Starting in amthematics:

The infinite in mathematics must be differentiated: we have the sequence - 0,1,2,3...; where each number is distinct. The same goes for infinite ordinals and cardinals.

Geometry is differentiated: take the infinite straight-line, even synthetically we see positions on the line; analytically of course it is made up of points. (Note, the use of analytic and synthetic has nothing in common with how these terms are introduced in epistemology).

These are the two modes of mathematics - algebra and geometry: thus it appears the infinite in mathematics, must be differentiated. Physics, of course is intimately tied with mathematics; but one notes that there is of course no infinities (but possibly, the possibility of one).

So turning to theology:

Spinozas God, as theorised in his Ethics, the uncaused, sole and neccessary substance is infinite and differentiated - the infinity of attributes & modes that he explicitly acknowledges.

The negative theology of mainstream Islam is of an Infinite One. Can one say that it is differentiated? There are the attributes of Allah - his names - Majesty, Greatness, Beauty etc that also manifest themselves in the human world. At the same time, it is affirmed that no comparison can be made with the things of the world (the created) with Allah (the uncreated).

And then to philosophy:

Kants noumena, the existent behind the phenomena, he states explicitly is non-differentiable; it is not that we cannot know the thing-in-itself, but the world-in-itself remains behind a veil. But is it infinite?

  • 1
    What does differentiated mean in this context? I don't know what you mean by saying the mathematical infinite "must" be differentiated. Transfinite cardinals and ordinals are historically contingent, they're only 140 or so years old. And the whole theory depends on the Axiom of Infinity. It's not a "truth about infinity." It's a truth about mathematical infinity. Big difference. So what do you mean exactly? Cantor thought his infinite numbers were the metaphysical infinite. But today nobody thinks that.
    – user4894
    Jun 3 '14 at 4:17
  • Infinity in mathematics doesn't have to be differentiated. Take f(x)=floor(1/x) for plural integer x. x=1 f(x)=1, remaining x 2..inf f(x)=0. Or take a constant function, f(x)=1`. ones all the way from minus to plus infinity, no differences, infinite number of them.
    – SF.
    Jun 3 '14 at 16:36
  • @SF: differentiation isn't just the differention of elementary calculus. There is also for example, the cellular [differentiation](en.wikipedia.org/wiki/Cellular_differentiation) of the tissues of the body from the originary single cell. Perhaps it would be simpler to think of distinguishing? Jun 3 '14 at 19:00

As commenters have pointed out, your question could use a lot of clarity concerning what is meant by "an infinite" and "differentiated". Nevertheless, I will push back on one of the examples you have given, namely that of the infinite line. Yes, the line must necessarily be understood to consists of different points - if it were just one point, well, it would be a point, not a line, and would not in any usual sense of the word be understood as infinite. However, I contend that the points that constitute the line are indistinguishable, or as you seem to put it, undifferentiated.

"Now," you would say, "surely the points on the line are differentiated? For we say that this point is at x=0, this one at x=1, etc." Well, we label them like such only after we choose an entirely arbitrary system of labels (namely, a coordinate system with an arbitrary point as its coordinate origin x=0). As a purely geometrical object, a line is a line even before we conceive of coordinate systems. We can understand a line without any reference to coordinate systems or equivalent notions - per Euclid, an infinite straight line is a "breadthless length...which lies evenly with the points on itself" and extends without cessation in either direction. This is also reflected in the fact that historically, the notion of a line predates the notion of a coordinate system.

To explicate this further, we could pick "this" point or "that" point as the coordinate origin x=0, and the line would look exactly the same. For, since the line is straight, we could only tell the two situations apart by measuring the (signed) distance from the origin to a point that we knew was "the same" in both cases; but before we imposed the coordinate system there was nothing to label said point by, and we couldn't possibly tell that it was the same in both cases!

Thus, I think that the sense of "an infinite" and "differentiated" used above, yes, an infinite can be undifferentiated.

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