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00:00the world we live in can seem pretty illogical the things people say the ways we behave the
00:12complex choices we have to make what's the quickest way to get home can I trust any of
00:21you where did you all come from that process of making sense of all this stuff of sorting
00:30between the truth and the nonsense comes down to one of the most simple and yet powerful tools
00:36ever created by humans logic yes yes there is definitely beauty and logic who would like to
00:46be bits of a computer in the building next door to me at work there's a door and there's a sign on it
00:53that says this door must be kept closed at all times let's just look at this in amazement really
01:00why did you build a door then is this sentence true or false philosophy maths science and language
01:08logic is the engine for all of them in fact it drives the fundamental process of reasoning itself
01:15I'm a professor of computer science computer scientists tend to think that logic is the
01:21bee's knees so it follows that I think logic is brilliant logic has inspired our greatest boffins
01:28it's given us transformational technologies and even made us question what it means to be human
01:39I want to see if there's any limit to what logic can do for us so join me it would be terribly logical
01:47not to
01:48logic is right at the heart of what I do around 15 years ago kind of by accident I created something
02:06that had a really big impact here on the trading floors of the city of London it was a computer
02:17program I called zip and it used logic to replicate this a centuries-old tradition of human traders
02:24supposedly vested with very special skills crammed into rooms shouting at each other
02:29it's ever so simple just a few logical inferences decisions and a little bit of maths it learns from
02:39its trading successes and failures its aim is to trade as profitably as possible in a fast-moving
02:45market where levels of supply and demand are shifting rapidly it turns out the zip built squarely on
02:55logic was impressively proficient at this trading lock in fact today in many markets billions or
03:03trillions of dollars worth of deals go through with no human intervention at all which is kind of mind-boggling
03:08every 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
03:21and every day scientists like me earn a living using logic to find solutions to all kinds of other
03:29real-world challenges so why am I not as rich as Bill Gates well I gave away the zip software for free and
03:38and looking back that was probably not my most logical move
03:46so what is logic what does being logical even mean
03:52a pint of lager please well all you need to explain it are three logicians and a boozer
03:58logic is actually all about the rules of correct reasoning let me tell you a joke three logicians walk
04:08into a bar the barman says gents would you three like a beer and the first logician says I don't know
04:16the second logician says I don't know and then the third logician says yes yes we would all like a beer
04:26okay so it's not exactly a side-splitting laugh-out-loud gag more of a chalking for nerds
04:33but what went on there well forgive me I'm going to analyze that joke to death
04:38remember the barman's question was would all three of you like a beer the key here is the all three bit
04:49if any one of those logicians doesn't want a beer then he'd be able to answer no that's because if one
04:56doesn't want a beer they don't all want one logician one does want a beer but he can't speak for the
05:02others so he has to say I don't know exactly the same goes for logician two then happily for logicians
05:11one and two magician three also wants a beer and so he correctly uses logical inference to arrive at
05:18the right answer to the question yes yes we would all like a beer at last logician three ends the
05:26torment because he can speak for everyone cheers to that the important thing to understand is that
05:35logic isn't knowledge logic doesn't create knowledge what it does is it gives us cast iron rules for
05:41how to organize and handle knowledge even so the quality of the conclusions you get out depends on
05:49the quality of the ideas that you put in time please gents it would be a funny old world if we
06:04followed the rules of logic all the time these days logic is studied and taught in academic
06:26institutions the world over its history stretches back two and a half thousand years to the age of the
06:32greek philosopher aristotle he created the first formal rules of logic that would govern good
06:39reasoning clear thought and reliable argument aristotle's most famous logical tool is the syllogism
06:47a syllogism is is a certain simple kind of argument consisting of three propositions and the
06:54first two propositions are the premises the things that we take for granted in the argument
06:59so for example all men are mortal socrates is a man those are our two premises and from them we draw the
07:20conclusion socrates is mortal aristotle's example is good logical reasoning first we take one premise or
07:29thing we know all men are mortal yes indeed i'm socrates then pair it with a second one i am a man
07:39then we figure out or infer that alas socrates is mortal that makes me sad
07:47if your premises are reliable and you follow aristotle's rules you get answers that are reliable too
07:55but aristotle's theory of the syllogism can deal with more complicated arguments that don't just have
08:04all in them but some in them and knots
08:08take all these into account and you find there are lots of ways to make a syllogism
08:16so if you multiply that up you find that there are 256 kinds of syllogism and aristotle identified 19 of
08:28these as being logically valid so that if the premises are true the conclusion has to be true as well
08:35and all the others of those 256 forms you can have true premises but a false conclusion so arguing in
08:44that way is fallacious those kinds of syllogisms are fallacies it's the old logical fallacy all cats
08:52have four legs my dog has four legs therefore my dog is a cat he's suffering from politicians logic
09:03this is just one of aristotle's fallacies it looks similar to good logic the premises are both true
09:09but the way they're organized means the reasoning is completely backwards and the conclusion bonkers
09:16something must be done this is something therefore we must do it but doing the wrong thing is worse
09:22than doing nothing doing anything is worse than doing nothing
09:31such was the power of aristotle's logic that scholars used and taught it but actually didn't do a great deal to
09:37change it for the next two thousand years but it wasn't just philosophers that were enamored of logic
09:45by the 19th century the public had fallen for it too for this our thanks must chiefly go to a
09:52mathematician who spent most of his life working at christ church in oxford charles dodson he's much
09:59better known by his pen name lewis carroll the mathematics books were mainly under his real name
10:07charles lutwitch dodson but he chose to use his pen name lewis carroll for the game of logic and symbolic
10:15logic clearly to give it a wider audience
10:19and the mad tea party the march hare says you must say what you mean and alice replies
10:49well i mean what i say it's the same thing you know hatter says you might as well say that i see what
10:58i eat is the same as i eat what i see dodson was so keen to introduce people to the delights of logic
11:10that he drafted a book initially called logic for ladies he was very conscious the girls in particular
11:18were not heard they were not given the chance to go to school very few had the opportunity of going to
11:24university they certainly weren't able to get a degree happily for us blokes dodson had a change of heart
11:33and logic for ladies was renamed symbolic logic together with the game of logic it did surprisingly
11:40well he felt that young people needed a tool to detect fallacious arguments that they might meet
11:48in books and magazines he wanted them to have the ability to detect that
11:55into a Bitches change of their own things that they Painter
12:10intrinsically anti-semitic I think Dodson would have wholeheartedly
12:22approved of today's most popular logic game Sudoku there's something really
12:27captivating about the fact that logic tells you that the answer must be in
12:31there but you need to apply logical reasoning to find it it can be really
12:35engaging but it can also be really frustrating and annoying too Charles
12:44Dodson had been the first person to popularize the idea of logical reasoning
12:48and critical thinking but for all its growing popularity logic itself was due
12:56for an upgrade in 1847 this groundbreaking book was published it's called the
13:04mathematical analysis of logic now this isn't logic for philosophers or puzzle
13:08fans the author of this book argues that the real purpose of logic is mathematics
13:13and this book was written by George bool born into a poor family in Lincoln
13:21rule mastered mathematics at a fantastically young age and by 20 he'd opened his own
13:28school balls big idea was that logic was actually closer to mathematics than
13:35philosophy all you needed to do was change the words in a logical argument to
13:39symbols and then it could be solved just like an equation he called it his calculus
13:45of reasoning first he demonstrated that the letters that we use in algebra to represent numbers can
13:56actually be used to represent whole classes of things in the real world so for instance we
14:01might have the class X of things that are fluffy and the class Y of things that bark second he
14:12introduced a set of operators for combining these classes of things the three most important ones
14:17are and or not and they're known as boolean operators in his honor so if we redraw our classes so they
14:24overlap the bit in the middle that's things that are fluffy and bark X and Y if we look at the
14:32whole area of the two circles well that's things that are either fluffy or they bark so that's X or Y and
14:38finally if we think about the area outside well they're neither fluffy nor barking so that's not
14:44X and not Y things that aren't fluffy and don't bark like me balls new mathematical logic reduces any
14:51logical problem to symbols that can be combined in new ways and there was one final and crucial
14:57innovation in balls new mathematical logic everything's either in or out statements are either true or false
15:07everything's either a one or zero for example if I were to ask my dog floss are you fluffy and do
15:16you bark she would have to bark yes taking one to mean yes and zero to mean no with ball we get this
15:25it was an entirely new form of logical reasoning seemingly anything could be boiled down to symbols
15:32and just two numbers and it's in my field that balls vision would prove transformative almost a
15:40century after his death his logic would become the language of computing my logical hero has to be George
15:47Boole Boolean logic is so simple yet so fundamental to explaining our world and even the world today which
15:56is full of complex systems that he could never have imagined and Boolean logic allows us to reason about
16:03them
16:03I think the application area and the use of logics has changed dramatically in the last 20-30 years with the
16:24advent of computer science and software systems because fundamentally these systems are about zeros and ones
16:31entities that map onto truth and falsity and what I think is just absolutely brilliant is that we go back to lots of
16:39the logical ideas invented and conceived over a hundred years ago before anyone imagined the systems that they'd be
16:47applicable to ball never knew it but thanks to him all computers today process their information as binary digits or bits
16:54with binary any number can be represented by combinations of ones and zeros 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 or subtract or
17:14multiply or multiply or divide or to compare to each other in fact any time you see a computer doing anything
17:21whether 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
17:35Boole's logic can be used for computing at their simplest computers work by passing bits of information ones and zeros through a circuit like the one that we're building here the most important parts of 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 that we could never do in our own
17:42heads they can all be worked out with something like this like this I'm going to use these guys to一些
17:49are the junctions where the bits of information are combined and passed on. These are called
17:56Boolean logic gates, and the way you order them determines exactly what the circuit can
18:01do. From simple addition to calculations that we could never do in our own heads, they can
18:07all be worked out with something like this. I'm going to use these guys and some very
18:14simple logic gates, and, not, and or, and a circuit that we've got out there in the school
18:21hall. And what this circuit's going to do is to add together two numbers to come up with
18:27one answer. Who would like to be bits of a computer? Come on up, and I'll give you out
18:36your shirts, okay? This one is a number one, which is for Ishmael. Where's Ishmael? They're
18:43not just pretending, they will be a computer. Charlie T, thank you very much for being
18:48an and gate. Normally, of course, computers work on electric currents. Our computer will
18:54be powered by kids, who will pass on their ones and zeros by either tagging the next kid
18:59in line for a one, or not tagging them for zero. It's time for the kids to take their places
19:07in our circuit. And, for the record, I've never tried this before. Okay, some of you
19:14are being and gates. Do you remember what an and gate has to do? The rule for ands is they
19:19only get a one to pass on if they're tagged on both shoulders. So now some of you are
19:24being or gates. Ors pass on a one if they're tagged on one or both shoulders. Some of you
19:31are being not gates. Nots are different. They get a one to pass on if they're not tagged. Numbers,
19:38you are the most important thing, because the whole circuit is about processing numbers. We're
19:42going to put these four bits into the circuit, which arranged like this represent two and three.
19:48Off you go. The bits of information have been inputted. They're relayed on by the first set
19:56of kids. If they're following their rules, only some should be carrying ones, while others won't.
20:04At each gate, the bits are combined and passed on. They're nearly there. At last,
20:10the output numbers are either tagged or not. So we've got a one, a zero and a one. A four,
20:18and a one, and that makes five. And the numbers we added at the start were a three and a two. So
20:25a three and a two moving through this circuit, with all of you just doing very simple things,
20:29being and or or not, ended up with a five this end. So you have calculated the right number.
20:41Today, all our computers are built using Boole's logic gates. Here we have 13. But a modern computer
20:48chip like this one might have 250 million. They're all doing exactly what these guys were doing,
20:53but an awful lot faster. We just did a simple sum here, but Boole heralded a new era for logic,
21:02in which reasoning about anything could be done in the language of maths.
21:06There are lots of different logics, because there's lots of different kinds of systems
21:12or worlds that we want to reason about. I've been applying logic to reason about a wide variety
21:19of complex systems. I've looked at communications for air traffic control systems, molecular biology.
21:27I've also looked at advanced telephony. But regardless of the application, all logics have
21:34one thing in common. Amongst all these logics, the unifying property is they're about axioms
21:41and rules. So the answer is unambiguous. We can automate the procedure of computing the answer
21:49in logics. But we still need to pose the question. Taking exactly those questions and automating the
21:59way we logically answer them requires what's known as an algorithm. It's the province of my very own
22:05breed of nerd, the computer programmer. And there's nowhere more important for today's generation of up
22:13and coming young programmers than this, the annual International Olympiad of Informatics, held this
22:20year in Brisbane, Australia. We're trying to find the best and the smartest students when it comes to
22:26computational thinking, algorithms and programming. On each competition day, everyone is set three
22:33questions, which must be answered within five hours. The easiest one, you just had a bunch of locked
22:40doors. And you had a bunch of switches. Each of the switches was connected to one of the doors. But you
22:47didn't know which switch was connected to which door. And what they asked for is to determine for each
22:54switch, which door it's connected to, and which position is the correct one.
22:59Johnny Ho is last year's champion. So there's a lot to live up to. But things aren't quite going his way.
23:05By now, I've actually solved all three, but I didn't actually solve them during the contest,
23:11because there's just a lot of pressure.
23:13We test the ability of students to come up with clever algorithms to solve algorithmic problems.
23:20They not only have to come up with the algorithm, but they have to write a computer program
23:25that runs the algorithm.
23:26Algorithms turn real-world problems into questions that logic can help us answer.
23:35If, for example, these guys wanted to spend their day off competition duties defining the group of
23:40all animals in a zoo that are marsupials, the first step of the algorithm could be to ask,
23:46of all the animals I see, which would I find in the wild in Australia?
23:53No. No. No. No. Yes. No. I dunno. Yes. Yes. Yes. Definitely not. Yes.
24:09Certainly not all of the yeses and don't knows will be marsupials, so the list can then be refined by
24:14asking which of these animals have pouches. And here there are options too. They could look in a book.
24:21They could ask Chris. He's an expert. Or they could crowdsource the question and go for the most
24:26popular answer. Each logical algorithm incurs a different cost in effort, time or accuracy.
24:34But, whichever way, they'll each get to an answer eventually.
24:37And there are certain situations where a good logical algorithm can be the difference
24:46between life and death.
24:47This is the Nats Control Centre in Swanwick, South East England. At any one time,
24:59around 100 air traffic controllers are responsible for 200,000 square miles of airspace over the UK.
25:06Delta 1, wonderful to ocean entry point.
25:09Landing over two million flights a year, it's perhaps surprising that until very recently,
25:15these folk did their job using brain power alone. But that's all changing. New automated
25:23algorithms have started to take on some of that responsibility for guiding the planes in our skies.
25:28The equipment now is talking to the aircraft. And so, whereas before, the human was reacting with
25:36the human, and obviously there were sometimes mistakes made, the computers can now double-check
25:41that interaction and provide a warning to the controller if anything is amiss. Equally, in terms
25:46of capacity, because it's reduced the amount of workload for the controller, we've seen capacity
25:50about 40% increase on some sectors, because the computers are doing some of the logical
25:55calculations and thinking on behalf of the human being.
26:00I think logics are really crucial as a tool for reasoning about the systems we use in our modern
26:07world. We are surrounded by these complex systems like air traffic control, railway signalling,
26:13the electricity grid.
26:14I think it's really important that we raise the next generation of users of these systems so that
26:27they know it's not magic. They also know that they have the tools of logic to understand and reason
26:33about the systems that they depend on, crucially, every single day of their lives.
26:37Back at the International Olympiad of Informatics, it's day two of the contest. The judges are looking
26:48for programmes to do logic that aren't just right, they have to be fast.
26:54So if you have an algorithm that is technically correct but it would take 100 million years to run,
26:58then you would score no points. If you have an algorithm that solves the same problem and runs in,
27:03say, five seconds, then you can score much higher points.
27:08I think the simpler an argument is, the more beautiful it is. So if it can be expressed in
27:14perhaps just 10 words, that argument would be pretty neat.
27:20Competition has finished. Thank you very much for your patience.
27:27It's an anxious wait for the final ranking.
27:30I think this competition is, in all its geeky glory, an amazing event.
27:39With the ability to implement their problem-solving talents in the language of computing,
27:43these kids are going to be the future of all things logical.
27:48First place winner of I.O.I. is Lijia Chen from China.
27:54In the end, it's a Chinese one, two, three.
28:09It's lucky the Brisbane competitors didn't have this problem to solve.
28:13It's one that no logical algorithm can cope with.
28:16All I want to know is, what do you think? Is this sentence true or false? Is it true or false?
28:23You can have this if it's false.
28:26The point is, if the sentence is false, then it's true.
28:30But if it's true, then it must be false.
28:34It's a paradox.
28:36But if it's false, it's true.
28:37My sign is inspired by the first known logical paradox from around 600 BC by the Cretan Epimenides of Knossos.
28:45Well, if you read the sentence that this sentence is false as its true meaning, then yes, it is false.
28:52Epimenides wrote, all Cretans are liars.
28:56But he was a Cretan, so was he lying?
28:58If so, then all Cretans aren't liars, in which case he would be telling the truth.
29:03It's a genius statement, isn't it?
29:07A paradox? Well done.
29:10Paradoxes are fundamental contradictions that logicians have puzzled over for centuries.
29:15They've been described as truth standing on her head to get attention, and for good reason.
29:23In the late 19th century, round about the same time that George Ball was developing
29:27logic or deduction as a branch of mathematics, paradoxes exactly like this
29:32became a really deadly serious matter.
29:35In fact, they came to threaten the very foundation of mathematics itself.
29:50The Austrian capital, Vienna, renowned for its music, elegance, legendary cafes,
29:57and exquisite cakes.
30:00But at the turn of the 20th century, it was also the place to be if you were interested in logic.
30:09Despite its grace and gentility, Vienna can lay justifiable claim,
30:13perhaps more than any other city, to being the birthplace of the modern.
30:17For it was here, in art, design, philosophy, science and psychology,
30:22that people most boldly challenged the tired conventions and assumptions of the 19th century.
30:31But what was modern?
30:32Was it about replacing religion and tradition with logical empiricism and pure reason?
30:38Or was it about admitting to a new uncertainty,
30:41the limits of our perceptions and the moral vacuum of the Freudian subconscious?
30:47Until this point, it could be argued that logic wasn't exactly a topic on everybody's mind.
30:53But here, it was at the forefront of this titanic clash.
30:57From the city's coffee houses to the University of Vienna itself, the struggle for modernity played out.
31:07In 1894, the university commissioned a great ceiling painting for their ceremonial hall.
31:12The theme was the victory of light over darkness, and it had separate panels celebrating the great
31:18achievements of the university's faculties of jurisprudence, of medicine, and of philosophy.
31:23Given the subject matter, it was perhaps unfortunate that the artist that they
31:28commissioned for these paintings was Gustav Klimt.
31:36In 1900, he presented them with philosophy, a depiction of naked men and women drifting
31:41trance-like in empty voids. It expressed anything but victory, certainty or optimism.
31:48Klimt's proto-modernist vision of philosophy was shocking to the people of Vienna,
31:55and deeply unsettling to the professors at the university.
32:01He was attacking everything that they stood for, and Klimt's paintings were rejected outright.
32:11Hidden away for 40 years, the original works were destroyed by the Nazis.
32:15These replicas were finally installed on the centenary of their ejection.
32:23Klimt's dark vision had seriously offended the growing academic aspiration that science
32:28and mathematics would provide us with complete knowledge, founded on absolute, provable truth.
32:36This was something it was hoped logic could provide.
32:39In mathematics, this problem of definitive truth, of certainty, had recently become all too real.
32:47No one had yet proven the most basic rules of mathematics.
32:52Those rules might say that 1 plus 2 equals 3, but without proof that they will never lead to a contradiction,
32:58you can never say for sure that 1 plus 2 might not also equal 20, or anything else for that matter.
33:07In the grip of uncertainty, a logic fever took hold.
33:13Ball's logic had already been adopted by the greatest logicians of the day, but there was a problem.
33:18His method was simply insufficient to describe all of maths.
33:24The race was on for a new and more complex logic.
33:31Over 20 years earlier, a German mathematician called Gottlob Frege had studied exactly this problem,
33:36and Frege's work ensured that logic was up to this search for certainty, which was unfolding right here.
33:42It was in Jena, Germany, in the late 19th century, that Gottlob Frege opened a new chapter in the story of logic.
33:58For him, there should be nothing, whether numbers or ideas, that could not be described and analysed using his new logical quantifiers.
34:06So, with his new mathematical logic, he could express ideas like everybody loves Frege, everybody loves somebody,
34:20there is somebody whom everybody loves, there is somebody whom no-one loves,
34:28and, alas, there is somebody whom Frege does not love.
34:37That somebody who Frege probably did not love was British philosopher Bertrand Russell,
34:42who, independently, was engaged in pretty much exactly the same project,
34:46using logic to firm up the foundations of mathematics.
34:50In 1902, Frege was just days from publishing the second volume of his magnum opus on logic,
34:57when he received a letter from Russell, and it was the kind of letter that any logician dreads receiving.
35:03Russell had spotted a big problem.
35:08Both men's logic relied on consistently describing sets of things.
35:12You can have the set of all even numbers, or, for that matter, the set of all mums,
35:19or the set of all dogs.
35:22Almost all sets aren't members of themselves, the set of dogs isn't itself a dog.
35:28So if you take the dog set and bundle it up together with all the other ones like it,
35:34you get the set containing all sets that are not members of themselves.
35:39But this is the set of all sets that don't contain themselves,
35:43and it doesn't contain itself, so this set should include itself.
35:49But then, if it does, then this is no longer the set of all sets that don't contain themselves,
35:54so it can't be part of itself.
36:00It's one of those logical paradoxes.
36:03Frege immediately wrote back to Russell.
36:06Dear colleague, your discovery of the contradiction has surprised me beyond words,
36:10and, I should almost like to say, left me thunderstruck,
36:13because it has rocked the ground on which I meant to build arithmetic.
36:18Your discovery is, at any rate, a very remarkable one,
36:21and it may perhaps lead to a great advance in logic, undesirable as it may seem at first sight.
36:29Russell now took on Frege's project with an even greater zeal,
36:33to develop an even more outrageously complex logic
36:37that would get round this problem with sets, and so be free of paradox.
36:41After nine years of toil, the monumental Principia Mathematica was published.
36:48It took over 360 pages to logically prove that one plus one equals two.
36:57It was never going to be a bestseller, but here it had a huge impact.
37:04It was magnificent.
37:05A whopping great bucketload of logical concrete poured right into the foundations of mathematics.
37:11Definitely a triumph, not a trauma, for philosophy.
37:21But the final word on logic would not come from Bertrand Russell.
37:30It was here that that project came to a dramatic conclusion,
37:34centred on a group of thinkers called the Vienna Circle.
37:41They were firmly pro-logic.
37:44For them, Russell's Principia Mathematica was manna from heaven.
37:49The Vienna Circle had people who inspired them.
37:52They were their idols.
37:53One was Albert Einstein.
37:56One was Bertrand Russell.
37:58And these were the most prominent scientists of the day.
38:02The interest shifted almost imperceptibly at first from the foundations of physics to the foundations of
38:23mathematics and to logic.
38:25It came almost against their will that this became the most prominent topic of the Vienna Circle.
38:37Once every two weeks, they would meet here in this actual room.
38:42It's now a working physics lab.
38:50But when they met here, they had one aim.
38:53And that was to purge philosophy of anything that was neither directly observable through scientific
38:58experiment or derivable through the laws of logic.
39:01This logical analysis of the meaning was the essential first step.
39:10And therefore, it was forbidden to talk about such concepts like God, for instance, or metaphysical
39:19statements about the thing in itself or whatever, because you could never find a sentence
39:27that could be verified in a scientific way.
39:30In fact, the Vienna Circle loathed the idea of metaphysics so much that when they met here,
39:36Rudolf Carnap, a former student of Frege's, appointed someone to shout M during their discussions
39:44at the hint of any illegitimate sentence. M stands for metaphysics. M!
39:51It's the logician's way of saying bollocks.
39:53Now, the thing is, he was saying M so much that they got sick of it.
40:00Instead, they had him shout non-M any time that someone actually said something that was legitimate.
40:06Nicht M!
40:10Despite the purity of their logical methods, the problem of uncertainty that had plagued logic
40:16likewise stalked the Vienna Circle.
40:18Something that may have also imprinted this young generation of Austrian scientists was a scandal
40:28that happened in 1913, when it was discovered that the head practically of the counter-espionage
40:35service was a spy. And you see, the task of a counter-spy service is actually to make sure that there are
40:46no spies around. But what happens when the head of that organization is a spy himself? This is a
40:53fundamental uncertainty. Yes, the secret service can work very well, but can you be sure that the secret
41:00service is not infected? And something similar is happening in mathematics. You make sure that there
41:09exists no contradictions. You build up big walls against uncertainty or so, but maybe within these
41:16big walls, there is a contradiction sitting.
41:23Contradiction bothered one man more than most, Kurt Gödel.
41:29Kurt Gödel was the most reclusive member of the Vienna Circle.
41:33He'd had the finest logical training that you could imagine.
41:41It was in one of Vienna's famed coffee houses in August 1930 that 24-year-old Gödel first revealed
41:47a discovery that would end forever the logical quest that Frege, Russell and the like had set themselves.
41:55Goodell was one of the few who definitely had read all of Russell's Principia.
42:03He knew that for any logical system to be the foundation of mathematics,
42:07it had to be both complete and consistent.
42:12Goodell told Carnap that by studying the Principia, he had come to the conclusion
42:17that in any logical system, you could either be consistent or complete, but you couldn't have both at the same time.
42:26In Russell's masterpiece, Gödel had discovered a contradiction that became known as incompleteness.
42:33This means that in mathematical logic, there are going to be some truths which,
42:38although true, can never be proven to be so.
42:41The result of Kurt Gödel about the limitations of mathematics and logics was a terrible blow
42:51to the optimism of the Vienna Circle, and some of the members took a long time to come to grips with it.
43:01The grand search for absolute, provable truth had hit the buffers.
43:06By the mid-1930s, the Vienna Circle was over.
43:15The rise of fascism and the looming threat of war meant its members fled, were expelled or killed.
43:22Kurt Gödel left Vienna for Princeton, where his own search for certainty also came to a tragic end.
43:32Gödel became convinced that someone might try to poison him.
43:36The only person he would trust to cook, and indeed to taste his food, was his wife.
43:42And when she fell ill, and was hospitalised, he starved.
43:47He literally reasoned himself to death.
43:50The fact that all systems of mathematical logic were limited, that we could never have complete
44:03certainty, signalled the end of an era for logic.
44:07But for one British logician, Alan Turing, Gödel's work was the inspiration he needed to launch,
44:16inadvertently, a new and entirely more practical logic revolution.
44:22Alan Turing was just 23 years old when he imagined something extraordinary.
44:32He called it a universal machine.
44:35The universal machine is an entirely imaginary hypothetical device, and yet it's one of the most
44:43influential machines ever in human history.
44:46The device Turing imagined could tackle any mathematical problem using a logical
44:53algorithm encoded in its own limitless memory.
44:56In 1936, Alan Turing published a paper in which he demonstrated, he proved, that you couldn't
45:05decide beforehand which mathematical problems the machine would be able to solve, and which
45:09would just cause it to run on and on and on forever.
45:13That there are some problems that are simply uncomputable was startling, and yet another blow for
45:19mathematics. But it was also the beginning of something entirely unexpected, and destined to
45:25cement logic's role in the modern world.
45:30It's an extraordinary, almost exquisite paradox, that in demonstrating that some things can't be proved
45:38using a logical machine, what Alan Turing did almost single-handedly launched a technology revolution.
45:46Turing's universal machine is what we today call the computer.
45:53While stationed here at Bletchley Park during the Second World War, Turing began to implement his
45:58abstract ideas as real logical hardware.
46:03Working with Gordon Welchman, Alan Turing developed this machine.
46:08It's called the bomb.
46:11It's a bit loud. It's a form of electromechanical computer, and its logical function was to decode
46:19the messages that the Germans were sending using their Enigma encryption machines.
46:28But then Turing's colleague, Tommy Flowers, went a step further.
46:32It was built to crack another German encryption machine called the Lorenz, and for the men and
46:46women who built and operated it, it was an astonishing achievement. It shortened the war.
46:51But I think it's special for another reason. You see, this is the world's first programmable
46:58electronic computer. It uses digital information, binary, the streams of ones and zeros that are in
47:05all modern computers. And these vacuum tubes down here, they're wired together to be our Boolean logic
47:11gates, which perform Boolean operations and calculations.
47:18Colossus might not look high-tech to us, but it's hard to express just how important it was.
47:24The significance of all this as a piece of human engineering is on a par with the pyramids or
47:33the printing press or steam power, and yet it was all top secret. All these developments of electronic
47:40programmable computers here at Bletchley Park were classified, and the details were only declassified
47:46in the late 1970s. After the war, Turing went on to help build some of the world's first stored program
47:56computers. At their core, it all comes back to logical reasoning.
48:08Think about this. We're all surrounded by things that rely on some kind of logical machine or code.
48:14The failure of logic to deliver foundational answers for mathematics nonetheless gave rise to
48:22one of the most significant achievements in all of science and engineering.
48:29It started with those huge secret single-purpose computers, and yet right from the very beginning,
48:36some folk were already imagining the next big thing.
48:39We're still finding out what logic will do, but everybody's got them. You got a logic in your house.
48:48It looks like a vision receiver used to, only has got keys instead of dials, and you punch the keys
48:53for what you want to get. It's hooked into the tank, which has the cars...
48:58In 1946, science fiction writer Murray Leinster imagined an impressive specimen of
49:04interconnected technology. He named it a logic. Relays in the tank take over,
49:10and whatever vision program snafu is telecasting comes on your logic screen.
49:17Or you punch Sally Hancock's phone, and you're hooked up with the logic in her house.
49:22Also, it does math for you, and keeps books, and acts as a consultant, chemist, physicist,
49:28astronomer, and d-leaf reader, with an advice to love-lorn thrown in. It's very convenient.
49:38Well, that's extraordinary. It's a great characterization of the web that wasn't yet born.
49:50The digital world we live in, the computers that surround us, at their base, are running
50:01Boolean logic. They're running, actually, electrical currents. 1s and 0s are the product of those
50:06electrical currents. But on top of that, there are layers on layers on layers of complexity.
50:12Operating systems, machine code, applications that we use every day, from word processors to
50:18spreadsheets, to the browsers we use. And when you have your Skype conversation with your aunt in
50:23Australia, you don't think of that interaction in terms of those 1s and 0s. But without them,
50:28without the underlying processing, none of this would work.
50:33Not only did logic launch the digital revolution, but it's also the tool we use to sort,
50:39search, and retrieve the information we want online.
50:42The World Wide Web we have today represents the largest information
50:47construct humanity has ever created. It's 20 years old, barely, and yet we have billions
50:54and billions of pages encapsulating knowledge and information from all of human culture and all
51:01of human history. The challenge is to organize this massive information, this complexity, and logic
51:08gives us some of the perfect tools to do that.
51:15With the World Wide Web of information, logic means we're all more interconnected and informed.
51:21But back in the city, the march of logical machines has come at a cost.
51:26And I don't mean all the traders are spending too much time on Facebook.
51:30In the year that I was born, there were 22 separate stock exchanges in the UK,
51:35and this is how business was done. Now, this place, the London Metal Exchange,
51:41is the last venue where traders still go face to face.
51:49First, technology squeezed out the need for traders to meet in person.
51:53And now it's the traders themselves who may be heading for extinction.
51:57Not long after I wrote it, IBM did some tests of the Zip trading algorithm,
52:04and not only did they confirm that it worked, they showed that it outperformed human traders.
52:16When it comes to pure logical reasoning, the computers tend to beat us, hands down.
52:20It's an old adage, but people in this business joke that soon,
52:27the only things you'll find on a trading floor will be a big computer, a man, and a dog.
52:35The big computer's there to do all the trading.
52:39The dog's there to make sure that no one touches the computer.
52:43And the man's job?
52:45On the trading floor of the future, the man's job is to feed the dog.
52:49The man's job is to feed the dog.
52:54Mind you, despite my role in inventing these black boxes,
52:58I'm grateful that there's still a human around to pull the plug, sometimes.
53:03The thing is, computers still need their logical algorithms to be written for them.
53:08So they might take our jobs, but we still have the upper hand.
53:13Yet ever since their invention, the question as to whether this will always be the case
53:18has been a matter of fierce debate.
53:29When the digital revolution was in its infancy,
53:31the possibility of computers developing human-like intelligence was the hottest topic in town.
53:38Could a machine ever think, using the rules of logic alone?
53:43Or is there more to us than that?
53:47In 1950, Alan Turing published another visionary essay.
53:52In it, he predicted that by the end of the century,
53:55a computer would be able to converse with a human, and the human wouldn't know the difference.
54:01In trying to achieve this, people in my field have created some truly amazing computing machines.
54:07This is my university's supercomputer.
54:14And although it's bigger and noisier than Colossus,
54:17for every one Lorenz cipher that machine could solve, this can solve over two million.
54:25It takes up the whole room.
54:26machines like this are the workhorses of today's data-centric research.
54:34All the switches, wires and logic gates have long since disappeared under the hood.
54:38Meaning that for TV, we have a habit of trying to pretend that this doesn't all look like a load of
54:45well, cupboards.
54:46Turing thought that by the time we developed computers as powerful as this,
54:57we would also be capable of programming a machine with sufficient rules of logical reasoning,
55:03that its intelligence would rival that of us humans.
55:07That was then, and remains now, a very controversial idea.
55:11We like to think of our intelligence as raising us to a level above the rest of the creation.
55:20We associate it with the idea, perhaps, of an immaterial soul,
55:25being not just one amongst other animals, but special.
55:29And what Turing was suggesting was that this special quality could belong to a lump of computing
55:37machinery, and it could reason just as well as we could, maybe even better.
55:46At Bletchley Park, Turing had sketched out algorithms for playing chess.
55:51At that time, the chessboard was dominated by some of the world's most brilliant strategic,
55:56logical, mathematical brains.
55:59And so it became the battleground for an entirely new challenge for logic,
56:04artificial intelligence.
56:09In 1997, the most famous public battle between man and machine took place.
56:15Gary Kasparov, the reigning chess world champion, had previously trounced IBM's chess-playing computer,
56:22Deep Blue.
56:24During their rematch, for the first time ever, he was beaten.
56:29Kasparov has resigned.
56:33When I see something that is well beyond my understanding,
56:37I'm scared. And that was something well beyond my understanding.
56:41It was front-page news the world over. People demanded answers.
56:46Was this purely logical intelligence equivalent or even superior to the human brain?
56:54In the past, people have tended to compare humans to the latest technology.
57:00So maybe the brain is like a clock, or maybe it's like a steam engine.
57:05Now, maybe it's like an electronic computer.
57:10What Turing would want to say, and I think correctly,
57:14is that there's something different about the equation of the brain with a computer.
57:19He put it that both a brain and a computer are information processing systems governed by logical rules.
57:29In theory, there should be logical rules out there that would capture the way we think.
57:36This was a very big idea, with profound, even troubling implications.
57:45If we knew those rules, then one day, theoretically,
57:49we could code a logical rendering of ourselves into a computer.
57:54All we'd need to reproduce all of human thought is logic.
58:01My view is that there remain uniquely human characteristics, arguably the best ones,
58:07like altruism or creativity or love.
58:11That computers aren't even close to having programmed within their repertoire of logical reasoning.
58:18No one has yet created a logical machine that's just like us.
58:22And arguably, that could take a very, very long time, if indeed it's possible at all.
58:29And yet, surely, we should marvel at what we have achieved with logic.
58:34Remember, we created the rules of logic to pin down the truth and certainty that would otherwise so easily evade us.
58:43We harnessed logic in machines, and in doing so, we placed the power of pure reason at our fingertips.
58:52Mind you, I'm still no good at Sudoku.
58:54One, two, one, two, three, four.
59:22Just let's follow our church.
59:24Bye.
59:30Bye.
59:33Bye.
59:44Bye.
59:45Bye.
59:48Bye.
59:48Bye.
59:50Bye.