We know that the set of fundamental and derived physical units can be structured as a vector space over the rational numbers. In the International System of Units the dimension of this space is $7$ ( the seven foundamental units form a basis). This number has some special physical significance or it is simply a result of historical events and practical conveniences?
$\begingroup$
$\endgroup$
2
-
1$\begingroup$ In order to have the structure of a vector space, it should be possible to provide a meaningful sum of vectors, including the basis vectors. How would you define it? $\endgroup$– GiorgioP-DoomsdayClockIsAt-90Feb 14, 2019 at 21:02
-
$\begingroup$ @GiorgioP: You can see : en.wikipedia.org/wiki/Buckingham_π_theorem#Proof $\endgroup$– Emilio NovatiFeb 15, 2019 at 14:06
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
This number of “independent dimensions” is physically meaningless and just a historical convention. For example, we don’t have to treat distance and time as independent dimensions. We could decide to measure both in seconds, where “1 second of distance” is the distance light travels in vacuum in one second. Physicists do this kind of thing all the time when they use “natural units” with $c=1$ and/or $\hbar=1$ and/or $G=1$ and/or $k=1$, etc.
$\begingroup$
$\endgroup$
There is no special significance to the number 7 of SI units. They are indeed a matter of practical convenience.
For example, we could easily live in physics without the unit ‘mol’, but it was added to the SI units, because it is convenient in chemistry.
Another example: We could introduce entropy as a dimensionless quantity, and thereby measure temperature in units of energy getting rid of the ‘Kelvin’.
All this would change nothing fundamental in the way we describe physics today.
