
Descartes 1
The Method of
Analysis
Descartes suggests in earlier works that in order to achieve results in
the sciences we need to take things apart into their simplest elements,
see how those elements are related to one another, and then recompose
them. This is very much the resolutocompositive method that Galileo
used in analyzing motion. It is also a method which Descartes and other
sixteenth and seventeenth century mathematicians believed the great
mathematicians of antiquity, Euclid, Archimedes, and Apollonius for
example, had used to make their discoveries, before these discoveries
were presented in the synthetic form. Descartes developed his own version
of this method. Many philosophers were impressed by the deductive form of
Euclidean geometry. Descartes was not. His account of the method of
analysis has to do with problem solving, with the method of discovering
the truths which are then presented deductively. For a more detailed
discussion of the method of analysis, you should look at Chapter 2 of John
Cottingham's book The Rationalists. The point here is that what
Descartes' analysis of dreams tells us is that we are moving both towards
that which cannot be doubted on the strongest possible skeptical
hypothesis and, at the same time, towards that which is most simple.
Under the pressure of the dream hypothesis, and adhering to the method
of doubt, Descartes tries to determine which things can be doubted and
which cannot. He remarks that "...while eyes, heads and hands may be
imaginary, it must be granted that some simpler and more universal things
are real  the "real colors" from which the true and false images in our
thoughts are framed." Besides these:
Things of this sort seem to include general bodily
nature and its extension, the shape of extended things, their quantity
(that is their size and number) the place in which they exist, the time
through which they endure, and so on.
So, real colors, extension, shape, quantity, time and place all are immune
to the doubts raised by the dream hypothesis. Just as he did for the
senses, Descartes has neatly partitioned claims made on the basis of the
imagination into two classes, one of which almost always turns out to be
false, and the other which we find hard to doubt.
Claims about the world about
us derived from dreams These claims almost
always turn
out to be false I am sitting by the fire (when in
fact I am
in bed asleep).  Claims about the world which depend on truths of
mathematics, and claims about time and space. These claims are almost
all going to turn out to be true.  Claims derived from the imagination 

Descartes now shows his concern about the implications of these
skeptical hypotheses for the sciences. He writes:
Perhaps we can correctly infer that, while physics, astronomy, medicine
and other disciplines that study composites are dubious, disciplines like
arithmetic and geometry, which deal only with completely simple and
universal things without regard to whether they exist in the world, are
somehow certain and indubitable. For whether we are awake or asleep, two
plus three is always five, and the square never has more than four sides.
This passage makes it quite explicit that we are moving from the complex
to the simple. This is good evidence that Descartes' method of analysis
is being applied here. So, besides those general features of bodily
nature, the truths of mathematics are largely immune to the doubts raised
by the dream hypothesis. How then could one call such truths as that
squares have four sides and that 2 +3 = 5 into question? Note just how
difficult this is. If it seems mad to doubt things perceived near by, it
is surely more difficult to genuinely call into question such truths as
that squares have fours sides.

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