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- The world we live in
can seem pretty illogical.

The things people say,

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the ways we behave,

the complex choices we have to make.

(people shouting)

What's the quickest way to get home?

Can I trust any of you lot?

Where did you all come from?

That process of making
sense of all this stuff,

of sorting between the
truth and the nonsense,

comes down to one of the most
simple and yet powerful tools

ever created by humans, logic.

- Yes, yes.

There is definitely beauty in logic!

- Who would like to be bits of a computer?

- [Children] Yay!

- In the building next door to me at work,

there's a door and there's
a sign on it that says,

"This door must be kept
closed at all times."

I just look at this in amazement, really?

Why did you build a door then?

- Is this sentence true or false?

Philosophy, maths, science and language,

logic is the engine for all of them.

In fact, it drives the fundamental process

of reasoning itself.

I'm a professor of computer science.

Computer scientists tend to think

that logic is the bee's knees.

So, it follows that I
think logic is brilliant.

Logic has inspired our greatest boffins.

- I'm Socrates!

- [Narrator] It's given us
transformational technologies.

- Delta 11, report your entry point.

- [Narrator] And even made us question

what it means to be human.

- Off with her head!

- I want to see if there's any limit

to what logic can do for us.

So, join me, it would be
terribly illogical not to.

(upbeat music)

Logic is right at the heart of what I do.

Around 15 years ago, kind of by accident,

I created something that had
a really big impact here,

on the trading floors
of the City of London.

It was a computer program I called ZIP,

and it used logic to replicate this,

a centuries-old tradition
of human traders,

supposedly vested with
very special skills,

crammed into rooms,
shouting at each other.

(people shouting)

It's ever so simple, just
a few logical inferences,

decisions, and a little bit of maths.

It learns from its trading
successes and failures.

Its aim is to trade as
profitably as possible

in a fast-moving market,

where levels of supply and
demand are shifting rapidly.

It turned out that ZIP,
built squarely on logic,

was impressively proficient
at this trading lark.

In fact today, in many markets,

billions or trillions of dollars'
worth of deals go through

with no human intervention at all,

which is kind of mind-boggling.

Every day, computer
programs, on their own,

do deals that determine
the cost of everything

from our fuel and food, to
the worth of our pensions.

It's pretty important stuff!

And, every day, scientists like me

earn a living using
logic to find solutions

to all kinds of other
real-world challenges.

So, why am I not as rich as Bill Gates?

Well, I gave away the
ZIP software for free.

And looking back that was
probably not my most logical move.

So what is logic?

What does being logical even mean?

- I'd like a pint of lager, please.

- [Narrator] Well, all
you need to explain it

are three logicians and a boozer.

Logic is actually all about
the rules of correct reasoning.

Let me tell you a joke.

Three logicians walk into a bar.

The barman says.

- Gents, would you three like a beer?

- And the first logician says.

- I don't know.

- The second logician says.

- I don't know.

- And then the third logician says.

- Yes, yes, we would all like a beer.

(laughing)

- Okay, so it's not
exactly a side-splitting,

laugh-out-loud gag, more
of a chortle for nerds.

But what went on there?

Well, forgive me, I'm going
to analyze that joke to death.

Remember, the barman's question was,

"Would all three of you like a beer?"

The key here is the all three bit.

If any one of those logicians
doesn't want a beer,

then he'd be able to answer no.

That's because if one doesn't want a beer,

they don't all want one.

Logician one does want a beer,

but he can't speak for the others,

so he has to say, "I don't know".

Exactly the same goes for logician two.

Then, happily for logicians one and two,

logician three also wants a beer,

and so he correctly uses logical inference

to arrive at the right
answer to the question.

- Yes, yes, we would all like a beer.

- At last!

Logician three ends the torment

because he can speak for everyone.

Cheers to that!

The important thing to understand

is that logic isn't knowledge.

Logic doesn't create knowledge,

what it does is it give us cast-iron rules

for how to organize and handle knowledge.

Even so, the quality of
the conclusions you get out

depends on the quality of
the ideas that you put in.

Time, please, gents!

- [All] 11 o'clock!

- [Narrator] It'd be a funny old world

if we followed the rules
of logic all of the time.

These days, logic is studied

and taught in academic
institutions the world over.

Its history stretches back 2,500 years,

to the age of the Greek
philosopher, Aristotle.

He created the first formal rules of logic

that would govern good reasoning,

clear thought and reliable argument.

Aristotle's most famous
logical tool is the syllogism.

- A syllogism is a certain
simple kind of argument

consisting of three propositions.

And the first two
propositions are the premises,

the things that we take for
granted in the argument.

So, for example, all men are mortal,

Socrates is a man.

Those are our two premises.

And from them, we draw the conclusion,

Socrates is mortal.

Aristotle's example is
good logical reasoning.

First, we take one
premise, or thing we know,

all men are mortal.

- Yes, indeed.

I'm Socrates.

- [Narrator] Then pair
it with a second one.

- I am a man.

- [Narrator] Then we
figure out, or infer that,

alas, Socrates is mortal.

- That makes me sad.

- [Narrator] If your
premises are reliable,

and you follow Aristotle's rules,

you get answers that are reliable, too.

- But Aristotle's theory of the syllogism

can deal with more complicated arguments

that don't just have all in
them but some in them, and not.

Take all these into account,

and you find there are lots
of ways to make a syllogism.

So, if you multiply that up,

you find that there are
256 kinds of syllogism.

And Aristotle identified 19 of these

as being logically valid,

so that if the premises are true,

the conclusion has to be true as well.

And all the others of those 256 forms,

you can have true premises
but a false conclusion,

so arguing in that way is fallacious,

those kind of syllogisms are fallacies.

- It's the old logical fallacy,
all cats have four legs.

My dog has four legs.

- Therefore, my dog is a cat.

(audience laughing)

- He is suffering from politicians' logic!

- [Narrator] This is just
one of Aristotle's fallacies.

It looks similar to good logic,
the premises are both true,

but the way they're organized

means the reasoning is
completely backwards,

and the conclusion, bonkers.

- Something must be done.

This is something,
therefore, we must do it.

- But doing the wrong thing
is worse than doing nothing.

- Doing anything is
worse than doing nothing.

- [Narrator] Such was the
power of Aristotle's logic

that scholars used and taught it,

but actually didn't do a
great deal to change it,

for the next 2,000 years.

But it wasn't just philosophers
that were enamored of logic.

By the 19th Century, the
public had fallen for it, too.

For this, our thanks must
chiefly go a mathematician

who spent most of his life

working at Christchurch, in Oxford.

Charles Dodgson.

He's much better known by
his pen name, Lewis Carroll.

- The mathematics books were
mainly under his real name,

Charles Lutwidge Dodgson,

but he chose to use his
pen name, Lewis Carroll,

for the game of logic and symbolic logic,

clearly to give it a wider audience.

- Explain yourself, child.

- [Narrator] Alice's
adventures may seem barmy

but, curiouser and curiouser,

she was actually up to
her eyeballs in logic.

- In the Mad Tea-Party,

the March Hare says, "You
must say what you mean."

And Alice replies "Well,
I mean what I say.

"It's the same thing, you know."

The Hatter says, "You
might as well say that,

"'I see what I eat' is the
same as, 'I eat what I see'."

- Got you!

Bottles don't talk!

- [Narrator] Dodgson was
so keen to introduce people

to the delights of logic,

that he drafted a book initially
called Logic For Ladies.

- He was very conscious
that girls in particular

were not heard,

they were not given the
chance to go to school,

very few had the opportunity
of going to university.

They certainly weren't
able to get a degree.

- [Narrator] Happily for us blokes,

Dodgson had a change of heart

and "Logic For Ladies" was
renamed "Symbolic Logic".

Together with "The Game Of
Logic", it did surprisingly well.

- He felt that young people needed a tool

to detect fallacious arguments

that they might meet
in books and magazines.

He wanted them to have the
ability to detect that.

- While Dodgson's intentions

would have made Aristotle
proud, some of his syllogisms

stand out today for the wrong reasons.

What are you meant to conclude?

No marks for saying,

"Victorian England was
intrinsically anti-Semitic".

I think Dodgson would have
wholeheartedly approved

of today's most popular
logic game, Sudoku.

There's something
captivating about the fact

that logic tells you the
answer must be in there,

but you need to apply
logical reasoning to find it.

It can be really engaging,

but it can also be really
frustrating and annoying, too.

Charles Dodgson had been the
first person to popularize

the idea of logical reasoning
and critical thinking.

But, for all its growing popularity,

logic itself was due for an upgrade.

In 1847, this ground-breaking
book was published.

It's called "The Mathematical
Analysis Of Logic".

Now, this isn't logic for
philosophers or puzzle fans.

The author of this book argues

that the real purpose
of logic is mathematics.

And this book was written by George Boole.

Born into a poor family in Lincoln,

Boole mastered mathematics
at a fantastically young age

and, by 20, he'd opened his own school.

Boole's big idea was that logic

was actually closer to
mathematics than philosophy.

All you needed to do was change the words

in a logical argument to symbols,

and then it could be solved
just like an equation.

He called it his calculus of reasoning.

First, he demonstrated that
the letters we use in algebra

to represent numbers can
actually be used to represent

whole classes of things in the real world.

So, for instance we
might have the class, X,

of things that are fluffy,

and the class, Y, of things that bark.

Second, he introduced a set of operators

for combining these classes of things

the three most important
ones are and, or and not,

and they're known as Boolean
operators in his honor.

So, if we redraw our classes
so that they overlap,

the bit in the middle,

that's things that are
fluffy or bark, X and Y.

If we look at the whole
of the two circles,

well, that's things that are
either fluffy or they bark.

So that's X or Y.

And, finally, if we think
about the area outside,

well, they're neither fluffy nor barking

so that's not X and not Y,

things that aren't fluffy and don't bark.

Like me.

Boole's new mathematical logic

reduces any logical problem

to symbols that can be
combined in new ways.

And there was one final
and crucial innovation.

In Boole's new mathematical logic,

everything's either in or out,

statements are either true or false,

everything's either a one or a zero.

For example, if I were
to ask my dog, Floss,

"Are you fluffy?" and "Do you bark?"

she would have to bark, "Yes!"

Taking one to mean yes
and zero to mean no,

with Boole, we get this.

It was an entirely new
form of logical reasoning.

Seemingly anything could
be boiled down to symbols

and just two numbers.

And it's in my field that Boole's vision

would prove transformative.

Almost a century after his death,

his logic would become
the language of computing.

- My logical hero has to be George Boole,

Boolean logic is so simple,

yet so fundamental to
explaining our world,

and even the world today, which
is full of complex systems

that he could never have imagined,

and Boolean logic allows
us to reason about them.

What a guy!

I think the application area

and the use of logics
has changed dramatically

in the last 20-30 years

with the advent of computer
science and software system.

Because fundamentally these systems

are about zeroes and ones,

entities that map onto truth and falsity.

And what I think is just
absolutely brilliant

is that we go back to
lots of the logical ideas

invented and conceived over 100 years ago,

before anyone imagined the systems

that they'd be applicable to.

- Boole never knew it but thanks
to him, all computers today

process their information
as binary digits or bits.

With binary any number can be represented

by combinations of ones and zeroes.

I'm gonna do an experiment.

Come on in.

So the cool thing about binary numbers

is that they're really easy
for computers to manipulate,

to add and subtract,

or multiply or divide or
to compare to each other.

In fact, any time you see
a computer doing anything,

whether it's adding two numbers together

or computing stock-market derivatives,

inside, it's using Boolean
logic to do just that.

I want to demonstrate how Boole's logic

can be used for computing.

At their simplest, computers work

by passing bits of
information, ones and zeroes

through a circuit, like the
one that we're building here.

the most important
parts are the junctions,

where the bits of information
are combined and passed on.

These are called Boolean logic gates,

and the way you order them

determines exactly what
the circuit can do.

From simple addition to
calculations we could never do

in our own heads,

they can all be worked out
with something like this.

I'm gonna use these guys,

and some very simple logic gates,

and, not and or,

and a circuit that we've got
out there in the school hall,

and what this circuit is gonna do

is to add together two numbers
to come up with one answer.

Who would like to be bits of a computer?

(children cheering)

Come on up, and I'll give
you out your shirts, OK?

This one is a number one.

Which is for Ishmael.

They're not just pretending,
they will be a computer.

Charlie T, thank you very
much for being an and gate.

Normally, of course, computers
work on electric currents.

Our computer will be powered by kids,

who will pass on their ones and zeroes

by either tagging the next
kid in line for a one,

or not tagging them for zero.

It's time for the kids to take
their places in our circuit.

And, for the record, I've
never tried this before!

OK, some of you are being and gates.

Do you remember what
an and gate has to do?

The rule for ands is they
only get a one to pass on

if they're tagged on both shoulders.

So, some of you are being or gates.

ors pass on a one if they're tagged

on one or both shoulders.

Some of you are being not gates.

nots are different.

They get a one to pass
on if they're not tagged.

Numbers, you are the most important thing,

'cause the whole circuit is
about processing numbers.

We're going to put these
four bits into the circuit,

which arranged like this,
represent two and three.

Off you go!

The bits of information
have been inputted.

They're relayed on by
the first set of kids.

If they're following their rules,

only some should be carrying ones.

While other's won't.

At each gate, the bits are
combined and passed on.

They're nearly there!

At last, the output numbers
are either tagged or not.

So. We've got a one, and zero and a one.

A four and a one and that makes five.

And the numbers we added at the start

were a three and a two.

So, a three and a twp
moving through this circuit,

with all of you just
doing very simple things,

being and or or or not,
ended up a five this end,

so you have calculated the right number!

Today, all our computers are built

using Boole's logic gates.

Here we have 13,

but a modern computer chip like this one

might have 250 million.

They're all doing exactly
what these guys were doing,

but an awful lot faster.

We just did a simple sum here,

but Boole heralded a new era for logic,

in which reasoning about anything

could be done in the language of maths.

- There are lots of different logics

because there's lots of
different kinds of systems

or worlds that we want to reason about.

I've been applying logic to reason

about a wide variety of complex systems.

I've looked at communications

for air-traffic control systems,

molecular biology, I've also
looked at advanced telephony.

- [Narrator] But, regardless
of the application,

all logics have one thing in common.

- Amongst all these logics,
the unifying property

is they're about axioms and rules

so the answer is unambiguous.

We can automate the procedure

of computing the answer in logics,

but we still need to pose the question.

Taking exactly those questions

and automating the way
we logically answer them

requires what's known as an algorithm.

It's the province of my
very own breed of nerd,

the computer programmer.

And there's nowhere more
important for today's generation

of up-and-coming young
programmers than this,

the annual International
Olympiad of Informatics,

held this year in Brisbane, Australia.

- We're trying to find the
best and the smartest students

when it comes to computational thinking,

algorithms and programming.

- [Narrator] On each competition day,

everyone is set three questions

which must be answered within five hours.

- The easiest one, you just
had a bunch of locked doors

and you had a bunch of switches,

each of the switches was
connected to one of the doors,

but you didn't know which switch

was connected to which door.

And what they ask for is to
determine, for each switch,

which door it's connected to

and which position is the correct one.

- [Narrator] Johnny Ho
is last year's champion,

so there's a lot to live up to,

but things aren't quite going his way.

- By now I've actually solved all three,

but I didn't actually solve
them during the contest

because there's just a lot of pressure.

- We test the ability of students

to come up with clever algorithms

to solve algorithmic problems.

They not only have to come
up with the algorithms,

but they have to write a computer program

that runs the algorithm.

- [Narrator] Algorithms
turn real-world problems

into questions that
logic can help us answer.

If, for example, these
guys wanted to spend

their day off competition
duties defining the group

of all animals in a zoo
that are marsupials,

the first step of the
algorithm could be to ask,

"Of all the animals I see,

"which would I find in
the wild in Australia?"

No.

Nope.

No.

Yes!

No.

I don't know.

Yes.

Yes.

Yes.

Definitely not.

Yes.

Certainly not all of the yeses

and don't-knows will be marsupials,

so the list can then be refined

by asking which of these
animals have pouches.

And here there are options, too.

They could look in a book.

They could ask Chris, he's an expert.

Or they could crowd-source the question

and go for the most popular answer.

Each logical algorithm
incurs a different cost,

in effort, time or accuracy,

but, whichever way, they'd each
get to an answer eventually.

And there are certain situations

where a good logical algorithm
can be the difference

between life and death.

This is the NATS control center,

in Swanwick, SE England.

At any one time, around
100 air-traffic controllers

are responsible for 200,000 square miles

of airspace over the UK.

- Delta 11, report your entry point.

- [Narrator] Landing over
two million flights a year,

it's perhaps surprising
that, until very recently,

these folk did their job
using brain power alone.

But that's all changing.

New automated algorithms
have started to take on

some of that responsibility

for guiding the planes in our skies.

- The equipment now is
talking to the aircraft,

and so whereas before the human
was reacting with the human,

and, obviously, there are
sometimes mistakes made,

the computers can now
double-check that interaction

and provide a warning to the controller

if anything is amiss.

Equally, in terms of
capacity, because it's reduced

the amount of workload for the controller,

we've seen capacity about
40% increase on some sectors,

because the computers are doing

some of the logical
calculations and thinking

on behalf of the human being.

- I think logics are really
crucial as a tool for reasoning

about the systems we
use in our modern world.

We are surrounded by these complex systems

like air-traffic control,

railway signaling, the electricity grid.

I think it's really important

that we raise the next generation

of users of these systems so
that they know it's not magic,

they also know that they
have the tools of logic

to understand and reason about the systems

that they depend on
crucially every single day

of their lives.

- [Narrator] Back at the
International Olympiad

of Informatics, it's
day two of the contest.

The judges are looking
for programs to do logic

that aren't just right,
they have to be fast.

- So, if you have an algorithm
that is technically correct

but will take 100 million years to run,

then you would score no points.

If you have an algorithm
that solves the same problem

and runs in, say, five seconds,

then you can score much higher points.

- I think the simpler an argument is,

the more beautiful it is.

So, if it can be expressed
in perhaps just 10 words,

that argument would be pretty neat.

- [Announcer] The
competition has finished.

Thank you very much for your patience.

- [Narrator] It's an anxious
wait for the final ranking.

I think this competition is,

in all its geeky glory, an amazing event.

With the ability to implement
their problem-solving talents

in the language of computing,

these kids are gonna be the
future of all things logical.

- [Announcer] The
first-place winner of IOI

is Lijie Chen from China.

(audience applauding)
(audience cheering)

- [Narrator] In the end, it's
a Chinese one, two, three.

It's lucky the Brisbane competitors

didn't have this problem to solve.

It's one that no logical
algorithm can cope with.

All I want to know is, what do you think?

Is this sentence true or false?

Is it true or false?

You can have this if it's false.

The point is, if the sentence
is false, then it's true.

But if it's true, then it must be false.

It's a paradox.

But if it's false, it's true.

My sign is inspired by the
first known logical paradox,

from around 600 BC, by the
Cretan Epimenides of Knossos.

Well, if you read the sentence
that this sentence is false,

as its true meaning,
then, yes, it is false.

Epimenides wrote, "All Cretans are liars,"

but he was a Cretan, so was he lying?

If so, then all Cretans aren't liars,

in which case, he would
be telling the truth.

- It's a paradox.
- A paradox! Well done!

Paradoxes are fundamental contradictions

that logicians have
puzzled over for centuries.

They've been described as

truth standing on her
head to get attention,

and for good reason.

In the late 19th Century,

round about the same time that
George Boole was developing

logical deduction as a
branch of mathematics,

paradoxes exactly like this

became a really deadly serious matter.

In fact, they came to threaten

the very foundation of mathematics itself.

The Austrian capital, Vienna,
renowned for its music,

elegance, legendary cafes
and exquisite cakes.

But, at the turn of the 20th Century,

it was also the place to be if
you were interested in logic.

Despite its grace and gentility,

Vienna can lay justifiable claim,

perhaps more than any other city,

to being the birthplace of the modern.

For it was here in art,
design, philosophy,

science and psychology, that
people most boldly challenged

the tired conventions and
assumptions of the 19th Century.

But what was modern?

Was it about replacing
religion and tradition

with logical empiricism and pure reason?

Or was it about admitting
to a new uncertainty,

the limits of our perceptions

and the moral vacuum of
the Freudian subconscious?

Until this point, it could be argued

that logic wasn't exactly
a topic on everybody's mind

but, here, it was at the
forefront of this titanic clash.

From the city's coffee houses

to the University of Vienna itself,

the struggle for modernity played out.

In 1894 the university commissioned
a great ceiling painting

for their ceremonial hall.

The theme was "The Victory
Of Light Over Darkness",

and it had separate panels

celebrating the great achievements

of the university's
faculties of jurisprudence,

of medicine and of philosophy.

Given the subject matter,
it was perhaps unfortunate

that the artist they
commissioned for these paintings

was Gustav Klimt.

In 1900, he presented
them with Philosophy,

a depiction of naked men and women

drifting trance-like in empty voids.

It expressed anything but
victory, certainty or optimism.

Klimt's proto-modernist
vision of philosophy

was shocking to the people of Vienna,

and deeply unsettling to the
professors at the university.

He was attacking
everything they stood for,

and Klimt's paintings
were rejected outright.

Hidden away for 40 years,

the original works were
destroyed by the Nazis.

These replicas were finally installed

on the centenary of their rejection.

Klimt's dark vision had seriously offended

the growing academic aspiration,

that science and
mathematics would provide us

with complete knowledge,

founded on absolute, provable truth.

This was something it was
hoped logic could provide.

In mathematics, this
problem of definitive truth,

of certainty, had recently
become all too real.

No-one yet had proven the most
basic rules of mathematics.

Those rules might say that
one plus two equals three.

But, without proof, that
they will never lead

to a contradiction, you
can never say for sure

that one plus two might not also equal 20.

Or anything else for that matter.

In the grip of uncertainty,
a logic fever took hold.

Boole's logic had already been adopted

by the greatest logicians of the day,

but there was a problem.

His method was simply insufficient
to describe all of maths.

The race was on for a new,
and more complex, logic.

Over 20 years earlier,

a German mathematician
called Gottlob Frege

had studied exactly this problem.

Frege's work ensured that
logic was up to this search

for certainty which was
unfolding right here.

♪ If I had it in my power ♪

- [Narrator] It was in Jena, Germany

in the late 19th Century
that Gottlob Frege

opened a new chapter
in the story of logic.

For him, there should be nothing,
whether numbers or ideas,

that could not be described and analyzed

using his new logical quantifiers.

♪ Everybody loves somebody sometime ♪

- [Narrator] So, with his
new mathematical logic,

he could express ideas
like, everybody loves Frege,

everybody loves somebody,

there is somebody whom everybody loves,

there is somebody whom no-one loves,

and, alas, there is somebody
whom Frege does not love.

♪ If I had it in my power ♪

- That somebody whom Frege
probably did not love

was British philosopher Bertrand Russell,

who independently was engaged
in exactly the same project,

using logic to firm up the
foundations of mathematics.

In 1902, Frege was just
days from publishing

the second volume of
his magnum opus on logic

when he received a letter from Russell,

and it was the kind of letter
any logician dreads receiving.

Russell had spotted a big problem.

Both men's logic relied

on consistently describing sets of things.

You can have the set of all even numbers.

Or, for that matter the set of all mums,

or the set of all dogs.

Almost all sets aren't
members of themselves.

The set of dogs isn't itself a dog.

So, if you take the dog set

and bundle it up together with
all the other ones like it,

you get the set containing all sets

that are not members of themselves.

But this is the set of all sets

that don't contain themselves,

and it doesn't contain itself.

So this set should include itself.

But then, if it does,
then this is no longer

the set of all sets that
don't contain themselves.

So, it can't be part of itself.

It's one of those logical paradoxes.

Frege immediately wrote back to Russell.

Dear colleague, your
discovery of the contradiction

has surprised me beyond words

and, I should almost like to say,

left me thunderstruck, because
it has rocked the ground

on which I meant to build arithmetic.

Your discovery is, at any
rate, a very remarkable one,

and it may perhaps lead to
a great advance in logic,

undesirable as it may seem at first sight.

Russell now took on Frege's project

with an even greater zeal,

to develop an even more
outrageously complex logic

that would get round
this problem with sets,

and so be free of paradox.

After nine years of toil,

the monumental Principia
Mathematica was published.

It took over 360 pages to logically prove

that one plus one equals two.

It was never gonna a best-seller,

but, here, it had a huge impact.

It was magnificent, a
whopping great bucket load

of logical concrete

poured right into the
foundations of mathematics.

Definitely a triumph, not
a trauma, for philosophy.

But the final word on logic

would not come from Bertrand Russell.

It was here that that project
came to a dramatic conclusion,

centered on a group of thinkers
called the Vienna Circle.

They were firmly pro-logic.

For them, Russell's Principia Mathematica

was manna from heaven.

The Vienna Circle had
people who inspired them,

they were their idols.

One was Albert Einstein,
one was Bertrand Russell.

And these were the most
prominent scientists of the day.

Their interest shifted
almost imperceptibly at first

from the foundations of physics

to the foundations of
mathematics and to logic.

It came almost against their will

that this became the most prominent topic

of the Vienna Circle.

Once every two weeks they would meet here,

in this actual room.

It's now a working physics lab

but, when they met here, they had one aim

and that was to purge philosophy

of anything that was
neither directly observable

through scientific experiment,

or derivable through the laws of logic.

This logical analysis of the meaning

was an essential first step.

Therefore, it was forbidden to talk about

such concepts like God, for instance,

or metaphysical statements

about thinking itself or whatever,

because you could never find a sentence

that could be verified
in a scientific way.

- In fact, the Vienna
Circle loathed the idea

of metaphysics so much
that when they met here,

Rudolf Carnap, a former pupil of Frege,

appointed someone to shout M!
- M!

- During their discussions,

at the hint of any illegitimate sentence.

M stands for metaphysics.

- M!

- It's the logician's equivalent
of saying, "Bollocks!"

Now the thing is, he
was saying "M!" so much

that they got sick of it.

Instead, they had him shout "Non-M"

any time that someone
actually said something

that was legitimate.

- Nicht M!

- Despite the purity of
their logical methods,

the problem of uncertainty
that had plagued logic,

likewise stalked the Vienna Circle.

- Something that may have also imprinted

this young generation
of Austrian scientists

was a scandal that happened in 1913

when it was discovered
that the head, practically,

of the Counter Espionage
Service was a spy.

And, you see, the task
of a counter-spy service

is actually to make sure that
there are no spies around.

But what happens when the
head of that organization

is a spy himself?

This is a fundamental uncertainty.

Yes, yes, the secret
service can work very well,

but can you be sure that the
secret service is not infected?

And something similar is
happening in mathematics.

You make sure that there
exists no contradictions,

you build up big walls
against uncertainty or so,

but maybe, within these big walls,

there is a contradiction sitting.

- Contradiction bothered one
man more than most, Kurt Godel.

Kurt Godel was the most reclusive member

of the Vienna Circle.

He'd had the finest logical
training that you could imagine.

It was in one of Vienna's
famed coffee houses,

in August 1930,

that 24-year-old Godel
first revealed a discovery

that would end, for
ever, the logical quest

that Frege, Russell and the
like had set themselves.

Godel was one of the few
who definitely had read

all of Russell's Principia.

He knew that, for any logical system

to be the foundation of mathematics,

it had to be both complete and consistent.

Godel told Carnap that,
by studying the Principia,

he had come to the conclusion
that, in any logical system,

you could either be
consistent or complete,

but you couldn't have
both at the same time.

In Russell's masterpiece,

Godel had discovered a contradiction

that became known as incompleteness.

This means that, in mathematical logic,

there are gonna be some
truths which, although true,

can never be proven to be so.

- This result of Kurt Godel

about the limitations of
mathematics and logics

was a terrible blow to the
optimism of the Vienna Circle,

and some of the members took a long time

to come to grips with it.

- The grand search for
absolute, provable truth

had hit the buffers.

By the mid-1930s, the
Vienna Circle was over.

The rise of fascism and
the looming threat of war

meant its members fled,

were expelled, or killed.

Kurt Godel left Vienna for Princeton,

where his own search for certainty

also came to a tragic end.

Godel became convinced that
someone might try to poison him.

The only person that
he would trust to cook

and, indeed, to taste
his food was his wife.

And when she fell ill and
was hospitalized, he starved.

He literally reasoned himself to death.

The fact that all systems
of mathematical logic

were limited, that we could
never have complete certainty,

signaled the end of an era for logic.

But for one British logician, Alan Turing,

Godel's work was the
inspiration he needed to launch,

inadvertently, a new

and entirely more
practical logic revolution.

Alan Turing was just 23 years old

when he imagined something extraordinary.

He called it a universal machine.

The universal machine is
an entirely imaginary,

hypothetical device, and yet,

it's one of the most
influential machines ever

in human history.

The device Turing imagined could tackle

any mathematical problem
using a logical algorithm

encoded in its own limitless memory.

In 1936, Alan Turing published a paper

in which he demonstrated,

he proved that you
couldn't decide beforehand

which mathematical problems

the machine would be able to solve,

and which would just cause it to run

on and on and on for ever.

That there are some problems
that are simply uncomputable

was startling, and yet
another blow for mathematics.

But it was also the beginning

of something entirely unexpected

and destined to cement logic's
role in the modern world.

It's an extraordinary,
almost exquisite, paradox

that, in demonstrating that
some things can't be proved

using a logical machine,
what Alan Turing did

almost single-handedly launched
a technology revolution.

Turing's universal machine

is what we today call the computer.

While stationed here at Bletchley Park,

during the Second World War,

Turing began to implement
his abstract ideas

as real logical hardware.

Working with Gordon Welchman,

Alan Turing developed this
machine, it's called the Bombe.

It's a bit loud!

It's a form of electromechanical computer,

and its logical function
was to decode the messages

that the Germans were sending,

using their Enigma encryption machines.

But then Turing's colleague,

Tommy Flowers, went a step further.

This is Colossus.

It was built to crack another
German encryption machine

called the Lorenz,

and, for the men and women
who built and operated it,

it was an astonishing achievement.

It shortened the war.

But I think it's special
for another reason.

You see, this is the world's first

programmable electronic computer.

It used digital information, binary,

the streams of ones and zeroes

that are in all modern computers.

And these vacuum tubes down here,

they're wired together to
be our Boolean logic gates,

which perform Boolean
operations and calculations.

Colossus might not look hi tech to us,

but it's hard to express
just how important it was.

This significance of all this,

as a piece of human engineering,

is on a par with the Pyramids,

or the printing press or steam power,

and yet it was all top secret.

All these developments of
electronic programmable computers

here at Bletchley Park were classified

and the details were only
declassified in the late 1970s.

After the war, Turing
went on to help build

some of the world's first
stored-program computers.

At their core, it all comes
back to logical reasoning.

Think about this, we're
all surrounded by things

that rely on some kind of
logical machine or code.

The failure of logic

to deliver foundational
answers for mathematics

nonetheless gave rise

to one of the most
significant achievements

in all of science and engineering.

It started with those huge,

secret, single-purpose computers,

and yet, right from the very beginning,

some folk were already
imagining the next big thing.

- [Man] We're still finding
out what Logics will do,

but everybody's got 'em.

You got a Logic in your house.

It looks like a vision receiver used to,

only it's got keys instead of dials

and you punch the keys
for what you want to get.

It's hooked into the tank, which has--

- [Narrator] In 1946,
science fiction writer

Murray Leinster imagined
an impressive specimen

of interconnected technology.

He named it a Logic.

- [Man] Relays in the tank take over

and whatever vision-program
SNAFU is telecasting

comes on your Logic's screen.

Or you punch Sally Hancock's Phone

and you're hooked up with
the Logic in her house.

Also, it does math for
you, and keeps books,

and acts as consulting
chemist, physicist, astronomer

and tea-leaf reader, with an
Advice To Lovelorn thrown in.

It's very convenient.

- Well, that's extraordinary!

It's a great characterization

of the web that wasn't yet born!

- The digital world we live in,

the computers that surround us,

at their base, are running Boolean logic.

I mean, they're running
actually electrical currents,

ones and zeroes are the product
of those electrical currents

but on top of that there are layers

on layers on layers of complexity,

operating systems, machine code,

applications that we use every day,

from word processors to spreadsheets,

to the browsers we use.

And, when you have your Skype conversation

with your aunt in Australia,

you don't think of that interaction

in terms of those ones and
zeroes but, without them,

without the underlying processing,
none of this would work.

- [Narrator] Not only did Logic launch

the digital revolution,

but it's also the tool we use to sort,

search and retrieve the
information we want online.

- The World Wide Web we have
today represents the largest

information construct
humanity has ever created.

It's 20 years old, barely,

and yet we have billions
and billions of pages

encapsulating knowledge and information

from all of human culture
and all of human history.

The challenge is to organize
this mass of information,

this complexity,

and logic gives us some of
the perfect tools to do that.

- With the World Wide Web of information,

logic means we're all more
interconnected and informed.

But, back in the City, the
march of logical machines

has come at a cost, and I
don't mean all the traders

are spending too much time on Facebook.

In the year that I was born,

there were 22 separate
stock exchanges in the UK,

and this is how business was done.

Now, this place, the
London Metal Exchange,

is the last venue where
traders still go face to face.

First, technology squeezed
out the need for traders

to meet in person.

And now it's the traders themselves

who may be heading for extinction.

Not long after I wrote it,

IBM did some tests of the
ZIP trading algorithm,

and not only did they
confirm that it worked,

they showed that it
out-performed human traders.

When it comes to pure logical reasoning,

the computers tend to beat us, hands down.

It's an old adage, but
people in this business joke

that soon the only things
you'll find on a trading floor

will be a big computer, a man and a dog.

The big computer is there
to do all the trading.

The dog's there to make sure

that no-one touches the computer.

And the man's job?

On the trading floor of the future,

the man's job is to feed the dog.

Mind you, despite my role in
inventing these black boxes,

I'm grateful that there's
still a human around

to pull the plug sometimes.

The thing is, computers still need

their logical algorithms
to be written for them,

so they might take our jobs,

but we still have the upper hand.

Yet, ever since their invention,

the question as to whether
this will always be the case

has been a matter of fierce debate.

When the digital revolution
was in its infancy,

the possibility of computers

developing human-like intelligence

was the hottest topic in town.

Could a machine ever think,
using the rules of logic alone?

Or is there more to us than that?

In 1950, Alan Turing published
another visionary essay.

In it, he predicted that,
by the end of the century,

a computer would be able
to converse with a human,

and the human wouldn't
know the difference.

In trying to achieve this,

people in my field have created

some truly amazing computing machines.

This is my university's supercomputer.

Although it's bigger and
noisier than Colossus,

for every one Lorenz cipher
that machine could solve,

this can solve over two million.

It's takes up the whole room!

Machines like this are the workhorses

of today's data-centric research.

All the switches, wires and logic gates

have long since disappeared under the hood

meaning that, for TV, we have
a habit of trying to pretend

that this doesn't all look like a load of,

well, cupboards.

Or a launderette.

Turing thought that,

by the time we'd developed
computers as powerful as this,

we would also be capable
of programming a machine

with sufficient rules of logical reasoning

that its intelligence would
rival that of us humans.

That was then, and remains
now, a very controversial idea.

- We like to think of our
intelligence as raising us

to a level above the rest of the creation.

We associate it with the idea
perhaps of an immaterial soul,

being not just one amongst
other animals, but special.

And what Turing was suggesting
was that this special quality

could belong to a lump
of computing machinery,

and it could reason just
as well as we could,

maybe even better.

- At Bletchley Park,

Turing had sketched out
algorithms for playing chess.

At that time, the chessboard was dominated

by some of the world's
most brilliant strategic,

logical, mathematical brains.

And so it became the battle ground

for an entirely new challenge for logic,

artificial intelligence.

In 1997, the most famous public battle

between man and machine took place.

Garry Kasparov, the reigning
chess world champion,

had previously trounced

IBM's chess-playing computer, Deep Blue.

During their rematch,

for the first time ever, he was beaten.

Kasparov has resigned!

(audience applauding)

When I see something that is
well beyond my understanding,

I'm scared.

And that was something well
beyond my understanding.

It was front-page news the world over.

People demanded answers.

Was this purely logical
intelligence equivalent,

or even superior, to the human brain?

- In the past, people have
tended to compare humans

to the latest technology.

So maybe the brain is like a clock,

or maybe it's like a steam engine,

now, maybe it's like
an electronic computer.

What Turing would want to
say, and, I think, correctly,

is that there's something different

about the equation of the
brain with a computer.

- [Narrator] He put it that
both a brain and a computer

are information processing systems,

governed by logical rules.

- In theory, there should
be logical rules out there

that would capture the way we think.

- This was a very big idea, with profound,

even troubling, implications.

If we knew those rules,
then one day, theoretically,

we could code a logical rendering

of ourselves into a computer.

All we'd need to reproduce
all of human thought is logic.

My view is that there remain
uniquely human characteristics,

arguably the best ones, like
altruism or creativity or love,

that computers aren't even
close to having programmed

within their repertoire
of logical reasoning.

No-one has yet created a logical machine

that's just like us.

And, arguably, that could
take a very, very long time,

if indeed it's possible at all.

And yet, surely, we should marvel

at what we have achieved with logic.

Remember we created the rules
of logic to pin down the truth

and certainty that would
otherwise so easily evade us.

We harnessed logic in machines

and, in doing so, we placed
the power of pure reason

at our fingertips.

Mind you, I'm still no good at Sudoku.

- [Announcer] One, two,
one, two, three, four.

(electronic music)