- Aug 26, 2011Um, hi. It's been quite a while since the last post of this discussion. I have no excuse; I just completely forgot about it.

That, of course, isn't necessarily a bad thing; it's given me some time to give my arguments a little more solid ground; I think I can articulate myself much better than I could a year ago. I just hope that interest in this discussion hasn't been completely lost in the time since.

So, let's get started again; rather than directly responding to anything in this post, let me just bullet-point where we seem to stand at the moment:

1) Sauwelios has presented me with an argument (reproduced at the bottom), consisting of a progression of eight syllogisms, which represents the infinity/nothingness problem.

2) After having resolved several issues of definition, my objections to this argument reduce to:

(2.1) Syllogism (5) is invalid. While it seems intuitively true (as I've stressed before) that a finite existence must be surrounded by something (or nothing), what we know about Riemannian geometry and General Relativity calls it into question, as a universe can be curved and distorted to the point that it, in effect, can surround itself, without having to be curved "within" anything. I gave three definitions of curvature, which are given in the order of most precise to least precise (and of least clear to most clear):

I: Curvature is a measure of the extent to which the metric tensor of a Riemannian manifold is locally non-isometric to a Euclidean manifold.

II: A space X is curved if and only if the distance between two arbitrarily close points in X cannot be related, by a one-to-one function, to the distance between two corresponding points in a Euclidean space.

III: If a "straight" and evenly-spaced grid can be constructed in a space, that space is Euclidean, and has zero curvature by definition. A curved space, then, does not allow this to be done.

Note that I've amended some of my language to make it somewhat more intelligible and relevant; for example, the first versions of II and III included a definition of the curvature metric as well, but I've dropped that now; it's not necessary to be able to measure curvature, we just need to be able to recognize it.

The point of providing these definitions is to provide a jumping-off point; these definitions can be used to prove that curvature implies extrinsic curvature (i.e. curvature "within" something), and forbids intrinsic curvature (i.e. curvature as an intrinsic property of the space-time), if it indeed does so.

(2.2) Syllogism (6) is also invalid. Nothing about being enclosed by something unthinkable implies that the object in question is unthinkable. For example, I'm currently surrounded (or enclosed) by a gas containing argon gas; that doesn't imply that my body contains argon gas in the same concentration (my body does contain trace amounts of argon, but that cannot be inferred merely from the fact that the atmosphere contains it). Of course, it could just be the property of being unthinkable that allows the inference to be made, but then it would have to be demonstrated that being enclosed by something unthinkable (such as not being enclosed by anything) implies that the object itself is unthinkable.

Sauwelios posed the question of whether I can conceive (e.g.) the Earth as not being enclosed by anything; my answer is that I can certainly conceive the Earth, by itself, without having to conceive of any boundary. In my imagined representation of the Earth, then, the boundary is not anything; it therefore is nothing, by definition.

3) It seems, then, that, starting here, the discussion should focus on two questions:

(3.1) Does the theory of general relativity (GR) and, in particular, the Riemannian geometry used by GR, imply that existence can be enclosed (intrinsically) by itself, given the definitions of "curvature" above, and given that GR implies that it can curve severely enough to enclose itself?

(3.2) Is an object enclosed by something unthinkable itself unthinkable?

I hope to revive this discussion here, especially since there hasn't been *any* activity on this group since.

~Ian

The infinity/nothingness argument:

1

a. To think is to form or have in the mind: http://www.merriam-webster.com/dictionary/think

b. The mind is finite: this is a fact of experience.

c. Whatever is thinkable is finite.

2

a. Whatever is thinkable is finite: see 1.

b. Infinity is not finite: by definition.

c. Infinity is not thinkable.

3

a. To think is to form or have in the mind: see 1.

b. Only *things* (i.e., existents) can be formed or had: forming and

having both require an object.

c. Whatever is thinkable is a thing.

4

a. Whatever is thinkable is a thing: see 3.

b. 'Nothing' is no thing: by definition.

c. 'Nothing' is not thinkable.

5

a. Existence is all that exists: by definition.

b. Existence is finite: suppose.

c. Existence is enclosed by 'nothing' as by a boundary.

6

a. Existence is enclosed by 'nothing' as by a boundary: see 5.

b. 'Nothing' is unthinkable: see 4.

c. Existence is unthinkable.

7

a. Existence is infinite: suppose.

b. Infinity is unthinkable: see 2.

c. Existence is unthinkable.

8

a. Existence is unthinkable as finite: see 6.

b. Existence is unthinkable as infinite: see 7.

c. Existence is unthinkable. - << Previous post in topic Next post in topic >>