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I have studied linear algebra and commutative algebra, there are two kinds of dimension there : the vector space's dimension and the Krull dimension.

Also, in physics, dimension is also a very intuitive concepts.

My question is : What is the nearest mathematical definition of dimension to the physical one ?

In the wikipedia page, they also list some mathematical types of dimension : Dimension

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Related (possible duplicate): What is a physical “dimension” - in the sense of “dimensional” analysis? –  Rahul Jun 17 '12 at 2:13
    
Thank you for that. –  N.Z.K Jun 17 '12 at 2:27
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One mathematical definition of dimension nearest to physical dimension is the Hausdorff dimension. It is consistent with the physical dimension in the sense it gives the dimension of the Euclidean space $\mathbb{R}^n$ to be $n$. The essential idea behind Hausdorff dimension is that you define the measure of a $d$-dimensional ball as $C r^d$. (en.wikipedia.org/wiki/Hausdorff_dimension) Once you have defined this, –  user17762 Jun 17 '12 at 2:46
    
you then try to fill your set with these $d$-dimensional balls. For instance, if you try to fill a rectangle with $d$ dimensional balls, you will get the measure of the rectangle to be $\infty$, for $d<2$ and $0$ for $d > 2$. Hence, the dimension of the rectangle is $2$. Similarly, for a cube you will find the the measure is $\infty$ for $d<3$ and is $0$ for $d>3$. Hence, the dimension of the cube is $3$. It also has its own set of wierdities. The Hausdorff dimension of Cantor set on the unit interval is $\ln(2)/\ln(3)$. You should look at the wiki link for a more formal definition. –  user17762 Jun 17 '12 at 2:48
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3 Answers

The answers and comments so far indicate that we are talking about two completely different kinds of "dimension" here:

  1. There is the notion of dimension of a real vector space $V$ or manifold $M$. This is an integer $d\geq0$ and has the same meaning in physics as in mathematics. The intuitive physical interpretation of $d$ is the "number of degrees of freedom" in the physical system under study. – In a space of dimension $d$ (infinitesimal) volumes scale like $\lambda^d$ under a linear scaling by a factor $\lambda>0$. This property can be used to envisage sets $S\subset{\mathbb R}^d$ whose "volume" scales like $\lambda^\alpha$ with a noninteger $\alpha\leq d$. This value $\alpha$ is called the Hausdorff dimension of $S$; but this is a dimension in a measure theoretical, not in a topological sense.

  2. Physical quantities have a "dimension" of length, time, degree Kelvin, etc. This dimension is not a number, but a quality. It's up to a physics member of the community to give an exact definition. Tentatively I would say that (at least in the realm of mechanics) the set of physical dimensions is the multiplicative abelian group generated by the three elements $L$ (for length), $T$ (for time) and $M$ (for mass). To any physical quantity an element of this group is associated. Two physical quantities can be sensefully compared or added only if the associated dimensions are the same. Furthermore, the dimension of a quantity determines how the numerical value of this quantity changes when the base units for $L$, $T$ and $M$ are changed.

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I think the appropriate way to feel about dimension is the analogous of degree of freedom. A physical system having $N$ degree of freedom usually can be described in phase space by a $N$ dimensional trajectory. Differential Physical systems can be described by different systems of equations, but we can always 'count' the degree of freedom in some sense, even if some occasions it is infinite.

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The relevant notion of dimension to physics as I understand it is the dimension of a manifold (e.g. a (3+1)-dimensional Lorentzian manifold). If the manifold is smooth (which is, as I understand it, the case in physics) then the dimension of a manifold agrees with the dimension of its tangent spaces, so morally it is the linear-algebraic notion that is relevant here.

(Of course, in nice cases the Krull dimension ought to also equal the dimension of Zariski tangent spaces. But as far as I know, the universe is not profitably modeled as an algebraic variety.)

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