Equivalence Principle
Table of Contents
If the preceding be true, it seems to me impossible that an infinite number is greater than another infinite.
On the contrary, the larger the number we reach, the more we recede from this property of infinity.
This is because the greater the numbers, the fewer [relatively] are the squares contained in them.
But the squares in infinity cannot be less than the totality of all the numbers. Hence, the approach to greater and greater numbers means a departure from infinity.
Thus from your ingenious argument we are led to conclude that the attributes “larger,” “smaller,” and “equal” have no place either in:
- comparing infinite quantities with each other
- comparing infinite with finite quantities
Lines and all continuous quantities are divisible into parts which are themselves divisible without end.
These lines are built up of an infinite number of indivisible quantities. This is because a division and a subdivision which can be carried on indefinitely presupposes that the parts are infinite in number.
Otherwise, the subdivision would reach an end.
If the parts are infinite in number, then they are not finite in size. This is because an infinite number of finite quantities would give an infinite magnitude.
Thus we have a continuous quantity built up of an infinite number of indivisibles.
But if we can carry on indefinitely the division into finite parts what necessity is there then for the introduction of non-finite parts?
We are able to continue, without end, the division into finite parts [in parti quante].
This is why quantity is composed of an infinite number of immeasurably small elements [di infiniti non quanti].
In order to settle this, is a continuum is made up of a finite or of an infinite number of finite parts [parti quante]?
Their number is potentially infinite but actually finite. It is:
- potentially infinite before division
- actually finite after division
This is because parts cannot exist in a body which is not yet divided or at least marked out.*
If this is not done we say that they exist potentially.
Superphysics Note
A line which is 20 spans long is not said to contain actually 20 lines each one span in length except after division into 20 equal parts.
Before division, it contains them only potentially.
When the division is once made, is the size of the original quantity increased, diminished, or unaffected?
It neither increases nor diminishes.
I think so too.
Therefore the finite parts [parti quante] in a continuum, whether actually or potentially present, do not make the quantity either larger or smaller.
But if the number of finite parts actually contained in the whole is infinite in number, they will make the magnitude infinite.
Hence, the number of finite parts, although existing only potentially, cannot be infinite unless the magnitude containing them be infinite.
Conversely, if the magnitude is finite it cannot contain an infinite number of finite parts either actually or potentially.
How then is it possible to divide a continuum without limit into parts which are themselves always capable of subdivision?
This distinction of yours between actual and potential appears to render easy by one method what would be impossible by another.
But I shall reconcile these matters in another way.
and as to the query whether the finite parts of a limited continuum [continuo terminato] are finite or infinite in number I will, contrary to the opinion of Simplicio, answer that they are neither finite nor infinite.
This answer would never have occurred to me since I did not think that there existed any intermediate step between the finite and the infinite, so that the classification or distinction which assumes that a thing must be either finite or infinite is faulty and defective.