Harold Brochmann
So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality.
Albert Einstein: Geometry and Experience
Part 1: Newtonian vs Recursive Functions
Here is a familar example of a recursive
function:
FUTURE VALUE OF INVESTMENT
Suppose we have $100.00 which is to be invested at 10% pa, compounded
annually. After one year we will have $110.00. The calculation
is done like this:
V1 = V0 x (1 + r) = $100.00 x (1 + 0.10) = $110.00
At the end of the second, third and fourth years our investment will have grown to $121.00, $133.10, and $146.41:
V2 = V1 x (1 + r) = $110.00 x (1 + 0.10) = $121.00 V3 = V2 x (1 + r) = $121.00 x (1 + 0.10) = $133.10 V4 = V3 x (1 + r) = $133.10 x (1 + 0.10) = $146.41
This is a Recursive function because it is carried out repeatedly; the input for each recursion being the output of the previous one. The function may be generalized like this:
Vn+1 = Vn x (1 + r)
The same result can be achieved using the Newtonian Function for calculating the value of the investment at the end of each year:
V1 = V0 x (1 + r)^1 = $100.00 x (1 + 0.10)^1 = $110.00 V2 = V0 x (1 + r)^2 = $100.00 x (1 + 0.10)^2 = $121.00 V3 = V0 x (1 + r)^3 = $100.00 x (1 + 0.10)^3 = $133.10 V4 = V0 x (1 + r)^4 = $100.00 x (1 + 0.10)^4 = $146.41
These two kinds of functions accomplish the same end and may be generalized by:
Vn = f (V0,r,n)
which simply says:
"The value after some number of compounding periods is a function of initial value, the interest rate and the number of compounding periods."
It doesn't say what
the function is; just what the parameters are.
Note that the graph is smooth. We say that this is a differentiable function
because small changes in input (such as the initial amount) lead to
small (and predictable) changes in output. For example:
$100.00 compounded at 10% for 10 years yields $259.37 $100.50 compounded at 10% for 10 years yields $260.67
... $0.50 change in input yields $1.30 change in output. Similarly,
increasing the rate of interest from 10% to 10.05% raises the future vale by only $1.19.
Finally,
If the investmenty is held for 3 months short of the ten year period, its value is reduced by only $6.10
....which may expressed this way:
if Vn = f (V0 , r ,n ) then Vn + DVn = f (V0 + DV0, r , n ) and also Vn + DVn = f (V0 , r + Dr, n ) and also Vn + DVn = f (V0 , r , n + Dn)
It is possible to calculate the 'slope' of the graph at any point.
We describe this by saying that this is a differentiable function.
In the classic Newtonian universe, the Laws of Nature are written as differentiable functions.
Mathematics is the language with which God wrote the Universe. - Galileo
This means that the classic Newtonian universe is deterministic: if you know The Laws, and you know present conditions, you can predict the future. This means that Man choose his future. The doctrine of free will.
THIS IS POWERFUL STUFF
Even the motions of
celestial objects can be predicted using differential equations.
This illustration is the output of a computer program that calculates
and plots the elliptical path of a 'planet' which is placed in
orbit around a 'star'. For simplicity's sake, the universe is
assumed to be 2-dimensional. In this simulation, the 'planet'
starts off at coordinates (0,200) and has an initial velocity
of 200 units in the 'x direction'.
At this point we pause for a moment to observe that it is possible
to calculate such a simple orbit using either classic newtonian
functions as well as recursive ones; and also that recursive
functions were impractical before we got computers.
We note that the orbit is elliptical and 'progressive'. It is also stable; which means that the 'planet's' velocity and periodic distance from the 'star' does not change appreciably, regardless of how long the simulation is allowed to run.
Here we see a graph
showing the instantaneous speed of the planet over time. Note
that it varies in what might be called a 'cyclic way'.
It is also possible to plot a graph showing the distance that
the planet is from the star at any time:
Finally, we draw a phase diagram of these two dependent variables. Here is a
graph of distance vs time:
The shape of this graph is called its attractor. Note
that the attractor is 'simple'.