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Urs Schreiber Says:

May 23rd, 2007 at 5:06 am

Concerning the issue of dimensionful and dimensionless “constants of nature” we once had a detailed discussion over at the n-Cafe: Dimensional Analysis and Dimensional Analysis and Coordinate Systems.

Maybe you find something of interest there. The discussion does address precisely the issue that you are struggling with here.

May 25th, 2007 at 2:58 am

But, as quoted above, I believe that “a unit is a convention.”

Exactly. And over there we like to say, equivalently: “A unit is a choice of isomorphism”.