Renè descartes

BRIDGES THROUGH HISTORY WITH MATHS RENÈ DESCARTES 1596 - 1650 FABIO BARABINO & BIANCA FANTINO Liceo Classico Statale “Cristoforo Colombo” GENOVA (ITALY) YEAR 2009-2010

Descartes’ life and the Scientific Revolution ,[object Object],[object Object],[object Object],[object Object]

Descartes’ work (1) ,[object Object]

Descartes’ work (2) ,[object Object],[object Object]

Biography (1596-1597) ,[object Object],His birth and his family ,[object Object]

1607 – 1616: his studies ,[object Object],[object Object]

1618: the meeting with Beeckman ,[object Object]

1620-1628: his travels ,[object Object],[object Object],[object Object],[object Object]

1629: Descartes returns to study ,[object Object]

1630 – 1640: Descartes quarrels with Beeckman and the Utrecht period ,[object Object],[object Object]

His works ,[object Object],the “ Treatise on the World ”, which he abandoned to publish because the astronomer Galileo Galilei was condemned by the Roman Catholic Church and he was scared for himself because he shared Galileo’s ideas, and the " Discourse on the Method ", which was published in 1637 , and there were also other books concerning both mathematics and philosophical.

1643–1650: The last years of his life ,[object Object],[object Object],[object Object]

The Scientific Revolution Descartes lived in a period in which in Europe it was started what the historians call “The Scientific Revolution ”. It was called “revolution ” because it entirely changed the way of studying the nature: in fact it was passed from the “ Aristotelian way ” to the “ mechanistic way”.

“ The Aristotelian Way” ,[object Object],Aristotle

“ The mechanistic way” ,[object Object]

Two important years: 1543 and 1687 (1) ,[object Object]

Two important years: 1543 and 1687 (2) ,[object Object],These two years were chosen because these two books represent the big turning points of the history of the science.

Descartes and the Rationalism ,[object Object]

Francis Bacon and the Empiricism ,[object Object],[object Object],[object Object],[object Object]

Galileo Galilei and the “experimental method ” ,[object Object],This method is composed by an empirical part , the experiments, and by a rationalist part , the mathematical demonstrations, and for this it is also called “ double method ”, and with it Galileo formulated some of the most important laws of the modern Physics.

[object Object],[object Object],[object Object],Descartes’ mathematics studies

The Algebraic studies ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The Cartesian rule of the signs (1) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],+ -

[object Object],The Cartesian rule of the signs (2)

Example This is a third degree equation and it has three solutions X = ? X = ? X = ? 1 2 3

- 6X 3 + 5X 2 +2X -1 = 0 ,[object Object],+X 1

-6X 3 + 5X 2 + 2X -1 = 0 ,[object Object],-X 2

-6X 3 +5X 2 + 2X - 1 = 0 ,[object Object],+X 3

To summarize: ,[object Object],+X 1 -X 2 +X 3

Something more… ,[object Object],[object Object],[object Object]

Some more examples (1) ,[object Object],Q.E.D.

Some more examples (2) ,[object Object],Q.E.D.

Be careful! ,[object Object],X V

Other basic rules ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The Geometrical studies ,[object Object],[object Object],[object Object],[object Object]

Anecdote ,[object Object],[object Object],!?

The Cartesian Coordinate System ,[object Object],[object Object],[object Object],[object Object]

Example ,[object Object],NB : To write correctly a coordinate of a point it must be put in bracket the x-point and then the y-point, separated with the semicolon, for example P(+2; -1,5).

Something more… ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],WHY IS IT IMPORTANT?

“ La Gèomètrie” ,[object Object]

The First Book ,[object Object]

The Second Book (1) ,[object Object],[object Object],[object Object],[object Object]

The Second book (2) ,[object Object],[object Object],And then he determined the normal (a perpendicular line) to a generic plane algebraic curve in a generic point and its tangent. 1) a parabola 2) an ellipse 3) an hyperbola 1) 2) 3)

The Third Book ,[object Object],[object Object],[object Object],[object Object],[object Object]

The last geometrical discovery ,[object Object],It is an algebraic curve defined by the equation x 3 +y 3 -3axy=0. It forms a loop in the first quadrant with a double point at the origin and asymptote x+ y +a=0 . It is symmetrical about y = x . The fame says that it was discovered when Descartes challenged Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines. Fermat solved the problem easily, something Descartes was unable to do.

Descartes’ Physics studies ,[object Object],[object Object]

His Physical Works ,[object Object],[object Object],[object Object],[object Object]

Descartes and Optics ,[object Object]

The Law of the Refraction of the light: ,[object Object]

[object Object],The incidental ray, the refracted ray and the normal to the surface in the point of incidence all lie in the same plane ; the ratio among the sine of the angle of incidence (formed by the normal and by the incidental ray) and the sine of angle of refraction (formed by the normal one and by the refracted ray), independently from the angle of incidence, is constant .

The hypothesis of the wavy nature of the light ,[object Object]

Descartes and Astronomy ,[object Object],[object Object]

Descartes and Physiology ,[object Object]

Descartes and Kinematics ,[object Object]

The Cartesian reference system ,[object Object]

Descartes Philosophy ,[object Object]

Philosophical works ,[object Object],[object Object],[object Object],[object Object],[object Object]

The Cartesian Method ,[object Object]

These laws are: ,[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Methodical Doubt: “Cogito ergo sum” ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

For Descartes the validity of " Cogito ergo sum " rests in this, that the doubt presents intuitively to the mind the subject who doubts, that is, the thinking substance . ,[object Object]

The Proof of the Existence of God ,[object Object],and the existence of other minds (the other Cogito ). So he analyzes the contents of Cogito ( the ideas), separating them depending on their origins : adventitious , innate and fictitious . He rejects the adventitious ones because they come from the senses (unreliable); he rejects the fictitious because they are invented . Only the innate ones seem to be reliable to find if a reality out of Cogito exists. Between these Descartes privileges the idea of God because, as a perfect idea, it cannot be created by the Cogito itself.

At this point Descartes introduces the proofs of the existence of God : ,[object Object],[object Object],[object Object]

The nature of God ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The Malignant Genius ,[object Object],So, with the proof of the existence of God, the hypothesis of a malignant genius falls of its own weight. Ah ah ah!! I’m a malignant genius!!!

The origin of ideas ,[object Object],[object Object],[object Object],[object Object]

The “ Res Extensa ” and the Cartesian World ,[object Object],1

THE DUALISM OF SUBSTANCES ,[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object]

ETHICS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

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