Welcome to the Endless Knot!
Today is part two of a three episode series
about the word Average, in which we look at
probability and property!
As we saw in the last video, the word average,
which comes from an Arabic word that means
‘blemish’ and ultimately from a Proto-Semitic
root that means “to be one eyed”, originally
referred damage to shipping and how averaging
out those losses was an early form of insurance.
We also saw how Ptolemy had a geocentric model
of the universe, and how his star charts were
passed on to medieval Europe by Islamic scholars,
and how the Islamic world also passed along
coffee to European society, where coffeehouses
became hives for business transactions, including
the Italian invention of contract insurance,
which led to insurance brokers in Lloyd’s
Coffeehouse in London to become the insurance
market Lloyd’s of London.
Well the other thing the insurance business
needed to really get going was a way of predicting
the likelihood or chances of unfavourable
events.
So we’re back to predicting the future again.
The word likely, by the way, which around
1300 had the sense of “having the appearance
of truth or fact” and from that gained its
sense of “probable” in the late 14th century,
comes from Old Norse likligr, replacing the
native Anglo-Saxon cognate geliclic.
The word chance, on the other hand, comes
through French from Latin cado “to fall,
die”, and when it entered English around
1300 it had the sense “something that takes
place, what happens, an occurrence”, in
other words how matters fall, but reminding
us I suppose of how the dice fall, and thus
became a synonym for probability.
And the word probability itself comes from
Latin as well, from probabilitas.
This in turn is cognate with the word prove
and comes from the Latin verb probare “to
make good, esteem, represent as good, make
credible, show” from probus “worthy, good,
upright, virtuous” from Proto-Indo-European
*pro-bhwo- “being in front”.
As for calculating probabilities, we again
have the Islamic world to thank.
For the first name in the history of probability
theory is Al-Kindi the 9th century Arab mathematician,
philosopher and, if you’ll pardon the pun,
all around polymath.
As a philosopher he adopted and adapted Greek
ideas, and as a mathematician he was the first
to use statistics and probability to decode
a cipher by working out what the letter frequencies
were.
But in addition to his kicking off the study
and use of probability and statistics, he
is probably best known for introducing Indian
numerals to the Islamic world, and thence
to Christian Europe, where they became known
as Arabic numerals.
The 12th to 13th century Italian mathematician
Fibonacci was the first to popularize the
so-called Arabic numerals in Europe through
his 1202 book Liber Abaci or Book of Calculation.
You probably know him from the Fibonacci numbers,
which he also introduced in that book.
It was an important and influential work,
and was the inspiration and one of the main
sources for the book Summa arithmetica or
Summary of arithmetic, by the 15th to 16th
century Italian mathematician Luca Pacioli,
itself containing a number of firsts.
It was the first description of double-entry
bookkeeping, useful I suppose for all those
later coffeehouse financial transactions,
which led to Pacioli often being referred
to as the “father of accounting and bookkeeping”.
But for our purposes Pacioli’s book is important
for another first, the first mention of the
problem of points, to which he, incorrectly,
offered a solution.
The problem of points can be explained thusly:
imagine two gamblers are playing a coin toss
game upon which is riding a monetary prize.
The game is to see who is the first to win
ten coin tosses.
But for some reason the game is interrupted
and the players want to figure out how to
fairly distribute the stakes between them.
Simple enough to divide it in half if they
were tied, but harder to work out if one had
a lead.
It’s clear in that case that one of the
players has a greater chance of winning than
the other, but what chance?
This problem kicked off the development of
probability theory and the maths to solve
problems of probability.
Our next stop in the history of probability
was one Gerolamo Cardano, who was inspired
by Pacioli’s work.
You see Cardano was another one of these polymath
types, working as a physician, but also a
part-time mathematician and inventor, inventing
for instance the combination lock.
He was also an avid and disreputable gambler
— you can see why he was so interested in
Pacioli’s probability work — and he was
often short on funds, keeping himself afloat
by gambling and playing chess.
He was thus the first to write systematically
about probability and games of chance, publishing
his Liber de ludo aleae or Book about Games
of Chance in 1539, which included not only
the mathematical treatment of probability,
but also ways to cheat like rubbing a card
you want to draw from a deck with soap.
He wrote about the use of expressing odds
as the ratio of favourable to unfavourable
outcomes, like there’s a 1 in 6 chance of
rolling a six with one die, and even worked
on figuring out the probability of rolling
a seven with two dice.
As a result of all this, Cardano is sometimes
referred to as “the gambling scholar”.
So we also have gambling to thank for probability
theory.
Speaking of gambling, the word gamble is related
to the word game, as in games of chance, coming
from Old English gamenian “to play, joke,
pun” ultimately from the Proto-Germanic
collective prefix *ga- plus *mann meaning
“person” giving a sense of “people together”.
Gamble probably gained its “b” by influence
from the otherwise unrelated word gambol,
as in a lamb gambolling.
I suppose you need good luck when gambling,
and luck is an odd word with an uncertain
etymology.
It probably comes from Middle Dutch luk, a
shortening of gheluk meaning “happiness,
good fortune” and cognate with modern German
Glück meaning “fortune, good luck”.
But where this word ultimately comes from
is entirely unknown.
Another unexpectedly luck-related word is
speed, which comes from Old English sped “luck,
prosperity, success”.
It comes ultimately from Proto-Indo-European
*spe- “to thrive, prosper”.
The sense of “quickness”, now the dominant
sense, didn’t emerge until late Old English,
but there is a remnant of the older meaning
in the expression Godspeed which actually
means “may God prosper you” or even just
“good luck” and has nothing to do with
quickness, though I’m sure God is very fast.
But getting back to the gambling scholar Cardano,
he was also into astrology — there’s predicting
the future again — and struck up a friendship
with fellow astrologer and Lutheran theologian
Andreas Osiander.
Osiander edited a number of Cardano’s books
and even received a dedication in one of them.
Another writer that Osiander edited, who didn’t
get along so well with him, was Nicolaus Copernicus.
You see in his De revolutionibus orbium coelestium
or On the Revolutions of the Celestial Spheres,
Copernicus challenged that old Ptolemaic geocentric
model of the universe, presenting instead
a solar system with the sun in the centre
and the planets in orbit around it and the
various moons in orbit around the planets.
Made more sense of the apparent movement of
the celestial objects.
But while editing, unbeknownst to Copernicus,
Osiander slipped in his own preface to the
book stating that it wasn’t meant to be
taken literally, it was just a mathematical
model.
Copernicus was furious but by then it was
too late and there was nothing that could
be done about it, and soon after Copernicus
died.
But coming back to Pacioli’s problem of
points, it was finally solved in 1654.
The problem came to the attention of a French
writer named Antoine Gombaud who is more commonly
known as the Chevalier de Méré.
He wasn’t actually an aristocrat, it was
just a name he invented for his dialogues,
but soon his friends started to refer to him
that way and the name just stuck.
In addition to being a writer, the Chevalier
de Méré was also a proficient gambler as
well as an amateur mathematician, but his
math skills weren’t up to solving the problem,
so he brought it to his friend Blaise Pascal.
Pascal was a child prodigy in mathematics,
making many discoveries while still a teenager.
In 1650 he had something of a religious epiphany
while suffering from ill health, and abandoned
mathematics, turning instead to religious
meditation and philosophy.
He eventually did return to mathematics but
died at the unfortunately young age of 39.
As a result of all this he is known as greatest
might-have-been of mathematics.
Well, he started corresponding with fellow
French mathematician Pierre de Fermat about
that problem of points which the Chevalier
de Méré brought to him.
Actually Fermat was a lawyer with no formal
mathematical training.
Indeed he didn’t even get onto mathematics
until he was in his thirties.
But unlike his friend Pascal his life was
long and mathematically productive.
Perhaps best known for Fermat’s Last Theorem,
his contributions to mathematics were so great
that he is often referred to as “the prince
of amateurs”.
Well between them in their correspondence,
Pascal and Fermat worked out two entirely
different ways of solving the problem which
produced the same results, and the methods
they developed became the backbone of probability
mathematics.
And in keeping with Pascal’s vacillation
between mathematics and religion and philosophy,
Pascal united these two interests in the realm
of probability, writing “We know neither
the existence nor the nature of God … Let
us weigh the gain and the loss in wagering
that God is.
Let us estimate these two chances.
If you gain, you gain all; if you lose, you
lose nothing.
Wager, then, without hesitation that He is”.
In 1657, just three years after Pascal and
Fermat created probability maths, the Dutch
astronomer and physicist Christiaan Huygens
wrote it all up in the first formal treatise
called De ratiociniis in ludo aleae or On
Reasoning in Games of Chance.
And it’s perhaps fitting that Huygens is
best known today as an astronomer, since probability
came to be very useful in that field.
For instance, years later another child prodigy
mathematician Carl Friedrich Gauss used the
method of least squares to accurately predict
the location of the dwarf planet Ceres from
only a few observations as data points.
Imagine you have a graph with just a few data
points on it.
The method of least squares allows you to
find the line of best fit for that scant data.
So once again we return to the effort to determine
the motion of celestial objects, just like
Ptolemy and Copernicus.
Another important early contributer to probability
and statistics was Thomas Bayes, a Presbyterian
minister by calling.
He is most famous for Bayes Theorem, which
basically allows one to accurately work out
the probability of an event based on prior
knowledge.
Bayes actually published very little on mathematics
during his lifetime, and it was up to French
aristocrat and scholar Pierre-Simon Laplace
to further develop Bayes’ Theorem.
And in a nice bit of interconnection, Laplace
had also tried but was unable to calculate
the orbit of the dwarf planet Ceres, a problem
which you remember Gauss solved.
One of Leplace’s students, Joseph Fourier
made important contributions to both mathematics
and physics, but it’s, oddly enough, his
work on heat transfer that interests us.
Fourier was very interested in heat, and is
credited with discovering the greenhouse effect.
You see, Fourier got into an academic argument
with Siméon Denis Poisson over the theory
of heat — Poisson was forced to retract.
However Poisson had more luck in his work
on probability theory, which included the
Poisson distribution, which allows one to
know the probability of a given number of
events occurring in a fixed interval of time,
exactly the sort of thing an insurance company
needs to know.
After marine insurance, the next type to develop
was property insurance, specifically fire
insurance.
Unfortunately it’s a bit of a shut-the-barn-door-after-the-horse-has-bolted
sort of thing, because what really pointed
out the need for fire insurance was the Great
Fire of London in 1666, in which more than
13,000 homes burned down.
The job of rebuilding fell to architect Christopher
Wren, also a sometime physicist and mathematician
whose scientific work was highly regarded
by our friend Pascal.
Clearly Wren observed the need for an insurance
office as he included in his rebuilding plans
a site for one.
Wren’s assistant was polymath Robert Hooke,
the trajectory of whose life ran from being
a penniless scientist, to a wealthy and admired
member of society, to eventually an old man
in ill health, jealous and bitter towards
his scientific contemporaries.
However, Hooke’s efforts as surveyor after
the Great Fire of London won him much acclaim.
In addition, Hooke worked on the problem of
timekeeping and celestial navigation.
You see in order to calculate longitude (how
far east or west you were) you needed to know
the time back home where all your star charts
were calibrated to.
If you take a reading of a star’s position
and find out how far out it is compared to
the star chart, you can work out how far east
or west you are to see the star in that position.
But you can’t use a pendulum clock at sea,
and spring driven clocks weren’t accurate
enough.
So Hooke invented a balance spring pocket
watch which was up to the challenge, and tried
to patent and develop the technology, but
was unable to finance it, no doubt adding
to the bitterness of his later years.
What’s more our astronomer friend Christiaan
Huygens independently came up with the same
idea some five years later.
But getting back to the insurance office,
the first one to be established was founded
by a man with an unusual middle name: Nicholas
If-Christ-had-not-died-for-thee-thou-hadst-been-damned
Barbon.
Yes that’s actually his legal middle name,
the practise of giving such hortatory middle
names being popular with the Puritans at the
time.
Well, Barbon and eleven associates founded
the Insurance Office for Houses located at
the back of the Royal Exchange, the first
fire insurance company, and soon other companies
were started.
And once those new insurance companies got
their hands on all of that new probability
math such as the Poisson distribution, insurance
companies could estimate how often claims
would come in and thus set their premiums
appropriately to average out their risks and
losses.
Thanks for watching!
I’ll be back soon with the final part in
our look at the word Average, in which we
investigate statistics and stock markets.
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