The Newtonian world is one of cause and effect. Objects move as they do, according to universal laws of physics.
So, if we know the current position of an object, and we know the forces acting on it,
we can predict the future positions of the object. This is what Newton's second law tells us.
So, we would say that the motion, in this case, is deterministic,
just like the iterated functions, and the dynamical systems,
or the differential equations we've studied, are deterministic.
If we know the starting point, and we know the rule that governs it, then the path is unique,
guaranteed to exist, and is determined just by knowledge of the starting value, the initial condition.
So, if this idea applies to one object, presumably it would apply to other objects, and why not to all objects in the universe,
and may be we don't know all the laws in the universe yet, but, in the Newtonian view, there are laws,
there is something that is determining all of this,
and so, if we extend this idea, we are led to the conclusion that the universe itself is deterministic,
- that the way things are going to be tomorrow is already determined.
Perhaps we don't know it, but it's a logical, inevitable consequence of the way things are today - there can ony really be one tomorrow.
So, if we extrapolate these ideas a little bit, we run into a lot of puzzles,
and these are famously summarised and stated in a passage by Laplace from the early 1800s.
So, Laplace writes: "We may regard the present state of the universe as the effect of its past and the cause of its future.
An intellect, which at a certain moment would know all forces that set nature in motion,
and all positions of all items of which nature is composed,
if this intellect were also vast enough to submit these data to analysis,
it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom;
for such an intellect, nothing would be uncertain and the future just like the past would be present before it's eyes."
This vast intellect is now often referred to as "Laplace's Demon";
he didn't use the terms when he wrote this, but that's the standard term for this idea now.
Laplace's Demon is often used to explore questions of free will.
If the universe is mechanistic and deterministic as Laplace is suggesting, then everything has already been determined,
it's an inevitable consequence from the current conditions today.
I'm certainly not a vast intellect capable of seeing the future as if it was the present,
but even if there is no vast intellect, that doesn't change the fact that the future has already been determined,
and the fact that the present was already been determined by the past,
and this collides with the idea we have free will, that we have choices,
and this all - at least to me - a little bit too much to think about,
I mean, I feel like I decided to, say, cook a spicy tofu dinner last night,
that that wasn't determined the day before, or in 1850, or in 750.
I feel like I have choices now, in what I'm saying to you, and I had choices in that, which I just said,
and it wasn't all determined, and that there really is a me in there,
and it just gets to be too much to ... I mean, who ... and I have to take a break, and...
...maybe I'm not deciding to take this break, maybe that was already determined ... [ ! ]
OK, I don't know anywhere near enough philosophy to know where to begin to think about free will.
It 's an interesting question, to be sure, but it's actually not why I bring up Laplace's Demon.
To me, what's most interesting about Laplace's Demon, for this course,
is that it very succinctly spells out much of what science aspires to, - to get as close to Laplace's Demon as possible.
Laplace's Demon sounds like a naive idea, and nobody I think, thinks it's anything that's attainable or achievable,
but it does sort of say the direction that science is trying to move, to try to get closer and closer to this ideal Laplace's Demon.
Now, doing so requires, I think, at least three things: First: careful measurements.
The whole idea of this deterministic view is that the future is determined by the current conditions,
so we need to understand the current conditions very well, so we need lots of good, careful data.
second of course, we need to know the laws, or rules, or equations that govern the system - or systems - we're interested in;
and third, we need sufficient computing power,
so that, given those laws, and given the initial conditions, we can predict what the future behaviour will be.
Behind all this, I think, is the idea that the world ought to be understandable,
that we should be able to make measurements, figure out laws, figure out rules - maybe even approximate rules,
and then use computing power to figure out what's going to happen, and then maybe change what's going to happen.
So, for one example, if one may be thinks about curing cancer, or finding a cure, or treatment for a particular type of cancer,
it seems like if we knew a little more biochemistry, understood a little more genetics, had more data,
knew exactly what was going on in the cell, and had sufficient computing power to
grind out and compute all sorts of different possibilities, that,
OK, maybe we won't completely cure cancer, but that we should be able to
come closer and closer to understanding cancer, and coming up with a better treatment.
In any event, I think that aspiring towards Laplace's Demon is a good description for much of what goes on in science.
The belief is that if only we could measure things more accurately,
or had a better understanding of the laws that govern the phenomena we're interested in
and had more computing power, we'd be getting closer and closer to the truth.
I don't know that this describes all of scientific activity, but I think it describes a lot of it,
and, one of the central ideas behind this is that there are laws, that there are rules, there is some determinism,
and so then we could ask: If we live in a rule base, deterministic world, what sort of things are possible? It sounds kind of boring.
In the first two units of this course, I've introduced two types of dynamical systems, and a dynamic system is just a type of rule.
So, we've studied iterated functions, and differential equations, and these types of dynamical systems - rules, or laws
- are very commonly used for describing physical, and all sorts of phenomena,
- differential equations in particular, they're almost the language of physics, and parts of engineering.
So, we can ask then, and this is the approach of dynamical systems:
given a class of dynamical systems - differential equations, or iterated functions -
What can these equations do? What type of dynamical behaviour do we see?
and, admittedly, the answers to this question - so far - have been a little bit dull.
We've seen fixed points, and we've seen behaviour that goes off to infinity, or negative infinity,
but all that will change in the next unit,
where we'll encounter surprisingly complex and interesting behaviour, along with chaos and the butterfly effect.
So, we'll see you next week in unit 3.