Philosophy 024: Big Ideas

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Philosophy 024 Big Ideas Prof. Robert DiSalle
(rdisalle_at_uwo.ca) Talbot College 408,
519-661-2111 x85763 Office Hours Monday and
Wednesday 12-2 PM Course Website
http//instruct.uwo.ca/philosophy/024/
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Lucretius (Roman, ca. 94 BCE- 51 BCE) Background
Ancient Greek atomism Leucippus (ca. 435 BCE),
Democritus (ca. 410 BCE) -space is an infinite
void -bodies are composed of atoms (indivisible
particles) moving freely in the void, colliding
and combining -qualitative properties that our
senses apprehend arise from quantitative
properties of insensible atoms, their motions and
combinations
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Epicurus (341-270 BCE) atoms naturally tend to
fall through the void, and collisions and
combinations result from spontaneous and random
swerves. The random swerves allow the
atomistic, materialistic view to be reconciled
with freedom of the will. Understanding the
material origins of human life frees human beings
from ignorance and fear of death. At death the
body and the soul dissolve into their atomic
parts. Therefore death is nothing to us for
that which is dissolved is without sensation and
that which lacks sensation is nothing to us.
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The infinite turns out to be the contrary of what
it is said to be. It is not what has nothing
outside it that is infinite, but what always has
something outside it A quantity is infinite if
it is such that we can always take a part outside
what has been already taken. On the other hand,
what has nothing outside it is complete and
whole. For thus we define the whole-that from
which nothing is wanting, as a whole man or a
whole box. What is true of each particular is
true of the whole as such-the whole is that of
which nothing is outside. On the other hand that
from which something is absent and outside,
however small that may be, is not 'all'. 'Whole'
and 'complete' are either quite identical or
closely akin. Nothing is complete (teleion) which
has no end (telos) and the end is a limit.
(Aristotle, Physics)
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Euclids Postulate 2 To produce a finite
straight line continuously in a straight
line. Aristotle My argument does not even rob
mathematicians of their study, although it denies
the existence of the infinite in the sense of
actual existence as something increased to such
an extent that it cannot be gone through for, as
it is, they do not even need the infinite or use
it, but only require that the finite straight
line shall be as long as they please.
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Lucretius the case for the infinitude of space

• If the universe has a boundary, there must be
  something outside of it to limit it. But nothing
  can be outside the universe.
• Suppose there is an edge, and you can stand there
  and throw a missile. If it goes forward, then
  there is no boundary and you are not at the edge.
  If it is blocked, then there is something outside
  and you are not at the edge.
• If the void were finite, then all matter would
  eventually collapse by its weight into the center.

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The elemental view of nature (Empedocles, 440
BCE) Material things are composed of four
elements (earth, air, fire, water, based on four
fundamental qualities (hot, cold, moist, dry)
(Isidore of Seville, 1472)
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The geocentric universe (from the Cosmographia of
Apianus, 1524)
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Copernicuss heliocentric model (1543)
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The heliocentric system in infinite space (Thomas
Digges, 1576)
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Understanding infinite sets 1-to-1
correspondence For two sets, F and G the number
of Fs is equal to the number of Gs if and only
if the Fs and the Gs are in 1-to-1
correspondence. (Frege, 1884) Subset of a set S
A set that has some of the members of S. Proper
subset subset that does not have all the members
of S. For finite sets, the whole is greater than
the part. An infinite set can have proper
subsets that are in 1-to-1 correspondence with
the whole set. (Hilberts Hotel) E.g. for every
natural number 1,2,3,. there is an even number.
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The table of all the rational numbers
1 2 3 4 5 6 7
8 9 . 1/2 2/2 3/2 4/2 5/2
6/2 7/2 8/2 9/2 . 1/3 2/3 3/3 4/3
5/3 6/3 7/3 8/3 9/3 . 1/4 2/4 3/4
4/4 5/4 6/4 7/4 8/4 9/4 . 1/5 2/5
3/5 4/5 5/5 6/5 7/5 8/5 9/5 . 1/6
2/6 3/6 4/6 5/6 6/6 7/6 8/6 9/6
. 1/7 2/7 3/7 4/7 5/7 6/7 7/7 8/7
9/7 . 1/8 2/8 3/8 4/8 5/8 6/8 7/8
8/8 9/8 . 1/9 2/9 3/9 4/9 5/9 6/9
7/9 8/9 9/9 .
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Counting the rational numbers a set is countable
if it can be placed in 1-1 correspondence with
the natural numbers.
1 2 3 4 5 6 7
8 9 . 1/2 2/2 3/2 4/2 5/2
6/2 7/2 8/2 9/2 . 1/3 2/3 3/3 4/3
5/3 6/3 7/3 8/3 9/3 . 1/4 2/4 3/4
4/4 5/4 6/4 7/4 8/4 9/4 . 1/5 2/5
3/5 4/5 5/5 6/5 7/5 8/5 9/5 . 1/6
2/6 3/6 4/6 5/6 6/6 7/6 8/6 9/6
. 1/7 2/7 3/7 4/7 5/7 6/7 7/7 8/7
9/7 . 1/8 2/8 3/8 4/8 5/8 6/8 7/8
8/8 9/8 . 1/9 2/9 3/9 4/9 5/9 6/9
7/9 8/9 9/9 .
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Can we count up all the real numbers? 1st number
N1.a1a2a3a4a5 2nd number N2.b1b2b3b4b5 3rd
number N3.c1c2c3c4c5 .and so on. First,
suppose that all of the real numbers are listed
in this way. Then, choose a?a1, b?b2, c?c3, etc.
(Assume in each case that the digit is neither 0
nor 9, to avoid the case 0.9991.000.) Now,
consider the number z 0.abc. Evidently z is
not on the list! (It differs from the first
number in the first decimal place, the second
number in the second decimal place, etc. )
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The continuum of real numbers is equivalent to
(in 1-1 correspondence with) any part of it.
A1
B1
A2
B2
A8
B8
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The dome is bigger than the disc, but the points
are in 1-1 correspondence each point (x,y z) on
the dome can be mapped (projected) onto the
point (x,y) beneath it.
(x,y,z)
(x,y)
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A paradox of the infinite (Russells paradox)
Ordinary set does not contain itself as an
element. E.g. The set of all natural numbers is
not itself a natural number, and so it is not an
element of itself. Extraordinary set does
contain itself as an element. E.g. The set of
all sets not mentioned in the Bible is itself not
mentioned in the Bible, so it is an element of
itself. Question Since extraordinary sets are
obviously quite bizarre, can we confine ourselves
to talking about all and only the ordinary sets?
I.e., can we talk about the set of all ordinary
sets?
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The set S the set of all and only ordinary sets,
i.e. the set of all sets that are not elements of
themselves. Question Is S an ordinary set? If S
is an ordinary set, then it contains itself as an
element, since it is supposed to contain all
ordinary sets. But that means that S is an
extraordinary set, since extraordinary sets
contain themselves as elements. So S is not the
set of ordinary sets, since it contains an
extraordinary set (itself). Cf. the Barber of
Alcalà he shaves all men who do not shave
themselves. So, does he shave himself or not?