Before we begin I'd like to draw your attention
to the picture that's being projected.
This picture was a tribute to Ada.
It was created by the children of Botley Primary School's code club.
It was arranged by Peter Lister.
Thank you very much, Peter.
Parents of a certain age will recognize that this was implemented in a system
called Minecraft.
Minecraft is a plot to steal the souls of all the 10-year-olds in the country and
I speak from experience, but
a lovely tribute by the children of Botley Primary School.
So we have a couple of talks.
>> [APPLAUSE] >> We have a couple of talks this
afternoon.
Firstly, it's my pleasure to introduce all the way from California,
Judith Grabiner from Pitzer College.
She's gonna talk about mathematics in culture.
Judith. >> Thank you.
I wanna thank Ursula for inviting me and asking me to speak about mathematics in
culture, and secondly I want to start by saying there will be an introduction.
So where's the device that advances the slides?
This is it.
Okay, this is it.
Okay, good.
>> [LAUGH] All right I want to argue today that geometry
interacts with all aspects of human thought and life.
Now, every mathematician has some sort of relationship with Euclid, and
in my token reference to Ada Lovelace, she began her studies of mathematics
learning geometry and told her tutor that she could only understand the proposition,
when she could see the figure in the air.
Now that's a very good line that could also serves for me as a transition.
Because geometry is long been thought of as the science of space.
So what is space?
Does space exist?
What's its nature?
The history of these questions involves a lot of important thinkers, and
artists, and really the entire universe and that's what I want to do today.
Now Euclid is special.
When we use terms like truth, and proof, we're channeling Euclid.
For 2,000 years, Euclid's geometry seemed to prove truths about
geometrical objects and thereby to achieve certainty.
Now, Plato thought that geometry was true and
certain because the subject matter was eternal and helped to draw the soul
from the imperfect world of change to the changeless eternal truth.
But Aristotle said, no, the truth and
certainty of geometry comes from the way it's put together, using logical
deductions from explicit self-evident assumptions and clear definitions.
Euclid probably looked over his shoulder at Aristotle's criteria
while he was writing The Elements.
And many important later thinkers believed that other subjects might come to
share the certainty of geometry if only they followed the same method.
For instance, Descartes said, you know, if we start with self-evident truths and
then proceed by logically deducing more and more complex truths from these.
Why, he says, in effect there's nothing we couldn't come to know.
Wow.
Following Descartes, Benedict Espinoza wrote a book,
On Ethics Demonstrated in Geometrical Order.
With explicitly labeled axioms and definitions, and
including theorems like this one quote God, or a substance consisting
of infinite attributes, necessarily exists, end quote.
He has several proofs, he closes them with a QED.
>> [LAUGH] >> Now the influence of the Euclidean
ideal on science is clear
from Isaac Newton's Mathematical Principles of Natural Philosophy.
As you see here, Newton called his famous three laws of motion axioms, and
he deduced even his law of gravity in the form of explicitly stated theorems.
And Newton famously said, it is the glory of geometry that from so
few principles it can accomplish so much.
One more example.
The American declaration of Independence is an argument
whose author tried to inspire faith in its certainty by using the Euclidean form.
Thomas Jefferson, who knew more of the mathematics of his time
than any other American president, began his argument thus.
>> [LAUGH] >> See, in the United States,
people wouldn't laugh at that.
>> [LAUGH] >> Jefferson began,
we hold these truths to be self evident, that all right angles are equal.
>> [LAUGH] >> Oh, oh no!
That's not what he said, but that's what it sounds like.
Another self evident truth in the Declaration is that if any
form of government fails to secure human rights,
it is the right of the people to get rid of it and set up a new government.
The Declaration asserts that King George's government does not secure these rights.
Then it says to prove this, let facts be submitted to a candid world.
And once the facts have proved this, then we've got if p then q.
p, and therefore, q.
Thank you.
>> [LAUGH] >> Computer scientists, I mean,
you're supposed to know this.
>> [LAUGH] >> All right,
once the facts have proved this, the actual Declaration that founded
the United States is stated explicitly as a conclusion of a logical argument.
Beginning with a therefore, which, in one of the original printings,
is italicized, as I have it here.
So in philosophy, theology, science, politics,
the idealized Euclidean model of reasoning has shaped conceptions of proof,
truth and certainty.
Okay, that was the introduction.
>> [LAUGH] >> Now, for
the first half of the actual talk, we'll look at Euclid's Elements,
at the idea of Euclidean space, and
some important related ideas and their wide ranging influence.
And we'll see that it lives up to its press notices.
But we'll also see that there was a problem.
And in the second half of the talk, we'll see that the problem and
its resolution have produced a non-Euclidean world, and
that this break with the past has had a comparably great impact.
Okay, so here we go with part one.
These are Euclid's postulates.
Hopefully the minimal set of self-evident truths from which
all the true results of geometry can be logically deduced.
Well postulates one through four are pretty straightforward.
I hope no one wants to argue with them, but look at number five.
Euclid's so-called parallel postulate
which you will observe does not mention parallels at all.
That one's complicated, so I think we'd better draw a picture,
sort of retain in your mind how that struck you when you saw it.
Here's a picture.
What postulate five means is if the two green lines are cut by a third line,
AB, such that angle A and angle B add up to less than two right angles,
then the two green lines eventually meet on that side.
Now I ask my students to vote on whether this postulate is self-evident.
And they overwhelmingly vote that it is not.
Some of them don't understand it after two runs through.
>> [LAUGH] >> And you gotta draw a picture.
The ancient Greeks agree with my students.
And that is the problem that I mentioned.
Okay well if this postulate was not self-evident,
then maybe it can be proved from the other postulates.
So the Greeks tried to do this For
reasons that the mathematically literate will recognize.
They did not succeed.
And so, also trying did the best mathematicians in the Medieval Islamic and
Jewish worlds.
Them mathematicians in Christian Europe on into the 19th century.
Including John Wallace,
whose views on this subject are actually in a poster out in the lobby.
On into the 19th century.
Pretty important famous people.
Now, while trying to prove postulate five, one thing that the Greeks proved
right at the beginning is the Euclid's postulate is logically equivalent.
The uniqueness of parallel lines.
That is, given a line, then a given point in the same plane,
there is only one parallel through that point to that line.
It's a common alternative postulate found in many textbooks, including the one by
John Playfair in the 18th century, hence it's called Playfair's axiom if you care.
Euclid defined parallel lines as lines in the same plain that never meet.
From his first four postulates,
he can prove a lot of cool stuff, like that parallel lines can be constructed.
But he needs that problematic 5th postulate, which he uses very rarely.
For his proof of one key theorem.
If two parallel lines are cut by a third line,
the alternate interior angles, like angle 3 and angle 6,
and the corresponding angles, like angle 2 and angle 6, are equal.
From that theorem you could peruse not only the uniqueness of parallels but
many other theorems that involve parallel lines
like that they're every where equi distant.
We expect this of decent parallel lines, or
that the sum of the angles to a triangle is to right angles.
But Euclid doesn't talk about space.
Nevertheless starting in the Renaissance people talk a lot about space.
An infinite homogeneous space that's the same in all directions
in which every point is just like every other point.
This is the space that Euclid's geometry happens in, and
the role of philosophy in describing that space turns out to be really huge.
The idea that space must be the same in all directions comes from what is called
the principle of sufficient reason.
For everything that is, there's a reason why it is as it is and not otherwise.
That sounds pretty trivial.
Turns out, not to be quite so trivial.
Principles at least as old as Archimedes, here's what Archimedes noticed with it.
Have a lever with equal weights and
equal distances from the fulcrum it has to balance.
Why?
Why not?
>> [LAUGH] >> Because there's no reason for it to go
down to one side or the other because the sides are completely equivalent and
that's why the symmetric lever necessarily misbalanced.
Furthermore, this principle of sufficient reason, explains why space is infinite.
[INAUDIBLE] Bruno said, space must be infinite,
because there's no reason for it to stop, at any particular point.
All the points are the same.
The greatest advocate of the principle of sufficient reason, was Liebniz.
Leibniz said it's the principle of sufficient reason and
the laws of logic that God used in making the universe.
God made the universe optimal in the best possible way.
Many scientific laws still use the language of optimization,
like that light is refracted in the least possible time.
Now for Leibniz sufficient reason gives us the simplest
scientific laws and the best possible laws and universe.
And if the principle of sufficient reason is true
the universe is transparent to reason.
God made it rationally so we humans can figure it out and
here is one striking example.
Consider Newton's first law, which was first formulated 50 years before Newton,
independently by Descartes and Pierre Gassendi.
And here's how they found it.
The first law of motion says, a body with no forces acting on it,
continues in a straight line at a constant speed.
Why?
Well, it goes in a straight line because every direction is the same,
so there's no reason for it to turn to the left or to the right.
It goes at a constant speed because every point's the same, so there's no reason for
it to prefer this point to this point and speed up to get there.
And you can see that the same argument can be made
why a body that is at rest with no forces acting on it stays where it is.
Okay, now back to parallel lines.
The great mathematician Joseph-Louis Lagrange,
decided that he could prove the uniqueness of parallels.
Logically equivalent to Euclid's 5th postulate
by using the principle of sufficient reason.
Okay?
Well, this is what he's trying to prove, the uniqueness of parallels.
Well wait a minute, who says you can use the principles of sufficient reason,
in a mathematical proof?
Lagrange says so.
He says it is just as obvious and true as are the laws of logic, and
I hope I've convinced in that little brief thing I said that it was all over since
in the 18th century principle sufficient reason.
I found this manuscript in which Lagrange does this.
I think it's really, really interesting.
He's going to prove that theorem.
And it's a proof by contradiction.
Given that we have one parallel through point P,
which we have here, suppose there were another parallel.
The original line through point P might look like this.
But by the principle of sufficient reason, there's no reason that the new parallel
line should be below the point P on the left and above it on the right.
It could equally well be drawn the other way.
So the principle of sufficient reason says, there must be another symmetric
parallel line that goes the other way as in this new diagram.
Okay, Lagrange repeats this exact same argument for
lines symmetric to his new parallels.
So for instance you get a third line on the other side of his new parallel, and
it looks like this.
And so on, lots and lots more, and I will show you Lagrange's own picture,
of the final situation, from his manuscript.
Which Lagrange says is absurd.
>> [LAUGH] >> So he concludes,
there can only be one parallel, QED.
>> [LAUGH] >> Now,
Lagrange never published this manuscript.
Maybe he came to see that when he used the principle of sufficient reason at
the beginning to produce the new first set of new symmetric parallels,
that he was assuming that, this one here,
he was assuming that the two parallel lines were everywhere equidistant and
thus assuming the very Euclidian nature space that he was trying to prove.
But the fact that a great mathematician like Lagrange got up
in the Public Institute of France and linked space being Euclidian
with Leibniz's principal of sufficient reason demonstrates how closely linked
sufficient reason was with the necessity of space being Euclidian.
Oh yes.
And space has to be real.
Newton insisted on that.
Why?
So he could distinguish real accelerations,
that is accelerations with respect to space, from the parent accelerations.
Real accelerations involve real forces.
So from real acceleration, like a falling apple or
a planet orbiting the sun, real acceleration require forces and
thus he can show that gravity is a real force.
Furthermore, that real space for Newton's physics turns out to be Euclidean.
Why?
Well, one reason is that you have to use parallelograms of force all over
Newton's physics.
And proving the properties of parallelograms,
requires Euclid's theory of parallels, and therefore, postulate five.
Well okay, that's Newton.
There's another guy.
Oops, wrong guy.
All right, I'm sorry.
I thought I had a picture of Leibniz again, but I didn't.
Leibniz disagreed with Newton about space.
Leibniz essentially said there is no such thing.
Space is just the relations between bodies.
But, Newton's view prevailed?
Why?
Listen to Leonhard Euler.
Euler says the straight line constant speed motion of a single body with no
forces acting on it, according to Newton's first law, that can't possibly
depend on where other physical bodies just happen to be at some particular time.
It's gotta be a straight line always, therefore a straight line not with where
all the bodies happen to be right this second but
with respect to something that doesn't change namely space.
So space has to be real and Euclidian.
Why didn't mathematicians of the 18th century want so
much to prove the parallel postulate?
Well precisely, because it was so
important in establishing the correct nature of space.
Not for just for geometry, but for all of science.
Rested upon it.
And now, to philosophy.
A super influential philosopher, Immanuel Kant, agreed with Newton
that space exists, agreed that we have to order our perceptions in space.
But Kant said space exists in our minds.
And we each have the same unique space in our minds.
And it turns out for Kant that this space too has to be Euclidean and
let me explain because he obviously never says this because it never occurs to him
that there is a non-Euclidean alternative.
All right, to argue Kant argued that we can come to no non-trivial truths
about non-material things, technically synthetic [INAUDIBLE] judgments.
Kant uses Euclid's proof that the sum of the angles of the triangle
is two right angles.
If it's been a long time since you did this,
I will remind you how Euclid's proof goes.
You start with a triangle.
Well you can't add the angles.
They're in different places.
So to do the proof, first thing you do is you extend the base to D,
and then you construct a line to the corner parallel to the opposite side.
So CE is parallel to AB.
And then you prove the equality of various angles in the new diagram.
To the angles around C, because the angles around C obviously add up to right angles.
This is the proof.
The equal angles are specified in this slide.
But it isn't the details of the proof that matter.
Kant's keep point is this.
The proof cannot work unless and until you make those constructions.
Okay.
Well, where'd you make those constructions.
Not on paper.
Geometry isn't about physical angles.
You made them in space he said.
Space in your mind.
So, space.
Now the proof needs the results that if two lines are parallel,
the alternate interior.
And corresponding angles are equal.
As I said before,
Euclid's proof of that theorem explicitly requires the fifth postulate.
So this theorem about the sum of the angles requires space to be Euclidian,
now Kant doesn't say this, but
he does say there's only one space just as there's only one time.
So for Kant, no alternative to Euclid seems conceivable.
Well, Kant's pretty heavy.
Let's pick somebody lighter.
Oops.
There you go.
One more philosopher witness to Euclidian space as truth.
Voltaire shared the widespread 18th century
idea that universal agreement was a marker for truth.
And he said of Voltaire.
There are no sects in geometry.
One doesn't say, I'm a Euclidean.
Demonstrate the truth, he says, and the whole world will be of your opinion.
Mathematics exemplifies this for Voltaire, and he wants ethics to do it too.
Voltaire wrote in my favorite Voltaire quotation,
quote, there is but one morality as there is but one geometry.
>> [LAUGH] >> And I said that in class once and
one of my students shot back wrong on both points Voltaire.
>> [LAUGH] >> The art and architecture
of the early modern period also reflect the Euclidian nature of space.
Art and architecture reinforce the Euclidian intuition of space in the minds
of everybody who look at the paintings, and lived and worked in the buildings and
public squares, from the Renaissance to well into the 19th century.
Perspective in art which is based on Euclid's geometry
helped make people literally see space as Euclidian.
And let me illustrate a bit.
One of the most important early perspective paintings of the Renaissance
here, the Trinity by Masaccio,
We are used to two dimensional pictures that look three dimensional.
Cuz we've got photography and television and iPhones and
all sorts of wonderful things like that.
In the Renaissance they didn't.
So a painting like this as the quotation from Vasari makes clear,
was incredibly exciting to them and the realistic
illusion of depth in Renaissance art comes explicitly from geometry.
Useful briefly to look at a couple Medieval works of art to appreciate
the difference between Medieval and Renaissance.
Picked a couple things I like.
The Bayotuss tapestry, the people are the same size as the castle.
This is Nicholas Verdun, The Crossing of the Red Sea,
well the people are as big as the whole Red Sea Which is also red.
This is wonderful art, but there's no convincing three-dimensionality.
Now on the Renaissance we're going to move into a different kind of space.
A pretty dramatic contrast.
Okay, the geometry used in creating Renaissance art is literally Euclidian.
It comes from Euclid's elements of geometry and Euclid's optics.
Here's a theorem from Euclid's optics not Euclid's diagram.
I think you see the truth of the result and
here is something from Euclid's elements.
If a straight line is drawn parallel to the base of a triangle it cuts the other
two sides proportionally.
That theorem is just essential
to the Renaissance geometric theory of perspective.
Here's how the geometry lets you see three dimensional reality on a flat canvas.
This is known as the Alberti construction.
What you've got there is a square floor divided into squares and
the plane of that floor is perpendicular to the plane of, in this case, the screen.
This photograph does the Alberti construction for
two parallel railroad tracks, perpendicular to the plane of the screen.
Now the checker board construction of Alberti's.
Let's take a look at it again, checker board, okay.
Was used by many Renaissance painters because you have to square it off
pavement and it emphasizes the perspective and also helps the artist's draw it.
So, this is supposedly a religious picture about Christ but
I look at it as an application or a checker board.
Here's another artist you've probably heard of Doing the same thing,
checkerboard here is on top.
These are artists who are also mathematicians, Piero, and
Leonardo DaVinci.
Moving north from Italy into Germany, here's another mathematician artist,
Arbuteur showing you precisely how to construct the image.
Well okay, you're a renaissance artist.
Do you have to master all this mathematical theory to draw on
three dimensions?
Not at all, just find the right books.
Schern is a student of Durer's and
he says, just look at my drawing and you'll know what to do.
Here's another German I like this picture by Count Johann
because it literally embodies Euclid's theorem about
disproportional sides of similar triangles.
One more Renaissance painting.
This is so great.
Okay, in some the objects in
the painting look the way an objective observer would actually see them.
In a rational order created by Euclid's geometry.
The art helps to teach to see our world as Euclidian.
Also architecture, teaches us to see our world as Euclidian.
Okay, whenever I talk about this I'm always in a room like this one.
Full of parallel lines.
Look at the sides of this auditorium.
Full of parallel lines which are everywhere equidistant,
which make equal right angles with the floor.
The banister kindly let's us look at the making equal
non-right angles with a transversal,
all with the properties that Euclid used to prove his fifth postulate.
This is the kind of room that you would design if you wanted to brainwash people
into believing that space has to be Euclidian.
>> [LAUGH] >> So, here's the 18th century world,
the world of sufficient reason.
It's symmetric, it's balanced, it's based on self evident and
necessary truths, it's embedded in Euclidian space.
We can figure it all out rationally by ourselves.
Euclid's geometry is the universally agreed upon model of
perfect intellectual authority.
That is the end of part one of this talk, now we'll blow it all out of the water.
>> [LAUGH] >> It's really amazing that
in the early 19th century, Gauss, Bolyai, and Lobachevsky, the three independent
inventors of the first non-Euclidean geometries, were able, in spite of all
this Euclidean brainwashing, to imagine that space could be other than Euclidean.
Now what they did was to realize that the absurd consequences.
The absurd consequences of denying Euclid's postulate.
Consequences like parallels are not unique.
Parallel lines that is lines that never meet do not
have to be everywhere equidistant.
These consequences are not absurd at all, but are truths.
Truths in some alternate counter intuitive reality.
Well let's stick for the moment to two dimensions, a surface.
Here's a modern non-Euclidean surface that obeys Euclid's first four postulates, but
disobeys his fifth.
Notice that the red and yellow shortest paths never meet their parallel.
They're both parallel to the blue one.
Notice that they intersect.
So we got two parallels through the same point.
And notice that those three parallel lines, the red, the yellow, and
the blue, are not everywhere equidistant.
The idea of non-Euclidean geometry was so revolutionary,
that Gauss never even got up the nerve to publish his work on the subject.
Non-Euclidean geometry required a paradigm change.
Realizing that the logical implications of denying Euclid's postulate
were not intellectual absurdities, as Lagrange had thought.
But part of an alternate reality.
And of course, the Lobachevsky inversion of non-Euclidean
geometry with space of negative curvature was not the only one.
There are lots more non-Euclidean manifolds, including some
of positive curvature as was explained to mathematicians notably by Riemann.
But, what about all that stuff that Kant said.
That space is in the mind, and that the properties of space are the same for
all human beings.
And that we order all our perceptions in space.
Kant's space was Euclidean, all right.
We can't possible order our perceptions in a non-Euclidean space, can we?
Yes said Hermann von Helmholtz.
Of course, in order to do this,
we have to divest ourselves of 2,000 years of Euclidian experience.
But if we can order our perceptions in a three dimensional non-Euclidian space,
Kant is wrong.
And if Kant is wrong says Helmholtz, then the postulates of geometry
are not dictated by the nature of the human intellect, or by logical necessity
whether space is Euclidian or not is a question for experience.
So I challenge, you let's see if through new experiences
you can learn to order your perceptions in a non-Euclidian three dimensional space.
Here we go.
You probably have a convex mirror on your car.
And there's a warning on it.
Warning!
The space you see in this mirror is not Eucledian.
>> [LAUGH] >> See the parallel lines on the top and
the bottom of the bookcase not being everywhere equidistant?
Now, yes, you can learn to order your perceptions and
touch a space if you have made it so far with your car.
I salute you for having done so.
But you might say, look, it's just a mirror.
It's an illusion, the world in the mirror.
Only our own world, the Euclidean world, can possibly be real.
Oh yeah, says Helmholtz?
Look imagine you're having a dialogue with the guy in the mirror.
You say my world is real, yours is distorted.
He says yeah how can you tell?
You say, well in your world when you get closer, and
closer to the mirror you grow bigger.
When you get farther and farther away from the mirror you get smaller.
This violates the obvious Euclidian fact that when you move something
around through space it stays the same size and shape.
Guy in the mirror says, really?
Yeah, let me try that.
So he takes a yard stick.
He comes up to the interface between his world and ours, and he measures himself.
I'm six feet tall.
And he goes to the back of the room, away from the mirror, and
he measures himself again.
No, you're wrong, he says, I'm still six feet tall.
And you say, well your yardstick changed its size also.
And he says, come on, Jack.
>> [LAUGH] >> Helmholtz says there is no geometrical
experiment that you can do that will decide the question of which one of
these worlds is the real one.
By the way, if that sounds like relativity theory, it's no accident.
There's a, I was gonna say a straight line but I shouldn't use that word.
There is a line of historical influence in German
philosophy from Helmholtz to Reinsma and then on to Einstein.
The British mathematician, philosopher W.K. Clifford
argued that the new geometry was a revolution like the Copernican revolution.
He said, it used to be that the aim of every scientific student of every subject
was to bring his knowledge of that subject into a form as perfect as that
which geometry had attained.
But no more.
Before Copernicus and
Clifford people thought they knew all about the whole universe.
Now we only know one small piece of it.
Likewise before non-Euclidian geometry the laws of space and
motion implied an infinite space, who's properties were always the same.
No more said Clifford.
Okay, space appears flat and continuous, but
only as far as we can explore, and no farther.
Well here is an even more radical view.
Also saw the new geometries as revolutionary, but
he disagreed with both Kant and Helmholtz.
If, as Helmholtz said, geometry comes from experience,
then geometry would not be an exact science.
Still, knows that we have more than one kind of space in our minds.
He's on the other side of the non-Euclidean revolution.
So the postulates of geometry are not synthetic a priori intuitions,
as Kant said.
They're not experimental facts as Hemholtz said,
and they aren't necessary self-evident truths as Voltaire and LaGrange said.
Geometrical axioms says are conventions.
Okay, so how should we decide which set of axioms to use,
those of Euclid, Lobochesky, or Riemann?
Says our choice among these conventions can be guided by experience, but
as long as we avoid contradictions, our choice remains free.
What are we to think of the question is Euclidean geometry true?
The question has no meaning.
We might just as well ask if the metric system is true and the old weights and
measures false.
One geometry cannot be more true than another, it can only be more convenient.
In the 20th century, it turned out that non-Euclidean geometries of the Riemannian
type, that's positively curved space were in fact more convenient because
Reman's type of geometry is what Einstein needed to do general relativity.
According to Einstein's theory of gravity,
the path of planets moving under gravity are shortest distances.
In a non non-euclidean manifold, light no longer travels in Euclidean straight
lines, but shortest distances as again in a non-euclidean space time manifold.
Is real space really Riemannian?
Would say that's what works.
You know though, if we're really interested in what works,
we also ought to study empirically how people actually perceive space and
even before non-euclidean geometry,
philosophers like Bishop Barkley pointed out that we don't see distance at all.
What we see are angles, and
we infer the geometry of what's out there from the angles we actually see.
I'm gonna give you a really easy example.
If I asked you about this,
you'd say oh look, three 90 degree angles are coming together, but
look at what you actually see, three 120 degree angles coming together.
Our visual space isn't the same as the space that we claim to see, and
the invention of non-euclidean geometry really made psychologists
think a lot about things like that, like Helmholtz.
Helmholtz did an experiment where he asked people in a dark room
to arrange little points of light along a table.
He wanted them to do two parallel lines going away from themselves.
So the people made the lines by the little points of light, and then he turned
the lights back on and you could see that the lines weren't parallel at all.
They curved away from the observer and so Helmholtz concluded, ha ha!
Space, our perception of space is Lobachevskian, but
there are psychologists who have done many more experiments, and it turns out
the space of visual perception isn't represented by any consistent geometry.
That's depressing.
Now this is all European material.
The cultural linguist Stephen Levinson has shown
that people in many different cultures have other ways of ordering their
perceptions than an external Euclidean space.
Some cultures do use the idea of a fixed coordinate system in space with
four cardinal directions as when you say the car is to the south of the building.
That's like Newton's framework.
But other cultures order their perceptions in space more in line with Levinson's
ideas, a relation between bodies.
Some from the individual point of view,
as if you say the car is to the right of the building.
You're in that okay.
Or, you can get the individual out of it entirely and just talk about the intrinsic
properties of the objects as if you say car is in front of the building.
In our own culture, GPS navigational systems are changing people's supposedly
innate intuitions of space from the Newtonian to the.
Once I asked a taxi drive in Maryland
if his GPS system had changed how he thought about space.
He was a smart guy.
He thought that was a really cool question, and I wrote down what he said.
Here's what he said.
I used to have the whole geography of Greater Baltimore in my head,
I don't any more.
Like when I take you somewhere, I'll turn left out of the airport,
I'll take the expressway to your exit, then I'll turn right.
When I leave you off, I'll just reverse that.
I'll turn left, I'll get on the express way,
I'll turn right at the airport, I'll get back, but I won't know where I've been.
>> [LAUGH] >> Well okay, let's get back to
the paradigm shift from Euclidean geometry to non-euclidean geometry.
I'll start with a classroom anecdote.
Got you!
Okay.
Once, I passed a convex mirror around in class and
one of my students looked at himself in the mirror and
he said this is distorted and somebody else said, you're a Euclidean chauvinist.
>> [LAUGH] >> So
what does modern theorists of culture have to say about this paradigm shift?
One especially interesting thinker on this topic was Jose Ortega y Gasset.
Ortega used the new geometries to argue that provincialism,
assuming that our own experience or values are universal, is wrong.
He sounds like Clifford here.
Euclidean geometry's provincial.
It was an unwarranted extrapolation of what was locally observed to the entire
universe.
Instead, Ortega says reality organizes itself to be visible from all view points.
Einstein's Theory of Relativity promotes the harmonious multiplicity
of all possible points of view, not just for Mathematics and
Physics, but also for Politics and Culture.
There's a Chinese perspective Ortega says, that is fully as justified as the Western.
Okay, now briefly to architecture.
Take Zaha Hadid, the first woman to win the Pritzker Architecture prize.
As an under graduate, she specialized in mathematics.
Notice we aren't in the Renaissance ideal city any more.
Smore.
Here's what she says about the world of the 21st century,
the most important thing is motion, the flux of things,
a non-euclidean geometry in which nothing repeats itself, a new order of space
and the new geometries inspire many other types of artistic freedom.
I'm gonna run by rapidly three examples from cubist art,
of what Ortega called reality organizing itself to be seen from all points of view.
Tea time.
I'm in England you know.
Look at the tea cup, the guitar, a portrait by Picasso.
Planes and angles from multiple points of views.
Notice the guy in the picture is not in a visually graspable
three dimensional space at all.
A different kind of geometrical influence on art, plaster models like
the ones in your building, this is all we got in Claremont, but you got lots more.
Mathematicians use these to teach and
study the geometry of non-euclidean surfaces.
Let me show you some art inspired by such models.
That's the world, the name of this one makes clear that it comes from geometry.
Remember the helix, the thing I just showed you?
It's just a lovely piece by Man Ray, it's just beautiful.
The plaster models that Man Ray studied
were the ones in the Pohanka Ray Institute.
Man gets his credit.
Pohanka Ray in Paris.
Man Ray was just blown away by them.
He said these shapes were so unusual,
as revolutionary as anything that is being done today in painting or in sculpture.
One more of many examples, I find this charming.
So, artists are influenced by the new geometry.
They cite it's authority and it's prestige on their own behalf,
they use it as part of their creation of self consciously modern art and
they help us see a different world.
Coming now to my last pair of slides, here's the first.
I found this searching Google image for hyperbolic paraboloid.
>> [LAUGH] >> And I like this picture a lot
because it shows us how much non-euclidean geometrically objects
had become part of the general culture already more than half a century ago.
And here's one that became famous much more recently in your own country.
Maroff is another hyperbolic paraboloid.
Okay so you're entitled to conclusion, conclusion.
Twice in history, first in Euclid's world and then in the non-euclidean world,
ideas about geometry have helped shaped art and architecture, and science,
and philosophy, and have helped shape the way people see and think about the world.
I hope that this talk, and
in fact this entire conference, help a wider public better understand
the intimate relationship between Mathematics and culture.
Thank you very much.
>> [APPLAUSE]