Vallicella at Maverick Philosopher has little discussion of sqrt(2) [square root of 2] in this post. It is related to his previous posts on the issue of actual vs. potential infinites. In the post Vallicella says:
Suppose you have a right triangle. If two of the sides are one unit in length, then, by the theorem of Pythagoras, the length of the hypotenuse is the sqr rt of 2 = 1.14159. . . . And yet the length of the hypotenuse is perfectly definite, perfectly determinate. If the points in the line segment that constitutes the hypotenuse did not form an infinite set, then how could the length of the hypotenuse be perfectly definite?
What I want to point is that the hypotenuse doesn’t have any particular number related to it. It is square root of 2, only in relation with the sides which *are taken to be* 1. The nature of the number to be a ratio is shown here… the number which we assign to the hypotenuse is about the ratio between it and the other sides. However, we could instead take the hypotenuse to be 1, and the sides would be then sqrt(1/2). Or we could take the sides to be 3, and the hypotenuse would be then 3sqrt(2), etc… We see that what doesn’t change is the ratio, and what is perfectly definite here is the ratio.
Now, is there anything infinite here? I don’t think so… the hypotenuse and the other sides of the right triangle have a certain and limited length. Also, such as they are, we can speak of the ratio they form, and hence speak of numbers in relation to them. But numbers are not merely sums – we might *try* to express the ratio of the hypotenuse and the side of the right triangle through a sum, but in the given case we will fail. And this failing, this impossibility to finish with the “translation” of this ratio from its “definiteness” into our numerical notation, IS what we have as infinity here. We just can’t succeed into finishing this translation *because it can’t be done*! Some things can’t be translated into aggregates or sums. That is because there is more to this world than aggregates (or sums).
This ratio also has relations to other ratios which can be expressed as ratio of whole numbers. It can be bigger than some of those ratios and smaller than others. And just because of it, there is possibility of getting as close as needed to this ratio by using ratios of whole numbers. (Of course the ratio of rational numbers also falls under ratio of whole numbers as a/b:c/d is nothing other but the a*d/b*c ratio of whole numbers).
Of course, we can continue this translation as much as we can, and only because it will never succeed, we might be inclined to say that the product of this translation is infinite. But if we search for an ‘actual infinity’, it will not show itself in the results of some repeating iteration, but what is just “imperfectly” expressed by this sum, is as actual, definite and finished in the form of the term sqrt(2) (*and all of different cases which share the ratio with the ratio between the hypotenuse of the right triangle, and it’s side*, or which can be showed as identical).
Seems to me, that if we want to talk about actual infinity, we can just talk about being infinite in relation to something else – i.e. that that former can’t be reduced to the later.