Research: Theory, Method, Practice
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| 1 Research: Theory, Method, Practice Lecture 2; Stefan Arnborg, KTH Why Greek science?? • Well studied and documented • Greek classicism shapes our way of seeing the world. • Greek society cruel: Slaves, Wars, Racism,Oppression of women (i.e., like Europe) Thales -585 Anaximander 611-547 Anaximenes -502 Pythagoras 570-508 Parmenides 510- Zenon 488- Empedokles 450 Herakleitos 540-480 Anaxagoras 500-428 Herodotos 425 Protagoras 420 Demokritos 460-370 Sokrates 469-399, Antisthenes Platon 428-348 Aristarkos Aristoteles 384-322 Arkimedes -300 E Euklides Appolonius Epikureos 342-270 Selevkos Epiktetus 50-125 Poseidonius
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Hipparkos Theory of Evolution • First account by Anaximandros, including sketch of natural selection • Based on mechanistic view, not Intelligent Design • Restated by Empedocles • Rejected by Aristotle as implausible. Teleological explanation. Important paradim shift (in ’wrong’ direction). 2 Modern theory of Evolution • Based on careful collection of supporting observations (many of which can also be found in Aristotle: Parts of animals) • Refutable by age of earth (Kelvin could not know about heating by radioactivity ) and lack of understanding of genetics (Mendel’s work had been unnoticed) • Still considered somewhat daring, but (almost) only remaining hypothesis. Greek Astronomy • Relied on Eastern knowledge (Persia, India,…) • Predict eclipses (Thales, 585 BC) • Sizes of earth, moon, the zodiac to within 1% • Size of sun : Aristarkos 180 times earth -> Heliocentrism as a plausible model • Poseidonius (teacher of Cicero): Size of sun 6000 tim es earth (50% low) Explanation of tidal water (sun, moon) - made possible tidal water tables Astronomy • Aristotle Hipparkus and Ptolemai geocentrists • Appolonius: Defined both conic sections and the epicycle system. … and in the west? • Copernicus: Sun might be the center because of its majestic appearance? (similar to Aristarkos quantified argument) • However, predictions based on heliocentrism inferior • It took more than 100 years before Kepler saved the heliocentric view by using Appolonius conic sections instead of his epicycles. If the heliocentricists had followed a scientific method, they should have rejected their hypothesis(Feyerabend). Aristarchus On the Sizes and Distances of Sun and Moon
3 Tycho Brahe’s system • The moon and sun circle around earth, but planets around the sun • Absence of stellar parallax indicates geocentrism • Also convenient and safe wrt church, • Which made Kepler a looser, undeservedly because his system is ’almost right’. Tycho Brahe: First ’Big Science’ • The construction of Uranienborg consumed a sizeable proportion of Danish State Income. • Tycho Brahe was the first (documented) ’Big Science’ performer • He had to motivate his needs by writing horoscopes for kings and their like • Todays big scientists also have to motivate their needs by guessing about the practical use of their expensive equipment Atomism • Not unique for Greek philosophers • Democrit, Leukippos, from observations of life cycles and chemical processes • Epikuros combined it with an ethics of no after-life, explicated in one of the great antique works of literature, Lucretius ‘ De Rerum Natura’, On the Order of Nature. The ‘Dark ages’ • Greek science and literature survived in the Byzantine and Muslim worlds • Applied to rational analysis of theological problems (Ibn Rushd), medicine (Ibn Sina), social science (Ibn Khaldun). • Grinding halt after destruction of Baghdad (1258) and conquest of Constantinople (1453) • Translated to Latin from Greek and Arabic (Plato, Aristotle) • Aristotle surpasses Plato as ‘the Philosopher’, treated as semi-god rather than human. • Scholasticism - fascinating, but not in line with course
4 Islamic Science • The first islamic law schools (ca 800), e.g., in Fez, developed the academic degree system and CV concept (Doctor’s degree, promotion and hat) which were taken over by Bologna and Padua, and still exist • Mufti -> professor of opinion (fatwa), mostly in law, • Faqih -> Master, licenced to practice profession • Muddaris -> Doctor, licensed to teach Islamic Scholars • Ibn Sina (Avicenna), ca 1000, practice based medicine (antibiotics, vaccines (inoculation). • Ibn Rushd (Averroes), ca 1200, precursor of scholasticism, mixing ‘axioms’ in the form of Quran statements with observations, deriving new truth by syllogism. Saved Aristotle. Ibn Khaldun (ca 1360): Muqqadimah • Politician, social scientist, historian, economist. • First statements of market theory, importance of stable institutions, property right, stable currency • First scientific Marxist (without political program): Power and wealth distribution depends on how production is organized • ‘Anyone can have ideas, but only through words and language can you convince’ Newton, (1642-1727) 1665 - Alchemy 1666 - Calculus 1667 - Fellow, Trinity College 1669 - professor 1682-4 Principia 1689 - Parlamentarian 1692 - Opticks 1696 - Royal Mint 1703 - Royal Society 1733 - Daniel and Apocalypse First modern or last ancient??
5 Modern Global University system: clever and delicate merge of enlightenment inspired French ‘Grandes Ecoles’ and German idealism, worked out by Wilhelm von Humboldt, Friedrich Althoff, and Eduard Spranger. Visualization • Visualize data in such a way that the important aspects are obvious - A good visualization strikes you as a punch between your eyes (Tukey, 1970) • Pioneered by Florence Nightingale, first female member of Royal Statistical Society, inventor of pie charts and performance metrics Three Inconvenient Germans • Karl Marx (1818-1883) Class, Organization of Production, Revolution, ’Making Hegel practical’ • Friedrich Nietzsche (1844-1900). Aesthetics revolutionized, existentialist and post-modernity icon • Sigmund Freud (1856-1938), discoverer Social Science for improvement • Naive positivism: Measure, analyze, find causes of ‘bad things’, remove causes
• Intervention: Document indicators before and after. Problems: Hawthorne, outcome definition, spill-over, confounders, ethics. 6 Social Science Agenda • Critical Theory: Find obstacles for emancipation, and implementation method. • Scientific validity? • Evidence based everything: • Study phenomenon (education, health care, civil service) as a preparation for intervention study Standard view on Math Phil • Mathematical results are certain • Mathematics is objective • Proofs are essential • Diagrams are unnecessary • Mathematics is safely founded in logic • Independent of senses • Cumulative, setbacks trivial • Computer proofs are kosher • Some exotic problems in math are unsolvable What is a good Math result? • Somewhat difficult to find • Fits into an existing paradigm (there are several), ’significant result’. • Correct if agreed to be correct by reviewers • Most results are forgotten - if there are errors, no-one finds them • Most accepted results continue to be correct. • However, acceptance is not proof of correctness Exemplar paradigms in math • Socrates in Plato’s Meno - arguments less formalized than ‘modern’ proofs. Similar methods applied, e.g., by Pytagoreans • Aristotle/Euclid: Rigor stepped up, exemplary until 1960:s
• Newton, Leibniz, Maxwell, Euler, Stokes new math rather confused, carried by community of practitioners (Wranglers) • Critizised by Bishop Berkeley: The Analyst. • Bolzano, Weierstrass, Cauchy, Dedekind: Foundations of ‘rigorous analysis’. Analysis ‘King’ of Math.
7 Cambridge Wranglers -Created the math you studied: Green, Stokes, Macauly, Routh Maxwell, Larmor, Cunningham, Dirac… -Competitive math examination aimed at ranking candidates for fellowships -- -Appointments for life with no particular duties -- often awarded at age 20-25 Foundational Crises • Hilbert last polymath: 23 centennium problems in 1900. Hilbert’s program. • Russell, Whitehead: Realize logical foundation: develop all of math within logic. • Surprise: Math and computation undecidable (Gödel, Turing). Several of Hilbert’s problems not solvable. • Constructivism/Intuitionism: Only what can be ‘intuited’ can be real. Scientific Computation • Computational Complexity (& Algorithms) • Math ‘educational’ crisis: interest waning, culture disappears (Matematikdelegationen). Zermelo-Fraenkel Set Theory with Axiom of Choice (ZFC) • Extensionality: Two sets are the same if they have the same members.
• Empty set: There is set with no element. • Pairing: for sets x and y there is a set containing x and y, and nothing else. • Union: for any set F there is a set containing every member of every member of F • Infinity: There is an infinite set, eg {{},{{},{{}}},…} • Axiom (schema) of specification: For every set x and property P, there is a set consisting of those members of x satisfying P. • Replacement: Zermelo-Fraenkel Set Theory with Axiom of Choice (cont) • Axiom of separation (definition): For every set x and property P, there is a set consisting of those members of x satisfying P (and only those). • Replacement: For a function f and subset of its range x, there is a set containing the image of x, {y:y=f(z) | z ∈ x} • Power set: For set x, there is a set consisting of the subsets of x • Regularity: Every non-empty set x contains an element y disjoint from it. • Axiom of Choice: Given a set x of mutually disjoint non-empty sets, there is a set containing exactly one element from each member of x
8 Zermelo-Fraenkel Set Theory • The safest and most accepted logical foundation of mathematics • Consistency of ZFC cannot be proven within ZFC • Consistency can be shown with forcing (Paul Cohen), as well as the independence of the Continuum Hypothesis (Hilbert’s first problem) and other somewhat subtle things Intuitionism/constructivism is computational • Building on the positive integers, weaving a web of ever more sets and more functions, we get the basis structures of mathematics. Everything attaches itself to number, and every mathematical statement ulötimately expresses the fact that if we perform certain computations within the set of integers, we shall get certain results. Even the most abstract mathematical statement has a computational basis. (Bishop & Bridges, 1985) This book argues that conceptual metaphor plays a central, defining role in mathematical ideas within the cognitive unconscious- from arithmetic and algebra to sets and logic to infinity in all of its forms: transfinite numbers, points at infinity, infinitesimals, and so on. Alan Turing Halting Theorem First result in computational complexity: It is not possible for a computer to decide whether or not a computer computation (with unbounded memory) will terminate. Prof by reduction: If such a method exists, a program can be constructed which must terminate and also must not terminate
9 The Art of Computer Programming D.E. Knuth. Started 1962. Vol 1: Fundamental Algorithms, 1968 Vol 2: Seminumerical Algorithms, 1970 Vol 3: Searching and Sorting, 1973 TeX, ….
Vol 4: Combinatorial Algorithms, Vol 5: Syntactical Algorithms Vol 6: Theory of Languages Vol 7: Compilers Algorithms • Measure performance asymptotically • Multiplication Example: as in school: • Smarter: Fourier transform, Multiplication lower bound: , since you must look at every input bit. • There is typically a (very) significant gap between lower and upper asymptotic bounds The Turing Machine Opening a combination lock is difficult Unless you know its combination, 10 61 78 20 12, you must try a billion combinations
10 Winners of Cipher Challenge 2001 RSA cryptosystem uses that primality is easy, but factorization is difficult (Rivest, Shamir, Amir, 1971) Produce two large primes and multiply them. Produce a pair of keys (E,D). With the product and E (public key) you can encrypt messages but you can only decrypt if you have D and the product, or know E and the factors RSA secrets are temporary! Year Largest prime Biggest factorization Long ago 10 digits 1957 1000 digits 1982 13400 digits 77 2005 8.7 million digits 155 digits One day your key will be factorizable! Or, this day may be tomorrow Or yesterday for a paranoic cryptographer With time, the chart of complexity classes has become embarrassingly complex. And it rests on unproved conjectures. Logics of knowledge and belief Games Combinatorial optimization Feasible problems Parallelizable problems 2006: 442 classes in the complexity zoo Zero-knowledge proofs • Graph 3-colorability : • Given graph (V,E), known by both p(rover) and v(erifier). Only p has access to a 3-coloring φ: V→{1,2,3} • In each round: p permutes colors, randomization π sends each π(i) in sealed envelope to v. v asks for two specific adjacent vertices i,j, and p unlocks them. Now v can verify φ(i)≠ φ(j). • v has probability ≥1/|E| to reveal a bluff in each round - if there is one 11 Zero-knowledge Fundamental tool for cryptographic protocol analysis: •Key exchange and verification •Digital Cash: Anonymity, check against multiple spending … •Voting: No cheating, anonymity, no selling of votes … =? 0100100010100010001111110 Yüklə 4,06 Mb. Dostları ilə paylaş:
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