One of the things about the thermometer definition is it explicitly states how to pick out temperature without telling you anything about temperature. I think that's a feature..........In the case of counter-factuals, when we're talking about "heat is the motion of molecules" vs. "heat is a caloric substance that goes from one object to the other", then I think both must be picking out the same things in the case of the first part, but I'm not sure about the latter part still. — Moliere
The example of the thermometer may be a key into Kripke's necessary a posteriori.
We may know an effect without needing to know its cause
I observe the number on a thermometer change. I don't know what is causing the number to change, but I name whatever is causing the number to change as heat. The number isn't heat, but what is causing the number to change is heat.
Some hypothesise that what is causing the number to change is the motion of molecules, ie, heat is the motion of molecules. Others hypothesise that what is causing the number to change is caloric, ie, heat is caloric. If one of these hypothesise becomes with time self-evidently true, it becomes an axiom. Society then accepts as given the axiom that heat is the motion of molecules.
Both the motion of molecules and caloric are referring to the same thing, heat in the world, even if we never actually know what heat in the world is. In practice, we don't need to know what is causing the numbers to change, we don't need to know what heat in the world is, all we need to know is the effect of heat in the world, the numbers changing on the thermometer.
As long as we know the effect of heat, we don't need to know what heat is.
Kripke is using the word "heat" in two very different ways
We can have a particular sensation, which we name the sensation of heat. We name the cause of this particular sensation heat. The word heat is being used in two different ways, one as a name of an effect, the sensation of heat, and the other as the name of its cause, heat. The effect is very different from the cause, though they share the same name.
Kripke refers to the sensation of heat. Page 185: "There is a certain external phenomenon which we can sense by the sense of touch, and it produces a sensation which we call “the sensation of heat.”"
Kripke also refers to heat as the motion of molecules. Page 170: “Heat is the motion of molecules.”
Heat as the sensation of heat in the mind
Kripke discusses possible connections between the sensation of heat and its cause: i) "So, it might be thought, to imagine a situation in which heat would not have been the motion of molecules" ii) "the motion of molecules but in which such motion does not give us the sensation of heat" iii) "Martians, who do indeed get the very sensation that we call “the sensation of heat” when they feel some ice which has slow molecular motion, and who do not get a sensation of heat—in fact, maybe just the reverse".
Heat as the cause in the world of the sensation of heat in the mind
Kripke discusses possible causes of our sensation of heat: i) "First, imagine it inhabited by no creatures at all: then there is no one to feel any sensations of heat" ii) "the judgment that heat is the motion of molecules would have been false."
Rigid designators
The motion of molecules in the world is a rigid designator. As this lectern is made of wood , this lectern is necessarily made of wood, similarly, as these molecules are in motion they are necessarily in motion.
A sensation of heat in the mind is a rigid designator. As this particular sensation of heat in the mind is this particular sensation, it is necessarily this particular sensation, whatever it is named, in that it could have been named "heat", "cold", "apple" or "The Eiffel Tower".
Heat in the world as the cause of a sensation of heat in the mind is a rigid designator. As this lectern is made of wood , this lectern is necessarily made of wood, similarly heat in the world as the cause of a sensation of heat in the mind is necessarily the cause of the sensation of heat in the mind, whatever it is named.
Kripke concludes that heat is necessarily the motion of molecules
He wrote: page 187: "To state the view succinctly: we use both the terms ‘heat’ and ‘the motion of molecules’ as rigid designators for a certain external phenomenon. Since heat is in fact the motion of molecules, and the designators are rigid, by the argument I have given here, it is going to be necessary that heat is the motion of molecules."
There are two possible meanings to heat is necessarily the motion of molecules
The motion of molecules in the world has one possible meaning, although heat has two possible meanings.
Meaning one: Heat in the world is necessarily the motion of molecules in the world. There may be heat in the world, and there may be molecules in motion in the world. Both the heat in the world and molecules in motion in the world are rigid designators, but it doesn't of necessity follow that there is a link between them. For example, both "Nixon" and "Caesar" are rigid designators in all possible worlds, but there is no necessary link between them.
Meaning two: Heat in the mind is necessarily the motion of molecules in the world. The sensation of heat in the mind is necessarily caused by heat in the world, but as there is no necessary link between heat in the world and the motion of molecules in the world, there is no necessary link between the sensation of heat in the mind and the motion of molecules in the world.
Conclusion
Heat is not necessarily the motion of molecules.