Epicycles, and neoclassical economics

How Ptolemy was smarter than Samuelson

Epicycles

Now term of condescension

Ancient Greek theory of Lunar and planetary motion

Replaced by Kepler’s model

Seen as classic example of baroque complexity in theory

But epicycle theory a model of science compared to neoclassical economics!

But

The epicycle theory was an early scientific theory

It made pretty exact predictions, and it was potentially falsifiable

In contrast, neo-classical value theory is far more baroque, makes only vague predictions and is unfalsifiable by empirical data.

What was epicycle theory

What problem did it try to solve

How well did it solve it

Why was it rejected

Lunar motion

Simple idea is that the Moon goes round in a circle centered on the Earth.

Two problems

1. Change in apparent size of the moon2. Variation in speed of movement against the fixed stars

Size

The Moon is very visibly closer to us at times, so it can not be going round in a perfect circle

Full Moon Super Moon 14% bigger

Speed

When it is closer to us, it is seen to move more rapidly against the background of the fixed stars

So the motion can not be uniform circular motion.

Off circle

Clearly the super moon at Perigee is inside the average circular distance

And at Apogee it is outside the average circular distance

The Greek astronomers needed a theory to account for this

EarthWhen super

moon

ApogeePerigee

Epicycle

This answers both problems

Allows the super moon to be closer

At this point the motion of the two circles adds, so the movement against

the stars is faster

At Apogee the epicycle makes it further and the rotations work against one another to slow the aparent motion against the stars.

Anticlockwise main orbit

Epicycle moves clockwise

Start of month

14 days later

Ptolemaic orbit

The blue line shows the orbital path of the moon according to Ptolemaic astronomy.

It is composed of uniform circular motion.

Epicycle rotates twice each main orbit.

Could be well calculated by astronomers using compasses and rulers, or by ancient mechanical computers.

Mechanical models

−Before Kepler, Ptolemy−Before Ptolemy came Hipparchus and Apollonius

Ptolemy’s epi-cycle model is well known, but it is equivalent to Apollonius's Cycle and Deferent Model

Comparison of Ptolemy and Apollonius

Hipparchus’s actual model?

In 1900 a group of sponge divers sheltering from a storm anchored off the island of Antikythera.

Diving from there they spotted an ancient shipwreck with bronze and marble statuary visible.

Further diving in 1902 revealed what appeared to be gear wheels embedded in rock. On recovery these were found to be parts of a complicated mechanism, initially assumed to be a clock. Starting in the 1950s and going on to the 1970s the work of Price established that it was not a clock but some form of calendrical computer.

Using X-rays, modern reconstructions have been built showing that it physically implemented Apollonius's model of the lunar orbit.

It used epicyclic gears

These are modern ones

Modern copy

The original machine dates from the 2nd century BC but modern reconstructions have been built.

I show a particularly beautiful one by Tania van Vark.

You turn the handle and get predictions of the position of the sun and moon in the sky and the dates of eclipses.

It emphasises how a scientific model is a microcosm emulating a macrocosm.

Flaws

There are two other points in the orbit where the moon should look far away, but does not.

Modern solution

Moon moves in an ellipse with same apogee and perigee as before.

Rotation follows Keplers law in equal time equal areas are swept out.

Hipparchian orbitKepler orbit

Earth

Epicycle puts moon too far at this point.

To scale

Previous diagrams exaggerated the degree of eccentricity of the lunar orbit

This is to scale,

At this scale the epicyclic orbit does not look bad

Ellipse also epicycle

If you have an epicycle with the same period as the main orbit, you get a perfect ellipse!

So the modern theory is not so different.

Problem is that the ellipse you get this way has the earth at its center not at one of the foci of the ellipse - so no perigee opposite appogee. Also gets speed against stars wrong.

Why is Newton better?

Economy of information

● For solar system need, angular momentum, radius and rotational phase for each planet at time 0, the laws of motion give us all the rest: 3 numbers per planet

● For Ptolemy we need at least 2 radii, 2 rates of rotation and 2 starting phases: 6 numbers per planet

○ That is twofold redundancy compared to Newton

Epicycles and economics II

Leibniz principle

A scientific law must perform data compression.

It must have fewer parameters or free variables than the observations that it has to account for.

The more data compression the better.

Leibniz claim

Given any set of data points on graph paper he claimed he could come up with a formula for a curve which went through the points.

Lets try that with the world oil market data from 2007 to 2016

Ignoring the years, there are 20 numbers here

yearoutput million barrel per day

$ price per barrel

2007 82.3 68.19

2008 83 94.34

2009 81.2 61.39

2010 83.3 78.06

2011 84 106.18

2012 86.2 109.08

2013 86.5 105

2014 88.7 97

2015 91.5 51

2016 92 41

As scatter plot

What is the mathematical relation between these points?

They look almost random, but perhaps there is a sort of curve to them.

Exact formula

It took me a little while but it is not hard, using Fourier series to come up with a formula that exactly fits the observed data for oil prices

Remember what a cosine means

Yellow line is =cos(Θ) + 0.2 cos(-2Θ)

θ

cos(θ)

Sum of 2 cosines

Ptolemaic price theory

The equation for P as a function

of q is a direct application of the

Ptolemy epicycle method.

First equation defines the

quantity range between 80 and

100 to be one great rotation, on

which are superimposed 9

epicycles to generate the

observed prices.

How many parameters

We have a general formula with 10 constants

Plus another couple in the definition of theta.

How many parameters

We have a general formula with 10 constants

Plus another couple in the definition of theta.

The original 20 numbers are now being rendered with 12 numbers plus quite a few maths symbols.

So in terms of explicit parameters used it is a pretty efficient formula

How many parameters

We have a general formula with 10 constants

Plus another couple in the definition of theta.

The original 20 numbers are now being rendered with 12 numbers plus quite a few maths symbols.

Is this a law? No because the formula is still too long and complicated

Samuelson

According to Wikepedia: American economist and the first American to win the Nobel Memorial Prize in Economic Sciences. The Swedish Royal Academies stated, when awarding the prize in 1970, that he "has done more than any other contemporary economist to raise the level of scientific analysis in economic theory"

Samuelson

According to Wikepedia: American economist and the first American to win the Nobel Memorial Prize in Economic Sciences. The Swedish Royal Academies stated, when awarding the prize in 1970, that he "has done more than any other contemporary economist to raise the level of scientific analysis in economic theory"

Author of standard textbook of economics.

Samuelson on price

Market price is explained as the result of two curves intersection, the S or supply curve and the D or demand curve.

Samuelson on price

Market price is explained as the result of two curves intersection, the S or supply curve and the D or demand curve.Lets look at this from Leibniz viewpoint.

Samuelson on price

Market price is explained as the result of two curves intersection, the S or supply curve and the D or demand curve.Lets look at this from Leibniz viewpoint.How much informat is required for the curves

Samuelson on price

Market price is explained as the result of two curves intersection, the S or supply curve and the D or demand curve.Lets look at this from Leibniz viewpoint.How much information is required for the curves.

Copy of D curve from Samuelson’s book, with same data points

Samuelson on price

Market price is explained as the result of two curves intersection, the S or supply curve and the D or demand curve.Lets look at this from Leibniz viewpoint.How much information is required for the curves.2nd order polynomial only approximately fits D curve so it is at least a 3rd order polynomial

Copy of Dcurve data points from Samuelson’s book

Samuelson on price

Market price is explained as the result of two curves intersection, the S or supply curve and the D or demand curve.Lets look at this from Leibniz viewpoint.How much information is required for the curves.2nd order polynomial only approximately fits D curve so it is at least a 3rd order polynomialNeeds at least 4 numbers to specify each curve.

This is gross overkill

● 2 curves to with a total of 8 parameters to explain 2 numbers

○ The supply price, price pair (12, 3)

● That is a fourfold redundancy!○ Ptolemy’s epicycle theory only had twofold redundancy

○ And the supply and demand curves are even more imaginary than

Ptolemy’s epicycles

● By successive observations Ptolemy was able to calculate the parameters of his epicycles

○ Samuelson’s curve pairs can never be actually measured

Impossible to estimate

Since we only have one data point, we can potentially draw lots of curves through it.

There are infinitely many pairs of curves you can draw that are compatible with the same actual observation.

These are all simply figments of Samuelson’s chalk on a blackboard. They are all entirely imaginary.

Samuelsons curves Another possible pair of curves for the same observation

S

D

Not operational

A curve and the corresponding function is only a scientific abstraction if there is an operational procedure for estimating the function.

To do that you need multiple observations, as many as there are parameters on your curve.

Not operational

A curve and the corresponding function is only a scientific abstraction if there is an operational procedure for estimating the function.

To do that you need multiple observations, as many as there are parameters on your curve.

But for each new observation Samuelson introduces a new curve.

According to Samuelson’s theory, for each price change at least one of the curves has shifted.

If we knew which one had shifted then we might begin to estimate the other one.

But since both of them may have shifted we are none the wiser.

Samuelson’s diagram

What curves shifted to produce this?

An economist if free to invent 20 curves intersecting at 10 points to give us these real observations.

An nobody can prove of disprove of their existence.

In this sense his curves are like invisible fairies or angels.

You can believe in invisible fairies, and nobody can disprove their existence.

But it is not science!

Compared to my joke ‘economic law’

My joke theory of oil prices requires 12 parameters to fit the data

Compared to my joke ‘economic law’

My joke theory of oil prices requires 12 parameters

With Sameulson’s theory we can go wild - I have tried to join as many points as possible with the same curves, but it would still require around 35 parameters to account for 10 observations.

This is pure mysticism, wizards inscribing pentagrams to impress apprentices.

Joke theory of price

Definition of random data:

Data whose mathematical description is more complex than the data itself. ( Chaitin )

Economies are stochastic systems.

Constructing theories like neoclassical economics that give no information economy is just a form of higher superstition.

Some things are just random