Imagine a bottle with no inside or a number bigger than infinity or parallel lines that meet. Welcome to the world of pure mathematics. NOVA offers a look into a wholly abstract, quirky world of mathematics.
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00:00Welcome to the world of pure mathematics, where geometries exist in many dimensions
00:10and numbers are bigger than infinity.
00:20It's a world where objects take on strange configurations, and if you thought that parallel
00:31lines never meet, you're in for a surprise.
00:37For over a decade, Bertrand Russell tried to find certainty through mathematics by reducing
00:43it to logic.
00:45In his massive work, Principia Mathematica, it took him 362 pages to prove that one plus
00:51one equals two.
00:55Twenty years later, another mathematician, Kurt Gödel, proved that mathematics would
00:59never be completely certain.
01:04Do the abstract objects of mathematics have anything to do with the real world?
01:09Is mathematics the key that unlocks the universe?
01:40These are some of the classical unsolved problems in mathematics.
01:49And they have resisted solution by the world's greatest mathematical minds.
01:56Solve any one of them, and you would achieve instant fame.
02:05Some of these problems have applications in the real world.
02:08Others are more abstract problems, made by mathematicians for mathematicians, and studied
02:13for their own intrinsic interest.
02:21Throughout history, mathematicians believed in the certainty of mathematics, that every
02:26problem, no matter how difficult, had a solution.
02:29But over the last 50 years, certain events have shaken this belief.
02:37Will there ever be solutions to these questions?
02:41Do the abstract problems of pure mathematics matter in the real world?
02:45Or are mathematicians in a world of their own?
02:59In the 18th century, Bach created compositions that had discernible mathematical structures.
03:11When Leonardo da Vinci drew the perfect proportions of the human body, they fit within a circle
03:15and a square.
03:23And ancient civilizations constructed massive monuments with mathematical precision.
03:32The astronomer Galileo said, to understand the universe, you must understand the language
03:36in which it's written, the language of mathematics.
03:43Mathematics possesses not only truth, but supreme beauty.
03:46It has been said that it is the work of the human spirit in its quest to comprehend our
03:51world.
03:55Some of the finest mathematicians of today agree.
03:59A mathematician really wants to understand things.
04:04And when you see a solution of a problem which improves our understanding, then you say this
04:10is a beautiful and important result.
04:12It has much greater certainty, I think, than any other branch of science.
04:17What gives you the certainty about mathematics is quite different from what you have in physics
04:21or in chemistry or in biology.
04:22We don't conduct experiments.
04:24So instead of that, we have logical arguments that lead from a hypothesis to a conclusion.
04:31Proof, which is the logical sequence of steps that enables you to deduce one thing from
04:35another one, is the glue that holds mathematics together.
04:40Mathematicians are apt to regard the physicists as a somewhat lower category of being who
04:46deal with knowledge that's somewhat certain, or at least fairly probable, whereas the mathematicians
04:53think that once something is proved, has been shown, the fact that it's true is certain.
04:58There's no question about it.
05:04Sir Isaac Newton, physicist, astronomer, and one of the world's greatest mathematicians.
05:10As a scientist, his revolutionary discoveries on the nature of light are a clear example
05:15of what is known as the scientific method.
05:21It was thought that white light was the purest form of color.
05:24Newton did an experiment where he passed sunlight through a prism and it separated into a spectrum
05:29of colors, disproving the hypothesis of the time.
05:33In science, experiments are the confirming evidence.
05:37In mathematics, however, proof, a logical succession of thoughts, is what's important.
05:42For example, if you agree A equals B and B equals C, then you must conclude A equals
05:48C. Experiments are necessary because the logic is irrefutable.
05:54This concept of proof has made mathematics almost entirely uncontroversial.
05:59It's common in all branches of mathematics, including arithmetic, which is the study of
06:04numbers, geometry, the study of shapes, and analysis, the study of infinity, which includes
06:13calculus.
06:16Numbers are the fabric of mathematics, but not all numbers are the same.
06:21Some are called prime numbers.
06:24Most numbers can be divided into smaller numbers.
06:28These in turn can be broken down.
06:31Ultimately, you are left with numbers that cannot be divided any further.
06:392, 3, 5, 7, 11.
06:44These are called prime numbers, and they crop up at random, getting rarer as numbers get
06:48larger.
06:50But how could you prove this pattern would continue?
06:53The method of proof is a legacy of the ancient Greeks.
06:56Euclid proved that there are an infinite number of primes.
07:00And now his proof is considered a model of mathematical thinking.
07:04It's final, that's right, it's final.
07:09Some Euclid's proof in number theory saying that there are infinitely many primes, this
07:13is final.
07:14It has been done 2,500 years ago, it's still good.
07:18Nobody ever challenged it.
07:21What Euclid didn't prove, nor anyone since, was this.
07:25Are there an infinite number of prime twins?
07:29These are pairs of primes which differ by 2, like 41 and 43, or 59 and 61, and so on.
07:42There's another unproved prime number problem called the Goldbach conjecture.
07:47It was in the 18th century that Christian Goldbach wondered if all even numbers could
07:52be obtained by adding up two primes.
07:55It's clearly true for small numbers.
07:59In fact, it's been verified by computers up to 100 million, and it never fails.
08:06So isn't that proof enough?
08:09Not for mathematicians.
08:11They point to the case of the Mertens conjecture.
08:14It too is about prime numbers and verified to 10 billion, which recently turned out to
08:19be wrong.
08:21As the British journal Nature reported, despite the apparently overwhelming evidence in its
08:26favor, the Mertens conjecture is false.
08:30Two mathematicians proved it would fail, sometime before it reached this enormous number, 10
08:35to the power 10 to the power 70.
08:38The proof is incredible because this number is, an order of magnitude totally inaccessible
08:43to direct computation.
08:46In fact, it's much bigger than even the number of atoms in the universe.
08:50To find proofs for these classical problems is extremely difficult, yet some amateur mathematicians
08:56have tried their hand at these complicated questions.
09:00In 1984, a British publication, The Guardian, claimed that an amateur had solved a famous
09:05problem in mathematics called Fermat's Last Theorem.
09:11The problem has an intriguing history.
09:14One night in 1631, so the story goes, the French mathematician Pierre de Fermat was
09:20perusing the writings of the ancient mathematician Diophantus.
09:28He was considering equations of different powers.
09:36For equations to the power of 2, there existed whole number values of x, y, and z, which
09:41balanced the equation.
09:50But this didn't seem to be true for higher powers.
09:54Equations to the power of 3, for example, didn't have whole number solutions.
10:02But could it be proved that there were no whole number solutions for all higher powers?
10:10That night, Fermat wrote in the margin that it could.
10:13I have discovered a truly marvelous proof of this, but unfortunately, the margin is
10:18too narrow to contain it.
10:21Fermat often left his theorems unproven like this, and they usually turned out to be right.
10:26So when these words were discovered after his death, mathematicians began a quest for
10:31the proof of what is known as Fermat's Last Theorem.
10:35The greatest mathematicians of history have sought the proof.
10:38Euler tried and failed.
10:41So did Lagrange.
10:46Even Gauss, the prince of mathematics, was defeated, as was Poincaré.
11:08Was this distinguished history of failure?
11:11Had the amateur mathematician Arnold Arnold really succeeded?
11:15He had not.
11:16Within a short period of time, mathematicians had found the error, and the Guardian had
11:20to apologize.
11:23The proof was wrong.
11:26That happens all the time, you know, when people make errors or jump at conclusions,
11:32you see.
11:33It's so painful to write down all the details of a proof, that at times you say, all right,
11:39this is obvious, and you go on.
11:42Usually that's where the error lies.
11:45When you see in the paper, it is obvious that, it's fishy.
11:53Usually you say, if there is an error, it's there.
11:57And I've been guilty of that myself, and I think everybody has been at times.
12:02It's so tedious to write down every single detail.
12:06Sometimes you jump over one, and sometimes that's where the error lies.
12:11You have to, really, to convince your colleagues that really you are right.
12:19And this, of course, requires a proof.
12:23Writing a proof is really making yourself at the level of the other people, so to speak.
12:32This prize money is offered as a reward to those who succeed in solving mathematical
12:36mysteries.
12:37Paul Erdős, a famous Hungarian mathematician, travels the world over, offering his own money
12:45and challenging other mathematicians to work on difficult problems.
12:48In fact, I was asked several times, what would happen if all your problems would suddenly
12:55be solved?
12:56How could you pay?
12:58I said I couldn't, but what would happen to the strongest bank if all the depositors
13:03would suddenly ask their money back?
13:05The bank would clearly be ruined, and I think it is much more likely that this will happen
13:11than all my problems will suddenly be solved.
13:22Mathematics can be done without proof.
13:27The Chinese devised calculating tables to shorten the process of certain computations.
13:37The Egyptians were doing complicated calculations as early as 3000 BC, but they didn't use proof.
13:44Their mathematics was practical and used for counting, for measuring land, and for weighing
13:49agricultural products like grain.
13:52The early Egyptian mathematics was essentially a cookbook sort of mathematics.
13:57It said something like, take the number 2, square it, add 3 to it, divide it by 3, and
14:06so on, and such and such will happen.
14:08You will get a certain result.
14:11Now there was no sense of generalization there.
14:15Always you find them referring to specific numbers, but it's clear that it's just as
14:20much a generalization as if you look in a cookbook and it says, take 2 cups of flour,
14:24add 1 quarter cup butter, and so forth.
14:26It's not referring to a specific 2 cups of flour.
14:30You can do it with any 2 cups of flour you find, and when it says, take 2, in effect
14:35it's saying, take any whole number and do this to it, and you will get a certain result.
14:422000 years after the Egyptians, a Greek mathematician is said to have produced the first mathematical
14:48proof.
14:49He proved that the diameter of a circle divides it into two equal parts.
14:56Aristotle later formalized 14 rules of proof, and these were the guiding principles employed
15:01by Euclid in his treatise on geometry.
15:06Euclid's book, Elements, is the most widely circulated book in human history after the
15:10Bible.
15:16A Euclidean proof proceeds in logical, orderly steps to demonstrate the proof of particular
15:22propositions.
15:23The assumptions are open to question, but the line of reasoning is irrefutable.
15:32Euclid succeeded in deducing every known theorem in geometry from just 10 axioms, statements
15:41which everyone held to be true, and they were unquestioned for more than 2000 years.
15:57But in the 19th century, mathematicians tried rewording some of them to see what might happen.
16:12Strange new geometries were the result, and these non-Euclidean geometries were just as
16:17valid as Euclid's.
16:19Geometry has never looked back.
16:22Today geometry deals with all possible shapes in all possible dimensions.
16:27For example, a hypercube, a rotating cube of the fourth dimension.
16:32If you live in one dimension on a line, just like a road, cars traveling on a road, then
16:40the geometry there is quite different from what it is when you got more degrees of freedom.
16:44If you're a ship on the sea, it's quite different from a car on a road.
16:47If you're an airplane in the sky, it's again quite different from a ship on the sea.
16:51You have more room in which to move around, and so new features can happen.
16:56New fields like topology contemplate bizarre objects, like the so-called Klein bottle.
17:03We usually think of a bottle as having both an inside and an outside, but a Klein bottle
17:08has only one side, one of the unexpected consequences of exploring the fourth dimension and beyond.
17:20The most important single idea in geometry is probably the idea about curvature, how
17:24things curve, and when you analyze that, you find that you need four dimensions in order
17:29to allow the curvature to have all its full ramifications.
17:31With more dimensions, nothing much alters, and on the other hand, having more dimensions
17:36enables you somehow to unwind things and become simpler.
17:39So it looks like the situations get more difficult between one, two, three, and four dimensions.
17:44In some way, they get easier afterwards.
17:46That means four dimensions looks like it's the most interesting or difficult, exciting
17:51dimension to be in.
17:52It happens to be the dimension of space-time, and somehow that seems to be extremely suggestive.
17:58The depth of ideas in geometry, in four dimensions, are related to the deep things in physics.
18:05The idea of the fourth dimension could be imagined this way.
18:09This straight line could be used to represent a one-dimensional object.
18:14This figure could be used to describe the second dimension.
18:20Add depth and this cube represents a three-dimensional object.
18:24Although the fourth dimension can't be easily visualized, consider that the cube also exists
18:29in time.
18:31Time is an example of the fourth dimension.
18:34Objects in four or more dimensions may really exist, but because they are not part of our
18:38ordinary experience, they're considered abstract.
18:41So you have to think of these as abstract objects.
18:44They were abstract, they have remained abstract, and they will always remain abstract.
18:48So from about 1840 on, mathematicians had to deal with objects which were not susceptible
18:55of having a visual approach.
19:00And that's what happened, that's why mathematics has become more and more abstract, because
19:06abstractions pile upon abstractions, and there doesn't seem to be any end to it.
19:18Do the abstract objects of mathematics correspond to real things, or are they merely inventions
19:24of the mathematical mind?
19:27Could they still be there even if there were no mathematicians to discover them, or do
19:31we invent them?
19:34The Greek philosopher Plato believed that all objects, no matter how abstract, exist
19:39in the universe, and that mathematics discovers them.
19:43This view is called Platonism.
19:45Indeed, widely different cultures have discovered the same mathematical truths independently,
19:51like Pythagoras' theorem.
19:52I am, mathematically, I am a Platonist, and I believe that mathematical entities do exist
20:01outside the mind of the people.
20:06Of course, such a statement is difficult to justify, but at least the psychological feeling
20:16is that really you discover things and you do not invent them.
20:22When they're talking to each other, they are Platonists, almost pure and simple.
20:28That is, they act as if they are talking about something that's actually there, and it has
20:33certain definite properties, and you can prove something about them, something quite explicit
20:38and definite.
20:39One mathematical object that seems to appear over and over again is the golden rectangle.
20:45You'll find it in Da Vinci's sketches and in architecture from classical to modern times.
20:54Why should the proportions of this rectangle be so satisfying?
20:58If you divide the height by the width, you get what is called the golden section, and
21:02this number crops up in the strangest places, like this symmetric expression.
21:07The longer you continue, the closer its value gets to the golden section, approximately
21:131.618.
21:17Another example of how mathematical ideas correspond to real objects is visible in the
21:21symmetries of nature.
21:24Symmetry is something which everybody understands intuitively.
21:31You have figures like circles, triangles, squares, beehives, crystals, all of which
21:38exhibit something which we think of as symmetry.
21:42But how to understand this mathematically seems a very difficult question.
21:46If you take a square and you can turn it to 90 degrees, it still looks like the same thing.
21:55You take an equilateral triangle and you rotate it to the right, about its middle, to the
21:59right number of angles, the points will move around and you'll again get the same figure.
22:03So symmetry has to do with the ways in which you can move things in such a way that they
22:09reproduce themselves.
22:11And that's the basic idea that mathematicians have tried to abstract.
22:13It doesn't matter what it is that's being involved, so long as you can consider the
22:17different possible ways of changing things around.
22:20And the change doesn't need to be thought of as a physical motion in space.
22:23If I have two brothers and I think of just interchanging them in their names, that would
22:28be a symmetry.
22:30This star illustrates the mathematician's way of describing symmetry.
22:36The numbers on the points of the star are paired with the numbers of their five positions.
22:41When the star is rotated to the right, the star looks the same, but the first point now
22:46corresponds to the second position.
22:50Mathematicians don't usually depend on images like this star.
22:53They visualize symmetry wholly in terms of numbers.
22:57In this way, they have been able to see symmetries in nature that we may not easily recognize.
23:07The mathematical study of symmetry is called group theory.
23:15It was founded by a young French student, Avariste Galois.
23:20It's a tragic story that begins one night in 1830.
23:25Galois hurriedly scribbled down his ideas in a letter to a friend.
23:29He had to write quickly because in the morning he was to fight a duel over a woman.
23:36The letter contained the seeds of group theory, what has become one of the most powerful theories
23:41in all mathematics.
23:45Galois was only 21 years old, and this was the last act in a tragic life.
23:51That morning he was killed in the duel.
23:57During the decades that followed, mathematicians have been able to perceive the many ways in
24:01which the universe is symmetrical.
24:04By the 19th century, mathematicians had entered a realm of infinite possibilities.
24:21The question of infinity has captivated the human imagination for thousands of years.
24:29Infinity is used to describe that which goes on forever.
24:33And that's characteristic of a group of numbers called irrational numbers.
24:38If you divide the circumference of a circle by its diameter, you get one of the most famous
24:42numbers in mathematics.
24:47It's called pi, and it crops up everywhere.
25:01Even in games of chance, if these lines are separated by the length of one pin, what are
25:06the odds that a pin will hit a line?
25:09The answer turns out to involve pi.
25:16For this series, if you were to continue it forever, it would add up to one quarter
25:20of pi.
25:30Throughout history, many have tried to measure it.
25:34The ancient Egyptians calculated pi to 3.16, and the Greeks, by the time of Archimedes,
25:41had established its value to 22 over 7.
25:47But that's only an approximation.
25:49In decimal form, pi is infinitely long, and its digits never repeat the same sequence
25:54of numbers.
25:55It's called an irrational number.
25:58By calculating pi to 39 decimal places, you have more than enough to measure the size
26:03of the universe with remarkable accuracy.
26:06Earlier this year, two Japanese scientists calculated it to more than 16.5 million decimal
26:11places.
26:15These are the last known digits of the longest calculated number in the universe.
26:22This is a prime number.
26:24It can be written in a special way, 2 to the power 31 minus 1.
26:31This abbreviation was invented by the French friar Mersenne, and you can see how it works
26:36using a checkerboard.
26:38Pile two checkers on the first square, doubling the checkers as you go.
26:47The numbers would soon get very big indeed.
26:52Two to the power of 64 would form a pile 37 million million kilometers high, enough to
26:59reach the nearest star.
27:04Now compare it to the number Brian Tuckerman found in 1971, and prove to be a prime number
27:10on his IBM 360 computer.
27:15But then, two 15-year-old high school students from California did better.
27:19They found this prime number.
27:23The current record holder is David Sluinski.
27:25His prime, 2 to the power 132,049 minus 1, is the largest known prime number.
27:33It took only a week to calculate it on a Cray supercomputer, using an ingenious shortcut
27:38method.
27:40Without this shortcut, such a calculation would have taken longer than the age of the
27:44universe.
27:52Imagine that all of you are in Hades, and I am the devil.
27:57I make the following proposition.
27:58I say, I've written down a positive integer on a slip of paper.
28:04A whole number, either 1, 2, 3, 4, 5, 6, 7, 8, and so forth.
28:09Every day, you're allowed one guess as to what that number is.
28:14If and when you guess the number, you go free.
28:18Professor Raymond Smullyan is using a classic story to illustrate a startling discovery
28:22by mathematicians that there are numbers bigger than infinity.
28:27In the 1880s, the German mathematician, George Cantor, developed a way of comparing the sizes
28:32of infinite sets of numbers by pairing them one by one.
28:39For example, take the set of whole numbers like 1, 2, 3, and so on.
28:44There is an infinite number of them.
28:48Compare it to a set of these numbers.
28:50Would a collection or set of every tenth number be only one-tenth as large?
28:55By pairing them off, Cantor proved that, in fact, both collections are the same size.
29:05Now I'm going to make the test much more difficult.
29:09This time, I have a victim, let's say a charming female whom I wish to keep forever.
29:16I want to make matters very difficult for her ever to escape.
29:19And so I say, this time, young lady, I'm thinking of a fraction.
29:24One whole number divided by another whole number.
29:27It could be like 5 over 3, or 3 over 7, or 2 over 9.
29:34Every day, you have one guess as to what it is.
29:38If or when you guess it, you go free.
29:44By pairing the set of fractions with the set of whole numbers, surprisingly, Cantor found
29:48they were the same size.
29:54But he discovered other sets of known numbers were larger.
29:58On a number line, there exist numbers that cannot be expressed as either whole numbers
30:03or fractions.
30:05There are more of them in the tiniest line segment than there are counting numbers in
30:09the universe.
30:25These other numbers are the irrational numbers, numbers like pi.
30:30And by pairing them off with the whole numbers, Cantor proved there would be some irrational
30:34numbers left over.
30:36The set of irrational numbers was genuinely bigger than infinity.
30:40It meant that beyond infinity, which he called Aleph-0, there were larger infinities, Aleph-1
30:47and Aleph-2, and still bigger ones beyond.
30:55Cantor's ideas were received with skepticism at first, but eventually they evolved into
31:00an important subject now called set theory.
31:07If we did not have the infinite, there would be almost no mathematics at present.
31:13We say we need it all the time.
31:15Well, of course, you know that there were philosophical preconceptions against the actual
31:22infinite.
31:23For us, this has disappeared, because the infinite for us has become just simply formal.
31:34We have an axiom of the infinite, but we don't attach to it a material meaning.
31:42We don't claim that there are infinitely many objects of some kind somewhere.
31:50That's why people, I think, were so reluctant to accept infinity, because they said if the
31:55infinite exists, there must be infinitely many things somewhere, and then they went
31:59into contradictions.
32:06Other mathematicians were inspired by Cantor's work and applied his new ideas to the whole
32:11of mathematics.
32:13Shortly after Cantor's revelations, this manuscript appeared, which few people could understand.
32:19Its author, Gottlob Frege, borrowed Cantor's definition of sets and tried something quite
32:23remarkable, to reduce the whole of arithmetic to something he believed more fundamental,
32:28logic, a subject which had been around since Aristotle.
32:32Now in Aristotle you have essentially the notion of syllogism, an example of which would
32:38be all men are mortal, Socrates is a man, therefore Socrates is mortal.
32:44And it was thought that all reasoning could be put in one or another form of syllogism,
32:51and that all reasoning even in mathematics was of this type.
32:56Now with Frege you have the first real understanding that all reasoning in mathematics is not of
33:03this sort, that there are kinds of reasoning there that need some other modes to describe
33:09it, and that these modes can be made very precise, and that given that precision, arithmetic
33:17will then be found to be part of logic.
33:21At this point Bertrand Russell enters the story.
33:25Well just as the second volume of Frege's masterwork on this subject was about to appear
33:31in print, Bertrand Russell, who at that time is a rather young and not that well-known
33:39philosopher, writes to Frege saying that he's read some of his work and found it very interesting,
33:46and by the way he's found this paradox which he's not able to solve.
33:53Russell's paradox concerns set theory, but can be told as a story about a librarian who
33:58is ordered to compile a catalogue of every book in her library.
34:05As she's finishing she's struck by a thought.
34:08Should she include the catalogue itself in the catalogue?
34:11It is after all a book.
34:17She decides not to.
34:21The National Librarian receives such catalogues from all the libraries in the country.
34:27Some yellow, where the librarian has listed the catalogue itself.
34:34Some blue, where they haven't.
34:36He now has the awesome job of compiling a master catalogue of the blue ones, the catalogues
34:42which don't list themselves.
34:46But on thinking about it he realizes it is impossible, because what does he do with the
34:50master catalogue itself?
34:52If he doesn't list it in itself then it will not be complete.
34:56But if he does it's an error, because then it's no longer a catalogue of catalogues which
35:00don't list themselves.
35:02Why should Russell have thought this paradox so important?
35:08It was because the most general way of thinking about any mathematical object was in terms
35:13of collections or sets of them.
35:15A catalogue of books is in principle no different from a set of numbers.
35:21Ironically the effort to be logical was leading mathematicians not to certainty as they had
35:26come to expect, but to uncertainty.
35:31The ideas of both logic and sets were so fundamental to mathematics that to run into such a contradiction
35:37at this level of mathematics was very worrying.
35:40The whole enterprise might be built on sand.
35:45Now Frege was absolutely devastated by this and regarded it as in essentially destroying
35:52his life's work.
35:54Frege and Russell then corresponded and Frege put forth various possibilities of a solution
36:00as Russell did as well, but Frege was never the same after that.
36:07Russell however remained optimistic that his paradox could be resolved and that logical
36:12certainty would be restored.
36:16For the next decade or so with Alfred North Whitehead he labored to produce the Principia
36:21Mathematica.
36:23This massive work sought to deduce all of mathematics from basic principles of logic.
36:31It takes a while to get going.
36:34Some 362 pages before they could prove that 1 plus 1 equals 2.
36:42Professor Gratton Guinness.
36:45Nobody had done anything on the scale of the detail that this Principia Mathematica constitutes
36:52and you have 2,000 pages of what looks like wallpaper most of the time.
36:57I mean, at times it's hardly a prose word on the page.
37:08And you must have had these mounds of manuscript all over the place.
37:12That sort of thing can happen.
37:13Oh dear, you make a slip proving proposition 47.275.
37:18Have you made the same slip anywhere else?
37:20I mean, you could easily spend a morning checking things like that.
37:24I can understand exactly how he must have broken in producing this thing.
37:30Russell himself only intermittently worked on it there afterwards.
37:34In fact, he said it broke him intellectually.
37:37He wasn't as sharp after it as he had been before.
37:42But was the scheme a success?
37:45What Russell and Whitehead do in Principia Mathematica is sort of get ready to do mathematics
37:51without ever really getting as far as doing some mathematics.
37:55And in a way, the work is like some vast overture to an opera which never got written.
38:05Russell himself wrote, I wanted certainty in the kind of way in which people want religious
38:10faith.
38:11I thought that certainty is more likely to be found in mathematics than elsewhere.
38:17And after some 20 years of very arduous toil, I came to the conclusion that there was nothing
38:22more that I could do.
38:32In France, the response to Russell's program of logic was a series of mathematical volumes
38:37published under the name Bourbaki.
38:41For centuries, France boasted one of the finest mathematical traditions in Europe.
38:46The line was broken when the outbreak of the First World War interrupted intellectual pursuits.
38:53After the war, a group of top-level French mathematicians met together to revive the
38:57tradition with an outrageously ambitious project.
39:01To rid all mathematics of any and all paradoxes, they named themselves Bourbaki.
39:06Bourbaki, as you know, was one of Napoleon's generals, I believe, and they took his name
39:12simply as a sort of pseudonym for this body of French mathematicians.
39:17Messieurs, je propose que nous commencions tout de suite par...
39:20But they work in a very interesting way.
39:21I was once actually attended one of their meetings, and they get together as a collective
39:25group and write their books together.
39:27And they go through the manuscripts page by page, discussing in great detail.
39:31They're very, very pedantic, one might say, very careful.
39:36And they've taken enormous pains over this.
39:39One might say almost too much sometimes, they take themselves perhaps a little too seriously.
39:46For Russell, logic was beyond suspicion.
39:51For Bourbaki, however, everything, including the age-old rules of logic, were held up for
39:56close scrutiny.
39:59They stressed the need for correct terminology and language.
40:04It was a very formal approach to mathematics.
40:12We French people, we have been brought up in tradition that we should respect the language.
40:18We should consider that the French language has been evolved through centuries to a point
40:26where it's a good tool and it has, really, qualities of its own.
40:31And to misuse it, we consider it as a kind of sin.
40:37But others, like René Tombe, express some skepticism of Bourbaki's influence.
40:42I was asked what is perhaps the main achievement of Bourbaki in mathematics.
40:52I would say it has made clear the distinction between the circle and the disc, and between
41:00the sphere and the ball.
41:02And that's very important in my view.
41:08The need for precision and rigor in language was of great interest to Bourbaki, and their
41:13published works are of some influence even today.
41:19Meanwhile, David Hilbert, one of the greatest mathematicians of the early 20th century,
41:24was also worried about paradoxes.
41:27Hilbert thought that the paradoxes had occurred because mathematicians were trying to describe
41:31mathematical ideas using ordinary language.
41:35To him, mathematics could only be properly described in abstract symbolic terms.
41:42It's a kind of thing completely wrong to say that I can imagine what 10 to the 10 means.
41:50For me, it's just simply a system of figures, but it has no actual meaning.
41:55Number three has a meaning, because I can immediately see three objects.
42:00But number 10 to the 10 has no meaning at all, except from the meaning you give it through
42:05the axioms.
42:08Hilbert's ideas can be understood if you compare mathematics with a game of chess.
42:14When humans play chess, they use their intuitions, they form mental pictures, they are emotional.
42:21But the game of chess itself can be described in completely abstract terms, so that a machine
42:26can play it.
42:30A microcomputer has no intuition or mental images about the pieces.
42:34The axioms of chess, the rules, are sufficient to define all the possible moves.
42:41And if the program's been written correctly, the machine will never break the rules.
42:48In the same way, Hilbert thought mathematics could be made into a kind of game.
42:53Its principles, he thought, could be expressed completely in terms of abstract symbols and
42:59manipulated mechanically.
43:02He hoped that by taking the meaning out of mathematics, by sticking rigorously to the
43:07rules, he would free it once and for all from paradox and contradiction.
43:14Historically, it's quite true that most mathematics come from questions or theories which had
43:24even a physical meaning.
43:26If you take partial differential equations, say for instance, alright, they are very abstract,
43:31but most of them come from concrete problems of physics or mechanics or that kind of thing.
43:39But once they have been translated into pure mathematics, deriving from an axiom system,
43:46their meaning is irrelevant.
43:48We have to work according to the assumptions we made on them, that's all.
43:54The price of rigor could be taking the meaning out of mathematics, a price that some mathematicians
43:59feel is just too high.
44:01Rigor has nothing to do with interest.
44:04There are a lot of interesting things which are not rigorous, and a lot of rigorous things
44:09which are devoid, really, of any interest.
44:13In fact, one of my favorite maxims is, in French, tout ce qui est rigoureux est insignifiant.
44:20Anything which is rigorous is meaningless.
44:24This is, in some sense, I would say, the philosophy of Hilbert's program, pushed to the extreme.
44:34You achieve rigor only through meaninglessness.
44:47By 1930, mathematicians from Russell to Hilbert were trying to restore certainty
44:52to mathematical reasoning.
44:54But a young mathematician was to shock them all by proving that it could never be done.
45:00In 1931, an Austrian mathematician, Kurt Gödel, published a theorem in logic
45:05which demolished Hilbert's program to resolve contradictions.
45:11Gödel's incompleteness theorem showed that mathematics would always remain plagued by paradoxes of a sort.
45:17There would always be questions that mathematics could not resolve.
45:21Consider this set of principles.
45:23Gödel proved that neither this set nor any other set would ever be completely adequate
45:28to decide all mathematical questions.
45:30Hilbert would never accomplish what he set out to do.
45:36This completely demoralized or undermined this whole program of laying the foundations of mathematics.
45:42Well, of course, there's a lot of discussion been going on ever since
45:45about what the foundations of mathematics are, how you set them up.
45:48But because this initial program failed, most working mathematicians take a more pragmatic attitude.
45:54They say, well, if we can't achieve ultimate certainty about mathematics by providing foundations,
45:58that's no reason for us to stop doing mathematics.
46:01Physicists get along quite happily, although their foundations are much shakier than ours.
46:05So most mathematicians go along quite happily with their mathematics,
46:09even though they know that in some deep ultimate sense the foundations are perhaps a little uncertain.
46:15Just as Einstein had transformed physics, Gödel changed mathematics forever.
46:20One example from the 19th century is the continuum hypothesis.
46:24Mathematicians have been unable to prove it true or false.
46:28It has a third status. It's undecidable.
46:31It's undecidable. You cannot prove the continuum hypothesis.
46:36You cannot disprove the continuum hypothesis from the present axioms of set theory.
46:41Now, some very formalistic kind of mathematicians say,
46:44well, that means the continuum hypothesis is neither true nor false.
46:48Well, I don't buy that, and most mathematicians wouldn't.
46:53We want to know whether the continuum hypothesis is true or not.
46:57Supposing if you build a bridge, and the physicists and engineers are getting together,
47:02they want to know. The next day, the army's going to march across the bridge.
47:06Will the bridge stand the weight, or won't it?
47:08It's not going to do them any good to be said,
47:10well, in some axiom systems you can prove that it will stand the weight,
47:14and in some axiom systems you can prove that it won't stand the weight.
47:17They want to know whether it really will stand the weight.
47:20And so-called mathematical Platonists or classicists, which most working mathematicians are,
47:26and I certainly am, regard the continuum hypothesis as definitely true or false.
47:31I just don't know which.
47:34Just because it's undecidable, the base of the present axiom,
47:37doesn't mean that it's neither true nor false.
47:39It means that the axioms, the present-day axioms of set theory,
47:42don't tell us enough about sets to decide the question.
47:48The situation is quite remarkable.
47:51Mathematicians might be working on problems that cannot be solved.
47:55They may simply be wasting their time.
47:58Well, in a sense, I should say for us,
48:01most mathematicians think it's quite beneficial,
48:04because you just simply tell them, don't go and try that problem.
48:08You're wasting your time.
48:11Because before that, lots of people were working on, say,
48:14this continuum hypothesis, and at the time thinking they would solve it,
48:19there was a French mathematician, not very good,
48:22who spent all his life working on the continuum problem
48:25and turning out new proofs all the time.
48:28And the proofs were examined and all were found wrong, of course.
48:31He did not desist, and he went on until the end of his career.
48:36Well, it's much better to have to know in advance that it's hopeless.
48:43As far as we know, these great unsolved problems are not undecidable,
48:47though several dozen others may be.
48:50Far from making mathematics certain,
48:53logic as we know it has exposed its limitations,
48:56and logic has had another unforeseen consequence.
49:03The use of logic is still perhaps our best means
49:06for arriving at answers to real and abstract questions.
49:10In fact, symbolic logic turned out to be the vital ingredient
49:13in the wartime development of what's become
49:16the most dynamic technology of our time,
49:19the computer.
49:23And just as the computer has changed everything else,
49:26it may change mathematics.
49:29Up until 1976, the four-color problem was unsolved.
49:35This century-old problem requires a proof
49:38that four colors are sufficient to color any map
49:41on a plane or sphere, so that no two adjacent countries
49:44have the same color.
49:47This must be proof, not just for the maps we see in atlases,
49:50but for any conceivable map that could ever be drawn,
49:53like this.
49:56And in fact, it has been proved that four colors are enough.
49:59What was new was that it was done on a computer.
50:02The computer participated by drawing thousands of configurations.
50:06But is this a proof?
50:11That's the first example, perhaps,
50:14of a computer being used in helping
50:17or improving something in mathematics.
50:20And it's pretty clear that in the advent of more and more powerful computers,
50:23this is going to happen more and more.
50:26Now, I think that there are advantages and dangers there,
50:29and on the one hand, it's clear that mathematicians shouldn't
50:32reject new tools, and if a computer can help them
50:35to get a better understanding of anything,
50:38they should obviously, and will, use it.
50:41On the other hand, if we get computers to come
50:44we don't achieve the same level of understanding.
50:47We simply say, the computer told us that this was true.
50:50First of all, one could ask whether, really,
50:53a proof made by computing on a computer
50:56is really a proof.
50:59Because, after all, if the computer makes a mistake,
51:02then the proof is mistaken.
51:05So, in that respect,
51:08proofs realized by computing ways
51:12should not be considered strictly as proofs.
51:15Add to my test.
51:18We're not simply in the business of mathematicians
51:21to get answers. We want to understand.
51:24And understanding is something that you can't
51:27do, can't achieve, unless you're actually involved
51:30in the process all the way through.
51:33If your computers take over large parts of that process,
51:36then, simply, human beings will be left out of the understanding process.
51:39The people who feed the things in at the beginning
51:42take the things out at the end.
51:45That, I think, is not, with us yet, a danger, seriously, in mathematics.
51:48But it's a potential danger.
51:51And with the glamour and attraction and obvious advantages of computers
51:54and the younger generation getting involved with them,
51:57one can see that it's a danger that's certainly on the horizon, if I may say so.
52:00Most mathematicians aren't concerned with just the solutions.
52:03They're interested in the problems themselves.
52:06It's not whether they're going to build a better automobile
52:09or a better computer.
52:12It's whether that is an important problem.
52:15In some sense, that is only really explicable
52:18to a mathematician.
52:21And mathematicians, particularly first-class mathematicians,
52:24are not interested in attacking problems, generally,
52:27unless they are of that character.
52:30Now, often, one of the things that makes them have that character
52:33is that some mathematicians have tried them,
52:36very good mathematicians, and have not been able to solve them.
52:39So you were mentioning six problems,
52:42Fermat's Last Theorem, Goldberg Conjecture,
52:45Riemann Hypothesis, and so forth.
52:48Now, what is it that's important about these things?
52:51One of the things that's very important about them
52:54is some damned good mathematicians have tried to solve them
52:57over a long period of time and failed.
53:00You need your reputation from then on.
53:03It doesn't matter.
53:06I mean, you can be a first-rank mathematician
53:09and only do that with one of the problems.
53:12And if you do nothing else for the rest of your life,
53:15Oxford will want you, Cambridge will want you,
53:18Harvard will want you, etc.
53:21These mathematicians of the future
53:24will have to live with the undecidable,
53:27with computer proofs,
53:30and with an ever more abstract mathematics.
53:33You said that the set, which you were saying was the set of all sets
53:36he had written in the book,
53:39then it would not be possible for him to have written that set in the book...
53:42Now, when you have mathematics becoming much more abstract,
53:45as has been the case in the last hundred years,
53:48then there is a very strong potential
53:51that some of it will become very recherché,
53:54some of it will become very abstruse and baroque.
53:57And what is to keep that from happening?
54:00In a way, the answer is probably nothing.
54:03Some of that is bound to occur.
54:06And the question is simply,
54:09how much of that will occur?
54:12And will it form a significant fragment
54:15of mathematics as a whole?
54:18I think, in a way,
54:21the answer to that
54:24is partly a very pragmatic one.
54:27The lack of funding alone
54:30will keep too much of that from happening.
54:33The other side of that
54:36is that what is actually important as mathematics
54:39and what is unimportant as mathematics
54:42is very time-relative.
54:45Despite the flaws,
54:49and the knowledge we possess,
54:52but it's manifestly a human enterprise.
54:55Mathematicians set the rules and decide the goals,
54:58even if, in the end,
55:01they seem unable to control
55:04exactly what they have created.
55:07Philosophically, it's quite important to know
55:10that mathematics, after all,
55:13is something which has to remain incomplete in some ways.
55:16We are quite satisfied with that.
55:19We don't want it to be complete.
55:22We are satisfied that it gives us
55:25a lot of very beautiful and important results.
55:28We don't care that it's not complete anymore.
55:31There are a lot of undecidable things.
55:34All right, we won't try to decide them.
55:37We'll look at other problems.
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