Arguments that Infinity is Incomprehensible

The SoupdishGalileo 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 his argument:4 and 10 contradict. Galileo is unclear about which of two conclusions we should draw. (1) We might think that an implicit premise of our argument is that we assumed we can reason about infinity. Right after this proof, Salviati says, "the infinite is inherently incomprehensible to us, as indivisibles are likewise" (38 [77]). (2) We might conclude (knowing what he says later) that there is no actual infinity of points in a volume (instead he might say there is only a potential infinity of points in a volume).

- There exists a 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 and cone).
- There is a cut of a plane through the soupdish that yields a point and a circle.
- A point and circle have the same volume (by 2 and 3).
- By definition a point has one point.
- By definition a circle has infinitely many points.
- A circle has more points than a point (by 5 and 6 and the observation that infinitely many is more than one).
- A volume X is larger than a volume Y if and only if X has more points than Y.
- The volume of the circle is greater than the volume of the point (by 7 and 8).
- A point and circle do not have the same volume (by 9).

Composing Indivisible: two indivisibles together cannot make a divisible7 and 8 contradict. We conclude that we cannot reason about indivisibles.

- Assumption for reductio: we can reason about indivisibles (or infinitesimals).
- If we could reason about indivisibles, then we can see that any number of points makes a line.
- Any number of points makes a line (by 1 and 2).
- Five five points will make a line (by 3).
- Any line can be divided evenly in half.
- The line made of five points can be evenly divided in half (by 4 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.

Comparing LinesGalileo 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 9 contradict. We conclude that there is no actual infinity of points in any line.

- Assumption for reductio: there are an actual infinity of points in any line.
- Suppose line AC is twice the lenght of its segment AB.
- AC has infinitely many points.
- AB has infinitely many points.
- Any infinity of points is the same size as any other infinity of points.
- AC and AB have the same number of points (by 3, 4, and 5).
- 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 7).
- AC does not have the same number of points as AB (by 8).

(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 functions4 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 (x
^{2}is on the naturals and onto the squares), and each square has a natural number as root (the inverse of x^{2}, namely the square root of y, is on the squares and onto the naturals).- 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 natural numbers.

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 13 September 2018.]