I'll end this unit by talking
a little bit about some ideas
from the history
of philosophy and science,
focusing on Newton and Laplace.
This isn't a definitive
or comprehensive history by any means.
I just want to highlight a few ideas
that will help set the stage
for the next unit,
when we encounter chaos
and the butterfly effect.
My main goal is not so much
to get a historical account,
but to look at some historical ideas
to help maybe elucidate,
or bring to the forefront,
some ideas about science, and how science
goes about explaining the world,
and some assumptions about the world,
that I think are still with us today.
And these are some ideas
that the discovery of chaos
and various related phenomena
cause us, I don't think
to throw out completely,
but maybe to re-examine
or reconsider a little bit.
In any event, our story
begins with Isaac Newton.
In 1687, Newton published
the Principia Mathematica,
in many ways the crowning achievement
of the Scientific Revolution.
So, there's a lot that Newton did in this,
but one of the main things he did was,
he laid out a theory of motion.
He explained why things move.
And this is most clearly embodied
in what's now known
as Newton's Second Law of Motion.
You've likely seen this before
if you've taken a physics
or physical science class.
So the equation says –
On the left-hand side we have: F⃗_{net}.
F is the forces, and net means total.
So we're looking at the total forces
that are acting on an object.
The arrow on top of F means
that force is a vector quantity.
When considering the total forces,
we need to take into account
not just how strong a push or pull it is,
but the direction of that push or pull.
And vectors are a mathematical way
for doing that.
The right-hand side
of the equation is: ma⃗.
a is acceleration.
It's the rate of change,
or the rate of change of the position,
or the rate of change of velocity –
how fast something's slowing down
or speeding up.
Like forces, acceleration
is a vector quantity.
In order to describe motion
I need to say not only the magnitude
of acceleration but also the direction.
And m is an object's mass.
So what Newton's second law says is
that if I know the total forces
acting on an object,
and I know its object's mass,
I can figure out its acceleration,
and from its acceleration
I can figure out how it will move.
The basic idea is
that forces determine motion
in the same sense that the functions,
iterative functions
and differential equations,
that we've been studying so far,
are deterministic.
So Newton's second law is a rule;
it tells us how things move
if you know the forces that act on them.
So the second law is –
I think of it as a theory of motion,
it says why things move.
Another key part of Newton's Principia
was that he put forth the idea
that the laws of physics are universal.
And in many ways I think this is the most
revolutionary part of Newton's work.
So he said that the laws of physics
that describe how an object falls
in Cambridge, England,
those laws are at the same in Brussels
or Beijing or Bogota, anywhere.
Then not only anywhere on the Earth,
anywhere in the Universe.
So the same law that describes
how an apple falls in England
describes how the Moon orbits the Earth
or how the Earth orbits the Sun.
So, as I said before,
Newton's Principia is in many ways
a crowning achievement
of the Scientific Revolution.
And to summarize in very brief strokes,
the idea of the Scientific Revolution
is that knowledge can be generated
through careful observation
and repeated experiment,
building on the work of others,
and that that knowledge is expressed
in a coherent
and consistent logical framework
that's often mathematical in nature.
So Newton in the Scientific Revolution
gives way to a period of intellectual
history called the Enlightenment,
that I think of
as spanning most of the1700's.
The Enlightenment is associated
with expanding individual rights
and democracy,
continued scientific advances,
a belief in the power of logic and reason
over authority and doctrine.
So during the Enlightenment
a scientific method and scientific
worldview starts to solidify.
And I sort of think of this
as a Newtonian framework.
So, some ideas about the world
and about science
that aren't directly contained
in Newton's Laws
but sort of flow naturally from them.
So, in the Newtonian framework,
the world is mathematical,
mechanistic and material.
It's mechanistic
because things happen for a reason,
objects move because forces act on them.
So motion can be explained,
things happen not just because,
but for particular reasons.
It's mechanistic in that the world is seen
as being made up of stuff,
material objects.
Even forces that act apparently
at a distance, like gravity,
were viewed as being transmitted
by a tangible physical medium.
So it's a world of things, of stuff,
in which things happen for a reason.
And lastly the world
is seen as being mathematical.
The laws of nature,
and maybe nature itself,
is seen as being mathematical.
And Galileo has a quote
that summarizes this idea really nicely.
So Galileo says: "Philosophy is written
in this grand book of the Universe,
which stands continually open to our gaze.
But the book cannot be understood
unless one first learns
to comprehend the language
and to read the alphabet
in which it is composed.
It is written
in the language of mathematics.
So although that quotation
is hundreds of years old,
I could imagine a physics professor
saying that today.
So in the Newtonian framework
the world is mechanistic,
material and mathematical.
The picture that emerges
is of the Universe as a giant clock,
or a giant machine.
Gears and levers pushing on each other,
making things happen,
making other things happen,
and then making other things happen,
but always following the laws of motion.
The Newtonian world
is one of cause and effect.
There are laws
and a certain orderliness out there.
And even if we don't understand
all of those laws,
the Newtonian world holds out the promise
that the world is fundamentally
an understandable and logical place.