BCA Semester II
Additional English Notes
Rene Descartes
1) What idea did
Descartes have when he watched the fly buzzing about in the corner of his
bedroom?
Ans:- When he was
lying on his bed one day, Descartes watched a fly buzzing about in his room. He
realized he could describe the fly’s position with just three numbers (because
the room was 3D). These numbers were the fly’s position in relation to the two walls
and its height. When we use this system for a graph, we describe points in
relation to a point called the ‘origin’, which is like the corner of a room.
The two walls coming away from the corner are the x and y axes. The x axis runs
along the bottom and the y axis goes up. If we wanted to tell someone where the
fly was, we’d give the x number (or ‘co-ordinate’) first, then the y
co-ordinate, and then the z co-ordinate (if there is one). It’s important we
always use the same order, so we know we are all communicating in the same
way.
2) What did
Pythagoras and Descartes have in common?
Ans:- Rene
Descartes and Pythagoras are similar because they are both philosophers. Each
have contributed to the math world. Pythagoras discovered the formula for the
Pythagorean Theorem and Rene Descartes invented analytical geometry. Descartes
combined Algebra and geometry together in a completely new way as Pythagoras
had done before him. The application of algebraic methods to geometry has been
called ‘The greatest single step ever made in the progress of the exact
sciences.
3) On what grounds
did Descartes criticize Galileo’s scientific method?
Ans:- Some
philosophers of science suggest that philosophical assumptions must influence
historical scholarship, because history (like science) has no neutral data and
because the treatment of any particular historical episode is going to be
influenced to some degree by one's prior philosophical conceptions of what is
important in science. However, if the history of science must be laden with
philosophical assumptions, then how can the history of science be evidence for
the philosophy of science? Would not an inductivist history of science confirm
an inductivist philosophy of science and a conventionalist history of science confirm
a conventionalist philosophy of science? I attempt to resolve this problem;
essentially, I deny the claim that the history of science must be influenced by
one's conception of what is important in science — one's general philosophy of
science. To accomplish the task I look at a specific historical episode,
together with its history, and draw some methodological conclusions from it.
The specific historical episode I examine is Descartes' critique of Galileo's
scientific methodology.
4) What was the
essential characteristic of scientific method as recommended by Francis
Bacon?
Ans:- Bacon has
been called the father of empiricism.[6] His works argued
for the possibility of scientific knowledge based only upon inductive reasoning
and careful observation of events in nature. Most importantly, he argued this
could be achieved by use of a skeptical and methodical approach whereby
scientists aim to avoid misleading themselves. Francis Bacon defined a method
more appropriate to the physical and natural sciences were the inductive
method. According to this method, the scientist collects masses of facts by
observation and experiment. From these, he is ten able to make generalizations,
and establish principles.
5) Why did
Descartes resolve to ‘divide each difficulty under examination into as many
parts as
possible’?
Ans:- What is
unknown can only be understood in relation to what is known; nothing is
completely unknown; for if it were, it could never be known. The prerequisite
to attain knowledge is that we are, from the start, is possession of all the
data required to find the truth. To find the truth there, firstly, must be in
every question something not yet known, otherwise the enquiry would be to no
purpose. Secondly, the not yet known must be, in some way, marked out;
otherwise our attention may tend to deviate towards something else. Thirdly,
the unknown can only be marked out in relation to something which is already
known. Thus, if we are asked to find what is the nature of a magnet, we already
know what is meant by those two words, ‘magnet’ and ‘nature’, and thereby we
are determined to enquire on these two words than on something else. But over
and above this, if the question is to be perfectly understood, we require that
it is made so completely determinate that we have no need to seek for anything
beyond what can be deduced from the (already known) data.
6) What is meant
by saying that Descartes’ action-plan for the conduct of scientific
investigation was ‘rigidly methodical’?
Ans:- In the first
half of the 17th century, the French Rationalist René Descartes used methodic
doubt to reach certain knowledge of self-existence in the act of thinking,
expressed in the indubitable proposition cogito,
ergo sum (“I think, therefore I am”). He found knowledge from tradition
to be dubitable because authorities disagree; empirical knowledge dubitable
because of illusions, hallucinations, and dreams; and mathematical knowledge
dubitable because people make errors in calculating. He proposed an all-powerful,
deceiving demon as a way of invoking universal doubt. Although the demon could
deceive men regarding which sensations and ideas are truly of the world, or
could give them sensations and ideas none of which are of the true world, or
could even make them think that there is an external world when there is none,
the demon could not make men think that they exist when they do not.
7) What did
Descartes and Socrates have in common?
Ans:-
Descartes feels
that thinking is evidence of existence as his metaphysics is the study of the
self. Unlike Socrates whose philosophy of ethics attempts to understand how man
relates to others be it person, place or thing. On the contrary, there is a
branch of philosophy that both Socrates and Descartes share, and that is
epistemology: the search for the origin, nature and materialization of
knowledge. It seems to me that Socrates and Descartes came to the same
conclusion concerning the origin of knowledge, which is that, the origin of
knowledge and all things coming forth from knowledge reside in God. Both
Descartes and Socrates alike are seekers however, ultimately do not claim any
knowledge of their own but owe their progresses and attainments of certain
knowledge to God. Socrates and Descartes exist more in mundane and subjective
realities. However, it is in a deeper faculty of their work where the
similarities exist. Both philosophers claim nothing of knowledge to be owned by
them but resolve to God as the sole owner of knowledge and in the branch of
philosophy which they share, epistemology, they agree that the origin of
knowledge is God.
8) What was
Descartes’ purpose in doubting even whether he existed?
Ans:- It
does not. Descartes claimed that one thing emerges as true even under the
strict conditions imposed by the otherwise universal doubt: "I am, I
exist" is necessarily true whenever the thought occurs to me. This truth
neither derives from sensory information nor depends upon the reality of an
external world, and I would have to exist even if I were systematically
deceived. For even an omnipotent god could not cause it to be true, at one and
the same time, both that I am deceived and that I do not exist.
If I am deceived, then at least I am.
Although Descartes’ reasoning here is best known in
the Latin translation of its expression in the Discourse, "cogito,
ergo sum" ("I think, therefore I am"), it is not merely
an inference from the activity of thinking to the existence of an agent which
performs that activity. It is intended rather as an intuition of one's own
reality, an expression of the indomitability of first-person experience, the
logical self-certification of self-conscious awareness in any form.
Skepticism is thereby defeated, according to
Descartes. No matter how many skeptical challenges are raised—indeed, even if
things are much worse than the most extravagant skeptic ever claimed— there is
at least one fragment of genuine human knowledge: my perfect certainty of my
own existence. From this starting-point, Descartes supposed, it is possible to
achieve indubitable knowledge of many other propositions as well.
9) How did
Descartes come to the conclusion that he had a mind?
Ans:- What
then, is this "I" that doubts, that may be deceived, that thinks?
Since I became certain of my existence while entertaining serious doubts about
sensory information and the existence of a material world, none of the apparent
features of my human body can have been crucial for my understanding of
myself. But all that is left is my thought itself, so Descartes concluded that
"sum res cogitans"
("I am a thing that thinks"). In Descartes’ terms, I am a substance
whose inseparable attribute (or entire essence) is thought, with all its modes:
doubting, willing, conceiving, believing, etc. What I really am is a mind .So
completely am I identified with my conscious awareness, Descartes claimed, that
if I were to stop thinking altogether, it would follow that I no longer existed
at all. At this point, nothing else about human nature can be determined with
such perfect certainty.
10)In what respect
is Descartes’ argument for the existence of Go a ‘circular’ one?
Ans:- AsAntoine Arnauldpointed out in an Objection
published along with the Meditations themselves,
there is a problem with this reasoning. Since Descartes will use the existence
(and veracity) of god to prove the reliability of clear and distinct ideas in
Meditation Four, his use of clear and distinct ideas to prove the existence of
god in Meditation Three is an example of circular reasoning. Descartes replied
that his argument is not circular because intuitive reasoning—in the proof of
god as in the cogito—requires
no further support in the moment of its conception. We must rely on a
non-deceiving god only as the guarantor of veridical memory, when a
demonstrative argument involves too many steps to be held in the mind at once.
But this response is not entirely convincing.
The problem is a significant one, since the proof
of god's existence is not only the first attempt to establish the reality of
something outside the self but also the foundation for every further attempt to
do so. If this proof fails, then Descartes’ hopes for human knowledge are severely
curtailed, and I am stuck in solipsism, unable to be perfectly certain of
anything more than my own existence as a thinking thing. With this reservation
in mind, we'll continue through the Meditations,
seeing how Descartes tried to dismantle his own reasons for doubt.
11)Describe the
contribution of Rene Descartes to the field of geometry?
Ans:-
La Géométrie (Geometry) is the ground
breaking work of Descartes in mathematics. It was published in 1637 as
one of the appendices of Discourse on the Method. In La Géométrie, Descartes first proposed that each point in two dimensions can be
described by two numbers on a plane, one giving the point’s horizontal location
and the other giving the vertical location. He thus invented the Cartesian coordinate system,
which forms the foundation of analytic
geometry. It also provides geometric interpretations for other branches
of mathematics, such as linear algebra, complex analysis, differential
geometry, multivariate calculus, group theory and more. In La Géométrie,
Descartes also introduced what later became the standard algebraic notation: using lowercase a, b and c for known quantities and x, y and z for unknown quantities.
DESCARTES IS
REGARDED AS THE FATHER OF ANALYTIC GEOMETRY Analytic geometry, also known
as Cartesian geometry after
Rene Descartes, is the study of
geometry using the Cartesian coordinate system. It allowed for the first
time the conversion of geometry into
algebra; and vice versa. Any algebraic equation can be represented on
the Cartesian plane by plotting on it the solution set of the equation. Also,
it allows transforming geometric shapes into algebraic equations. Analytic
geometry is widely used in physics and engineering, and also in aviation,
rocketry, space science and spaceflight. It is the foundation of most modern fields of geometry, including
algebraic, differential, discrete and computational geometry. Analytic geometry
is by far Descartes’ most important
contribution to mathematics. He is widely considered as the father of analytic geometry.
Or
This answer for
Rene Descartes contributions on geometry (Any One)
RENÉ DESCARTES
(1596–1650), philosopher and mathematician, is of course universally, regarded
as the inventor of the method of co-ordinates in geometry; hence the common
name for them, Cartesian co- ordinates. Yet, if it were a question of priority,
a good claim could be put forward for Fermat (1601–1665), a contemporary, and
an even greater mathematician; for, though Fermat's work “Ad locds pianos et
solidos isagoge” was not
published until much later (1679), it was certainly conceived and peir haps
written before 1637, the date of publication of Descartes' “Géométric”;
moreover, the method of co-ordinates comes out much more clearly in Fermat, and
his analytical geometry generally is much more like ours than Descartes' is.
Fermat's share in the new discovery nevertheless remained unknown until quite
recent times, and the whole credit was given to Descartes by no less learned a
geometer than Chasles, who, in a eulogy which now seems exaggerated, speaks of
Descartes' doctrine as “prolem sine matre creatam” and one “of which no germ
can be found in the writings of the ancient geometers.” Anticipations of the
method of co-ordinates are, however, as is now well known, to be found in
Archimedes and Apollonius of Perga; the latter stated, for example, in words (without
symbols), the fact that the locus of a point satisfying the equivalent of an
equation of the first degree in two unknowns is a straight line, and the form
in which he states the fundamental property of each of the three conies is the
exact equivalent of the Cartesian equation referred to any diameter and the
tangent at its extremity as axes. Descartes' actual achievement -and it was
momentous enough-was to remove the impasse to which Greek geometry had come
through want of notation, by introducing into geometry the unrestricted use of
all the resources of algebra (then recently introduced into France from Italy)
as a recognized and even indispensable auxiliary.
Descartes method-
4 rules (Deductive Method)
Rule 1 “Never to accept anything for true which I did not clearly know to be
such; that is to say, carefully to avoid precipitant and prejudice, and to
comprise nothing more in my judgment than what was presented to my mind so
clearly and distinctly as to exclude all grounds of doubt.” – Descartes
Prejudices are a
product of information which has been imparted on us. We have all learned much
in our years of schooling but have we gained knowledge? Knowledge is beyond
doubt, but in whatever we have learned there is no single matter on which wise
men agree upon. There is no such matter which is not under dispute. “He who
entertains doubt on many matters in no wiser than he who has never thought of
such matter”. It is better not to study at all than to occupy ourselves with
objects so difficult that, owing to inability to distinguish true from false,
we may be obliged to accept the doubtful as certain. In such inquiries there is
more risk of diminishing our knowledge than of increasing it. No modes of
knowledge which are probable are acceptable; only that which is perfectly known
and in respect of which doubt is not possible can be considered knowledge. All
such matter which involves “probable opinions” is to be ruled out as a base to
acquire “genuine knowledge”. What is then genuine knowledge, what is known to
be beyond doubt? In Descartes’ words “Accordingly, if we are representing the
situation correctly, observation of this rule confines us to arithmetic and
geometry, as being the only science yet discovered.”
Rule 2 “To divide each of the difficulties under examination into as many
parts as possible, and as might be necessary for its adequate solution.” – Descartes
What is unknown can only be understood in relation to what is known;
nothing is completely unknown; for if it were, it could never be known. The
prerequisite to attain knowledge is that we are, from the start, is possession
of all the data required to find the truth. To find the truth there, firstly,
must be in every question something not yet known, otherwise the inquiry would
be to no purpose. Secondly, the not yet known must be, in some way, marked out;
otherwise our attention may tend to deviate towards something else. Thirdly,
the unknown can only be marked out in relation to something which is already known.
Thus, if we are asked to find what is the nature of a magnet, we already know
what is meant by those two words, ‘magnet’ and ‘nature’, and thereby we are
determined to enquire on these two words than on something else. But over and
above this, if the question is to be perfectly understood, we require that it
is made so completely determinate that we have no need to seek for anything
beyond what can be deduced from the (already known) data.
Rule 3 “To conduct my thoughts in such order that, by commencing with objects
the simplest and easiest to know, I might ascend by little and little, and, as
it were, step by step, to the knowledge of the more complex; assigning in
thought a certain order even to those objects which in their own nature do not
stand in a relation of antecedence and sequence.” – Descartes
The third rule can
be understood as a process of “analysis and synthesis”; a digging to the bottom
rock (analysis) and a reconstruction of the structure from the bottom
(synthesis). It is about distinguishing simple things from the more complex
ones and arranging them in such an order so we can directly deduce the truths
of one from the other. This rule admonishes us that all information can be
arranged in certain series, not classified as categories, but in order in which
each item contributes to the knowledge of those that follow upon it. There are
two different relation which can be found while digging: the ones at the rock
bottom (the absolute ones) and the on the way to the bottom (the relative ones).
Absolute is that
which possesses in itself the pure and simple nature of that which we have
under consideration, i.e. whatever is viewed as being independent, cause,
simple, universal, one, equal, like, straight, and such like. These are the
simplest and easiest to apprehend and they serve to find the relative one; the
search is always from the bottom to the top. The relatives share some
properties with the absolutes, since they are deduces from them, yet they
involve in its concept, over and above the absolute nature, certain other
characters i.e. whatever is said to be dependent, effect, composite,
particular, multiple, unequal, unlike, oblique, etc.
The above rule
requires that these relatives should be different from one another, and the
linkage and the natural order of their interrelations be so observed, that we
may be able, starting from that which is nearest to us (as empirically given),
to reach to that which is completely absolute, by passing though all
intermediate relatives.
Rule 4 “To make enumerations so complete, and reviews so general that I might
be assured that nothing was omitted.” –
Descartes The
search for knowledge is not easy, many are the question which will have to
answered, some will be known and some not. Some of the truths which we have
been seeking of are not immediately deduced from the primary self evidencing
data; this deduction sometimes involves series of connected terms arranged in a
sequence. The process is long and this is why it is not easy for the mind to
remember all the links which it did to conduct us to the conclusion. Continuous
movement of thought is required to remedy this weakness of memory. One should
run over each link several times and this process should become so continuous
that while intuiting each step it simultaneously passes to the next one; this
process should be repeated until the mind learns to pass from one step to the
other, so quickly, that almost none of the step seem to exist independently but
the whole process seems a “whole”. This process of deduction should nowhere be
interrupted, for even if the smallest link misses the chain brakes and
certainly the truth will escape from us.
This whole process
relies on enumeration. In all questions there is something (however minute or
however negative) which may escape us, and only by enumeration can we be
conscious of leading a correct induction. Only by the means of enumeration can
we be assured of always passing a true and certain judgment on whatever is
under investigation.
Additional English Notes
Rene Descartes
1) What idea did
Descartes have when he watched the fly buzzing about in the corner of his
bedroom?
Ans:- When he was
lying on his bed one day, Descartes watched a fly buzzing about in his room. He
realized he could describe the fly’s position with just three numbers (because
the room was 3D). These numbers were the fly’s position in relation to the two walls
and its height. When we use this system for a graph, we describe points in
relation to a point called the ‘origin’, which is like the corner of a room.
The two walls coming away from the corner are the x and y axes. The x axis runs
along the bottom and the y axis goes up. If we wanted to tell someone where the
fly was, we’d give the x number (or ‘co-ordinate’) first, then the y
co-ordinate, and then the z co-ordinate (if there is one). It’s important we
always use the same order, so we know we are all communicating in the same
way.
2) What did
Pythagoras and Descartes have in common?
Ans:- Rene
Descartes and Pythagoras are similar because they are both philosophers. Each
have contributed to the math world. Pythagoras discovered the formula for the
Pythagorean Theorem and Rene Descartes invented analytical geometry. Descartes
combined Algebra and geometry together in a completely new way as Pythagoras
had done before him. The application of algebraic methods to geometry has been
called ‘The greatest single step ever made in the progress of the exact
sciences.
3) On what grounds
did Descartes criticize Galileo’s scientific method?
Ans:- Some
philosophers of science suggest that philosophical assumptions must influence
historical scholarship, because history (like science) has no neutral data and
because the treatment of any particular historical episode is going to be
influenced to some degree by one's prior philosophical conceptions of what is
important in science. However, if the history of science must be laden with
philosophical assumptions, then how can the history of science be evidence for
the philosophy of science? Would not an inductivist history of science confirm
an inductivist philosophy of science and a conventionalist history of science confirm
a conventionalist philosophy of science? I attempt to resolve this problem;
essentially, I deny the claim that the history of science must be influenced by
one's conception of what is important in science — one's general philosophy of
science. To accomplish the task I look at a specific historical episode,
together with its history, and draw some methodological conclusions from it.
The specific historical episode I examine is Descartes' critique of Galileo's
scientific methodology.
4) What was the
essential characteristic of scientific method as recommended by Francis
Bacon?
Ans:- Bacon has
been called the father of empiricism.[6] His works argued
for the possibility of scientific knowledge based only upon inductive reasoning
and careful observation of events in nature. Most importantly, he argued this
could be achieved by use of a skeptical and methodical approach whereby
scientists aim to avoid misleading themselves. Francis Bacon defined a method
more appropriate to the physical and natural sciences were the inductive
method. According to this method, the scientist collects masses of facts by
observation and experiment. From these, he is ten able to make generalizations,
and establish principles.
5) Why did
Descartes resolve to ‘divide each difficulty under examination into as many
parts as
possible’?
Ans:- What is
unknown can only be understood in relation to what is known; nothing is
completely unknown; for if it were, it could never be known. The prerequisite
to attain knowledge is that we are, from the start, is possession of all the
data required to find the truth. To find the truth there, firstly, must be in
every question something not yet known, otherwise the enquiry would be to no
purpose. Secondly, the not yet known must be, in some way, marked out;
otherwise our attention may tend to deviate towards something else. Thirdly,
the unknown can only be marked out in relation to something which is already
known. Thus, if we are asked to find what is the nature of a magnet, we already
know what is meant by those two words, ‘magnet’ and ‘nature’, and thereby we
are determined to enquire on these two words than on something else. But over
and above this, if the question is to be perfectly understood, we require that
it is made so completely determinate that we have no need to seek for anything
beyond what can be deduced from the (already known) data.
6) What is meant
by saying that Descartes’ action-plan for the conduct of scientific
investigation was ‘rigidly methodical’?
Ans:- In the first
half of the 17th century, the French Rationalist René Descartes used methodic
doubt to reach certain knowledge of self-existence in the act of thinking,
expressed in the indubitable proposition cogito,
ergo sum (“I think, therefore I am”). He found knowledge from tradition
to be dubitable because authorities disagree; empirical knowledge dubitable
because of illusions, hallucinations, and dreams; and mathematical knowledge
dubitable because people make errors in calculating. He proposed an all-powerful,
deceiving demon as a way of invoking universal doubt. Although the demon could
deceive men regarding which sensations and ideas are truly of the world, or
could give them sensations and ideas none of which are of the true world, or
could even make them think that there is an external world when there is none,
the demon could not make men think that they exist when they do not.
7) What did
Descartes and Socrates have in common?
Ans:-
Descartes feels
that thinking is evidence of existence as his metaphysics is the study of the
self. Unlike Socrates whose philosophy of ethics attempts to understand how man
relates to others be it person, place or thing. On the contrary, there is a
branch of philosophy that both Socrates and Descartes share, and that is
epistemology: the search for the origin, nature and materialization of
knowledge. It seems to me that Socrates and Descartes came to the same
conclusion concerning the origin of knowledge, which is that, the origin of
knowledge and all things coming forth from knowledge reside in God. Both
Descartes and Socrates alike are seekers however, ultimately do not claim any
knowledge of their own but owe their progresses and attainments of certain
knowledge to God. Socrates and Descartes exist more in mundane and subjective
realities. However, it is in a deeper faculty of their work where the
similarities exist. Both philosophers claim nothing of knowledge to be owned by
them but resolve to God as the sole owner of knowledge and in the branch of
philosophy which they share, epistemology, they agree that the origin of
knowledge is God.
8) What was
Descartes’ purpose in doubting even whether he existed?
Ans:- It
does not. Descartes claimed that one thing emerges as true even under the
strict conditions imposed by the otherwise universal doubt: "I am, I
exist" is necessarily true whenever the thought occurs to me. This truth
neither derives from sensory information nor depends upon the reality of an
external world, and I would have to exist even if I were systematically
deceived. For even an omnipotent god could not cause it to be true, at one and
the same time, both that I am deceived and that I do not exist.
If I am deceived, then at least I am.
Although Descartes’ reasoning here is best known in
the Latin translation of its expression in the Discourse, "cogito,
ergo sum" ("I think, therefore I am"), it is not merely
an inference from the activity of thinking to the existence of an agent which
performs that activity. It is intended rather as an intuition of one's own
reality, an expression of the indomitability of first-person experience, the
logical self-certification of self-conscious awareness in any form.
Skepticism is thereby defeated, according to
Descartes. No matter how many skeptical challenges are raised—indeed, even if
things are much worse than the most extravagant skeptic ever claimed— there is
at least one fragment of genuine human knowledge: my perfect certainty of my
own existence. From this starting-point, Descartes supposed, it is possible to
achieve indubitable knowledge of many other propositions as well.
9) How did
Descartes come to the conclusion that he had a mind?
Ans:- What
then, is this "I" that doubts, that may be deceived, that thinks?
Since I became certain of my existence while entertaining serious doubts about
sensory information and the existence of a material world, none of the apparent
features of my human body can have been crucial for my understanding of
myself. But all that is left is my thought itself, so Descartes concluded that
"sum res cogitans"
("I am a thing that thinks"). In Descartes’ terms, I am a substance
whose inseparable attribute (or entire essence) is thought, with all its modes:
doubting, willing, conceiving, believing, etc. What I really am is a mind .So
completely am I identified with my conscious awareness, Descartes claimed, that
if I were to stop thinking altogether, it would follow that I no longer existed
at all. At this point, nothing else about human nature can be determined with
such perfect certainty.
10)In what respect is Descartes’ argument for the existence of Go a ‘circular’ one?
Ans:- AsAntoine Arnauldpointed out in an Objection published along with the Meditations themselves, there is a problem with this reasoning. Since Descartes will use the existence (and veracity) of god to prove the reliability of clear and distinct ideas in Meditation Four, his use of clear and distinct ideas to prove the existence of god in Meditation Three is an example of circular reasoning. Descartes replied that his argument is not circular because intuitive reasoning—in the proof of god as in the cogito—requires no further support in the moment of its conception. We must rely on a non-deceiving god only as the guarantor of veridical memory, when a demonstrative argument involves too many steps to be held in the mind at once. But this response is not entirely convincing.
The problem is a significant one, since the proof
of god's existence is not only the first attempt to establish the reality of
something outside the self but also the foundation for every further attempt to
do so. If this proof fails, then Descartes’ hopes for human knowledge are severely
curtailed, and I am stuck in solipsism, unable to be perfectly certain of
anything more than my own existence as a thinking thing. With this reservation
in mind, we'll continue through the Meditations,
seeing how Descartes tried to dismantle his own reasons for doubt.
11)Describe the
contribution of Rene Descartes to the field of geometry?
Ans:-
La Géométrie (Geometry) is the ground
breaking work of Descartes in mathematics. It was published in 1637 as
one of the appendices of Discourse on the Method. In La Géométrie, Descartes first proposed that each point in two dimensions can be
described by two numbers on a plane, one giving the point’s horizontal location
and the other giving the vertical location. He thus invented the Cartesian coordinate system,
which forms the foundation of analytic
geometry. It also provides geometric interpretations for other branches
of mathematics, such as linear algebra, complex analysis, differential
geometry, multivariate calculus, group theory and more. In La Géométrie,
Descartes also introduced what later became the standard algebraic notation: using lowercase a, b and c for known quantities and x, y and z for unknown quantities.
DESCARTES IS
REGARDED AS THE FATHER OF ANALYTIC GEOMETRY Analytic geometry, also known
as Cartesian geometry after
Rene Descartes, is the study of
geometry using the Cartesian coordinate system. It allowed for the first
time the conversion of geometry into
algebra; and vice versa. Any algebraic equation can be represented on
the Cartesian plane by plotting on it the solution set of the equation. Also,
it allows transforming geometric shapes into algebraic equations. Analytic
geometry is widely used in physics and engineering, and also in aviation,
rocketry, space science and spaceflight. It is the foundation of most modern fields of geometry, including
algebraic, differential, discrete and computational geometry. Analytic geometry
is by far Descartes’ most important
contribution to mathematics. He is widely considered as the father of analytic geometry.
Or
This answer for
Rene Descartes contributions on geometry (Any One)
RENÉ DESCARTES (1596–1650), philosopher and mathematician, is of course universally, regarded as the inventor of the method of co-ordinates in geometry; hence the common name for them, Cartesian co- ordinates. Yet, if it were a question of priority, a good claim could be put forward for Fermat (1601–1665), a contemporary, and an even greater mathematician; for, though Fermat's work “Ad locds pianos et solidos isagoge” was not published until much later (1679), it was certainly conceived and peir haps written before 1637, the date of publication of Descartes' “Géométric”; moreover, the method of co-ordinates comes out much more clearly in Fermat, and his analytical geometry generally is much more like ours than Descartes' is. Fermat's share in the new discovery nevertheless remained unknown until quite recent times, and the whole credit was given to Descartes by no less learned a geometer than Chasles, who, in a eulogy which now seems exaggerated, speaks of Descartes' doctrine as “prolem sine matre creatam” and one “of which no germ can be found in the writings of the ancient geometers.” Anticipations of the method of co-ordinates are, however, as is now well known, to be found in Archimedes and Apollonius of Perga; the latter stated, for example, in words (without symbols), the fact that the locus of a point satisfying the equivalent of an equation of the first degree in two unknowns is a straight line, and the form in which he states the fundamental property of each of the three conies is the exact equivalent of the Cartesian equation referred to any diameter and the tangent at its extremity as axes. Descartes' actual achievement -and it was momentous enough-was to remove the impasse to which Greek geometry had come through want of notation, by introducing into geometry the unrestricted use of all the resources of algebra (then recently introduced into France from Italy) as a recognized and even indispensable auxiliary.
Descartes method-
4 rules (Deductive Method)
Rule 1 “Never to accept anything for true which I did not clearly know to be
such; that is to say, carefully to avoid precipitant and prejudice, and to
comprise nothing more in my judgment than what was presented to my mind so
clearly and distinctly as to exclude all grounds of doubt.” – Descartes
Prejudices are a
product of information which has been imparted on us. We have all learned much
in our years of schooling but have we gained knowledge? Knowledge is beyond
doubt, but in whatever we have learned there is no single matter on which wise
men agree upon. There is no such matter which is not under dispute. “He who
entertains doubt on many matters in no wiser than he who has never thought of
such matter”. It is better not to study at all than to occupy ourselves with
objects so difficult that, owing to inability to distinguish true from false,
we may be obliged to accept the doubtful as certain. In such inquiries there is
more risk of diminishing our knowledge than of increasing it. No modes of
knowledge which are probable are acceptable; only that which is perfectly known
and in respect of which doubt is not possible can be considered knowledge. All
such matter which involves “probable opinions” is to be ruled out as a base to
acquire “genuine knowledge”. What is then genuine knowledge, what is known to
be beyond doubt? In Descartes’ words “Accordingly, if we are representing the
situation correctly, observation of this rule confines us to arithmetic and
geometry, as being the only science yet discovered.”
Rule 2 “To divide each of the difficulties under examination into as many
parts as possible, and as might be necessary for its adequate solution.” – Descartes
What is unknown can only be understood in relation to what is known;
nothing is completely unknown; for if it were, it could never be known. The
prerequisite to attain knowledge is that we are, from the start, is possession
of all the data required to find the truth. To find the truth there, firstly,
must be in every question something not yet known, otherwise the inquiry would
be to no purpose. Secondly, the not yet known must be, in some way, marked out;
otherwise our attention may tend to deviate towards something else. Thirdly,
the unknown can only be marked out in relation to something which is already known.
Thus, if we are asked to find what is the nature of a magnet, we already know
what is meant by those two words, ‘magnet’ and ‘nature’, and thereby we are
determined to enquire on these two words than on something else. But over and
above this, if the question is to be perfectly understood, we require that it
is made so completely determinate that we have no need to seek for anything
beyond what can be deduced from the (already known) data.
Rule 3 “To conduct my thoughts in such order that, by commencing with objects
the simplest and easiest to know, I might ascend by little and little, and, as
it were, step by step, to the knowledge of the more complex; assigning in
thought a certain order even to those objects which in their own nature do not
stand in a relation of antecedence and sequence.” – Descartes
The third rule can
be understood as a process of “analysis and synthesis”; a digging to the bottom
rock (analysis) and a reconstruction of the structure from the bottom
(synthesis). It is about distinguishing simple things from the more complex
ones and arranging them in such an order so we can directly deduce the truths
of one from the other. This rule admonishes us that all information can be
arranged in certain series, not classified as categories, but in order in which
each item contributes to the knowledge of those that follow upon it. There are
two different relation which can be found while digging: the ones at the rock
bottom (the absolute ones) and the on the way to the bottom (the relative ones).
Absolute is that
which possesses in itself the pure and simple nature of that which we have
under consideration, i.e. whatever is viewed as being independent, cause,
simple, universal, one, equal, like, straight, and such like. These are the
simplest and easiest to apprehend and they serve to find the relative one; the
search is always from the bottom to the top. The relatives share some
properties with the absolutes, since they are deduces from them, yet they
involve in its concept, over and above the absolute nature, certain other
characters i.e. whatever is said to be dependent, effect, composite,
particular, multiple, unequal, unlike, oblique, etc.
The above rule
requires that these relatives should be different from one another, and the
linkage and the natural order of their interrelations be so observed, that we
may be able, starting from that which is nearest to us (as empirically given),
to reach to that which is completely absolute, by passing though all
intermediate relatives.
Rule 4 “To make enumerations so complete, and reviews so general that I might
be assured that nothing was omitted.” –
Descartes The
search for knowledge is not easy, many are the question which will have to
answered, some will be known and some not. Some of the truths which we have
been seeking of are not immediately deduced from the primary self evidencing
data; this deduction sometimes involves series of connected terms arranged in a
sequence. The process is long and this is why it is not easy for the mind to
remember all the links which it did to conduct us to the conclusion. Continuous
movement of thought is required to remedy this weakness of memory. One should
run over each link several times and this process should become so continuous
that while intuiting each step it simultaneously passes to the next one; this
process should be repeated until the mind learns to pass from one step to the
other, so quickly, that almost none of the step seem to exist independently but
the whole process seems a “whole”. This process of deduction should nowhere be
interrupted, for even if the smallest link misses the chain brakes and
certainly the truth will escape from us.
This whole process
relies on enumeration. In all questions there is something (however minute or
however negative) which may escape us, and only by enumeration can we be
conscious of leading a correct induction. Only by the means of enumeration can
we be assured of always passing a true and certain judgment on whatever is
under investigation.
