# The Eve Before Calculus : Fermat

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# René Descartes

René Descartes (31 March1596 – 11 Feb1650), the 17th century French mathematician who invented XYZ Cartesian Analytical Geometry.

He also invented the ‘Methodology’. His scientific philosophy ‘Cogito ergo sum’ (Je pense donc je suis / I think therefore I am / 我思故我在) influenced Issac Newton later.

# Descartes 6th Meditation

Descartes 6th Meditation

Body is by nature divisible.

If so and if Mind and Body are one and the same,
then Mind is also divisible.

However, the Mind is entirely indivisible.
It follows that the Mind and Body are not the same.

# Probability by 2 Great Friends

Today Probability is a “money” Math, used in Actuarial Science, Derivatives (Options) in Black-Scholes Formula.

In the beginning it was “A Priori” Probability by Pascal (1623-1662), then Fermat (1601-1665) invented today’s “A Posteriori” Probability.

“A Priori” assumes every thing is naturally “like that”: eg. Each coin has 1/2 chance for head, 1/2 for tail. Each dice has 1/6 equal chance for each face (1-6).

“A Posteriori” by Fermat, then later the exile Protestant French mathematician De Moivre (who discovered Normal Distribution), is based on observation of “already happened” statistic data.

Cardano (1501-1576) born 150 years earlier than Pascal and Fermat, himself a weird genius in Medicine, Math and an addictive gambler, found the rule of + and x for chances (he did not know the name ‘Probability’ then ):

Addition + Rule: throw a dice, chance to get a “1 and 2” faces:
1/6 +1/6 = 2/6 = 1/3
(Correct: 1 & 2 out of other six faces)

Multiplication x Rule: throw two dices, chance to get a “1 followed by a 2 ” faces : 1/6 x 1/6 = 1/36
(Assume 1st throw does not affect or influence the 2nd throw: independent events)

Pascal never met Fermat personally, only through correspondences (like emails today), but Pascal regarded with respect Fermat as superior in Math than himself.

Fermat and Descartes were not so. Descartes openly criticized Fermat as a second class mathematician. Both independently discovered Analytical Geometry, but Descartes scored the credit in ‘Cartesian’ coordinates.

Fermat never published any books in Math. As a successful judge in Toulous, he spent his free time as an amateur mathematician, especially in Number Theory. He showed his Math discoveries to friends in letters but never provided proofs. Hence the Fermat’s Last Theorem made the world mathematicians after him (Gauss, Euler, Kummer, Sophie Germaine, Andrew Wiles…) busy for 380 years until 1994.

Fermat died 3 years after Pascal. Another Modern Mathematics was being born – Calculus – in UK by Newton and Germany by Leibniz. Probability was put in the back seat over-taken by Calculus.

# Quotations

1. Issac Newton: Hypotheses non fingo (I frame no hypotheses)

2. Gauss: Pauca sed matura (Few but ripe)

3. Descartes: Bene vixit qui bene latuit (he has lived well who has hidden well.)