Early Greeks & Aristotle
II. The Method of Exhaustion (Eudoxus and Archimedes)
Euclidean Geometry: Two notions of the infinite:
(a) infinite divisibility of line segments (b) infinite extendability of line segments 

lead to paradoxes of infinitely small and infinitely big 
BUT: Early Greeks tended to avoid talk of the infinite. In geometry, all objects are really 

finite (like natural numbers: any one is finite; together are all
infinite) 
Example: Method of Exhaustion (Eudoxus 408355 BC) as used by Archimedes to prove area of circle = πr^{2}. 
Let C be a circle with radius r.
For each natural number n, let be a regular polygon inscribed in C.  

n equal sides and n equal angles 
Divide into n congruent triangles  
Let
= base of triangle THEN: area of triangle = 
Now visualize C as 
a polygon with infinitely many infinitely small sides.
SO: When n = ∞:
The height of each (infinitely thin!) triangle in is identified with the radius of C (and the base of each triangle in is very, very small... infinitely small!). 
SO: area of C = area of
, when n = ∞
Problems
(1) What does it mean to multiply by an infinitely
small amount (
when n = ∞)? (Can’t be same as multiplying by 0!) (2) What is a polygon with infinitely many infinitely small sides? (3) As n goes to infinity, approximates C, but also C*: 
What
does it mean to say C is what 
Archimedes’ Solution
Proved 2 claims:
Claim I: There is a regular polygon as close in area to C as you care to
specify.
i.e., For any arbitrary small area ε, there is always a number n such that differs  
in area from C by less than ε.) 

not true for C* 
Consequence: The area of C is at most πr^{2}.
Claim II: The area of C is at least πr^{2}.
Consequence of I and II: The area of C is exactly πr^{2}
Significance of Archimedes’ Solution: No mention of infinity!
III. Aristotle
only world there is for Aristotle 

Empiricist: Platonic Forms are in the physical world. 
Relevant Question: Is anything in nature infinite? Aristotle’s Answer:
actual infinite: that whose infinitude
exists at some point in time Note: For A., this is literally the distinction: Time is infinite, but not space 
Aristotle’s Response to Zeno’s Paradoxes:
Achilles and the Tortoise
The distance between Achilles and the Tortoise is only potentially infinitely
divisible; it is not
actually infinitely divisible. And there is no contradiction in claiming that a
finite length is
potentially infinitely divisible.
or:
To travel a potentially infinitely divisible distance, Achilles needs a
potentially infinite time. And
there is nothing wrong in claiming he has such a time available.
Aristotle’s infinite: “the untraversible”
Something is infinite if, taking it quantity by
quantity, we can always take something outside.”
“It is not what has no part outside it that is infinite,
but what always has some part outside it.”
Under Moore’s reading, Aristotle rejects the “metaphysically infinite” and
adopts the “mathematically infinite”.
Problem for Aristotle: What about the infinite past?  
Already traversed? So actually infinite? 
Wittgenstein’s Story:
Suppose we come across a man saying “... 5, 1, 4, 3.”, who then proceeds to tell
us that he has just finished reciting π backwards for all past eternity. Why
does
this strike us as impossible, whereas someone who just starts reciting π
forwards
and will continue for all future eternity does not (given that we concede the
possibility of living forever).