Why is the meter considered a basic SI unit if its definition depends on the second?

Within the SI system

Let’s first assume the SI system as given. Within this system, the characterising feature of base units is that they are arithmetically independent of each other, i.e., one cannot be obtained from the other using arithmetic operations and real numbers. For example, the hertz arithmetically depends on the second: $1\,\text{Hz} = 1\,\text{s}^{-1}$. By contrast, you cannot write down an equation that has $1\,\text{m}$ on the left-hand side and something that does not explicitly or implicitly contain the unit $\text{m}$ on the right-hand side – the metre is arithmetically independent from the other SI base units.

Choice of the unit system 1: Keeping arithmetics

Given some arithmetic relationship between units, what units you choose as base units is arbitrary, as long as they are arithmetically independent in your system. E.g., if you take the SI system and replace the ampère with the coulomb as a base unit, you still end up with a valid unit system. However, you cannot just add the coulomb to the set of base units and still obtain a valid unit system, as it would then be arithmetically dependent on the ampère and the second. The SI system using the ampère instead of the coulomb as a base unit is due to history.

Choice of the unit system 2: Different arithmetics

What units are arithmetically dependent (and thus the number of base units) is historically grown as well, based on what were well-established experimental facts. A few examples to illustrate this:

• Had the finite speed of light been a pervasive phenomenon to mankind since the dawn of time, we might have incorporated this strict relationship in our thinking and unit system, always equating a length with the time it takes light to travel that length and never using separate units for length and time.

• If the relationship between charge, current, and time were not so straightforward, we might as well have defined arithmetically independent units for each of them and later found the empirical relationship:

  $$\text{[unit of current]} = k \frac{\text{[unit of charge]}}{\text{[unit of time]}},$$

  where $k$ is some constant. Instead we chose our units such that $k=1$, establishing an arithmetic relationship.

• On practically relevant scales, the equality of gravitational and inertial mass is very well established experimentally and a pervasive phenomenon. Therefore we use the same unit to measure both inertial and gravitational mass. Should it turn out that there is a difference between the two, the SI unit system (and any other current unit system) would not be well equipped to describe the pertaining phenomena. If there were a clear difference between the two, we would probably not use the same unit for them.

• In many areas of physics, natural unit systems are used that equate natural constants (such as $c, \hbar, …$) to 1. You can view this as a notational parsimony: If writer and reader know what quantity is measured and what unit system is used, there is no need to write down the quantity. You can also view this as introducing new arithmetical equivalences and thus reducing the number of base units in the unit system.

You are confusing two concepts which are actually independent: that of being a base unit and that of the unit definition.

The starting point is that of system of quantities, which is a set of equations (e.g., Newton's law, Maxwell's equations in the rationalized form) specifying the relationships between all the quantities of interest.

Among all the quantities of a system, one can conventionally choose a set of base quantities with the property that each quantity of this set cannot be expressed as a function of the other base quantities, only. That is, I cannot express a length as a function of time, only.

A base unit is then a unit conventionally chosen for a base quantity. The independence between the base units implies that I cannot write

$$\mathrm{m} = f(\mathrm{s}),$$

with $f$ containing only dimensionless constants. However, there can surely be relationships involving the metre, the second and other quantities with dimension, and you can actually use those relationships to define a base unit.

Therefore, the definition of a base unit can depend on other base units, if such a definition is convenient for an accurate realization of the unit, but the definition of a base unit cannot depend on other base units, only.

Remark. Note that the way the unit metre is defined involves a shift of paradigm with respect to the classical way of defining units. This shift of paradigm will become standard in the upcoming revision of the SI (2018).

In the classical paradigm, base units are defined independently from the fundamental constants. If $K$ is a fundamental constant and $[K]$ is its, possibly derived, unit, then the numerical value of $K$ is determined through an experiment as

$$\{K\} = \frac{K}{[K]}.$$

In the forthcoming revision, all base units will be defined in a way similar to that of the metre, so that seven fundamental constant will have an exact numerical value. If $K$ is a fundamental constant, its numerical value $\{K\}$ is set by definition and $[K]$ is indirectly obtained as

$$[K] = \frac{K}{\{K\}}.$$

The metre depends not only on the definition of the second, but also on the definition of the speed of light as the quote says.

• You could define the second and the metre-per-second (maybe give it a new name in that case so it doesn't "sound" derived) as SI base units and the metre as a derived unit, or
• you could define the metre and the second as base units and the metre-per-second as the derived one.

In any case you'll need two definitions because you want to cover both the dimension of time and that of space.

The choice might be more historical than fundamental (the metre was first - in the original definition the length of a (more or less arbitrarily chosen) specific platinum bar stored in Paris, France just like the kilogram.)

We might all be able to agree that the most fundamental base units would be the metre, second, Coulomb, gram, number-of etc. But as you state yourself, the Coulomb is not chosen to define charge, but rather Ampere defines current, the gram is not defining mass, but rather kilogram is, and the amount of substance is not a simple number-count, but rather defined with the mole.

The SI system is weighed down by convention, tradition and history in this matter. It's point is not really to define things the most fundamental way - rather exhaustively (and of course precisely).