Arguments that Infinity is Incomprehensible
Galileo identifies a stucture formed of two interpentrating
solids with the following property: a plane perpendicular to
their common axis will cut two solids (a cone and a strangely
shaped ring) with the same volume. Here is a reconstruction of
6 and 12 contradict. We conclude that there is no actual
infinity of points in a volume.
- Assumption for reductio: there are an actual
infinity of points in any volume.
- There exists a geometric figure with the features
of the soupdish.
- Any plane cut perpendicular to the axis of the
soupdish yields two solids of the same volume (a ring
- The ultimate cut of a plane through the soupdish
yields a point and a circle.
- By 2 and 3, the ultimate cut of a plane through
the soupdish yields two figures with the same volume.
- By 5 and 4, a point and circle have the same volume.
- By definition a point has one point.
- By definition a circle has infinitely many points.
- A circle has more points than a point (by 7
and 8 and the observation that infinitely many is
more than one).
- Volume is composed of points; and a volume A is larger
than a volume B if and only if A has more points than B.
- The volume of the circle is greater than the volume
of the point (by 9 and 10).
- A point and circle do not have the same volume
Right after this proof, Salviati says, "the infinite is inherently
incomprehensible to us, as indivisibles are likewise" (38 (77)).
Composing Indivisible: two indivisibles together cannot make a divisible
7 and 8 contradict. We conclude that we cannot reason about
- Assumption for reductio: we can reason about
indivisibles (or infinitesimals).
- Any line can be divided evenly in half.
- If we could reason about indivisibles, then
we can see that any number of points makes a line.
- (By 1 and 3,) any number of points makes a line.
- (By 4,) five points will make a line.
- The line made of five points can be evenly
divided in half (by 2 and 5).
- Then the middle point (the third point) of the
line made of five points can be divided in half.
- No point can be divided in half, including
the middle point (the third point) of the line made
of five points.
Galileo compares two lines, one longer than the other.
The proof is more elegant if we assume a line and a segment
of that line.
5 and 8 contradict.
We conclude that there is no actual infinity of points in any
- Assumption for reductio: there are an actual
infinity of points in any line.
- Suppose line AC is twice the lenght of its
- AC has infinitely many points.
- AB has infinitely many points.
- AC and AB have the same number of points
(namely, infinitely many) (by 3 and 4).
- If one line is twice as long as the other,
then that line has twice as many points.
- AC has twice as many points as AB (by 6).
- AC does not have the same number of points
as AB (by 7).
(NOTE: up to this point, Galileo has made a very important
assumption about the relation between length and the number of points
in line or circle; and about volume and the number of points in a
volume. This is required for the soupdish and the lengths arguments
to have contradictory conclusions. Can you identify this assumption?)
Squares and other functions
4 and 8 contradict. We conclude we were wrong to suppose that there
is an actual infinity of natural numbers or of their squares.
- Assumption for reductio: there is an actual infinity
of natural numbers and an actual infinity of their squares.
- If there is a one-to-one correspondence between
all the natural numbers and all the squares, they
are the same quantity.
- There is a one-to-one correspondence between the
natural numbers and the squares, because each
natural number is the root of a square
(x2 is on the naturals and onto the
squares), and each square has a natural number as
root (the inverse of x2, namely the
square root of y, is on the squares and onto the
- The natural numbers and the squares are the same quantity
(by 2 and 3).
- If one group is a proper subset of the other, that group
has a smaller quantity.
- The squares are a proper subset of the naturals, because
every square is a natural number but not every natural number
is a square.
- The squares have a smaller quantity than the naturals
(by 5 and 6).
- The natural numbers and the squares are not the same
quantity (by 7).
Galileo also makes an argument about how the squares become fewer and fewer.
I'm skipping that here. But his point is that it is even more absurd to find
that the squares and the naturals are the same size (as we did in step 4 above)
because as we count through the squares we see we missed more and more of the
Based on these arguments, Galileo concludes that we cannot quantify
infinities, and so should avoid talking about infinities and also
infinitesimals. He proposes an approach that might seem similar to
Aristotle's notion of potential infinities. Instead of infinity, we
should say that a line, for example, has any number of points; or that
we can identify or make use of each natural number.
Guiding questions as we look ahead
- Refering to infinity is useful! For example, Calculus
appears to refer to infinitesimals. Is it enough to say
just that "we can always find another" or that "there are
at least as many points as any number we choose"? Also,
is it reasonable to ask how many there are when we can
always find another or when there are at least as many as
any number we choose?
- Have we discovered here truly a limit to human reason,
or can we find some way to talk about infinite quantities?
[Revised 5 Feb 2017.]