Colours Clouds and God_3of4_Is God a Number
Is God a Number? Maths that mimic the Mind (1999).
"Is God A Number?" is an account of the science of mathematics and its connection to mind and consciousness. If mathematics underpins the elegant precision of the macroscopic and microscopic worlds, is there a Master Mathematician as well? This fascinating film examines the computational paradigms being used to model human consciousness and to quantify reality, from Euclidean geometry to fractal transform algorithms. Oxford mathematician Sir Roger Penrose, quantum physicist Reverend John Polkingorne, compression technology expert Michael Barnsley, and physiologist Horace Barlow seek to understand how the brain functions--and grope for evidence of a guiding force. The film looks at the mystery of consciousness, whilst exploring the links between mathematics, the mind and the physical, observable universe. Computer graphics enhance this exploration of inner and outer space.
"Is God A Number?" is an account of the science of mathematics and its connection to mind and consciousness. If mathematics underpins the elegant precision of the macroscopic and microscopic worlds, is there a Master Mathematician as well? This fascinating film examines the computational paradigms being used to model human consciousness and to quantify reality, from Euclidean geometry to fractal transform algorithms. Oxford mathematician Sir Roger Penrose, quantum physicist Reverend John Polkingorne, compression technology expert Michael Barnsley, and physiologist Horace Barlow seek to understand how the brain functions--and grope for evidence of a guiding force. The film looks at the mystery of consciousness, whilst exploring the links between mathematics, the mind and the physical, observable universe. Computer graphics enhance this exploration of inner and outer space.
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00:00For centuries, astronomers and mathematicians have gazed in awe and awe-inspired awe at
00:30the heavens, seeking to discover if there's a master plan.
00:36From the finite to the infinite, from the smallest subatomic particles to the entirety
00:41of the universe in which we live, there is evidence of the work of a master mathematician.
00:48For many of our greatest natural philosophers, like Sir Isaac Newton, to study the mechanics
00:54of nature was to study the work of the Creator.
00:58There could be no difference between science and religion, because there was only one truth.
01:04The universe, and all the ways in which it works, was an act of God, not some random
01:09act of chance.
01:11And at its heart was mathematics.
01:24Every atom in our bodies was created in a nuclear cauldron, within a star, millions
01:37of years ago.
01:39It's true to say that we are the children of the stars.
01:47We are all living beings.
01:49We are born, we exist, and then we die.
01:53But what makes our presence remarkable is our consciousness.
01:58We are aware of our own existence.
02:03Despite all our great scientific achievements, though, we cannot explain our own consciousness.
02:09In fact, the brain operates automatically.
02:13Its intricate processes seem to defy analysis.
02:17How does the brain receive, process and store the enormous amounts of information it has
02:23to deal with?
02:25And how does the brain interpret this data into feelings, like sadness or joy?
02:32Just as it's used in examining the way the universe works, so mathematics is becoming
02:37a major tool in understanding how the brain functions.
02:42The consequences for mankind are staggering and will influence our whole appreciation
02:47of the power of mathematics and the understanding of our consciousness itself.
02:54Michael Barnsley is a world-renowned mathematician.
02:58His work has led him to gain a deep understanding of the relationship between mathematics and
03:03the brain, the human mind and its perception of the physical world.
03:08As we have tried over the centuries to describe precisely and accurately what things are,
03:16to try to get to the bottom of things, so mathematics has emerged as our most exact,
03:22precise, logical description of the physical, observable universe.
03:29But even mathematics is limited in its power.
03:33There are things that mathematics cannot describe.
03:37And one of these things is consciousness.
03:41So, what do leading mathematicians and scientists already think about consciousness?
03:48We know very little about what kind of structures produce consciousness.
03:53We know that human brains do, but what it is in a human brain that makes it conscious
03:58is something quite mysterious to us.
04:00There's a tremendous gap between any talk about neural networks, the firing of synapses.
04:05It's interesting and important that that is.
04:07And our simplest mental experiences of, say, feeling hungry or perceiving a patch of pink.
04:12And at the moment, we haven't the slightest idea of how to bridge that gap.
04:15Consciousness is not something which you can understand in terms of simply computational models.
04:22There's a prevalent view these days that computers, if they get to a certain level, would be conscious.
04:27I don't think that's right at all. I think there's something else involved.
04:30And the reason I believe this has to do with our understanding of mathematics.
04:34And the understanding of mathematics can be shown to be something which is simply not a purely computational activity.
04:41There's something else involved.
04:43We know an awful lot about when people are conscious and when they're not conscious.
04:47But I don't think there's any scientific explanation for the actual experience I have
04:53when, for example, I look at flowers and see colors, shapes that tickle my fancy.
04:59All those subjective aspects, I don't think there's any scientific explanation for that.
05:06What Michael Barnsley has now discovered, though,
05:09is how to make computers actually see images in a way that mimics the human brain.
05:16Our brains produce images of far higher resolution than the signals recorded by our eyes.
05:22Barnsley's mathematics mimic our minds by capturing the fundamental patterns of nature.
05:28His work has already resulted in a radical approach to interpreting and delivering images with computers.
05:35His mathematics could revolutionize how we store, retrieve,
05:39and even how we fundamentally think about manipulating information electronically.
05:45Barnsley's discoveries in data compression technology cleverly define reality in mathematical terms,
05:51delivering pioneering theories and major practical applications.
05:55His work was used to compress the 7,000-plus colour images in the CD-ROM multimedia encyclopaedia Encarta.
06:04Michael Barnsley's fantastic voyage of discovery into mathematics
06:08will enhance not only the development of computers,
06:12but also our understanding of our consciousness,
06:17maybe even bringing us one step closer to knowing the mind of the master mathematician.
06:26But how far can mathematics go now in helping us in our interpretation of reality
06:31and in our quest to make sense of our world?
06:35Let's look first at the nature of space.
06:39What is space made of? How do we describe this nothing stuff?
06:44Not the ferns, but the spaces between, where the air is and where it is not,
06:50these smaller and smaller bits of space between the fronds.
06:54It seems limitlessly vast and infinitely small.
07:00How do I connect this nothing, this empty space out here, this part of reality,
07:05with the inner space of my intuition, of my imagination,
07:09that place where the dreams occur and one sees those things that are not of this world,
07:15but in the mind where abstract ideas happen?
07:18How do we think of this stuff?
07:20How do we connect it to the inner space of our minds?
07:25Well, the way I like to look at these issues is really in terms of three worlds
07:30and the three mysteries which connect the worlds.
07:34One of the worlds is the physical world,
07:36the world of physical objects like trees and so on.
07:39And there is a mystery about why it is that physical objects behave
07:44so precisely in accordance with mathematical laws.
07:48The more deeply we look into the behaviour of things,
07:51the more we find that it's mathematics which governs the behaviour of things
07:56and the precision of that mathematics is absolutely extraordinary.
08:00This rock, what is it that I am touching?
08:03What is it made of?
08:05Atoms. And what are they made of?
08:08Mainly empty space with tiny electrons orbiting nuclei made of protons and neutrons.
08:15These things are both vibrant energy like light and simultaneously particles
08:21bound together by a mathematical equation called the Schrodinger equation.
08:27As we make more and more precise experiments to try and discover
08:31and say very precisely what matter is made of,
08:34we bump into the language of mathematics.
08:37Real things seem to become abstract concepts,
08:40seem to become the stuff of consciousness itself.
08:45Then there's the mystery of why it is that mentality arises
08:51when you have the right kinds of physical structure.
08:53Human brains are physical objects, yet they are also objects.
08:58Human brains are physical objects, yet they seem to evoke this third world,
09:04the mysterious mental world.
09:06And it's that mental world whereby we access the world of mathematical absolutes.
09:12There's a world that we explore through our conscious mathematical abilities,
09:18a world of already existing entities, I think,
09:20things like the Mandelbrot set, very complicated things
09:23that surely didn't come into existence when human beings first began to think about them,
09:27which have always been there in some sense,
09:29and we have access to it through our conscious powers of thinking,
09:32a particular sort of thinking, of course, the precise thinking of mathematics.
09:36Some people might take the view that even those mathematical notions
09:40are things which emerge out of mentality.
09:42So we have, in a sense, a rather mysterious triangle,
09:45the triangle of how it is that the physical world
09:48seems almost to emerge out of the mathematical world,
09:51how it is that our mentality seems to emerge out of the physical world,
09:55and the third mystery, how it is that the mathematical world
10:00almost seems to emerge out of our mentality.
10:03So here we have a kind of paradox, the three things,
10:05each one seeming to come from the one before.
10:08I'd like to present it in this way as a kind of additional mystery, if you like,
10:12just to emphasise that there is something deep there that we really don't understand.
10:16Maybe there's just one thing which has these different aspects to it.
10:21More than 2,000 years ago, the learned Greeks conceived
10:25that true reality was mathematical in nature.
10:29Euclidean geometry, expounded in the 4th century BC,
10:34provides a theory of the physics of space.
10:38It provides a very accurate description of the world in which we live.
10:43Euclid reasoned on figures drawn in the imagination, or on papyrus.
10:48He began by thinking about statements so obvious and intuitive
10:53that no-one would dispute them.
10:55They were called axioms.
10:57They were really obvious stuff.
10:59Statements like...
11:01Given two points, there is a line that joins them.
11:05A line can be prolonged indefinitely, endlessly, without limit.
11:12If you have a line and a point not on it,
11:14then there exists a line through the point that does not meet the first line,
11:18called a parallel line.
11:22Really obvious, right? And we can all agree on them.
11:25They provide the glue that holds our models of reality together,
11:29make a given model self-consistent and able to make predictions.
11:34Using these axioms, the ancient Greeks made an imagined,
11:38perfect mathematical space of points and lines.
11:42They imagined a plane, flat, a mirror smooth,
11:46a microscopically thin, endless, vast sheet going on forever and ever.
11:52This flat, abstract place is called the Euclidean plane.
11:56It is stretchable.
11:58We can shrink or expand it without limit.
12:02This magical plane made possible the introduction of the idea of measurement.
12:08Here is a simple measuring stick.
12:10Think of it as a line segment in the Euclidean plane.
12:15Then we can shrink the measuring stick to produce two smaller sticks,
12:19each one a half the length of the original.
12:23The measuring stick is made of two smaller copies of itself.
12:28Space is subdivided, yet still the same.
12:31We can picture this process repeated over and over.
12:35Using this device, we can express distances in the Euclidean plane.
12:40We have a way of giving addresses, that is, coordinates,
12:44pairs of decimal numbers to points in the plane.
12:47This is the basis of measurement itself,
12:50the measurement of length and indeed of most things that we can measure.
12:55What is exciting to me is how, in reading the scale on a measuring stick,
13:01we are able to make an objective connection
13:04between the outer reality and the Euclidean plane
13:07using the coarse, blurry, finite resolution window that is the eye.
13:14Using the Euclidean plane, we are able to make predictions about reality.
13:20Let us think for a moment about an obvious sort of prediction that we can make.
13:26Space is made of points.
13:28Where is a point in the real world?
13:31Euclidean geometry predicts that we can specify a point,
13:35a specific location, to infinite precision.
13:40Here is the satellite view of where I am standing.
13:43It gives my coordinates as longitude 84.43, latitude 30.95.
13:52Here we have zoomed in closer and closer still.
13:56Now you can see me.
13:58The point is right here where I am standing.
14:01What are its coordinates now?
14:03Longitude 84.43, 794.
14:08Latitude 30.95, 692.
14:13But where exactly are we talking about?
14:16This point right here.
14:19We are zooming in closer and closer,
14:21dividing up space and specifying the point to greater and greater precision.
14:27But there are limits to the power of our instruments
14:30to magnify and observe the smallest particles in space.
14:34However, in the abstract world of the Euclidean plane,
14:38the mathematician can go on and on forever,
14:42specifying the exact position of a point with greater and greater accuracy.
14:48The reason why many of the things predicted by Euclidean geometry
14:53are logically true is profound.
14:56It is that little bits of space are like big bits of space
15:00if you magnify them up or move them around.
15:04Space is the same everywhere at all sizes.
15:09To capture this idea, we need the idea of a transformer.
15:14This is a new idea.
15:16It's not an electrical transformer.
15:19A transformer is a revolutionary concept in mathematics.
15:23It's a virtual entity.
15:25This means it can only exist in our minds.
15:29It's not real. We can't touch it.
15:32But its consequences in equations allows us to do amazing things.
15:38It can move objects around the Euclidean plane in very special ways.
15:44This transformer moves everything on which it acts
15:48closer and closer towards a fixed point which doesn't move.
15:52At the same time, the transformer makes the object it's moving progressively smaller.
15:59This is the basic principle behind transformers.
16:04Transformers are very special and surprising.
16:07Here's a different transformer.
16:09The repeated application of this transformer causes the initial leaf
16:13to be rotated and shrunk along a spiral path ending at the fixed point.
16:20This transformer deforms a butterfly's wing,
16:23turns it upside down, makes it smaller and moves it.
16:27Repeated application of the transformer makes a beautiful pattern.
16:33Amazingly and stunningly, with only a few of these transformers,
16:38we are able to describe geometrical objects
16:41and make deep predictions about reality.
16:44Indeed, one point of view is that it is these transformers
16:48that form the basic geometrical elements
16:51rather than the lines, triangles and so on.
16:56Here's another example.
16:58Look at this line.
17:00Along comes a transformer.
17:03It shrinks the line, creating a new one.
17:06It's exactly the same as the original, except it's smaller.
17:11Now, this transformer makes another new line.
17:15We are going to discover something wonderful that we can do with these two transformers,
17:20the red one and the blue one.
17:22Now what happens when we adjust these transformers?
17:26We get the original line out of the two small ones.
17:30It's the same as the two transforms of itself.
17:36Is this deep?
17:38Yes, this is what enabled our measuring stick to work.
17:41We can add up lengths.
17:43Note that one cannot actually see a line segment
17:47any more than one can see a distant star.
17:50But we can make synthetic pictures
17:53to examine and approximate ever closer to a line segment
17:57using instruments such as computer graphics devices.
18:01Geometrical objects are in many ways more certain than any star
18:06because of their timelessness, their precision, their universality
18:11and our detailed knowledge of them.
18:15A very practical prediction about reality was Pythagoras' theorem.
18:20It is one of the great and timeless discoveries of those old Greeks.
18:25It will remain true 2,000 years in the future
18:29regardless of our changing models of reality.
18:32It was built from the axioms, using the glue of logic and reason.
18:38Let's see.
18:40Three units from this first point to here,
18:43and at right angles to it, four units to this tip.
18:47So how far is it from the first point to the tip?
18:51Well, our scientific prediction, using Pythagoras' theorem,
18:56is five units because three squared plus four squared equals five squared.
19:01And our prediction is correct. How amazing!
19:05Well, the idea of using a model to make predictions
19:09that you then go along and test is the basis of science.
19:13Using Pythagoras' theorem and other predictions,
19:16the ancient Greeks validated their first science.
19:20They succeeded in establishing a connection
19:24between the inner world of the imagination and the outer reality.
19:31We've grasped it. We have a model.
19:34This stuff out here, this emptiness, is described inside our heads.
19:39We have an inner space like the outer reality.
19:43We have sensed reality through the clumsy instruments of our vision
19:47and got it inside us.
19:49Our abstract inner ideas match our perceptions of reality.
19:56As it was perfectly logical, Euclidean geometry seemed to offer
20:00the perfect way of looking at the world in ancient times.
20:04However, as a deeper understanding of mathematics developed,
20:08Euclidean geometry proved not to be a perfect match to reality.
20:12Albert Einstein had another idea.
20:16Using very sophisticated measuring techniques,
20:19space was found to be gently curved.
20:22In fact, during a total eclipse of the sun in the First World War,
20:26scientists, working together from nations actually at war,
20:30proved Einstein's theory was correct.
20:33Here, the quest for universal truth transcended human failings.
20:38Today, we know that space is slightly warped.
20:43But such mismatches between physical reality
20:46and the Euclidean abstract reality
20:48do not make either place any less real.
20:52Way back, over 2,000 years ago,
20:55the Greeks thought that their geometry was the major key
20:59towards understanding the language of the universe.
21:02The Greeks presumably thought this was what geometry was.
21:06And if you like, the reason they thought that
21:09was because space, in fact, accords with this geometry
21:13to an extraordinarily precise degree.
21:16So the fact that they somehow thought this was the only geometry
21:20was related to the fact that they were trying to reveal
21:23something out there in three-dimensional space,
21:27which they realised did accord to this precise mathematical notion
21:33that they were developing.
21:35The Greeks were excited by the abstract place they had found.
21:40The Pythagoreans thought that the physical world
21:43was ordered according to pleasing mathematical relationships
21:47and developed a mystical view of reality.
21:50I think of Pythagoras as a wizard, drawing pentagons in the sand
21:55and singing out the magical significance of numbers.
21:59He believed that all was mathematics, equation and number.
22:04How extraordinarily modern this idea has turned out to be.
22:08Most of modern physics validates this idea.
22:11When you try to pin down what reality is made of,
22:14it gets away from you.
22:16It more and more seems as if all material things
22:19and our very selves are made of mathematical stuff.
22:26Look at this computer graphics image.
22:28What is it made of?
22:30Essentially it's made of lots of little triangles
22:33with imaginary light bouncing off them.
22:36And the computer expresses all of this in numbers.
22:39Pythagoras would be leaping around with delight.
22:43The Greek philosopher Plato talked of the realm of the unchanging forms.
22:48He conceived the observable world as an imperfect image
22:52of a realm of unobservable and unchanging forms,
22:56eternal, changeless and incorporeal.
22:59He believed in the world of mathematical objects,
23:02that lines and triangles are real but perceived by the mind.
23:07It was one of the great things that Plato achieved
23:10was to realise the distinction
23:12between the mathematical idea of geometry,
23:15which was something out there in a sense,
23:18its sort of absolute notions,
23:20and the physical world.
23:24Partly because Euclid's geometry was considered to be the only geometry,
23:28no-one bothered to question or examine many of its details.
23:33Towards the end of the 19th century, another view emerged,
23:37and Euclidean geometry is now regarded
23:40as one example of possible geometries.
23:44During the 20th century,
23:46mathematicians discovered many new denizens
23:49of the realm of unchanging forms.
23:52Many mathematical professor types
23:54started to see in the world of the imagination
23:58beautiful shapes and forms on the Euclidean plane.
24:02The first pictures of these objects
24:05started to appear in about 1980 on computer screens.
24:09So out of the imaginative minds of mathematicians
24:13came a brand-new world,
24:15a virtual world,
24:17based upon numbers and equations.
24:21But as a consequence, they also created a new art form,
24:25something in the computer age
24:27that inspires us with feelings of extraordinary wonder.
24:32This leaf is actually a series of numbers.
24:37This is an amazing arabesque pattern.
24:40And this biological cell?
24:42Numbers again.
24:44As is this graph,
24:46just like the ebb and flow of the stock market prices.
24:49This one looks like a galaxy.
24:52These amazing images are very diverse.
24:57Many of these new geometrical things have the property
25:01that when you magnify them up,
25:03they reveal more and more intricate detail,
25:06not boring old line segments.
25:08Look at this curve.
25:10As we zoom in on it, we see it remains wiggly and complicated.
25:14Now, many of these geometrical objects
25:17have become as transforms of themselves,
25:20just like the line segment and the triangle.
25:26In 1978, Mandelbrot named these things fractals.
25:30But in 1982, a mathematician called John Hutchinson
25:34discovered something about triangles and lines
25:37that I don't think those old Greeks,
25:40or anyone else since, for that matter, knew.
25:43And what he discovered applies to these fractals as well.
25:47To explain what he discovered,
25:49get hold of your blue and red transformers.
25:52I start with a spider.
25:54First I apply the blue transformer to the spider
25:57to make this smaller spider here.
26:00Then I apply the red one
26:02to make this shrunken spider here.
26:05Then I add up the two transformed spiders
26:08to get the result.
26:10It's a spider monster!
26:12Now I again use the two transformers,
26:15this time on the spider monster.
26:17The result is another horrible thing, more like a millipede.
26:21Let's keep going.
26:25We arrive at the line segment.
26:27When it is transformed by the red and blue transformers,
26:30it doesn't change.
26:35Now what happens if instead of starting with the spider,
26:38we start with an elephant?
26:40Watch.
26:43As we repeatedly apply the two transformers,
26:46the elephant turns into the same line segment.
26:52What Hutchinson discovered
26:54was that if you have any set of transformers,
26:57you can always make one,
26:59and only one geometrical object out of them.
27:02The blue one and the red one always make a line segment,
27:06exactly the same one.
27:08Other sets of transformers make other things,
27:11some ordinary, some extraordinary.
27:15Let's put this to the test.
27:17Here's a set of transformers.
27:19What picture will they make?
27:23It looks biological.
27:25And we can magnify it.
27:28Any set of transformers is a formula
27:30for a unique and possibly fantastic geometrical object.
27:34Here, transformers make a leafy pattern.
27:38What happens when we apply a yellow and green transformer
27:41to the spider?
27:44First we apply them once.
27:46Look, a strange shape is emerging.
27:48Apply them again, and again.
27:51Oh! Oh, surprise, surprise!
27:54Our friend, the right-angled triangle.
27:57Again, it makes no difference to the final picture
28:00whatever we start from.
28:04Look at this galaxy fractal.
28:07It's made up of two transforms of itself.
28:10One transform moves the galaxy towards this point
28:13and also shrinks it.
28:16The other one does this.
28:18Now throw away the galaxy, but keep the two transformers.
28:23Now let's return to our spider
28:25and apply these two transformers.
28:27Again, shrink and rotate.
28:30Shrink and rotate.
28:33Then add up the result.
28:36Repeat the process, and we get back to the galaxy fractal.
28:42So a complex shape can be expressed simply with transformers.
28:47These can be applied to other shapes, such as flowers.
28:51Repeated application of the transformers
28:54reconstitutes the complex shape.
28:58We now have a powerful mechanism
29:00to represent extremely complicated things by simpler things.
29:04Think of the fantastic power this gives us in the virtual world.
29:11So what's the big deal?
29:13It's this.
29:15Any set of transformers
29:17make a unique, special, new geometrical object.
29:20The objects may look very complicated,
29:23but the transformers are all you need to describe them perfectly.
29:27One of the things that I discovered
29:29was that if, upon looking at an object in the real world,
29:33you could see some transformed copies of the object inside the object
29:38so that it is covered up with the transformed copies,
29:41then you could use the transformers
29:43to make a mathematical model of the object,
29:46sort of lift it out of the real world
29:49and put it on the Euclidean plane in idealised form.
29:53Here's an example.
29:55Let's take a close look at this fern.
29:58It's not hard to see that there are four transformers which make it up.
30:03The whole is made of all of this top part,
30:06which looks like a shrunken fern.
30:08This frond on the lower left,
30:10this frond on the lower right,
30:12and the stem.
30:14The stem is made by an extreme sort of transformer
30:17that shrinks all of space
30:19The stem is made by an extreme sort of transformer
30:21that shrinks all of space sideways.
30:25Now throw away the fern and keep the four transformers.
30:30What will appear when these transformers
30:32are repeatedly applied to this rectangle?
30:36After one application, we get four rectangles,
30:39one of which is very thin.
30:41After two applications of the transformers,
30:44we get this picture.
30:46And if we apply them again, and again, and again,
30:50we get this amazingly beautiful result.
30:53It's this almost magical mathematical fern
30:56that will live in the Euclidean plane forever,
30:59and of course, in our imaginations.
31:02So what conclusion can we draw from our experiment?
31:07The original fern equals this transform of the fern,
31:10plus this transform of the fern,
31:13plus this transform of the fern,
31:15and finally, plus this transform of the fern.
31:20This is a revolutionary way of representing objects
31:23and a mechanism for creating entirely new ones
31:26in the virtual world.
31:28It could also be how our brains construct reality
31:32inside our minds.
31:35The transformers in our brains
31:37interpret the raw input arriving from our senses
31:41to create the physical, observable universe
31:44which we experience.
31:46The shapes, colours, textures, sounds and aromas,
31:51down to even the very basic, solid and separate
31:54appearance of things.
31:56That's the theory.
31:58But what can we do with this mathematics now?
32:02Knowing only these four transformers,
32:05this picture can be passed on to the future.
32:08I like to think of some old professor 2,000 years from now
32:13showing our construction of the fern.
32:16The real world will have changed,
32:19but the mathematics will not.
32:22But this mathematical fern and this real fern
32:25are both pictures of objects that belong in different realms,
32:28one called reality and the other called Euclidean geometry.
32:32They are different.
32:34The Euclidean one is cleaner and simpler.
32:37You will never know this specific, exact, real fern.
32:40Already it is fading and changing.
32:43But to the extent that you can know it precisely,
32:46it will share in common with the abstract one
32:49that the language in which it must be described
32:52is a mathematical one.
32:54But there is much about the real fern that we can never know.
32:58Indeed, that is true even of the Euclidean fern,
33:02because mathematics always contains statements
33:05that we can neither prove nor disprove.
33:09Mathematics is incomplete,
33:11requiring more and more input from human minds
33:15to decide what is true and what is false.
33:20Our models of reality are always incomplete.
33:25Believe it or not, that's a theorem, Godel's theorem.
33:30There was a view being put forward in the early part of the century
33:35that somehow all mathematical thinking
33:37could be phrased in terms of what are called formal systems.
33:41Roughly speaking, it means that you could remove
33:45the necessity of understanding from mathematics.
33:48You mechanise the entire thing.
33:50It's all a question of following rules.
33:52And what Godel showed is that that's simply not true.
33:56Logical proof is a very limited category.
33:59Most interesting things are things that we have reasons for believing in
34:03but are not logical certainties.
34:05In fact, Kurt Godel, who was the great logician of the 20th century,
34:09showed that you couldn't even prove
34:11the logical consistency of arithmetic.
34:14We are not simply machines
34:17following according to pre-assigned rules.
34:20There is something in our insight, in our consciousness, I would say,
34:24in our ability to understand things
34:27which enables us to transcend rules, no matter what those rules are.
34:31So, mastering mathematics
34:34is more than simply understanding a set of rules.
34:37It's also about applying our perceptions of reality
34:41to mathematical models in order to achieve specific aims.
34:47Let us discuss how mathematics can describe real pictures.
34:52I'm talking about the pictures that fill the field of vision
34:56as we look out on the real world.
34:59Think of one frozen in time, like a photograph,
35:03but in the mind's eye.
35:06How do we describe these colourful pictures?
35:09They are seemingly more complex than triangles and fern images.
35:13Real pictures are very special, very particular.
35:17They are very far from random.
35:20Real pictures are rich in details and textures.
35:24The bark on that tree,
35:26the coruscations on the cloud and the ripples on the water.
35:29How would you describe your vision
35:32to a person who had lived all of his life in a dark cave,
35:36with no knowledge of the banks of trees,
35:39the sunlit rivers and the sky?
35:42What special properties do pictures have?
35:46Real pictures usually contain edges.
35:49Here is an edge of the tree line.
35:52Here's the edge of a sawtooth palmetto leaf.
35:55And here's the edge of the water.
35:58Pictures also contain smooth regions, like the clear sky,
36:02the white of a flat cloud, or still water.
36:05And lastly, there are textures.
36:08Mud textures, lichen textures, tree textures.
36:14To precisely describe these edges, smooth regions and textures,
36:18imagine a picture on the Euclidean plain
36:21with all of its colours painted on it.
36:24It's stretchable, like the skin of a balloon.
36:27Suppose that our transformer can pick up a colourful copy
36:30of what is in a region of the picture,
36:33and shrink and move the region.
36:35Any picture can be modelled with transformers like these.
36:39First an edge.
36:41Here, this edge of rock and sky.
36:44This transformer picks it up, transforms it and puts it back.
36:48We can make all of the edges like this.
36:51The edges are transforms of themselves.
36:54Now this smooth piece here.
36:56Look at the blue, so steady and bright.
36:59Using a transformer, let us take a part of the blue and shrink it down.
37:04It superimposes almost perfectly on part of the sky.
37:09The sky is made of transforms of itself.
37:12Look at this part of a rock.
37:14It is smooth too, and looks like a special transform of itself.
37:22We use our special transformers to make copies of textures
37:26and then move them around in a real picture.
37:29The texture of this sand is transformed to this one,
37:32then to this one, and to this one.
37:37Look at this tree trunk.
37:39It's like this one, this one and this one.
37:43And this petal is like this one and this one.
37:48Putting all this together,
37:50suppose that we have got the transformers for a butterfly picture.
37:54Then we can use them to make a mathematical picture.
38:01Imagine a magical photocopy machine.
38:04It works on colourful Euclidean plain substance instead of paper.
38:09Put any colourful picture on top, any one you like.
38:14All of the transformers that we kept go to work on it.
38:17Each part of the transformed picture
38:20comes from somewhere in the one on top of the copier.
38:23Now take the picture that comes out of the copier
38:26and put it back on top.
38:28Then make a specially transformed copy of it.
38:31Then put that copy on top and make a copy of it.
38:35And again, and repeat, and repeat forever.
38:40The result, created on the colourful Euclidean plain,
38:44is a unique mathematical picture,
38:46a model of the original from which we got the transformers.
38:50The final picture is always the same.
38:53It's described precisely,
38:55uniquely specified by the transformers and them alone.
39:00Like the right-angled triangle.
39:02Only this time, it's a full-colour picture
39:05that can be magnified endlessly.
39:07It belongs to the realm of unchanging forms.
39:12It is made by mimicking the functioning of our brains.
39:29What we have just done is make an intuitive description of a theorem.
39:34It enables us to make mathematical pictures
39:37whose relation to real pictures
39:39is analogous to the relation of the mathematical fern to a real fern.
39:45But when you actually examine a real photograph closely,
39:49you find it is made of printed dots or chemical emulsion.
39:53Indeed, it is not possible to capture pictures
39:56of the physical observable universe at infinite resolution.
40:00When we try to look very closely at reality,
40:03we can no longer see details
40:05because of the finite wavelength of light
40:08and the trembling of the atoms.
40:11Even if our pictures of reality are fuzzy,
40:14the human mind is still able to produce images
40:18of far higher resolution
40:20than those physically recorded by the retina.
40:24What are the factors that limit the resolution
40:27of the images sent from our eyes to our brains?
40:31There are three possible factors limiting the resolution of the eye.
40:35That is the quality of the image received at the back of the eye.
40:39One is that it's the quality of the image
40:42provided by the lens and the cornea.
40:45The other would be that it's actually limited
40:48by the wave nature of light,
40:50in the same way that the resolution of a telescope is limited.
40:54And the third is that it might be limited
40:57by the separation and size of the photoreceptors
41:01which pick up the image.
41:03For example, if I see a person far away in a field,
41:07I know it's a man, and I know he has two eyes.
41:10I can perhaps make out a vague indication
41:13that he's not mutilated in some way.
41:15So I know he's got all the details there.
41:18And so I think I see him with two eyes and five fingers and so on,
41:23even though there's no way in which one could deduce
41:26that he has five fingers and two eyes
41:29from the actual details of the image.
41:31We are constantly filling in details from past experience
41:36to complete the incomplete evidence of our current experience.
41:41So what is a real physical picture?
41:44In the end, it is in the mind, for it is there that we see.
41:49We see the world one way, insects see it another,
41:53and artists in yet other ways.
41:56A bee can see ultraviolet, and flowers look very different to them.
42:01A dragonfly sees with a compound eye, like lots of little eyes,
42:05and the visions of artists are amazing.
42:08Is reality glowing light as Turner saw it?
42:13My point is this.
42:15In the end, our pictures of reality depend on our own consciousness.
42:20So what can we say precisely and carefully about that?
42:24We often think we see details in an image
42:27when, in fact, we're supplying these with our brain.
42:30For example, I look over there and I see a tree
42:33with lots of fine leaves on it,
42:35but when I look back, I still see the leaves,
42:39even though the quality of the image in the periphery of the field of vision
42:43is nowhere near good enough to enable me to see those leaves now.
42:47But I still see them.
42:49My brain is filling in the details from past experience
42:52and from what it knows is there.
42:55The mysteries of vision and consciousness are linked.
42:59Can mathematics explain the inner world
43:02wherein we are aware of mathematical pictures,
43:05abstraction, love and poetry?
43:09If you take me apart, you will find that I'm made up of,
43:12ultimately, quarks and gluons and electrons.
43:15But you would have destroyed me by the act of taking me apart.
43:18So I'm very much more than just a collection of quarks, gluons and electrons,
43:22though from the point of view of an elementary particle physicist,
43:25which is what I am, that's how I would think about myself.
43:28But there's much more to me than that.
43:30Just as there's much more to a beautiful picture, for example,
43:33than just a collection of specks of paint of known chemical composition.
43:36We can learn lots of interesting things by taking things apart,
43:39but there are many things that we can only learn
43:41by looking at things in their wholeness and their totality.
43:44The very phenomenon of consciousness is something like this too,
43:48that you cannot reduce it to the behaviour of a lot of individual units.
43:53There's something holistic about the way that it operates.
43:57There's something holistic involved in consciousness.
44:01Actually, science still has no description
44:04of what consciousness is or how it works.
44:08There certainly is no mathematical theory of it.
44:12Human consciousness is a sort of transcending element of human beings.
44:17We're part of the physical world, but we're more than that.
44:20Pascal once said that we are thinking reeds.
44:23We are just tiny, insubstantial things compared to the stars,
44:27but we are greater than the stars because we know them and ourselves
44:30and they know nothing.
44:32So consciousness is pointing us in the direction that there is more to the world
44:35than simply the physical world that science describes.
44:39And I think if we push that thought further,
44:42that leads us to the idea of a divine mind and a divine purpose
44:45that is behind all that rich, many-layered experience.
44:50There seems to be a very deep union between mathematics and the physical world.
44:55Now, as a mathematician, I tend to view mathematics
44:59as something which is out there, absolute, independent of ourselves,
45:03yet we have access to it with our understanding, our mental abilities.
45:08And that again is a mystery, how it is we have somehow access to this world,
45:13this absolute world, timeless world, independent of ourselves,
45:17which somehow has been governing the way the physical world operates
45:21long before there were any human beings around.
45:24As we study the physical world, we see a world of great rational beauty,
45:27rational transparency, a world shot through with signs of mind, one might say.
45:32And to me it's a very attractive explanation,
45:34that there is indeed the capital M Mind of God the Creator
45:38that lies behind that wonderful rational beauty.
45:41So you see, when we try to get to the bottom of what reality is,
45:46we end up with a mystery of our own minds.
45:50What makes us so very special in nature
45:53is that we are aware of our own being,
45:56our position in time and in space, and that we will die.
46:01Every time we study nature, we learn more about our own limitations
46:06and how much more we need to know.
46:09And that's what makes us so special in nature.
46:14We learn more about our own limitations and how much more we need to know.
46:20But while we are alive, we have a consciousness and intellect
46:24that can ask fundamental questions about our own existence,
46:27as well as a passion to uncover the great and profound laws that mould our universe.
46:33There may be many forces and many hidden worlds yet to be discovered.
46:38That's the challenge to scientists and mathematicians.
46:43Perhaps mankind does have a special purpose in the evolution of the universe,
46:48where we may be unique and where our role may be only just beginning.
46:53Could our quest for total understanding of our brains
46:57and the mechanisms of our consciousness
47:00allow us into the mind of the master mathematician?
47:06I think there's a very deep human desire to understand the world,
47:10actually a God-given desire, and I think it's one that we should seek to fulfil.
47:15There may be limits to our understanding, but we only find that out by pushing the limits.
47:19And it would be deeply satisfying and illuminating, I'm sure,
47:22to understand consciousness and even to understand profoundly the nature of mathematics.
47:28We want to know what our role in the world is, how we got here,
47:33what's going to happen to us after death and things like that.
47:37So I think it's the same curiosity drives people to religions as drives me to science.
47:47We have come on a voyage of discovery and exploration,
47:51a voyage in which we have discovered the power and limits of mathematics to describe reality.
47:57My own exploration has left me more and more awed
48:01by the intricate interlocking of scientific method, reason and measurement.
48:06I am stunned by all we have learned and continue to learn over all these years.
48:11But the reasonableness of all these things, these laws of physics,
48:16that are agreed by us mathematical scientists, does not surprise me and nor should it you,
48:23because our understanding of reality is necessarily self-consistent.
48:28Inconsistent, contradictory observations cannot possibly be part of our understanding.
48:35That is, the reasonableness of the physical observable universe
48:40is a self-evident, automatically true thing, like survival of the fittest.
48:45The fascinating thing to me is not that we can make reasonable models of the physical observable universe,
48:53but that it is all so beautiful and can seem so to us who are in it and part of it.
49:01On our journey we have seen something of hard mathematical reality,
49:06as well as the ephemeral character of the real world.
49:10There are logical and scientific limits to our most precise knowledge.
49:15Mathematical science, magical and mysterious as it is, is not enough.
49:21Boundaries are created by the structure of reason itself,
49:26the puzzle of our own consciousness and the unreachableness of infinite precision.
49:36There may well be limits to what we can know, but we can only find those limits by pushing up against them.
49:41I'm entirely in favour of human beings trying to understand as much as they can about the world.
49:47I think they should do so persistently but humbly.
49:51And probably the most difficult thing for us to understand is ourselves.
49:58I believe that at the heart of our scientific knowledge of reality,
50:02there is space for philosophy, free will and God.
50:06We are back with the wise men of Greece all those years ago.
50:10Much has been learned and as much as ever remains unknown, the domain of the heavens.
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