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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?](http://math.stackexchange.com/q/61710/856)2012-06-17
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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 $C r^d$. (http://en.wikipedia.org/wiki/Hausdorff_dimension) Once you have defined this,2012-06-17
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    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.2012-06-17

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