We are conscious to live in a 3 dimension universe. Maybe it has more than 3 spatial dimensions (like the 10 mentioned in the string/M-theory) but that is not my concern here. My question is the following.

Is this number of (at least) 3 related with our cognitive process? I mean: in a cognitive process, there are 3 terms:

  1. observing subject;

  2. observed object;

  3. link.

Implies that to observe the universe we necessarily build a link. This link must have numerous degrees of freedom and this is not possible in a 2-D universe.


closed as off-topic by John Rennie, Neuneck, Danu, Kyle Kanos, BMS Jun 19 '14 at 15:56

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  • $\begingroup$ First off I would like to say that there are four accepted dimensions: the three of spacial and one of time. So ya... $\endgroup$ – Gummy bears Jun 19 '14 at 10:13

It is mathematically possible to map $\mathbb{R}^3$ bijectively to $\mathbb{R}$. So, you could in principle define laws of physics in one dimension and describe the whole universe on one axis. However, that is not meant by dimensionality.

The dimension is always directly entwined with the metric of a space. The metric tells you, how far apart are any two points in your space. To determine the dimension of the space you live in, take any point. Then count all the points (integrate, if your space is continuous) that are closer than a certain small distance $\varepsilon$. The set of those points are called an $\varepsilon$-Sphere. Now, compare the volumes $V_\varepsilon$ of many $\varepsilon$-Spheres and check if there is a relation $V_\varepsilon = c \varepsilon^d$, where $c$ is some constant. If there is, $d$ is the dimension of your space. As it happens, in our space, the volume of a sphere is $V=\frac{4}{3}\pi r^3$, so our space is three dimensional. If the volume of spheres were $\pi r^2$ our space would be two-dimensional (we would live in Flatland.

  • $\begingroup$ +1 for motivating dimensionality in metric spaces without making use of orthogonality. However, I suppose the OP was headed in a more metaphysical direction. Mathematical sidenote: you can also define dimension w/o refraining to metric in topological spaces. $\endgroup$ – Jonas Greitemann Jun 19 '14 at 11:07
  • $\begingroup$ Probably. What I had in mind was a not necessarily continuous space, like a graph, where you still can define dimensionality if you take the shortest path as distance between two nodes. $\endgroup$ – M.Herzkamp Jun 19 '14 at 11:11

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