Is 3+1 spacetime as privileged as is claimed? I've often heard the argument that having 3 spatial dimensions is very special. Such arguments are invariably based on certain assumptions that do not appear to be justifiable at all, at least to me. There is a summary of arguments on Wikipedia.
For example, a common argument for why >3 dimensions is too many is that the gravitational law cannot result in stable orbital motion. A common argument for <3 dimensions being too few is that one cannot have a gastrointestinal tract, or more generally, a hole that doesn't split an organism into two.
Am I being overly skeptical in thinking that while the force of gravity may not be able to hold objects in stable orbits, there most certainly exist sets of physical laws in higher dimensions which result in formation of stable structures at all scales? It may be utterly different to our universe, but who said a 4D universe must be the same as ours with one extra dimension?
Similarly, isn't it very easy to conceive of a 2D universe in which organisms can feed despite not having any holes, or not falling apart despite having them? For example, being held together by attractive forces, or allowing certain fundamental objects of a universe to interpenetrate, and thus enter a region of the body in which they become utilized. Or, conceive of a universe so incomprehensibly different to ours that feeding is unnecessary, and self-aware structures form through completely different processes.
While I realise that this is sort of a metaphysical question, is 3+1 dimensions really widely acknowledged to be particularly privileged by respected physicists?
 A: I would like to share my view on this issue.
I think some answers with the word "anthropic" need not to be dismissed, but could be interpreted them in a deeper sense.
Anthropic should not be something derogatory, "just humans", as if we were not part of universe, instead perhaps it could be treated as concepts like "inertial frame of reference" are treated. A measure, a way to measure, a point of view, a frame of reference.

An imagination exercise:
Suppose one day a networking software is self aware.
Then is make some self replicas, and they ask themselves :
"Why we are on layer 7 of the OSI model?" 
"Does it have something special?"
One of them would say "Because we can't live in lower layers then if
  the universe would be lower layered we wouldn't be asking things like
  this"
Another might say : "To live in layer 7, previous layer must exist to
  allow us, but, think on layer 0, our conversation are ultimately
  travelling through a cable for example, then we are at the same time,
  layer 0, layer 1, ... layer 7, the universe is not layer 7!!, 
  its one or all layer at same time, depending "who" is measuring it,
  we can see it till layer 7, but the top we see doesn't mean it's the whole
  that exist, perhaps there are higher layers than 7, and lower than 0,
  that are forbidden to us, and can't be known at all"

I think 3D+1 is the top that our natural senses are aware of, with technology we could know or suspect other dimensions,  as far we know,  "conscious beings" can't rise in lower dimensions, but that perhaps is a prejudice, because whatever we call 3D+1 perhaps can be parsed in just 1D! (similar as in the above story), so we should review our statements, of course beings could exist in higher dimensions too (if they do not exist already, they would).
A single matrix in a paper although is within a 3D+1 it could contain higher dimensions, of course a matrix in a paper is not conscious, but nobody knows if a computer program will be aware of itself someday, that day, it will "live" and even "measure" a higher dimension, and again as the matrix in the paper, we would know that it coexist in a lower dimension too.
It's a very interesting topic, I've asked about this before, you could read the answer to that question too
what are dimensions?
Regards
A: It is the minimum dimension required for the Weyl Tensor $C_{abcd}$to exist in the decomposition of the (completely covariant) Riemann Curvature Tensor $R_{abcd}$. That is kind of privileged. Or else, there would be no gravity in a vacuum (and thus, no long distance gravity, and no orbits, no free-fall)! And if it were any more, the gravity would weaken too quick (the inverse square law would become the inverse cube law, etc.)
A: The answer is no - and many of these reasons can be chalked up to a deficiency of imagination as to what sorts of possible physical laws there could be in other dimensions. For example, the idea that life is only possible with at least 3 dimensions is actually demonstrably false. Two-dimensional space has been explicitly shown to be sufficient by considering systems such as two-dimensional cellular automata: in particular, Conway's "Game of Life" can be considered to describe "a universe", and it supports self-replicating systems in it, which could be considered one, albeit broad, way to define life. They don't eat and process food (metabolism), which might disqualify it in the eyes of some, at least those favoring a very strict definition based only on Earth and our own universe since what constitutes "life" is rather subject to dispute, but that's because they don't need to - there is no strict "conservation of energy" in this universe. Regarding things not being able to avoid each other or the organism having spatially separate parts, it gets around this because separated components can "communicate" with each other by exchanging particles ("gliders"), so it's more like a swarm of smaller separate components (although technically you could consider our universe to be similar if you consider atoms to be comprised separate components though quantum mechanics problematizes things somewhat with its "fuzzy" nature and its interpretation in non-experimentalist terms so as to "describe a universe on its own terms" in the same was as the Conway's rules, is not clear). It's not sure though if Conway's universe can spontaneously generate (abiogenesis) life like ours can (this depends on a number of questions involving the fate of an infinite random grid which is in a sense the "most likely" starting condition if it were taken to be a naturally-existing universe), however, but it shows at least that you don't need 3 dimensions, and in fact don't even need to have a continuous space time, for something that exhibits at least one of the most distinctive features of life.
Regarding other points mentioned like how that forces don't provide orbits in >3 dimensions, this is based on again naive straightforward extensions of our own physics. In particular, the two long-range forces in our universe, gravitation and electromagnetism, follow an "inverse square" law meaning that the force is proportional to $r^{-2}$ where $r$ is the separation of the objects in question. Such inverse square laws support stable Kepler orbits - this is a relatively simple problem (nowadays!) any physics student will encounter in the course of their training. The reason the universe operates on inverse square laws is these forces can be conceived of in terms of field lines, and more generally the exchange of virtual particles in quantum field theory, which can be thought of as emitting a kind of radiation, and omnidirectional radiation creates a constant flux through a spherical surface, and surface area of a sphere goes as $r^2$ because of the three-dimensional nature of space. Straightforwardly generalizing this to a four-dimensional case would produce inverse-cube forces (as the radiation flux passes through a hyperspherical surface and its surface-volume (not area!) goes up as $r^3$), and these have no stable orbits. (Indeed it may even be that quantum mechanics is not able to save the atom from collapse, these are called "super-singular" potentials, though I've not personally tried to solve the Schrodinger equation to see how it behaves.)
But in reality, one could imagine more drastic changes to physics that would invalidate these concerns. One could be that forces are carried by a different mechanism than virtual particle radiation (though this requires gutting quantum theory). Then perhaps you could have $r^2$ forces. Another one would be if that objects emitted a repulsive force in addition to an attractive one. This would create a place where the forces would balance, and you could have stability. (E.g. you could have two electromagnetic-like forces, one acting in the opposite way the other does and dependent each on their own kinds of "charge", which could be different, and this could structuralize atoms. I suspect the Schrodinger equation for such a complicated scenario will not be solvable analytically though and I won't even try. I also don't know how the energy levels will structure, and I suspect such an "orbital" will be more like a shell, as in a real hollow spherical surface, where the "electron's" (or whatever it is now) probability wave fills up the "ditch" in the potential around the nucleus.) Still other options might exist like that "particles" are actually innately extended bodies and not points, and can carry other forms of information on them (maybe different parts of their surface are "colored" differently, for lack of a better word and visualization, and differently "colored" parts interact in different ways), which would prevent their total collapse or that of a structure built from them.
The only thing that might privilege it is our very specific structure of physics, but if you're going to be that specific then you might as well say 3-dimensional space is just part of that and thus it's kind of trivial. So I really think it's down to lack of imagination; there is absolutely no reason at all a Universe, even one with life, can be built in a different dimensionality.
A: If you look at Static forces and virtual-particle exchange (Wikipedia) you'll see a line of reasoning that doesn't seem to depend on the number of space-like dimensions, yet still arrives at an inverse square law.  I realize this isn't exactly a rigorous QED calculation (for which I feel far too stupid) but it makes me reconsider my former belief in non-privilege.  If d = 3 is the only case that allows both radiation and conservation of energy, then that's just... wow.
A: I am not sure that being in 3+1-D is a privilege. Actually, all the troubles with Feynmann integrals come from 4D. Secondly, the QFT is integrable only in 2+1-D. From the mathematical point of view, the 4D differentiable manifolds are most problematic.
On contrary, I also heard that if the space is not 3D then the signal cannot be transmitted, but at the moment I don't know the proof. This is significant, since without signal transmission, our world has a bigger problem than to be able to tie a knot from a string.  
A: I think that many different dimensions and metric signatures have their specific “privileges”. More general, different geometries in a broader sense, and, even more general, different underlying mathematical structures (such as fields other than ℝ) also could be models for space-time of some alternative physics. But is was just (necessary for me) philosophical preface.
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One time and PDEs
What is special for Lorentzian manifolds with their (locally) one temporal and three spatial dimensions? First of all, a metric signature with one temporal dimension but some (one or more) spatial dimensions is something very special (I deliberately ignore the question of mathematical sign, whether $t^2$ is positive and $x^2$ negative or versa — it doesn’t make difference where the time is distinguished). There are two cases where a Cauchy problem can be solved for degree-2 partial differential equation, for a reasonably broad class of initial and boundary conditions. In $t^2 - x^2$-like metrics hyperbolic differential equations live. Notoriously, the other case are parabolic differential equations that are degree-2 by space but degree-1 by time and correspond to Galilean time; so it is also one-time-many-space-dimensions universe. In parabolic case, of course, there is no non-degenerate quadratic metric.
What is special with Cauchy problem? It is a natural formulation of evolution problem. We specified an initial state of the field, we specified boundary conditions, and we can predict evolution. And even without boundary conditions hyperbolic equations (but not parabolic) admit a solution in a cone-like domain, of space-time points where initial conditions traverse all the past cone. Hyperbolic PDE is the only case that allows exact prediction (is certain spacetime domain) in spite of spatially-bounded knowledge of initial conditions.
For more that one time solutions will not be unique. For a “time”, geometrically, no different from space, a solution will not always exist.
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Specificity of 3 + 1
Let’s think we proved that exactly one time dimension is a requisite. Why is special to have exactly 3 spatial dimensions, D = 3? In the case of quadratic metric (corresponding to abovementioned hyperbolic PDEs) the answer is simple: orthogonal group is the Lorentz group. Its unity component is isomorphic to Möbius group. The universal cover of said unity component is SL(2, ℂ) – it is very convenient for quantum field theory and other applications. 
The case of D = 1 is inconvenient for numerous reasons (not only symmetry-related). In the case of D = 2, apart of not having the full geometric SU(2), we’d have more types of quantum statistics than two types that we have in our universe (fermions and bosons). We could have particles with arbitrary angular momentum; it’s IMHO not for good. But photons couldn’t have helicity. What all this quantum stuff is for, indeed? Although D = 2 can be, in principle, habitable, it unlikely will be a quantum world.
What about D > 3, indeed? Geometrical gains are insignificant. There are some theories that requires extra dimensions but… in 4+ dimensional spaces we should have more than 2-component spinors. It is an unnecessary complication, isn’t it?
A: Science fiction writer (but also published physicist) Greg Egan has put quite a bit of work into investigating a universe with 4+0 dimensions: Orthogonal. Some of it is quite ingenious, eg. assuming a compact universe guarantees that the (modified) wave equation doesn't have exponentially growing solutions and time appears, without the -1 in the spacetime metric, as the local gradient of entropy.
A: No. While there are some arguments for why 3 spatial dimensions are a good place to live in, the answer to the question why our universe has 3 large spatial dimensions is presently not known. 
Karch & Randall wrote a paper on the issue some years back: http://xxx.lanl.gov/abs/hep-th/0506053 They consider some higher dimensional space filled with objects of different dimensions that have some interactions among each other and argue that 3 dimensional ones are among those most likely to dominate. It's an argument though that is not widely accepted due to the assumptions they have to make for this to work.
A: If we can control physics to our liking, there may be a few other possibilities, but we still seem privileged.
Lets look at these parameters:

*

*Number of space dimensions.

*Number of time dimensions.

*Dimension of the worldlines.

And here is the best guesses for different spacetimes:
3+1, 1D world lines. We are here, with 1D worldlines tracing out curves in 3+1 spacetime.
1+3. We are also here. There is no way to differentiate between $m+n$ and $n+m$.
4+0 and 2+2, 1D world lines. These allow closed timelike curves. Causality feels very important in terms of avoiding all sorts of strange paradoxes, so it seems hard for sentient life to exist in a universe that doesn't enforce causality. Both of these also have issues with stability since it is possible to generate an arbitrary amount of mass or energy from nothing. Nevertheless, Greg Egan has explored both of these in his novels.
1+1, 2+1. Planets don't have gravity in 1+1 or 2+1, but 2+1 can still have a big bang with cosmic inflation. Perhaps life floats freely in space filled with gas and dust? How would two neuron axons cross without mixing up the signals? Conways life can do so with timing, i.e. using "stoplights" at intersections. However, having evolution do so at the scale of intelligent life could be insurmountable; 2D space has low fertility so to speak.
4+1 (and 5+1, etc). In 3+1 forces would follow inverse cube law. This makes orbits unstable. Electron orbitals also are unstable; this is called the falling to center problem. But with extensive tweaking of forces and elementary particle masses it is possible to mix attractive and repulsive Yukawah potentials at various scales to allow stable atoms forming solids and liquids on a stable planet/star. So stability is not insurmountable like the 4+0 case, but achieving it sacrifices parsimony.
3.5+1: Are fractal dimensions possible? This has been explored for quantum field theory. For 3.5 spatial dimensions you would have inverse 2.5 law instead of inverse square law. Orbitals would be stable (they are stable for anything below inverse cube). The surface of planets would be 2.5 dimensional: with $n$ buildings within 1 km you could expect $4\sqrt{2} n = 5.66n$ within 2 km. There is no need to cross streets for d=2.5. This is all wonderful, except that non-integer dimensions would (I think) make space itself fractal: there would be "bubbles" of "non-space" at all length scales. This would prevent momentum from existing: any travelling wave immediately "hits" these bubbles and scatters in all directions. You couldn't throw a ball, shine a laser or see distant objects, get caught in a cyclone, or even have sound. Light diffuses instead of propigates, illuminating both the "day" and "night" side of your planet almost equally. Away from the bubbles are regions (on all scales) where space is "denser". City-scale "dense zones" are prime real-estate since you can fit more buildings within a 1km distance. Planets would be "glued" to large dense zones where more mass can be compressed into less "distance". Put your brain in a head-sized dense zones and your neurons pack more tightly; in general anything that moved would have to keep reconfiguring itself as the space it was in changed. The lack of momentum is bad for fertility and the difficulty of incorporating general relativity raises issues of parsimony.
3&4+1: One could have a semi-compact dimension along with 3 non-compact dimensions. Suppose the extra dimension was 1000km long. Particles moving in the 4th dimension would "wrap around" and return to the origin after travelling 1000km. Forces such as gravity are inverse-square for inter-planetary scales and higher but transition to  inverse-cube at shorter distances. Matter would need a Yukawah-like stabilization. Keeping the extra dimension at an "interesting" scale takes a huge parsimony cost.
3+2, 2D worldlines: Instead of worldlines, what about world planes? There would be 3 spatial dimensions left over for 3+2. This exotic-sounding situation may not actually be distinguishable from our own universe. Consider a classical, Newtonian test particle in a gravity well. At a point in time there would be two 3D velocity vectors: $v_{1}(t_a,t_b) = dx/dt_1, v_{2}(t_a, t_b) = dx/dt_2$. You also have gravitational forces: $\ddot v_1, \ddot v_2$. Neither the two forces nor the two velocities need be the same. However, you can choose a time direction: $t=\alpha t_1+\beta t_2$ and then evolve the dynamics of the system along that time direction. Doing so would be no different than having a single time direction. Timepoints outside of the timeline are in parallel universes and what happens there does not affect the timeline itself. I am 99% sure this argument would generalize to relativity and quantum field theory and we will get 3+2(2D) = 4+1(1D).
In summary: There are several fundamental "privileges". Causality allows "people" to have a "history" that is safe from paradoxes. Stability prevents the highest entropy state from being reached instantly. Life then steps in and moves things toward equilibrium, extracting energy in the process. Complex life is much easier if the physics have good fertility. Parsimony reduces the need for fine tuning, which means that it is more likely a "random" set of dimensionless physical constants allows life to exist. If we desire all these "privileges", (what is indistinguishable from) 3+1 is the winner. However, parsimony is not as necessary as the other criteria so 4+1, 5+1, etc are not ruled out either.
A: 3 dimensions of space are special, because this is the lowest number of dimensions, where a random walk doesn't return to it's origin with certainty (probability = 1), see
http://mathworld.wolfram.com/PolyasRandomWalkConstants.html
Similarly I think one time dimension is special, simply because less than one would mean no evolution at all, and more than one would lead to instabilities of all kinds. 
