Descartes - York University · 2 Math 1700 – Descartes 4 Descartes a convert to Copernicus Wrote a book about the Copernican system (The World) akin to Galileo's, but suppressed

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DescartesDescartes

The Man Who Would Be The Man Who Would Be AristotleAristotle

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RenRenéé DescartesDescartes•• 15961596--16501650

Born in Touraine, FranceBorn in Touraine, FranceEducated by Jesuits in Educated by Jesuits in traditional Aristotelian traditional Aristotelian philosophy. philosophy. Took a law degree, but Took a law degree, but decided that real knowledge decided that real knowledge came from experience, so he came from experience, so he became a soldier to be became a soldier to be around around ““realreal”” people.people.•• Joined the Dutch army and then Joined the Dutch army and then

later moved to the Bavarian later moved to the Bavarian army.army.

•• Apparently was a well respected Apparently was a well respected officer.officer.

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Descartes gives up on soldiersDescartes gives up on soldiers

After some years in the army, After some years in the army, Descartes decided that Descartes decided that ““realreal”” people people didndidn’’t know much either.t know much either.He retired from the army to devote He retired from the army to devote himself to thinking about himself to thinking about mathematics and mechanics, which mathematics and mechanics, which he believed would lead to true he believed would lead to true knowledge.knowledge.

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Descartes a convert to CopernicusDescartes a convert to Copernicus

Wrote a book about the Wrote a book about the Copernican system (Copernican system (The The WorldWorld) akin to Galileo's, ) akin to Galileo's, but suppressed its but suppressed its publication when Galileo publication when Galileo was condemned by the was condemned by the Inquisition.Inquisition.It was not published until It was not published until after his death.after his death.

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A Dutch immigrantA Dutch immigrantSettled in Holland Settled in Holland where he had more where he had more intellectual freedom intellectual freedom than in France.than in France.In 1649 moved to In 1649 moved to Stockholm to join the Stockholm to join the court of Queen court of Queen Christina of Sweden, Christina of Sweden, where, after a few where, after a few months, he caught months, he caught pneumonia and died.pneumonia and died.

Descartes, at right, tutoring Queen Christina

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DescartesDescartes’’ DreamDreamBack when Descartes was being a soldier, he Back when Descartes was being a soldier, he spent one winter night in quarters with the spent one winter night in quarters with the Bavarian army on the shore of the Danube, Bavarian army on the shore of the Danube, November 10, 1619. November 10, 1619. The room was very hot. Descartes reported The room was very hot. Descartes reported having three feverish dreams during the night. In having three feverish dreams during the night. In these, he said later, he discovered the these, he said later, he discovered the ““foundations of a marvelous new science,foundations of a marvelous new science,”” and and realized that his future career lay in mathematics realized that his future career lay in mathematics and philosophy.and philosophy.He pondered this for nine more years before He pondered this for nine more years before finally taking action, leaving the army and finally taking action, leaving the army and settling in Holland to think and write for the next settling in Holland to think and write for the next 20 years.20 years.

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Undertook to build a new Undertook to build a new systematic philosophysystematic philosophy

In 1628 decided to create a new In 1628 decided to create a new system of philosophy based on system of philosophy based on certainty (to replace Aristotle).certainty (to replace Aristotle).Certainty meant mathematics.Certainty meant mathematics.DescartesDescartes’’ goal was to replace goal was to replace AristotleAristotle’’s common sense system s common sense system with something organized like Euclid.with something organized like Euclid.

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DescartesDescartes’’ Principles of PhilosophyPrinciples of Philosophy

Published in 1644Published in 1644Organized like Euclid.Organized like Euclid.•• Sought to find a Sought to find a

starting place, a starting place, a certainty, which he certainty, which he would take as an would take as an axiom, and build axiom, and build up from that.up from that.

•• All his assertions are numbered and justified, All his assertions are numbered and justified, just like Euclidjust like Euclid’’s propositions.s propositions.

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The Principles of PhilosophyThe Principles of Philosophy

Part 1: Of the Principles of Human Part 1: Of the Principles of Human KnowledgeKnowledge•• 1. That whoever is searching after truth 1. That whoever is searching after truth

must, once in his life, doubt all things; must, once in his life, doubt all things; insofar as this is possible.insofar as this is possible.

•• 2. That doubtful things must further be 2. That doubtful things must further be held to be false.held to be false.

•• ......

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Cogito, ergo sumCogito, ergo sum

Part 1: continuedPart 1: continued•• 7. That it is not possible for us to doubt 7. That it is not possible for us to doubt

that, while we are doubting, we exist; that, while we are doubting, we exist; and that this is the first thing which we and that this is the first thing which we know by philosophizing in the correct know by philosophizing in the correct order.order.

Accordingly, this knowledge, I think, Accordingly, this knowledge, I think, therefore I am [cogito, ergo sum] is the first therefore I am [cogito, ergo sum] is the first and most certain to be acquired by and and most certain to be acquired by and present itself to anyone who is present itself to anyone who is philosophizing in correct order.philosophizing in correct order.

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Dualism assertedDualism asserted

Part 1: continuedPart 1: continued•• 8. That from this we understand the 8. That from this we understand the

distinction between the soul and the distinction between the soul and the body, or between a thinking thing and a body, or between a thinking thing and a corporeal one.corporeal one.

Note that this follows immediately after his Note that this follows immediately after his ““cogito, ergo sumcogito, ergo sum”” assertion.assertion.

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The two worldsThe two worlds

Descartes assertion divides the world Descartes assertion divides the world into two totally separate into two totally separate compartments:compartments:•• Res cogitansRes cogitans –– the world of the mind.the world of the mind.•• Res extensaRes extensa –– the world of things that the world of things that

take up space.take up space.

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Res cogitansRes cogitans

The world of the mind.The world of the mind.Descartes wrote extensively about Descartes wrote extensively about this, what is now considered his this, what is now considered his psychological and/or philosophical psychological and/or philosophical theory.theory.•• The main point for science is that it The main point for science is that it

does not directly affect the physical does not directly affect the physical world.world.

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Res extensaRes extensa

The world of extension, i.e., the The world of extension, i.e., the physical world, was, for Descartes, physical world, was, for Descartes, totally mindless.totally mindless.•• Therefore Therefore purposepurpose had no place in it.had no place in it.

Res extensaRes extensa obeyed strictly obeyed strictly mechanical laws.mechanical laws.•• Compare AristotleCompare Aristotle’’s natural motion.s natural motion.

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Motion in Res ExtensaMotion in Res Extensa

Part II: Of the Principles of Material Part II: Of the Principles of Material ObjectsObjects•• 36. That God is the primary cause of 36. That God is the primary cause of

motion; and that He always maintains motion; and that He always maintains an equal quantity of it in the universe.an equal quantity of it in the universe.

This is the This is the principle ofprinciple of conservation of conservation of motionmotion –– there is a fixed quantity of motion there is a fixed quantity of motion in the universe that is just transferred from in the universe that is just transferred from one thing to another.one thing to another.

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Inertial motionInertial motion

Part II: continuedPart II: continued•• 37. The first law of nature: that each thing, as 37. The first law of nature: that each thing, as

far as is in its power, always remains in the far as is in its power, always remains in the same state; and that consequently, when it is same state; and that consequently, when it is once moved, it always continues to move.once moved, it always continues to move.

This is the This is the principle of inertia, principle of inertia, which, along with which, along with conservation of motion, asserts that motion is a conservation of motion, asserts that motion is a natural thing requiring no further explanation. natural thing requiring no further explanation. Compare this to Aristotle, for whom all motion Compare this to Aristotle, for whom all motion required an explanation.required an explanation.

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Projectile motionProjectile motion

Part II: continuedPart II: continued•• 38. Why bodies which have been thrown 38. Why bodies which have been thrown

continue to move after they leave the continue to move after they leave the hand....having once begun to move, hand....having once begun to move, they continue to do so until they are they continue to do so until they are slowed down by encounter with other slowed down by encounter with other bodies.bodies.

Descartes here disposes of AristotleDescartes here disposes of Aristotle’’s s antiperistasisantiperistasis problem. A projectile keeps problem. A projectile keeps moving because it is natural that it do so.moving because it is natural that it do so.

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Straight line motionStraight line motionPart II: continuedPart II: continued•• 39. The second law of 39. The second law of

nature: that all movement is, nature: that all movement is, of itself, along straight lines; of itself, along straight lines; and consequently, bodies and consequently, bodies which are moving in a circle which are moving in a circle always tend to move away always tend to move away from the centre of the circle from the centre of the circle which they are describing.which they are describing.

Anything actually moving in Anything actually moving in a circle is always tending to a circle is always tending to go off on a tangent. go off on a tangent. Therefore the circular motion Therefore the circular motion requires a cause.requires a cause.

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Relentless MechanismRelentless Mechanism

Inertial motion was natural. Inertial motion was natural. Pushes and pulls transferred motion Pushes and pulls transferred motion from one body to another. from one body to another. Everything in Res extensa worked Everything in Res extensa worked like a machine (e.g. windmill, like a machine (e.g. windmill, waterwheel, clock).waterwheel, clock).Forces were occult Forces were occult –– i.e. came from i.e. came from another world, therefore forbidden as another world, therefore forbidden as an explanation.an explanation.

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Vortex TheoryVortex Theory

Where Where (Aristotelian) Logic (Aristotelian) Logic leads.leads.If natural motion If natural motion was in straight was in straight lines, why did the lines, why did the planets circle the planets circle the Sun?Sun?

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Vortex Theory, 2Vortex Theory, 2

Answer: They are Answer: They are pushed back pushed back toward the centre toward the centre by all the invisible by all the invisible bits that fill the bits that fill the universe.universe.•• The universe is The universe is

spherical and full.spherical and full.

Think of water in a Think of water in a bucket.bucket.

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Living bodies are machinesLiving bodies are machines

The soul belongs to Res cogitans. The soul belongs to Res cogitans. Anything in the physical world must Anything in the physical world must be mechanical.be mechanical.All living things are merely complex All living things are merely complex machines. machines. •• Animals were mere machines, no matter Animals were mere machines, no matter

how much emotion they appeared to how much emotion they appeared to show.show.

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The Human Body as a MachineThe Human Body as a Machine

Living bodies were Living bodies were merely very merely very complicated complicated systems of levers systems of levers and pulleys with and pulleys with mechanisms like mechanisms like gears and springs.gears and springs.

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AutomataAutomataFrench clockmakers French clockmakers produced toy produced toy automata that made automata that made the mechanical body the mechanical body conceivable.conceivable.•• The monk kicks his The monk kicks his

feet, beats his chest feet, beats his chest with one hand, waves with one hand, waves with the other, turns his with the other, turns his head, rolls his eyes, head, rolls his eyes, opens and shuts his opens and shuts his mouth.mouth.

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The Human ConditionThe Human Condition

Since human being had souls and Since human being had souls and also had volition, there must be also had volition, there must be some communication for them some communication for them between Res cogitans and Res between Res cogitans and Res extensa.extensa.But how is this possible if the worlds But how is this possible if the worlds are totally separate?are totally separate?

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The Pineal GlandThe Pineal Gland

In DescartesIn Descartes’’ time, anatomists had time, anatomists had discovered a tiny gland in the human discovered a tiny gland in the human brain for which they knew no purpose.brain for which they knew no purpose.•• It was not known to exist in the brains of other It was not known to exist in the brains of other

animals. (It does.)animals. (It does.)•• This was the This was the Pineal GlandPineal Gland (it was shaped like a (it was shaped like a

pine cone).pine cone).

Aha!, thought Descartes. This is the seat Aha!, thought Descartes. This is the seat of communication for the soul and the of communication for the soul and the body.body.

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The Pineal Gland in actionThe Pineal Gland in actionDescartesDescartes’’ idea idea was that the pineal was that the pineal gland received gland received neural neural transmissions from transmissions from the body, the body, communicated communicated them to the soul, them to the soul, which sent back which sent back instructions to the instructions to the body.body.

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God the clockmakerGod the clockmaker

Descartes, the JesuitDescartes, the Jesuit--trained trained philosopher and lifelong Catholic, philosopher and lifelong Catholic, saw Godsaw God’’s role as being the creator s role as being the creator of the universe and all its of the universe and all its mechanisms.mechanisms.•• God, the Engineer.God, the Engineer.•• This became a popular theological This became a popular theological

position for scientists.position for scientists.

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The Analysis of Res ExtensaThe Analysis of Res Extensa

Among DescartesAmong Descartes’’ most useful most useful contributions to science were the contributions to science were the tools he developed for studying the tools he developed for studying the physical world. physical world. Most important among these is the Most important among these is the development of a new branch of development of a new branch of mathematics: Analytic Geometry.mathematics: Analytic Geometry.

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Analytic GeometryAnalytic Geometry

A combination of geometry, taken from A combination of geometry, taken from Euclid, and algebra, taken from Arab Euclid, and algebra, taken from Arab scholars, and traceable back to ancient scholars, and traceable back to ancient Egypt.Egypt.•• Geometry was generally used to solve Geometry was generally used to solve

problems involving lines and shapes.problems involving lines and shapes.•• Algebra was most useful for finding numerical Algebra was most useful for finding numerical

answers to particular problems.answers to particular problems.

Descartes found a useful way for them to Descartes found a useful way for them to work together.work together.

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Cartesian CoordinatesCartesian Coordinates

The extended world The extended world can be divided into can be divided into indefinitely smaller indefinitely smaller pieces.pieces.Any place in this world Any place in this world can be identified by can be identified by measuring its distance measuring its distance from a fixed from a fixed (arbitrary) beginning (arbitrary) beginning point (the origin) point (the origin) along three mutually along three mutually perpendicular axes, perpendicular axes, x, x, y,y, and and z.z.

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Analytic GeometryAnalytic GeometryGeometric figures and Geometric figures and paths of moving bodies paths of moving bodies can be described can be described compactly with Cartesian compactly with Cartesian coordinates.coordinates.A circle:A circle:xx22 + y+ y22 = 10= 1022 = 100= 100This is a circle of radius This is a circle of radius = 10.= 10.Every point on the circle Every point on the circle is a distance of 10 from the centre.is a distance of 10 from the centre.By the Pythagorean theorem, every point (x, y) on By the Pythagorean theorem, every point (x, y) on the circle makes a right triangle with the x and y the circle makes a right triangle with the x and y axes.axes.

(6,8)

6

8

x

y

The graph of the circle x2 + y2 = 100.

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Capturing Projectile Motion in an Capturing Projectile Motion in an equationequation

Descartes’ coordinate system applied to projectiles.

By recording the horizontal motion of a ball rolling off a table on the x-axis and recording its vertical motion along the y-axis, Descartes could plot points along the ball’s path. He then found that he could express the curve along which all these points lay in terms of the relationship between each point’s x and y values, that is, as an equation. In this graph, the ball could lie anywhere along the curve y=kx2.

y

x

(y1,x1)

(y2,x2)

(y3,x3)

y=kx2

y1

y2

y3

x1 x2 x3

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The Discourse on MethodThe Discourse on Method

DescartesDescartes’’ revolutionary amalgamation of revolutionary amalgamation of algebra and geometry was published as an algebra and geometry was published as an appendix to his best known single work, appendix to his best known single work, the the Discourse on Method of Rightly Discourse on Method of Rightly Conducting Reason in the Search for Truth Conducting Reason in the Search for Truth in the Sciences, in the Sciences, published in 1637.published in 1637.Unlike the later Unlike the later Principles of Philosophy, Principles of Philosophy, which he wrote in Latin, the which he wrote in Latin, the Discourse on Discourse on Method Method was written in French and was was written in French and was intended for a general audience. intended for a general audience.

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The Discourse on Method, 2The Discourse on Method, 2The The DiscourseDiscourse is itself not a formal philosophical is itself not a formal philosophical treatise (though it is the work of Descartes that is treatise (though it is the work of Descartes that is most studied by philosophy students), but an most studied by philosophy students), but an autobiographical account of how Descartes autobiographical account of how Descartes arrived at his philosophical viewpoint, intended as arrived at his philosophical viewpoint, intended as a preface for the three works that followed.a preface for the three works that followed.•• It, like the It, like the Principles of PhilosophyPrinciples of Philosophy contains the contains the

argument from argument from ““I think, therefore I am.I think, therefore I am.””•• Now, the Now, the DiscourseDiscourse is studied extensively and the three is studied extensively and the three

appendices, which were intended to be the main subject appendices, which were intended to be the main subject matter, are ignored completely.matter, are ignored completely.

The three appendices are The three appendices are La La DioptriqueDioptrique (about (about light and optics), light and optics), Les Les MMééttééoresores (about the (about the atmosphereatmosphere——meteorology), and meteorology), and La La GGééomoméétrietrie..

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La La GGééomoméétrietrieIn fact, the original In fact, the original La La GGééomoméétrietrie was was written in a written in a confusing and confusing and disorganized way, disorganized way, with proofs only with proofs only indicated, with the indicated, with the excuse that he left excuse that he left much out much out ““in order in order to give others the to give others the pleasure of pleasure of discovering for discovering for themselves.themselves.””

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La La GGééomoméétrietrie, 2, 2This shortcoming was This shortcoming was remedied by the Dutch remedied by the Dutch mathematics professor, mathematics professor, FransFrans van van SchootenSchooten, who , who translated translated La La GGééomoméétrietrieinto Latin and added into Latin and added explanatory commentary explanatory commentary that itself was more than that itself was more than twice the length of the twice the length of the original original La La GGééomoméétrietrie..•• It was the Latin version that It was the Latin version that

became the standard text became the standard text that established analytic that established analytic geometry in the universities geometry in the universities of western Europe.of western Europe.

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La La GGééomoméétrietrie, , 33Some of the innovations of Some of the innovations of La La GGééomoméétrietrie::It introduced the custom of using the It introduced the custom of using the letters at the end of the alphabet, letters at the end of the alphabet, x, y, z,x, y, z,for unknown quantities and those at the for unknown quantities and those at the beginning, beginning, a, b, c, a, b, c, ……,, for constants.for constants.Exponential notation: Exponential notation: xx22, y, y33, , etc., was etc., was introduced.introduced.Products of numbers, e.g. Products of numbers, e.g. xx2 2 oror abcabc, , were were treated as just numbers, not necessarily treated as just numbers, not necessarily areas or volumes, as was done in Greek areas or volumes, as was done in Greek geometry.geometry.

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La La GGééomoméétrietrie, , 44

We think of Cartesian coordinates as perpendicular axes, We think of Cartesian coordinates as perpendicular axes, but in but in La La GGééomoméétrietrie, , they were merely two lines that met they were merely two lines that met at an arbitrary angle, but then defined any point on the at an arbitrary angle, but then defined any point on the plane (or three lines, defining any point in space).plane (or three lines, defining any point in space).In the above diagram, the horizontal line from the vertex to In the above diagram, the horizontal line from the vertex to the first diagonal line is arbitrarily given the value 1. The the first diagonal line is arbitrarily given the value 1. The first diagonal has value first diagonal has value aa and the horizontal line from the and the horizontal line from the vertex to the second diagonal has value vertex to the second diagonal has value bb. Then, Descartes . Then, Descartes shows that the length of the second diagonal line is shows that the length of the second diagonal line is abab. .

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The Mechanical PhilosophyThe Mechanical PhilosophyThough it is NewtonThough it is Newton’’s systematic account s systematic account of celestial mechanics that really of celestial mechanics that really established the mechanical viewpoint, established the mechanical viewpoint, DescartesDescartes’’ works were the vanguard of works were the vanguard of the new mechanical philosophy whereby the new mechanical philosophy whereby the educated public began to think of the educated public began to think of Nature as a large machine that ran on Nature as a large machine that ran on mechanical principles which could be mechanical principles which could be expressed in mathematical laws.expressed in mathematical laws.Quoting Descartes: Quoting Descartes: ““the rules of the rules of mechanicsmechanics……are the same as those of are the same as those of nature.nature.