On Two New Sciences, Galileo, 1638:
Salviati: These difficulties arise because we with our finite mind discuss the infinite, attributing to the latter properties derived from the finite and limited. This, however, is not justifiable; for the attributes great, small, and equal are not applicable to the infinite, since one cannot speak of greater, smaller, or equal infinities. An example occurs to me which I shall refer to your consideration, Signor Simplicio, since it was you who started the discussion. I take it for granted that you know which numbers are squares and which are not.
Simplicio: I'm aware of the fact that a square number arises through the multiplication of any number by itself; for example, 4 and 9 are square numbers formed from 2 and 3.
Salviati: Excellent. You remember also that just as the products are called squares, the factors, that is, the numbers which are multiplied by themselves, are called sides or roots. The remaining numbers, which are not formed from two equal factors, are called non-squares. If then I state that all numbers, squares and non-squares taken together, are more numerous than the squares taken alone, that is an obviously correct proposition, is it not?
Simplicio: It cannot be denied.
Salviati: If now I ask you how many squares are there, one can answer with truth, just as many as there are roots; for every square has a root, every root has a square, no square has more than one root, no root more than one square.
Simplicio: Entirely correct.
Salviati: Again, if I ask how many roots are there, one cannot deny that they are just as numerous as the complete number series, for there is no number which is not the root of some square. Admitting this, it follows that there are just as many squares as there are roots, since they are as numerous as the roots and every number is a root. Yet we said at the outset that all numbers are more numerous than all squares, since the majority of the former are non-squares. Indeed, the more numbers we take, the smaller is the proportion of squares ; for up to 100 there are 10 squares, that is, one tenth are squares ; up to 10000, one hundredth; up to 1000000, only one thousandth. Still up to an infinitely large number, granting we can conceive it, we were compelled to admit that there are just as many squares as numbers.
Simplicio: What is to be our conclusion?
Salviati: I see no escape except to say: the totality of numbers is infinite, the totality of squares is infinite, the totality of roots is infinite; the multitude of squares is not less than the multitude of numbers, neither is the latter the greater; and finally, the attributes equal, greater and less are not applicable to infinite, but solely to finite quantities.
2.
- Your turn.
- I don't know. I'm expected to outdo Galileo?
- Yes. What do you have to say?
- The infinite is an idea, but not an idea about the world.
- What else can it be about?
- About both us and the world, about something we do in the world.
- What?
- Operate a machine of thinking. We take what we have and add one. Then take that and add one. We instruct ourselves to continue doing this. The infinite is a sort of recipe for action.
- A program.
- Yes. We can follow a recipe to construct an infinite series of odd numbers, like we can for all numbers. We imagine that the odd infinite must be smaller than the all number infinite because the all number series also includes the even numbers which also are infinite. Imagine we count at the rate of one unit per second.
- We operate the mental machine once per second.
- Yes. We don't see a larger or smaller infinite. We don't see a thing, "the infinite" at all. Ideas are collected experiences we see all together when we stop acting and rest. Infinites, continual action by recipe, cannot be ideas, cannot be seen.
- Then what are we doing when we talk about larger and smaller infinites?
- We imagine that the counting in our mind is shown in a movement in space. Each time we count one more we move a little forward. It looks like the set of all numbers is moving forward more than the set of odd or even numbers. When we get to 2 for all numbers, we have taken two steps, but for the even or odd numbers, only the first.
- We seem to be packing more movement and distance covered in the same infinite counting?
- Yes. Counting odd numbers and even numbers and squares is slower, covers less distances.
- So when we talk about bigger and smaller infinites we are really comparing speed of constructing infinite series.
- Right. Now this has some rather amazing implications for social life.
- Here we go.
- Social roles both provide security and are alienating. They provide security by giving us a sense of power, the power to do repeatedly what is done in our particular role. Social role is a kind of infinite. We imagine how we could "operate" our role on whatever the world throws at us, always adding one more instance of successful performance. On the other hand, social roles are alienating. We imagine that if we had no particular role at all, were instead all roles, we'd be like the set of all numbers not only odd, even, or squares, we'd be "larger infinities", we'd get further quicker, we'd cover more ground in life.
- This reminds me of the paradox, Zeno's arrow. In one second it will hit the target. In half a second it is half way there, in a quarter second more it gets closer, in an eighth of a second more, closer still, in a sixteenth of second more, closer still. We can operate this machine of adding ever smaller periods of time, and the arrow seems to never get to the target. Are you saying something similar?
- When we first choose a social role, we are like the arrow traveled half way.
- I see that. Like odds or evens or squares.
- Imagine then we take on further specificity of social role. Asian, female, Christian, homosexual student life, for example, the subject of a movie I saw today. Each new role seems to be adding to life, but halves the ground covered, like odd numbers are half of whole numbers. The more specific the roles we take on, the smaller our infinite, and that makes us feel alienated. Our power is increasing but life is shrinking.
- Like the arrow, really we're getting nowhere.
2.
- Your turn.
- I don't know. I'm expected to outdo Galileo?
- Yes. What do you have to say?
- The infinite is an idea, but not an idea about the world.
- What else can it be about?
- About both us and the world, about something we do in the world.
- What?
- Operate a machine of thinking. We take what we have and add one. Then take that and add one. We instruct ourselves to continue doing this. The infinite is a sort of recipe for action.
- A program.
- Yes. We can follow a recipe to construct an infinite series of odd numbers, like we can for all numbers. We imagine that the odd infinite must be smaller than the all number infinite because the all number series also includes the even numbers which also are infinite. Imagine we count at the rate of one unit per second.
- We operate the mental machine once per second.
- Yes. We don't see a larger or smaller infinite. We don't see a thing, "the infinite" at all. Ideas are collected experiences we see all together when we stop acting and rest. Infinites, continual action by recipe, cannot be ideas, cannot be seen.
- Then what are we doing when we talk about larger and smaller infinites?
- We imagine that the counting in our mind is shown in a movement in space. Each time we count one more we move a little forward. It looks like the set of all numbers is moving forward more than the set of odd or even numbers. When we get to 2 for all numbers, we have taken two steps, but for the even or odd numbers, only the first.
- We seem to be packing more movement and distance covered in the same infinite counting?
- Yes. Counting odd numbers and even numbers and squares is slower, covers less distances.
- So when we talk about bigger and smaller infinites we are really comparing speed of constructing infinite series.
- Right. Now this has some rather amazing implications for social life.
- Here we go.
- Social roles both provide security and are alienating. They provide security by giving us a sense of power, the power to do repeatedly what is done in our particular role. Social role is a kind of infinite. We imagine how we could "operate" our role on whatever the world throws at us, always adding one more instance of successful performance. On the other hand, social roles are alienating. We imagine that if we had no particular role at all, were instead all roles, we'd be like the set of all numbers not only odd, even, or squares, we'd be "larger infinities", we'd get further quicker, we'd cover more ground in life.
- This reminds me of the paradox, Zeno's arrow. In one second it will hit the target. In half a second it is half way there, in a quarter second more it gets closer, in an eighth of a second more, closer still, in a sixteenth of second more, closer still. We can operate this machine of adding ever smaller periods of time, and the arrow seems to never get to the target. Are you saying something similar?
- When we first choose a social role, we are like the arrow traveled half way.
- I see that. Like odds or evens or squares.
- Imagine then we take on further specificity of social role. Asian, female, Christian, homosexual student life, for example, the subject of a movie I saw today. Each new role seems to be adding to life, but halves the ground covered, like odd numbers are half of whole numbers. The more specific the roles we take on, the smaller our infinite, and that makes us feel alienated. Our power is increasing but life is shrinking.
- Like the arrow, really we're getting nowhere.
