Why no thing is finite
There is a natural tendency to equate the finite with the definite or the rational. The infinite is the indefinite and the irrational. Within the rationalist tradition there is an ill-concealed suspicion of the infinite, to the point of denial. It is my contention that the rationalist must embrace the infinite.
A rational world is necessarily relational, and in such a world the finite is only a sometimes-useful fiction. The qualification of rationality can be made stronger to be that of conceivability or even possibility. The finite is thereby logically impossible.
I can say that two plus three is five, but that statement has no meaning outside the context of the natural numbers that are necessarily infinite, I would contend. Constructivists would argue otherwise about numbers. In the empiricist-positivist tradition, they would say that the only numbers that exist are the ones that have been experienced. This raises the fallacy of there being a largest number, which is inconceivable.
Consider the analogous problem of figure and ground. We are constantly referring to objects that have been abstracted from some context. It is hardly controversial to state that objects can only have a contextual meaning. The empiricist would claim that the implied context is only the sum of our individual finite experiences with a given sort of object.
I would contend that the notion of meaning necessarily invokes the infinite. To understand something is to understand that thing's possibilities. To understand possibility is to understand the unlimited.
Our historical problems with the infinite are likely derivative of the theological insistence that the primary distinction between creature and creator is precisely that between the finite and infinite. Thusly have the religious authorities kept us in our finite place.
Reconsider the original conflation of the rational, the conceivable and the possible. The grain of sand on my finger ought to epitomize the finite. Modern physics would certainly suggest otherwise. But even in a Newtonian context, that grain of sand could not be formally defined or described in any finite manner. The casual description as the grain of sand on my finger, requires only seven words, and there it is apparently existing all by its finite self. But nothing does. Every 'finite' object has to be somewhere at sometime, and that immediately begs an unlimited regress. A grain of sand in a universe by itself could not be anything like this one here. It would be like nothing at all. It would be inconceivable. My argument runs in a circle, an infinite circle, as all arguments do.