Research: Theory, Method, Practice



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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

 

100


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



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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

of the unconscious

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 (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



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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





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