The Joy of Logic (2013) Movie Script
1
(upbeat music)
(horns honking)
The world we live in can seem pretty illogical.
The things people say,
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 pro to-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)
(upbeat music)
(horns honking)
The world we live in can seem pretty illogical.
The things people say,
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 pro to-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)