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The intuitional definition for number of degrees of freedom is following: it is the minimal amount of numbers which allows us to describe the system's configuration correctly.

For example, for dot in our space it is enough to set 3 real numbers. But we know, that the line of real numbers is isomorphic to 3D space, so we could desribe the place of a dot using only one number with this isomorphism!

So, how does it comply with the fact, that a dot has 3 degrees of freedom?

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  • $\begingroup$ Related physics.stackexchange.com/q/72229 $\endgroup$ – leonbloy Feb 20 '19 at 19:21
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    $\begingroup$ There is no recursive isomorphic mapping from the line to 3d space, no algorithm to get one from the other. $\endgroup$ – Wolphram jonny Feb 20 '19 at 21:59
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There are lots of different kinds of "isomorphism," depending on what category we're talking about. When the line of real numbers and 3D space are both regarded as smooth manifolds in the usual way, they are not isomorphic to each other as smooth manifolds. In other words, regarded as objects in the category of smooth manifolds, they are not isomorphic to each other. In that category, "isomorphism" means "diffeomorphism." They're not isomorphic to each other in the category of topological spaces, either, at least not when they are equipped with their usual topologies. In that category, "isomorphism" means "homeomorphism."

When people count degrees of freedom, they're doing that counting in the context of whatever mathematical structures are important for the model's definition — which usually includes at least a topological structure so that continuity can be defined, and usually includes a smooth structure so that derivatives can be defined.

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I am not sure, but I suppose, that we, of course, could use this one number to describe our system, but it is very bad way to deal with system, cause the set of states of this system does not allow the vector space structure, so our theory becomes really unlinear.

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