A Mathematical Mystery Tour (1985)

Name: A Mathematical Mystery Tour (1985)
Uploaded: 2025-09-06T15:07:18+00:00
Duration: 56 min 40 s
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Documentary, A Mathematical Mystery Tour (1985)

Mathematical #Mystery Tour
"A Mathematical Mystery Tour" is a 1985 episode of the PBS science series #NOVA, which first aired on March 5, 1985.
The episode explores the abstract and often counterintuitive world of pure mathematics, introducing viewers to concepts such as geometries in multiple dimensions, numbers larger than infinity, and parallel lines that can meet.
It features interviews with prominent mathematicians and historians of the time, including discussions on foundational mathematical theories and philosophical questions about the nature of mathematics.

The documentary delves into several major unsolved problems and deep mathematical ideas, such as Fermat's Last Theorem, the Goldbach Conjecture, the Riemann Hypothesis, the P=NP problem, and the Continuum Hypothesis.
It also examines the historical development of mathematics, touching on figures like Euclid, Bertrand Russell, and Kurt Gödel, and explores the impact of Gödel's Incompleteness Theorems, which demonstrated that no axiomatic system including arithmetic can be both complete and consistent.
The episode highlights the tension between the perceived certainty of mathematics and the reality of its inherent limitations.

The program was written and produced by Jon Palfreman, with narration by Don Wescott, and featured advisory input from mathematicians like Dr. Keith Devlin and Dr. Gregory H. Moore.
It was hosted by Bob Gardner, who has since praised it as one of the best popular-level programs on pure mathematics and its methods.
The #documentary #Mathematical

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00:00Welcome to the world of pure mathematics,

00:07where geometries exist in many dimensions,

00:10and numbers are bigger than infinity.

00:20It's a world where objects take on strange configurations.

00:25And if you thought that parallel lines never meet,

00:32you're in for a surprise.

00:39For over a decade, Bertrand Russell tried to find certainty

00:42through mathematics by reducing it to logic.

00:45In his massive work, Principia Mathematica,

00:48it took him 362 pages to prove that 1 plus 1 equals 2.

00:55Twenty years later, another mathematician, Kurt Gödel,

00:58proved that mathematics would never be completely certain.

01:04Do the abstract objects of mathematics

01:06have anything to do with the real world?

01:09Is mathematics the key that unlocks the universe?

01:25Do the abstract objects,

01:26do the abstract objects,

01:27do the abstract objects.

01:28The world sees the absolute right and the abstract objects

01:29of the science.

01:31The The Life of Mastemates

01:34These are some of the classical unsolved problems in mathematics.

01:44And they have resisted solution by the world's greatest mathematical minds.

01:50solution 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

02:14for 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, or are mathematicians

02:47in a world of their own?

03:00In 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

03:37in 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 world.

03:55Some of the finest mathematicians of today agree.

03:58A 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.

04:16What gives you the certainty about mathematics is quite different from what you have in physics

04:20or in chemistry or in biology.

04:21We 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 another

04:36one, is what is the sort of glue that holds mathematics together.

04:40Mathematicians are apt to regard the physicists as a somewhat lower category of being, who deal

04:46with 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.

05:25Newton 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.

05:34In 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 C.

05:50Experiments aren't 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 numbers.

06:05Geometry, the study of shapes.

06:10And analysis, the study of infinity, which includes calculus.

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.

06:31But ultimately you are left with numbers that cannot be divided any further.

06:37Two, three, five, seven, eleven.

06:44These are called prime numbers and they crop up at random, getting rarer as numbers get larger.

06:50But how could you prove this pattern would continue?

06:53The method of proof is a legacy of the ancient Greeks.

06:57Euclid proved that there are an infinite number of primes.

07:00Even now his proof is considered a model of mathematical thinking.

07:06It's final.

07:07That's right.

07:08It's final.

07:09Euclid's proof in number theory saying that there are infinitely many primes, this is final.

07:14It has been done 2,500 years ago.

07:17It'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 two, 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 be obtained

07:53by 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 Merton's conjecture.

08:14It, too, is about prime numbers and verified to 10 billion, which recently turned out to be wrong.

08:21As the British journal Nature reported, despite the apparently overwhelming evidence in its

08:26favor, the Merton's conjecture is false.

08:30Two mathematicians proved it would fail, sometime before it reached this enormous number, 10 to the power

08:3610 to the power 70.

08:38The proof is incredible because this number is an order of magnitude totally inaccessible to 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.

08:54But some amateur mathematicians have 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 perusing

09:21the writings of the ancient mathematician Diophantus.

09:29He 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 balance

09:42the 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?

10:09That night, Fermat wrote in the margin that it could, I have discovered a truly marvelous

10:15proof of this.

10:16But unfortunately, the margin is too narrow to contain it.

10:21Fermat often left his theorems unproven like this, and he usually turned out to be right.

10:26But when these words were discovered after his death, mathematicians began a quest for the

10:31proof 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.

10:45Even Gauss, the prince of mathematics, was defeated.

10:50As was Poincaré.

11:08This distinguished history of failure, had the amateur mathematician Arnold Arnold really

11:13succeeded?

11:14He had not.

11:15He had not.

11:16Within a short period of time, mathematicians had found the error, and the guardian had

11:21to apologize.

11:24The proof was wrong.

11:26That happens all the time, you know.

11:29Well, people make errors or jump at conclusions, you see.

11:33It's so painful to write down all the details of the 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.

11:52Usually you say, if there is an error, it's there.

11:57I'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, and this, of course,

12:20requires a proof.

12:22Writing a proof is really making yourself at the level of the other people, so to speak.

12:30Sometimes prize money is offered as a reward to those who succeed in solving mathematical mysteries.

12:39Paul Erdos, a famous Hungarian mathematician, travels the world over, offering his own money,

12:45and challenging other mathematicians to work on difficult problems.

12:49In fact, I was asked several times, what would happen if all your problems would suddenly be solved?

12:56How could you pay?

12:57I said I couldn't, but what would happen to the strongest bank if all the depositors would suddenly ask their money back?

13:04The bank would clearly be ruined, and I think it is much more likely that this will happen than all my problems will suddenly be solved.

13:13Mathematics can be done without proof.

13:26The 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.

13:43Their mathematics was practical and used for counting, for measuring land, and for weighing agricultural products like grain.

13:51The 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 so on, and such and such will happen.

14:08You will get a certain result.

14:10Now, there was no sense of generalization there.

14:14Always you find them referring to specific numbers, but it's clear that it's just as much a generalization

14:21as if you look in a cookbook and it says, take 2 cups of flour, add 1 quarter cup butter, and so forth.

14:26It's not referring to a specific 2 cups of flour.

14:29You can do it with any 2 cups of flour you find.

14:32And when it says, take 2, in effect it's saying, take any whole number and do this to it,

14:39and you will get a certain result.

14:42Two thousand years after the Egyptians, a Greek mathematician is said to have produced the first mathematical proof.

14:48He proved that the diameter of a circle divides it into two equal parts.

14:54Aristotle later formalized 14 rules of proof, and these were the guiding principles employed by Euclid in his treatise on geometry.

15:03Euclid's book, Elements, is the most widely circulated book in human history after the Bible.

15:16A Euclidean proof proceeds in logical, orderly steps to demonstrate the proof of particular propositions.

15:23The assumptions are open to question, but the line of reasoning is irrefutable.

15:31Euclid succeeded in deducing every known theorem in geometry from just 10 axioms.

15:37A straight line is whatever lies evenly with the points itself.

15:40Statements which everyone held to be true.

15:43And they were unquestioned for more than 2,000 years.

15:46But in the 19th century, mathematicians tried rewording some of them to see what might happen.

16:02Strange new geometries were the result.

16:14And these non-Euclidean geometries were just as valid 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.

16:34If you live in one dimension on a line, just like a road, cars travelling on a road,

16:38then the geometry there is quite different from what it is when you've got more degrees of freedom.

16:43If you're a ship on the sea, it's quite different from a car on a road.

16:47If you're an aeroplane in the sky, it's again quite different from a ship on the sea.

16:50You have more room in which to move around.

16:53And so new features can happen.

16:55New fields like topology contemplate bizarre objects, like the so-called Klein bottle.

17:02We usually think of a bottle as having both an inside and an outside.

17:07But a Klein bottle has only one side.

17:10One of the unexpected consequences of exploring the fourth dimension and beyond.

17:16The most important single idea in geometry is probably the idea about curvature, how things curve.

17:25And when you analyse that, you find that you need four dimensions in order to allow the curvature to have all its full ramifications.

17:31In more dimensions, nothing much alters.

17:34And on the other hand, having more dimensions enables you somehow to unwind things and become simpler.

17:38So it looks like the situations get more difficult between one, two, three and four dimensions.

17:43In some way, they get easier afterwards.

17:45That means four dimensions looks like it's the most interesting or difficult, exciting dimension to be in.

17:52It happens to be the dimension of space-time.

17:54And 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 in time.

18:30Time is an example of the fourth dimension.

18:33Objects in four or more dimensions may really exist, but because they are not part of our ordinary 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 of having a visual approach.

19:00And that's what happened, that's why mathematics has become more and more abstract,

19:05because abstractions pile upon abstractions, and there doesn't seem to be any end to it.

19:11Do the abstract objects of mathematics correspond to real things, or are they merely inventions of the mathematical mind?

19:26Would they still be there even if there were no mathematicians to discover them?

19:31Or do we invent them?

19:34The Greek philosopher Plato believed that all objects, no matter how abstract, exist in the universe, and that mathematics discovers them.

19:42This view is called Platonism.

19:44Indeed, widely different cultures have discovered the same mathematical truths independently, like Pythagoras' theorem.

19:52I am mathematically, I am a Platonist, and I believe that mathematical entities do exist outside the mind of the people.

20:04Of course, such a statement is difficult to justify, but at least the psychological feeling is that really you discover things and you do not invent them.

20:21When they're talking to each other, they will, 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 certain definite properties,

20:34and you can prove something about them, something quite explicit and definite.

20:39One mathematical object that seems to appear over and over again is the golden rectangle.

20:44You'll find it in Da Vinci's sketches, and in architecture, from classical to modern times.

20:53Why should the proportions of this rectangle be so satisfying?

20:57If you divide the height by the width, you get what is called the golden section.

21:01And this 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 1.618.

21:16Another example of how mathematical ideas correspond to real objects is visible in the symmetries of nature.

21:28Symmetry is something which everybody understands intuitively.

21:31You have figures like circles, triangles, squares, beehives, crystals, all of which exhibit something which we think of as symmetry.

21:41But how to understand this mathematically seems a very difficult question.

21:47If you take a square and you can turn it through 90 degrees, it still looks like the same thing.

21:54If you take an equilateral triangle, and you rotate it to the right, about its middle, to the right number of angles,

22:00the points will move around, and it will again get the same figure.

22:03So, symmetry has to do with the ways in which you can move things such a way that they reproduce themselves.

22:10And that's the basic idea that mathematicians have tried to abstract.

22:14It doesn't matter what it is that's being involved, so long as you can consider the different 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 and their names, that would be a symmetry.

22:29This star illustrates the mathematician's way of describing symmetry.

22:35The 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 corresponds to the second position.

22:48Mathematicians 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.

23:03The 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 in all mathematics.

23:43Galois was only 21 years old, and this was the last act in a tragic life.

23:50That morning he was killed in the duel.

23:56During the decades that followed, mathematicians have been able to perceive the many ways in which the universe is symmetrical.

24:03By the 19th century, mathematicians had entered a realm of infinite possibilities.

24:13The question of infinity has captivated the human imagination for thousands of years.

24:28Infinity is used to describe that which goes on forever.

24:32And that's characteristic of a group of numbers called irrational numbers.

24:37If you divide the circumference of a circle by its diameter,

24:40you get one of the most famous numbers 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,

25:06what are the odds that a pin will hit a line?

25:08The answer turns out to involve pi.

25:16Or this series.

25:17If you were to continue it forever, it would add up to one quarter of pi.

25:30Throughout history, many have tried to measure it.

25:32The ancient Egyptians calculated pi to 3.16, and the Greeks, by the time of Archimedes, had established its value to 22 over 7.

25:46But that's only an approximation.

25:49In decimal form, pi is infinitely long, and its digits never repeat the same sequence of numbers.

25:54It's called an irrational number.

25:57By calculating pi to 39 decimal places, you have more than enough to measure the size of the universe with remarkable accuracy.

26:05Earlier this year, two Japanese scientists calculated it to more than 16.5 million decimal places.

26:12These are the last known digits of the longest calculated number in the universe.

26:18This is a prime number.

26:24It can be written in a special way.

26:262 to the power 31, minus 1.

26:29This abbreviation was invented by the French friar, Mersenne.

26:35And you can see how it works using a checkerboard.

26:38Pile two checkers on the first square, doubling the checkers as you go.

26:422 to the 2.

26:432 to the 3.

26:442 to the 4.

26:46The numbers would soon get very big indeed.

26:492 to the power of 64 would form a pile 37 million million kilometers high, enough to reach the nearest star.

27:04Now compare it to the number Brian Tuckerman found in 1971 and proved to be a prime number on his IBM 360 computer.

27:12But then, two 15-year-old high school students from California did better.

27:19They found this prime number.

27:22The current record holder is David Slowinski.

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 method.

27:39Without this shortcut, such a calculation would have taken longer than the age of the universe.

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.

28:03A whole number, either 1, 2, 3, 4, 5, 6, 7, 8, and so forth.

28:08Every day, you allow one guess as to what that number is.

28:13If and when you guess the number, you go free.

28:16Professor Raymond Smullyan is using a classic story to illustrate a startling discovery by mathematicians that there are numbers bigger than infinity.

28:26In the 1880s, the German mathematician George Cantor developed a way of comparing the sizes of infinite sets of numbers by pairing them one by one.

28:35For example, take the set of whole numbers like 1, 2, 3, and so on.

28:42There is an infinite number of them.

28:45Compare it to a set of these numbers.

28:49Would a collection or set of every tenth number be only one tenth as large?

28:54By 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 it that is 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 and when you guess it, you go free.

29:44By pairing the set of fractions with the set of whole numbers,

29:46surprisingly, Cantor found they 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

30:01as either whole numbers or fractions.

30:04There are more of them in the tiniest line segment

30:07than there are counting numbers in the universe.

30:09These other numbers are the irrational numbers, numbers like pi.

30:29And by pairing them off with the whole numbers, Cantor proved there would be some irrational numbers left over.

30:34The set of irrational numbers was genuinely bigger than infinity.

30:40It meant that beyond infinity, which he called aleph zero,

30:44there were larger infinities, aleph one, and aleph two, and still bigger ones beyond.

30:50Cantor's ideas were received with skepticism at first, but eventually they evolved into an important subject now called set theory.

31:03If we do, if 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 infinite.

31:21For us, this has disappeared because the infinite for us has become just simply a formal kind of,

31:35we have an axiom of the infinite, but we don't attach to it a material meaning.

31:41I mean, we don't claim that there are objects, infinitely many objects of some kind somewhere.

31:48That's why people, I think, were so reluctant to accept infinity,

31:54because they said if the infinite exists, there must be infinitely many things somewhere,

31:58and then they went into contradictions.

32:00Other mathematicians were inspired by Cantor's work and applied his new ideas to the whole of 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 remarkable.

32:24To reduce the whole of arithmetic to something he believed more fundamental, logic,

32:29a subject which had been around since Aristotle.

32:32Now, in Aristotle, you have essentially the notion of syllogism,

32:37an example of which would be all men are mortal, Socrates is a man, therefore Socrates is mortal.

32:45And 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 this sort,

33:04that there are kinds of reasoning there that need some other modes to describe it,

33:10and that these modes can be made very precise, and that, given that precision,

33:17arithmetic will then be found to be part of logic.

33:20At this point, Bertrand Russell enters the story.

33:24Well, just as the second volume of Frege's master work on this subject was about to appear in print,

33:33Bertrand Russell, who at that time is a rather young and not that well-known philosopher,

33:40writes to Frege saying that he's read some of his work and found it very interesting,

33:45and, by the way, he's found this paradox which he's not able to solve.

33:50Russell's paradox concerns set theory, but can be told as a story about a librarian who was ordered to compile a catalogue of every book in her library.

34:01As she's finishing, she's struck by a thought. Should she include the catalogue itself in the catalogue? It is, after all, a book.

34:16She decides not to.

34:20The National Librarian receives such catalogues from all the libraries in the country.

34:26Some, yellow, where the librarian has listed the catalogue itself.

34:31Some, blue, where they haven't.

34:36He now has the awesome job of compiling a master catalogue of the blue ones, the catalogues which don't list themselves.

34:44But on thinking about it, he realizes it is impossible, because what does he do with the master 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 don't list themselves.

35:02Why should Russell have thought this paradox so important?

35:05It was because the most general way of thinking about any mathematical object was in terms of collections or sets of them.

35:16A catalogue of books is, in principle, no different from a set of numbers.

35:20Ironically, the effort to be logical was leading mathematicians not to certainty, as they had come to expect, but to uncertainty.

35:28The ideas of both logic and sets were so fundamental to mathematics that to run into such a contradiction at this level of mathematics was very worrying.

35:41The whole enterprise might be built on sand.

35:44Now, Frege was absolutely devastated by this and regarded it as, in essentially, destroying his life's work.

35:52Frege and Russell then corresponded, and Frege put forth various possibilities of a solution, as Russell did as well.

36:02But Frege was never the same after that.

36:06Russell, however, remained optimistic that his paradox could be resolved, and that logical certainty would be restored.

36:13For the next decade or so, with Alfred North Whitehead, he labored to produce the Principia Mathematica.

36:23This massive work sought to deduce all of mathematics from basic principles of logic.

36:28It takes a while to get going.

36:33Some 362 pages before they could prove that 1 plus 1 equals 2.

36:42Professor Grattan Guinness.

36:45Nobody had done anything on the scale of the detail that this Principia Mathematica constitutes.

36:51I mean, 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 he must have had these mounds of manuscript all over the place.

37:12That sort of thing can happen.

37:14Oh, dear, you make a slip proving Proposition 47.275.

37:17Have you made the same slip anywhere else?

37:21I mean, you could easily spend a morning checking things like that.

37:24I can understand exactly how it must have broken in producing this thing.

37:29Russell himself only intermittently worked on it thereafter.

37:34In fact, he said it broke him.

37:36Intellectually, he wasn't as sharp after it as he had been before.

37:42But was the scheme a success?

37:43What 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.

38:04Russell himself wrote,

38:05I wanted certainty in the kind of way in which people want religious faith.

38:11I thought that certainty is more likely to be found in mathematics than elsewhere.

38:16And after some twenty years of very arduous toil,

38:20I came to the conclusion that there was nothing more that I could do.

38:23In France, the response to Russell's program of logic was a series of mathematical volumes published under the name Bourbaki.

38:39For centuries, France boasted one of the finest mathematical traditions in Europe.

38:44The line was broken when the outbreak of the First World War interrupted intellectual pursuits.

38:51After the war, a group of top-level French mathematicians met together to revive the tradition,

38:58with an outrageously ambitious project, to rid all mathematics of any and all paradoxes.

39:03They named themselves Bourbaki.

39:09Bourbaki, you know, was one of Napoleon's generals, I believe,

39:11and they took his name simply as a sort of pseudonym for this body of French mathematicians.

39:17Monsieur, je propose que nous commencions tout de suite...

39:20They work in a very interesting way.

39:21I was once actually attended one of their meetings,

39:23and they get together as a collective group and write their books together.

39:27And they go through the manuscripts page by page, discussing in great detail.

39:30They're very, very, well, pedantic, one might say, very careful.

39:36And they've taken enormous pains over this.

39:39One might say almost too much sometimes.

39:41They've taken 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,

39:56were held up for close scrutiny.

39:57They stressed the need for correct terminology and language.

40:03It was a very formal approach to mathematics.

40:07We, French people, we have been brought up in tradition that we should respect the language.

40:17We should have considered that the French language has been evolved through centuries to a point where it's a good tool and it has really qualities of its own.

40:31And to misuse it is very, we consider it as a kind of sin.

40:35But others, like René Thaume, express some skepticism of Bourbaki's influence.

40:42Merci, et je vous plaît de m'excuser.

40:44If I were asked what is perhaps the main achievement of Bourbaki in mathematics, I would say it has made clear the distinction between the circle and the disc and between the sphere and the ball.

41:01And that's very important in my view.

41:07The need for precision and rigor in language was of great interest to Bourbaki, and their published works are of some influence even today.

41:15Meanwhile, David Hilbert, one of the greatest mathematicians of the early 20th century, was also worried about paradoxes.

41:26Hilbert thought that the paradoxes had occurred because mathematicians were trying to describe mathematical ideas using ordinary language.

41:35To him, mathematics could only be properly described in abstract symbolic terms.

41:40It's a kind of thing completely wrong to say that I can imagine what ten to the ten means.

41:50For me, it's just simply a system of figures, but it has no actual meaning.

41:56Number three has a meaning, because I can immediately see three objects.

42:00But number ten to the ten has no meaning at all except from the meaning you give it through the axioms.

42:06Hilbert's ideas can be understood if you compare mathematics with a game of chess.

42:13When humans play chess, they use their intuitions, they form mental pictures, they are emotional.

42:20But the game of chess itself can be described in completely abstract terms, so that a machine can play it.

42:26A microcomputer has no intuition or mental images about the pieces.

42:33The axioms of chess, the rules, are sufficient to define all the possible moves.

42:39And if the program's been written correctly, the machine will never break the rules.

42:44In 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 manipulated mechanically.

43:00He hoped that by taking the meaning out of mathematics, by sticking rigorously to the rules, he would free it once and for all from paradox and contradiction.

43:12Historically, it's quite true that most mathematics come from questions or theories which had even a physical meaning.

43:26If you take partial differential equations, say for instance, all right, they are very abstract, but most of them come from concrete problems of physics or mechanics or that kind of thing.

43:38But once they have been translated into pure mathematics, that is deriving from an axiom system, their meaning is irrelevant.

43:48We have to work according to the assumptions we made on them. That's all.

43:53The price of rigor could be taking the meaning out of mathematics, a price that some mathematicians feel is just too high.

44:01Rigor has nothing to do with interest.

44:03There are a lot of interesting things which are not rigorous and a lot of rigorous things which are devoid really of any interest.

44:13In fact, one of my favorite, Maxime, is in French, tout ce qui est rigoureux est insignifiant.

44:19Anything which is rigorous is meaningless. And this is in some sense the, I would say, the philosophy of the Hilbert's, of Hilbert's program pushed to the extreme.

44:33You achieve rigor only through meaninglessness.

44:36You achieve rigor only through meaninglessness.

44:40By 1930, mathematicians from Russell to Hilbert were trying to restore certainty to mathematical reasoning.

44:54But a young mathematician was to shock them all by proving that it could never be done.

44:58In 1931, an Austrian mathematician, Kurt Gödel, published a theorem in logic which demolished Hilbert's program to resolve contradictions.

45:11Gödel's incompleteness theorem showed that mathematics would always remain plagued by paradoxes of a sort.

45:18There would always be questions that mathematics could not resolve.

45:21Consider this set of principles.

45:24Gödel proved that neither this set nor any other set would ever be completely adequate to decide all mathematical questions.

45:31Hilbert would never accomplish what he set out to do.

45:36This is completely demoralized or undermined this whole program of laying the foundations of mathematics.

45:43Well, of course, there's a lot of discussion going on ever since about what the foundations of mathematics are and how you should set them up.

45:48But because this initial program failed, most working mathematicians take a more pragmatic attitude.

45:55They say, well, if we can't achieve ultimate certainty about mathematics by providing foundations,

45:59that's no reason for us to stop doing mathematics.

46:02Physicists get along quite happily, although their foundations are much shakier than ours.

46:06So most mathematicians go along quite happily with their mathematics,

46:10even 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.

46:21One example from the 19th century is the continuum hypothesis.

46:25Mathematicians have been unable to prove it true or false.

46:29It has a third status. It's undecidable.

46:32It's undecidable.

46:34You cannot prove the continuum hypothesis.

46:35You cannot disprove the continuum hypothesis from the present axioms of set theory.

46:40Now, some very formalistic kind of mathematicians say, well, 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, they want to know,

47:03the next day the army is 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, well, 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.

47:16They want to know whether it really will stand the weight.

47:20And so-called mathematical Platonists or classicists, which most working mathematicians are, I certainly am,

47:27regard the continuum hypothesis as definitely true or false, we just don't know which.

47:32Just because it's undecidable on the base of the present axiom doesn't mean that it's neither true nor false.

47:39It means that the axioms, the present-day axioms of set theory don't tell us enough about sets to decide the question.

47:47The 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 it's, I should say for us, most mathematicians think it's quite beneficial,

48:04because just simply tell them, don't go and try that problem.

48:08You're wasting your time.

48:10Because before that, lots of people were working on, say, his continuum hypothesis,

48:16and at times thinking they would solve it.

48:20I was a French mathematician, not very good, who spent all his life working on the continuum problem

48:25and turning out new proofs of the time, and 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.

48:37Well, it's much better to have to know in advance that it's hopeless.

48:41As far as we know, these great unsolved problems are not undecidable, though several dozen others may be.

48:50Far from making mathematics certain, logic as we know it has exposed its limitations,

48:55and logic has had another unforeseen consequence.

49:03The use of logic is still perhaps our best means for arriving at answers to real and abstract questions.

49:10In fact, symbolic logic turned out to be the vital ingredient in the wartime development

49:15of what's become the most dynamic technology of our time, the computer.

49:19And just as the computer has changed everything else, it may change mathematics.

49:30Up until 1976, the four-color problem was unsolved.

49:36This century-old problem requires a proof that four colors are sufficient to color any map on a plane or sphere,

49:44so that no two adjacent countries have the same color.

49:46This must be proved, not just for the maps we see in atlases,

49:51but for any conceivable map that could ever be drawn, like this.

49:56And in fact, it has been proved that four colors are enough.

50:00What was new was that it was done on a computer.

50:03The computer participated by drawing thousands of configurations.

50:07But is this a proof?

50:08That's the first example, perhaps, of a computer being used in improving something in mathematics.

50:19And it's pretty clear that with the advent of more and more powerful computers, this is going to happen more and more.

50:23I think that there are advantages and dangers there, and on the one hand, it's clear that mathematicians shouldn't reject new tools,

50:32and if a computer can help them to get better understanding of anything, they should obviously, and will, use it.

50:38On the other hand, if we get computers to come and verify certain things for us, then we don't achieve the same level of understanding.

50:45We simply say the computer told us that this was true.

50:49First of all, one could ask whether really a proof made by computer, by computing on a computer, is really a proof.

50:58Because, after all, if the computer makes a mistake, then the proof is mistaken.

51:03So, in that respect, proofs realized by computing ways should not be considered strictly as proofs.

51:16And to my taste.

51:18We're not simply in the business of mathematicians to get answers.

51:22We want to understand.

51:24And understanding is something that you can't do, can't achieve, unless you're actually involved in the process all the way through.

51:31If your computers take over large parts of that process, then simply human beings will be left out of the understanding process.

51:38They will simply be the people who feed the things in at the beginning, take the things out at the end.

51:41And that, I think, is a, it's a, it's a, not with us yet a danger, seriously, in mathematics.

51:46But it's a potential danger.

51:48And with the grammar and attraction and obvious advantages of computers and the younger generation getting involved with them,

51:54one could see that it's a danger that's on the, certainly on the horizon, if I may say so.

51:58Most mathematicians aren't concerned with just the solutions. They're interested in the problems themselves.

52:03It's not whether someone else thinks that they're important or not. It's not whether they're going to build a better automobile or a better computer.

52:11It's whether that is an important problem in some sense that, that is only really explicable to a mathematician.

52:18And the mathematicians, particularly first-class mathematicians, are not interested in attacking problems, generally, unless they are of that character.

52:29Now, often, one of the things that makes them have that character is that some mathematicians have tried them, very good mathematicians, and have not been able to solve them.

52:38So you were mentioning six problems, you know, Fermat's Last Theorem, Goldberg, Conjecture, Riemann Hypothesis, and so forth.

52:47Now, what is it that's important about these things? One of the things that's very important about them is some damned good mathematicians have tried to solve them over a long period of time and failed.

52:57So that if you can solve them, and you're a mathematician, you have made your reputation from then on.

53:02It doesn't matter. I mean, you can be, you can be a first-rank mathematician and only do that with one of the problems.

53:09And if you do nothing else for the rest of your life, Oxford will want you, Cambridge will want you, Harvard will want you, etc.

53:16So four does not belong to the set listed on page four. So four does go into that set.

53:22These mathematicians of the future will have to live with the undecidable, with computer proofs, and with an ever more abstract mathematics.

53:30Well, you said that the set, which you were saying, was the set of all sets he had written in the book.

53:36Yeah.

53:37Then it would not be possible for him to have written that set in the book because...

53:40Now, when you have mathematics becoming much more abstract, as has been the case in the last hundred years, then there is a very strong potential that some of it will become very recherché,

53:55some of it will become very abstruse and baroque.

54:01And what is to keep that from happening?

54:06In a way, the answer is probably nothing.

54:08Some of that is bound to occur.

54:10And the question is simply, how much of that will occur?

54:13And will it form a significant fragment of mathematics as a whole?

54:20I think, in a way, the answer to that is partly a very pragmatic one.

54:25The lack of funding alone will keep too much of that from happening.

54:31The other side of that is that what is actually important as mathematics and what is unimportant as mathematics is very time relative.

54:43Despite the flaws, the mathematical tradition has led to the most certain body of knowledge we possess.

54:49But it's manifestly a human enterprise.

54:53Mathematicians set the rules and decide the goals.