How does a mechanism reduce 'the number of independent assumptions we need to make'?

The book only italicised; I added the majuscules.
Source: pp 226-227, Think: A Compelling Introduction to Philosophy (1 ed, 1999) by Simon Blackburn

[...] As with Boyle's law, we can say that while it is all we have got, we know something about the system. But we do not really understand WHY it is behaving as it does. Why should pressure vary inversely with volume? If it always does, why does it always do so? And why should constancy of temperature be important?
These questions were answered by providing a model in a more robust sense. The kinetic theory of gases sees gases as volumes of molecules in motion. Pressure is the result of the impact of these molecules on the walls of the container. The molecules speed up with increased temperature. Once a gas is seen like this, we have a mechanism, and given suitable assumptions, the empirical laws such as Boyle's law can be derived from the nature of the mechanism.
Finding a mechanism does not bypass the problem of induction. The continued uniform behaviour of items in a mechanism is a projection or extrapolation from what we have found so far, just as much as anything else. But it reduces the number of independent assumptions we need to make. A few stable features of things, and reliable interactions between them, might explain others. If we take the stable features for granted, we can explain the others in terms of them. These represent the EXPLANATORY and SIMPLIFYING ideals of science.

Inexperienced in the natural sciences, I do not understand the bolded. Suppose that as one equation, Boyle's Law (hereafter BL) is regarded as one independent assumption. Then any reference or use of BL means BL assumed true.
So how does a mechanism reduce 'the number of independent assumptions we need to make'?

• You may be touching on the problem of regress in rule following, but I can not quite tell. Like following a rule does not amount to interpreting its meaning every time it is followed applying a law does not amount to assuming it true every time it is applied, they are simply followed or applied, habitually. The assuming occurs only once. – Conifold Jan 31 '16 at 20:36

Consider we had 4 separate things, A B C and D. We do some experimentation on three characteristics we are interested in, x, y, and z. We find the physical evidence leads us to the following independent assumptions:

A exhibits x         A exhibits y         A exhibit z
B exhibits x         B exhibits y         B exhibit z
C exhibits x         C exhibits y         C exhibit z
D exhibits x         D exhibits y         D exhibit z

Now if we believe there is some mechanism M, which causes some things to behave in a certain way, such as exhibiting x y and z, we might work from the following independent assumptions:

All things operating via mechanism M exhibit x y and z.
A operates via mechanism M
B operates via mechanism M
C operates via mechanism M
D operates via mechanism M

We have now reduced the number of independent assumptions from 12 to 5, by using a mechanism to make each assumption "do more."

In the gas laws like Boyle's law, there is an assumption that all gas particles operate via the same mechanism, the behavior of gas particles. This lets us make a handful of assumptions about how gas particles behave, rather than having to worry about how each of the trillions of gas particles might have their own behavior.

If you make a bunch of observations about particle behavior and explain them with laws, and then observe gas behavior and come up with Boyle's Law, and then realize that Boyle's law is just a consequence of the particle laws, you are no longer assuming Boyle's law. You can still use it, but since it's a consequence of other laws (which you are assuming, and had to the whole time, because you needed to explain particles too, not just gases), it's not an independent assumption.

As another example, you could have a "rainbow law" about when and how rainbows appear. But it turns out that our knowledge of optics (regarding how light behaves when it passes through different materials) tells us that light passing through small droplets of water of roughly the right size must generate a rainbow. So you don't need to assume the rainbow law(s).

Note that Blackburn states "empirical laws such as Boyle's law can be derived from the nature of the mechanism".

Thus, molecular theory suffices to explain and replace a number of physical laws and theories of which Boyle's law is just one example; replacing them all with a single theory. In this way, it "reduces the number of independent assumptions we need to make" by making independent laws/theories derivable from a single theory or mechanism.

Perhaps the best example of such a reduction could be found in the successful formulation "theory of everything". Such a theory would replace our existing theories of the four forces with a single theory.

The quoted passage explains the principle to unify several different explanations by one single explanation. Often the concepts of a certain explanation can be reduced to the concepts and laws of a more fundamental explanation. This principle is named reduction of theories.

The example is to reduce the concepts pressure and temperature from Boyle's law of thermodynamics to the concepts energy and impluse of molecules and to apply Newtonian mechanics. Hence to reduce Boyle's law to mechanics, or more general: To reduce themodynamics to statistical mechanics based on Newtonian mechanics.

It is always a big step in science when seemingly different phenomena can be understood and explained as different aspects of one single phenomenon.

Examples: Magnetism and electrostatics are unified by electrodynamics, optics becomes part of Maxwell electrodynamics, mass and energy are equivalent according to E = m*c**2, weak and electrodynamic interaction are united to electroweak interaction, time and space are combined to spacetime, etc.