"infinity more of square roots" and "infinity of square roots"

Question

I have two texts in one mathematical book:

And its matrix counterpart I does indeed also have -I and I as square roots (we call a matrix S the square root of A if S^2=S*S=A). But I also has an infinity more of square roots!

and a little further there is the text:

Even more astonishing than an infinity of square roots, perhaps, is the fact that it is possible to have two nonzero matrices whose product is zero!

In both sentences there is "an infinity of square roots", but in first one there is "more" included. Can you tell me why? And what does "more" mean in first one?

Answer 1

By replacing infinity with a finite number the meaning becomes clear:

1. There are three more square roots.
2. There are three square roots.

The first sentence means, that in addition to the one square root mentioned earlier, there are three further square roots, meaning there are four square roots total. So the two sentences disagree on the total number of square roots.

In your case, the number at hand is infinity, not three. So the first sentence says there are infinity plus one square roots, while the second says there are "only" infinity. But this is not a contradiction: in mathematics, infinity plus one is equal to infinity.

Tangentially, I don't think either phrase is particularly idiomatic. As a PhD student in a math-adjacent field, I would write

There are infinitely many (more) square roots.

The nice part of this construction is that "infinitely many" is an adjective, just like "three". This makes it easy to describe things with finite or infinite quantity using the same sentence structure.

Answer 2

I think it's important to note that infinity is not a number, but a symbol. It is a symbol that represents the infinite. In this way, the text that you have quoted is incorrect. The correct sentences would be:

But I also has infinitely more square roots!

Even more astonishing than infinitely many square roots, perhaps, is ...

Because infinity is not a number, it doesn't make sense to have "infinity of" anything. To answer why it was written with more, I will rewrite the first sentence without:

But I has infinitely many square roots!

Notice that without more, it must lose the also here. Because your quoted sentence is comparing to the previously proposed set of [I, -I], the additional information that there are infinitely many square roots can be considered to be more.