Can Variance of Variances Define Higher Dimensions?

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SUMMARY

The discussion centers on the relationship between dimensionality and variance, proposing that as one progresses from zero dimensions (a point) to higher dimensions, the concept of variance evolves. The author suggests that moving from one point to two introduces the dimension of difference, while three points introduce a difference of differences, or variance. The inquiry extends to whether a variance of variances could exist in higher dimensions, potentially linking variance to time and exploring its independence from change and spatial dimensions.

PREREQUISITES
  • Understanding of basic dimensionality concepts (0D, 1D, 2D, 3D)
  • Familiarity with statistical variance and its implications
  • Knowledge of the concept of time in physics
  • Basic grasp of multiverse theories in cosmology
NEXT STEPS
  • Research the mathematical foundations of dimensionality in geometry
  • Explore advanced statistical concepts, particularly higher-order moments
  • Study the implications of time in physics, focusing on the Arrow of Time
  • Investigate multiverse theories and their relation to dimensionality
USEFUL FOR

Philosophers, physicists, mathematicians, and anyone interested in the intersection of dimensionality, variance, and the nature of time.

Vors
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I think I'll ask it this way. Here goes...

1 point = zero dimensions.
2 points = 1, which is line; 3 points, shape; 4 minimum to have volume.

Somewhere, or rather, when, time applies, most preferably at that case in which dimensionality, quite plainly, is. My question follows a hunch that supposes a possible elegance or symmetry that's beyond me.

From 1 point to 2 we go from no dimensions to the dimension of difference.
From 2 points to 3 do we not see a difference of differences, i.e. variance? (Keep in mind that I am accounting for any change in difference here as a kind of point, as well. Time has equal application here, particularly in the establishment of dimension, itself.) If so, then would it follow that a variance of variances comes next? This is where my eyes cross.

Assuming my train of logic is still on its track (appologies. my first post.), I'll ask my final:

What then? If we can draw a comparison between variance and time (which is what I'm attempting to do), then would a variance of variances fit into one of the progressively higher dimensions? Can variance fit into space independent of time, independent of change? (or is this where we get into the concept of multiverses?)
 
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Those are some good thoughts, but I think a problem might be that viewing any given dimension as time-like - as a form of change, as you say - is somewhat arbitrary. It works conceptually but the universe we perceive doesn't appear to work like that. I think a dimension would need to have an attribute like the http://en.wikipedia.org/wiki/Arrow_of_time" to feasibly be construed as change / time-like.
 
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