Rationale for the seven base dimensions

diman is based on the seven base dimensions for the quantities; mass, length, time, electric current, thermodynamic temperature, amount of substance and luminous intensity. Therefore, one may think of these seven names as the dimension names. But, they are quantities not dimensions. So,

What is a dimension?

This is best defined by McNish as

It is simply a tag we attach to a quantity in an equation expressing some physical law… [D]imensions are only symbols of an elementary algebra, involving neither addition nor subtraction. To ask what are the true or natural dimensions of a quantity makes no more sense than to ask what is the true or natural word for goldfish. [McNish1957]

Dimension vs Quantity Units

Since dimensions are represented by symbols, they follow the rule of algebra. However, units (eg. meter is a unit for the quanity length) are not bound by the rules of an algebra. This is an important concept for dimensional analysis.

Due to this reason we can have a minimum number of dimension to create a dimensional system such that we can relate arbitrarily chosen chosen into one coherent system. But, limiting the units to certain "absolute" units from which other units are derived would be problem specific.

If one could just assign a magnitude to a single quantity and consider this as the "absolute" unit we might be able to derive a system of units (consistent around this absolute unit) to measure all other quantities by arbitrarily assigning values to several physical constants. But, in practice measuring the units for many of the quantities in this system will require difficult experiments that cannot be performed accurately. McNish says,

The magnitudes of quantities are not determined by nature, but by the units we have arbitrarily chosen for our measurement system. [McNish1957]

Why seven elemental dimensions is a good start?

McNish’s argument for it is

[F]or the simplest mechanical quantities we need four dimensions to avoid ambiguities, five, if we include solid angle; that if we set one of these quantities equal to unity we can get along with four. But having one quantity equal to unity we cannot set another equal to unity without creating an ambiguity. Thus, unity itself becomes like a dimension, so again we may say we have five. Heat and electromagnetism add at least two more required dimensions. So I might venture to say that we should have seven elemental dimensions, at least, but I do not know, because I do not comprehend all of physics. One, of course, may get along with fewer dimensions if he will tolerate some ambiguities. [McNish1957]

The seven base dimensions for diman

Quantity	Dimension (Notation)
mass	[M]
length	[L]
time	[T]
electric current	[A]
thermodynamic temperature	[K]
amount of substance	[mol]
luminous intensity	[cd]

Quantity

Dimension (Notation)

mass

[M]

length

[L]

time

[T]

electric current

[A]

thermodynamic temperature

[K]

amount of substance

[mol]

luminous intensity

[cd]

The seven base dimensions for diman is based on the International System of Units, SI units [BIPM2020].

A part of the secret of analysis is the art of using notation well. - Leibniz

This is implemented in diman as

=> (pprint base_dimensions)
[{:quantity "mass", :dimension "[M]"}
{:quantity "length", :dimension "[L]"}
{:quantity "time", :dimension "[T]"}
{:quantity "electric current", :dimension "[A]"}
{:quantity "thermodynamic temperature", :dimension "[K]"}
{:quantity "luminous intensity", :dimension "[cd]"}
{:quantity "amount of substance", :dimension "[mol]"}]

Derived Dimensions

The standard_formula contains some well-known dimensions derived from the base_dimensions.

=> (pprint standard_formula)
[{:quantity "volume", :dimension "[L^(3)]"}
{:quantity "frequency", :dimension "[T^(-1)]"}
{:quantity "velocity", :dimension "[L^(1)*T^(-1)]"}
{:quantity "acceleration", :dimension "[L^(1)*T^(-2)]"}
{:quantity "force", :dimension "[M^(1)*L^(1)*T^(-2)]"}
{:quantity "mass density", :dimension "[M^(1)*L^(-3)]"}
{:quantity "energy", :dimension "[M^(1)*L^(2)*T^(-2)]"}
{:quantity "work", :dimension "[M^(1)*L^(2)*T^(-2)]"}
{:quantity "amount of heat", :dimension "[M^(1)*L^(2)*T^(-2)]"}
{:quantity "pressure", :dimension "[M^(1)*L^(-1)*T^(-2)]"}
{:quantity "stress", :dimension "[M^(1)*L^(-1)*T^(-2)]"}
{:quantity "catalytic activity", :dimension "[mol^(1)*T^(-1)]"}
{:quantity "charge", :dimension "[A^(1)*T^(1)]"}
{:quantity "capacitance", :dimension "[M^(-1)*L^(-2)*T^(4)*A^(2)]"}
{:quantity "inductance", :dimension "[M^(1)*L^(2)*T^(-2)*A^(-2)]"}
{:quantity "resistance", :dimension "[M^(1)*L^(2)*T^(-3)*A^(-2)]"}
{:quantity "conductance", :dimension "[M^(-1)*L^(-2)*T^(3)*A^(2)]"}
{:quantity "magnetic flux density", :dimension "[M^(1)*T^(-2)*A^(-1)]"}
{:quantity "electromotive force", :dimension "[M^(1)*L^(2)*T^(-3)*A^(-1)]"}
{:quantity "power", :dimension "[M^(1)*L^(2)*T^(-3)]"}
{:quantity "magnetic flux", :dimension "[M^(1)*L^(2)*T^(-2)*A^(-1)]"}]

Notice that the derived dimensions are in a sense the dimensional formula for respective quantity. Hence, the name standard_formula.

References

BIPM (2020). Base unit definitions. Retrieved from the Base units page.

McNish, A. G. (1957, April 1). Dimensions units and standards. Physics Today, 10(4), 19. 10.1063/1.3060330