Article in New Scientist

[indolikaa]

She-it. Considering my 38" waistline abhors exercise in all forms, who am I to talk?

It all depends on how tall you are. I have a 34" waist and I'm only 12" tall

What are you, a shaved poodle?!


Actually, for the official dvdrhelp.com record, I am 5'-8" with my socks off.

[Capmaster]

She-it. Considering my 38" waistline abhors exercise in all forms, who am I to talk?

It all depends on how tall you are. I have a 34" waist and I'm only 12" tall

What are you, a shaved poodle?!


Actually, for the official dvdrhelp.com record, I am 5'-8" with my socks off.

Shaved poodle? No. People have described my build as "not unlike a manhole cover"
Seriously, we're about the same height. I miss that 30" waist though. I had it for years and years until I crossed some magic threshold and it didn't matter what I ate, the waistline started enlarging. If I could only get my pecker to do that I'd be a bit happier about it

[northcat_8]

That argument is incorrect. It can be argued that it approaches 2 as the number of terms approaches infinity. Saying that you can never reach infinity is somewhat false. We are not talking about physicality here. We are talking about pure maths. There is a whole branch of mathematics looking at different forms of infinity.

In that summation, as the number of terms approaches infinite, the value of x approaches 2. If the number of terms = infinite, x = 2.

Denying the above is denying that calculus works.

It could be argued that since the value would continue to get infinitely closer to 2 but never equal 2 the value of the function will continue to increase and would then be considered infinite since no exact value of the increasing function can be determined.

UM, NO. If you don't ever want to reach infinity (and you can in maths) then the value will infinitesimally (different from infinite) less than two. That is still a finite number. It is a real value between 1 and 2.

Your point that the value would be 2 when X = infinity, is theoretical mathematic logic (or as some call it, common sense). Since the value would be so close to 2 it would be indistinguishable from 2, therefore equal to 2. However, since the exact value of infinity can never be reached, the exact value of 2 could never be reached, since infinity is not actually a number but rather a concept. That theory is similar to the theory that parallel lines meet at infinity.

Functions can approach a certain value as the independent variable approaches infinity but the exact value can never be obtained, even though the number would only increase by infinitesimal amounts it would still increase. and no matter how small of increments, the function would still be increasing and since our domain is infinite, the function would increase infinitely. The wording of "approaches infinity" is misleading because no number actually approachs infinity because infinity is not actually a number or place.

By your logic, in the function y = 1 / x it would be possible to have x = 0.

Our debate is taking place in 2 different mathematical environments. I am debating from the stand point of real numbers and you are debating from the stand point of theoretical.

In real numbers, there will be an asymptote at 2. While in theoretical mathematics, the value would get so close to 2 that the number is indistinguishable from 2 and therefore be 2. Since the number of terms would have to = infinite in order for X = 2. It is important to realize that infinity is neither a number or a place. When we say that "X approaches infinity", we are saying that X is becoming arbitrarily large. When we say "the limit is infinity", we are saying that the terms are becoming arbitrarily large, not that they are approaching some number. So by definition, the number of terms could never equal infinity, therefore the function could never equal 2.

I am aware of the different branches of mathematics and the fact that you are dealing in the theoretical field. I just jumped in the post to play devil's advocate, not to delve into deep mathematical logic and theory.

[vitualis]

My analogy isn't quite correct because we are a THREE dimensional being on the 2D SURFACE of Earth. As we have a higher physical dimension (height) than the surface of Earth, we can perceive the curvature. However, since the Earth is so much bigger than our "height", we "approximate" a 2D being.

I'm not sure your analogy between the curvature of space and that of the earth is really valid. The curvature of earth is not an anomaly, but rather the normal shape of the globe resulting from gravity acting upon it for so long. It isn't even a true sphere. But there's nothing mysterious about the earth looking flat from our perspective. Viewed from the moon the earth is obviously spherical. Viewed from an ant hill it's obviously flat.

Um... I think you are missing my point.

WE can observe the curvature of a two dimension surface (e.g., surface of a sphere or the Earth) because we are three dimensional beings. Being a 3D being, we can observe the world around us in 2 (and a bit) dimensions so we can actually easily directly see the curvature (or lack of) of a surface.

A 2D being living in a 2D surface can only observe the world around it in 1 (and a bit) dimensions and cannot tell the curvature of it's "space" (if one exists). For a 2D being, it would be very difficult for it to tell whether it actually lived on a flat plane (i.e., infinite universe) or on a surface of a sphere (i.e., a finite universe). If the "2D" universe was relatively small or had a strong curvature, the 2D being might notice some strange phenomena that may make it question whether it's subjective experience of living in a flat plane was actually true... For example, objects moving parallel to each other will gradually come together or drift apart.

Now, we aren't 2D beings so that's why I used the idea of the Earth as it is so much bigger than us when observed from the GROUND. A human on the surface of the Earth approximates a 2D being so that it makes it easier to understand how a 2D being living of a sphere would experience its universe.

If you prefer, imagine how an ANT views the Earth. From it's perspective, the Earth may as well be a flat plane.

As for it not being "mysterious", I believe there are still people who believe in the "flat Earth" hypothesis...

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Now, bump this up a physical dimension...

If our space-time (at least 3 physical dimensions + time) is also curved, we as 3D beings (just like the 2D being on a surface of a sphere) cannot directly observe the curvature of "space-time" either. As far as we are concerned in our empiric experience, space-time is "flat" (i.e., we could go endlessly in all three orthogonal dimensions). However, if space-time WERE curved, then we can predict some interesting phenomena ... and as I mentioned before, something like gravitational lensing (because according the Einstein gravity is curvature in space-time). This phenomena has been observed and is real!

Curvature of space isn't simply a matter of perspective ....it's an unproved theory among many theories about the nature of our universe.

As I said before, curvature of space-time has nothing to do with perspective. I was trying to give (obvious a bad!) analogy on trying to understand what it is like. Also, it isn't an unproved theory. There is a hell of a lot of data to support it including clear observational data. You are really talking about Einstein's General theory of relativity here...

The various "unproved" theories are cosmological hypotheses on the SHAPE of the universe. Space-time curvature definitely exists (assuming general relativity is right of course).

Best regards.

[indolikaa]

Dr. Tam, did you ever consider astrophysics as your major?