Two

the fall and rise of representation:

modernity and computation

Goals

Last week:Understand the development and worldview of contemporary art

Today:Understand the development and worldview of contemporary mathematics

Thursday:First programming assignment

Our modern age

The west has changed radically in the last 200 years Shift of work from people to machines Shift of people from rural areas to cities Shift of wealth and power from Church, State, and Aristocracy

to capitalist and corporation

1776-1792Democracy revivedSteam engine invented¼ million die in French revolution5% of US population in cities

1843 Dickens writesA Christmas Carol

1914-191810M die in WWI

20M die of fluNative American population falls below ¼ million

Russian revolutions

1945ENIAC

200412-13hr workday (including children)

Nostalgia for StalinPATRIOT II

75% of US population in cities

1993The web

1974 Watergate1975 End of Vietnam war Start of Microsoft Khmer Rouge kill 20% of the Cambodian population

1967-1969Student riotsWoodstockMoon landingFreedom Of Information Act

1925-1933Stalin, Mussolini, Hitler take power

Worldwide depression1867-1895Marx writes Das Kapital

183312-13hr workday

(including children)

1938-194555M die in WWII20M in USSR aloneChild labor laws enacted

Modernity

Western society underwent massive changes Deterioration of monarchies Loss of secular power of the church Depletion of agrarian life Urbanization Rise of industrial capitalism Revolution

The sense of unprecedentedness and uncertainty about the future became known as modernity

Political reaction to modernity

Modernity produced both fear and hope Fear

ReactionariesChange was bad, let’s return to the good old days

ConservativesChange is bad, let’s keep things as they are

Hope Progressives

Change is good, let’s keep going Revolutionaries

We don’t have enough change

And in some quarters Despair

The world is an awful place, let’s retreat into domains of pleasure(Art, liquor, opium…)

It’s all bullshit, including Art

Modernity and representation

Modernity involved a loss of authority Of the state

Monarchy no longer seen as inherent Democracy not really trusted either

Of the Church

This lead to a search for new sources of truth Science Art Return to the good old days Will to power

In both science and art, this lead to a reexamination of representation and meaning

Logical positivism

Science to the rescue! The Cambridge Moral Science Club

Bertrand Russell John Maynard Keyes Ludwig Wittgenstein

Interlude

Of math, logic, and rabid rationalism

Arithmetic in antiquity

Arithmetic was developed to support commerce First written records

date from Babylon (3000BC)

However, the most sophisticated systems were developed by the Greeks

Greek mathematics

Emphasized the axiomatic method Start with a small number of truths everyone found “obvious”,

called axioms Only accept assertions that can be proven (shown to follow as

logical consequences of those axioms)

This made geometry the most important branch of mathematics in Greece (and in the middle ages) Math wasn’t yet sophisticated enough to prove much about

arithmetic Algebra hasn’t yet been invented Proofs in geometry were pictures

Pythagoras

6th century BC

Classified numbers according to the geometrical shapes they suggested Hence the term “square”

Developed a quasi-religion around numbers

Weird dude

The Pythagorean Theorem

Pythagoras proved The square of the long side of a

right triangle Has the same area as the squares

of the two other sides together

Proof was through geometric construction

Let you “see” the proof But the arguments behind it were

justified by the axioms of geometry

Important in analytic geometry and therefore, in computer graphics

Euclid

Summed up all of contemporary geometry in one book, The Elements (300BC) Adopted a single set of minimal axioms Proved all theorems using only those axioms Remained the standard text in geometry for

2000 years (Think of the royalties!)

The middle ages

After the Greeks, the West continued to study arithmetic and geometry, but made few innovations

In the East, however, many innovations were made The Hindus developed full positional notation and the concepts

of zero and negative numbers The Islamic world preserved the works of the Greeks and

imported the work on the Hindus al-Khwārizmi (9th century CE, 3rd century Islamic)

Published Kitab al-jabr w’al Muquabalah(The Book of Restoring and Balancing)

Source of the Western term algebra Provided explicit procedures for solving certain classes of problems

Hence his name is the source of the Western term algorithm

The Renaissance

Greek philosophy reimported to the West from the Islamic world

Hindu and Islamic developments also imported

René Descartes (1596-1650)

Sought to put philosophy on a firm intellectual footing

Imported the methods of mathematics

Start from a small number of “obviously true” statements

Only accept statements that can be proven deductively

Grew into the philosophical movement called rationalism

Often credited/blamed for the birth of modern Western philosophy

Those stupid

scholastics …

Descartes’ cogito

Descartes looked for an axiom he could believe in philosophy Argued we could only find these by adopting a detached frame of mind

Invented the first paranoid VR fantasy Do you exist?

Well, gee, I think so, but maybe I’m just dreaming Does God exist?

Well, gee, I think so, but maybe there’s really just an evil demon who’s making me dream all this

Do I exist? Ha! I got you! How could I ask the question, if I didn’t exist in the first

place?

Cogito, ergo sum: I think, therefore I am

Cartesian dualism

Mechanical theories of physiology were starting to develop around this time

This led to a problem: is the body alive or is it a machine?

Descartes also argued that the world consisted of two fundamentally different types of things

Thinking things (res cogitans, the mind/soul) Spatial things (res extensa, the body/matter)

This was a big change because there were only two kinds of things Everything that wasn’t mind was matter The Church liked it because of the doctrine of animal autonomism

(dogs don’t have souls so they don’t go to heaven) But it was also the beginning of modern mechanistic theories of the

world

The mind/body problem

In Descartes’ theory, the mental and physical plains were causally independent Important to theology because otherwise the

atemporal world of the soul is contaminated by the temporal world of matter

But wait, then how does the mind affect the body (or know about the world)? Descartes answer: the pineal gland No really. I’m serious. Western philosophy still hasn’t recovered from this

problem

Analytic geometry

Descartes introduced the method of co-ordinates to specify points

Merged geometry with arithmetic

Described space (geometry) Using numbers

(arithmetic)

Allowed theorems from each to be used for the other

(6.27, 4.66)

(8.63, 3.02)

4.66-3.02 =

1.64

8.63-6.27 = 2.36

√(2.362 + 1.642 ) = 2

.87

Analytic geometry the position, sizes, and shapes of objects in space to be quantified

Issac Newton (1642-1727) and others could then quantify changes of position,i.e. the laws of motion

Replaced Aristotelian physics with a new mathematical physics

Drove further developments in mathematics such as “the calculus”

I really wanted to do theology, but I my rationalist friends would

make fun of me

The development of Newtonian physics

The development of modern rationalism

The axiomatic method had spread from geometry to

Arithmetic Philosophy, metaphysics, and theology Physics and chemistry

Why stop there? Why not ethics, politics, aesthetics? The rationalists believed that

eventually all problems were ultimately susceptible to human reason

Gottfried Wilhelm Leibniz (1646-1716) Co-inventor of calculus Believed we could someday develop a

calculus of thought that could express

All ideas All valid deductions

“Gentlemen, let us calculate!”

To be an Enlightenment scientist, you need a good head of hair

The Enlightenment (18th century)

Combined Humanism (from the renaissance) which believed in

The value of Human reason The individual The here and now, rather than just the afterlife

That people’s characters are a product of environment (nurture) rather than nature

Rationalism Sick of hearing about this yet?

Empiricism Argued truth is learned not just from introspective reason, but by observation

and experiment

Advanced the causes of science and democracy Paved the way for …

Modernity

Western society underwent massive changes Deterioration of monarchies Loss of secular power of the church Depletion of agrarian life Urbanization Rise of industrial capitalism Revolution

The sense of unprecedentedness and uncertainty about the future became known as modernity

Logical Positivism

Rationalism to the rescue! Let’s place all knowledge on a firm, rigorous foundation

Get rid of all the bad metaphysical speculation Come up with a set of axioms of science, math, ethics, politics,

etc. that all humans can agree on Use logic to end war, poverty, and strife Rest content in the knowledge of a job well done

The Cambridge Moral Science Club Bertrand Russell John Maynard Keyes Ludwig Wittgenstein

Formalism in mathematics

But first, maybe we should finish mathematics

Formalism: movement to put all of mathematics on a single, solid, foundation Define all mathematical systems axiomatically For each system show that

All true statements about the system can be proven (completeness)

No false statements can be proven (consistency) Show that the truth of statements can be determined

through some definite procedure

Metamathematics

If we want to do this right, we need to be rigorous about defining What our axioms are (duh)

But this is still hard because people have legitimate disagreements What kinds of deductions are valid from a given set of premises

This is harder Descartes got into trouble on this one People seem to legitimately disagree

What we mean by a definite procedure No one really had any idea about this one

Q: how do we do this? A: use more math!

Define logic as a mathematical system with its own axioms and rules Define the notion of procedure mathematically

A common language for the axioms

Modern mathematics is built on set theory (Georg Cantor, 1874)

Mathematics reasons about objects

Some of those objects are sets of objects (including other sets)

There’s a special set with no elements, called the empty set

Examples of sets The students in this class The set of points on the plane The set of points in a line The integers:

Z = { n | n isn’t a fraction } The even numbers:

E = { n | n/2 is in Z }

Very controversial

Functions

The most important kind of mathematical object is the function Functions are used to represent dependence or change

One or more inputs (mathematical objects) called arguments An output or result For each possible input, specify an output Like a big table

Examples Position of an object

Input: time (a number) Output: position (a point in space)

Addition Inputs: two numbers Output: one number (the sum)

Most mathematics involves studying the properties of functions

Set theory was controversial Definition:

Two sets A and B have the same size if They can be put in one-to-one correspondence I.e. if you can find a function from A to B such that

Every element of A is mapped to an element of B Every element of B has exactly one element of A mapped to it

Theorem:The set of even integers has the same size as the set of all integers

Proof: For each integer n, map it to the even integer 2n Every integer n has an even integer (2n) Every even integer m has exactly one integer mapped to it

(m/2)

Many mathematicians never accepted set theory Basic disagreements over what axioms were considered

“obvious” Lead to the development of intuitionism

Henri Poincare: “Set theory is a disease from which mathematics will soon recover”

Why does everybody hate me?

A common logic

Logic is the study of what can be inferred from facts

George Boole realized that the words “and”, “or” and “not” act like functions

Take the objects “true” and “false” as arguments

Return “true” or “false” as values The “and” function returns

True, if its arguments are true False, otherwise

Gottlob Frege extended Boole’s work to develop the first modern logic

Suppose “Ian is blond” = true “Ian is a nerd” = true “Ian is president of Latvia” = false

Then “Ian is blond” and “Ian is a nerd” =

true “Ian is blond” and “Ian is president

of Latvia” = false “Ian is blond” or “Ian is president

of Latvia” = true

A common set of axioms:Principia Mathematica (1902)

Russell and Whitehead Defined all mathematics in

terms of set theory It started with:

The empty set exists Two sets can be unioned

And used a variant of Frege’s logic to derive Arithmetic Functions … Calculus

Scary…

I’m an enlightened

liberal rationalist

Russell’s paradox

Let S = { s such that s s } The set of all sets that do not contain themselves as members

Is S a member of itself? Oops …

Russell’s paradox meant set theory was broken Came up with a fix, but it was considered ugly

shit.

Ludwig Wittgenstein Born to a very wealthy industrial familiar

Highly educated in the arts and sciences Loved science but distrusted modernity Wanted to prove art and poetry were the only

media that could express the truths of morality and the human condition

Gave away all his inherited wealth

Went to Cambridge to work with Russell on a better version of the Principia

Developed it in the Tractatus Logico-Philosophicus

Make a simpler, more powerful theory Showed that there were necessarily true facts

that couldn’t be expressed in the logic

Became the darling of the Logical Positivists, even he radically disagreed with their goals

I have David Bowie’s

cheekbones

Hilbert’s problems

Steps toward the goals of formalism

Hilbert’s 10th problem:Is there a definite procedure to find All integer roots

Of any integer polynomial?

2x2-18=0 x=3

Gödel's theorem

No mathematical system powerful enough to express

arithmetic Can be both Consistent, and Complete

Worse, You can never prove the

consistency or completeness of a system

Using the system itself

Oh, poop.

Gödel coding

A system for translating arbitrary text into numbers Used to prove his theorem,

but also … The basis for modern

digital data representation

Source text: “A=B” Letter codes:

a=1, b=2, …, z=26, ==27 Gödel-coded text:

012702

Back to Hilbert’s 10th problem

A math undergrad at Cambridge named Alan Turing decided to work on Hilbert’s 10th problem

He never did solve it

But the first thing he had to figure out was what “definite procedure” really meant

Formalizing problem solving

Math is very good at formalizing problems But it wasn’t very good at applying its own

techniques to the process of doing mathematics What does it mean (mathematically)

to follow a procedure? How do you know if you did it?

This is where Turing’s real contributions came

The Turing machine

Infinite memory tape Divided into cells Each cell can store

one letter CPU

Moves along tape Reads cell Chooses how to

Rewrite cell Move left or right

Running a Turing machine

Write the input on the tape

Start the CPU running CPU writes the

answer on the tape on top of the input

And halts

Programming a Turing machine

CPU could be in different states On each step it changed to a

new state So the CPU did table lookup:

For each possible state And each possible letter on

the tape Look up

What new state to go to What letter to write Which way to move the head

Different CPUs for different tasks

Back to Hilbert’s 10th problem

The machine gave Turing a way to formally specify procedures That meant he could prove

theorems about them Before trying to find out

whether the 10th problem was solvable

He decided to try a simpler question Are any problems

unsolvable?

The halting problem

Given: The lookup table for a

Turing machine An input on the Turing

machine’s tape Determine whether

the machine will infinite loop Don’t even worry if it

gives the right answer

Baby steps

The halting problem is too hard to think about

So Turing started with the question: Can you make a Turing

machine that Just prints what another

Turing machine would print On a given input

Turing’s universal machine

Does there exist a Turing machine that

Given: A description of a CPU’s

lookup table And the initial contents of its

tape Will determine

The contents of the tape when the TM halts

Yes! Turing called this the Universal

Machine

Who cares?

The Universal Machine was the first interpreter

The Universal Machine was the first programmable device

The Universal Machine was the first digital (binary) device

Even though it was never built per se

Back to the halting problem

Turing had shown that it was possible for a program to Represent another program To simulate the other program

Then he showed it was possible to construct a program that had a representation of itself

This is turns out to be a real problem …

The revenge of Russell’s paradox:The Undecidability theorem

Theorem:There can in principle be no program that tells whether an arbitrary program halts on an arbitrary input

Proof: Assume it existed Then you could make a program P that

Run the program that tests whether a program halts, feeding it P and P’s input as its inputs

If the testing program says P shouldn’t halt, it immediately halts Otherwise, it runs forever

Does P halt? Oh poop.

Positivism’s failure was our gain

The undecidability theorem showed that there were problems that couldn’t be solved through formal methods The grand vision of positivism was doomed

However, the project of formalism lead to the development of the key concepts behind modern computing

Computers are Formalism made flesh Autistic little positivism machines

I’m probably out of time, but …

more on the life of Alan Turing

Alonzo Church

Nobody cared about Turing’s work, except …

Alonzo Church, who had just developed a system for describing calculations called the “lambda calculus” … Basis for LISP, Scheme,

and Meta

Turing became Church’s postdoc

The Church-Turing hypothesis

Church and Turing proved their formalizations of computation were equivalent Turing machines can run lambda calculus Lambda calculus programs can simulate Turing

machines

They hypothesized that all sufficiently powerful formalizations of computation are equivalent

The Enigma

The Nazi’s invaded Poland, and eventually France England was at war

But the Germans were using a new mechanical code device called Enigma

And the Allied intelligence community couldn’t break the German codes

Code wheels

Enigma used a complicated set of mechanical wheels to represent codes and messages

The wheels moved as the message was encoded

Changing the code

Breaking the code required looking at

All possible sequences Of all possible wheels

Humans couldn’t do it

Bletchley Park

The British government collected a group of mathematicians at Bletchley Park to try to break the Enigma

The Turing Bombe

Turing build a machine to search through sequences of possible codes Mechanically Reliably And very quicky

Enigma was broken

Coventry

The German’s codes were broken but they didn’t know it

The breaking of Enigma was one of the most closely guarded secrets of the war

It was useless if the Germans knew the code was broken

One day, Winston Churchil found that the city of Coventry was to be bombed …

After the war: computer science

Turing went on to do pioneering for in the development of electronic digital computers

The imitation game

In a landmark paper on philosophy of mind, Turing proposed that

thought itself is computational

The Turing test Chat room Computer and human Can the human tell if she’s

talking to a computer?

The cure

Things were going great There was just one problem Turing was a homosexual

A felony in England in the 1950s

Turing was given a choice Prison (like Oscar Wilde) Injections of female hormones

The cure

The injections had horrible side-effects Migraine headaches Intense mood swings Clinical depression Unnatural growth of breasts

But it was the only way he could continue his work

Later work

Turing’s later work was on morphogenesis, the study of how biological shapes develop

Computational biology was largely ignored for 40 years

But is now a major research thrust for the nation

The end

Alan Turing was found dead on June 8 1954 Cyanide poisoning, source

unclear

An apparent suicide Although conspiracy

theories exist