
head start. As Achilles is much
faster, he will soon overtake
the tortoise and win the race.
Or will he?
46. Conclusion
• The process of discovering new knowledge is
rarely straightforward.
• The process of resolving paradoxes can provide
insight into an idea or even lead to a new
discovery.
• The concept of truth is complex.
46