The Joy of Logic

The world we live in can seem pretty illogical.

The things people say,

the ways we behave,

the complex choices we have to make.

HE SHOUTS

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?

ALL CHEER

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!

It's given us transformational technologies...

Delta 11, report your entry point.

..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.

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.

CLAMOUR

It was a computer programme 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.

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.

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.

-LAUGHTER

OK, 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 analyse that joke to death.

TAPE SCROLLING

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 1 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 2.

Then, happily for Logicians 1 and 2,

Logician 3 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 3 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 organise 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!

HE SIGHS

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.

Then pair it with a second one...

I am a man.

Then we figure out, or infer,

that, alas, Socrates is mortal

That makes me sad.

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.

He is suffering from politicians' logic!

This is just one of Aristotle's fallacies.

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

but the way they're organised

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.

LAUGHTER

BELLS RING

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 enamoured 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.

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!

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.

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 popularise

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".

DOOR SLAMS

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 honour.

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 1 or a 0.

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

"Are you fluffy?" AND "Do you bark?"

she would have to bark, "Yes!"

RUFF!

Taking 1 to mean "yes" and 0 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 0s and 1s,

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 1s and 0s.

I'm going to 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,

1s and 0s, through a circuit, like the one 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 going to 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 going to do

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

Who would like to be bits of a computer?

ALL CHEER

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

This one is a number 1. 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 1s and 0s

'by either tagging the next kid in line for a 1 -

'or not tagging them for 0.'

CHEERING

'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 1 to pass on

'if they're tagged on both shoulders.'

So, some of you are being OR gates.

'ORs pass on a 1 if they're tagged on one or both shoulders.'

Some of you are being NOT gates.

'NOTs are different.

'They get a 1 to pass on if they're not tagged.'

Numbers - you are the most important thing,

cos the whole circuit is about processing numbers.

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

'which arranged like this, represent 2 and 3.'

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 1s.

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 1, and 0 and a 1.

4 and 1, and that makes 5.

And the numbers we added at the start were a 3 and a 2.

So, a 3 and a 2 moving through this circuit,

with all of you just doing very simple things,

being AND or OR or NOT, ended up a 5 this end,

so you have calculated the right number!

WILD CHEERING

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.

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.

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.

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.

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 centre, in Swanwick, south-east 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.

Landing over 2 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 signalling, 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.

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 ten words,

that argument would be pretty neat.

'The competition has finished. Thank you very much for your patience.'

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 going to be the future of all things logical.

'The first-place winner of IOI is...

'Lijie Chen from China.'

CHEERING AND APPLAUSE

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 1 + 2 is 3.

But, without proof, that they will never lead to a contradiction,

you can never say for sure that 1 + 2 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... #

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 analysed

using his new logical quantifiers.

# Everybody loves somebody sometime... #

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 1 + 1 = 2.

CHEERING, APPLAUSE, FIREWORKS POP

It was never going to a best-seller,

but, here, it had a huge impact.

It was magnificent, a whopping great bucketload 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,

centred 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...

M!

..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 organisation 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 going to 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 hospitalised, 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,

signalled 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,

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

THE BOMBE WHIRRS AND CLICKS

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 1s and 0s 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.

'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...'

In 1946, science fiction writer Murray Leinster imagined

an impressive specimen of interconnected technology.

He named it a Logic.

'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 characterisation 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,

1s and 0s 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 1s and 0s but, without them,

without the underlying processing, none of this would work.

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 organise 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.

DOG BARKS

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 2 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!

APPLAUSE

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.

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 otherwise would 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...

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