There are several books on methodology that I should read to get to the bottom of this. However, after reading everyone’s replies, I would like to attempt to boil it down this way (a certain theme is repeated for clarity):
Rothbard points out the difference between human action, of which we do know the underlying cause (motivated man), and the rest of nature, of which we cannot know the underlying cause, because nature (the universe) is unmotivated.
In nature, we only see “facts”, perhaps Rothbard means “effects” (?). For example, given we can only see the “effect” of gravity, then Fg = Gm1m2/r^2 is the appropriate use of symbology to explain the effect we are seeing. Regarding Rothbard’s use of the term “meaningless”, what meaning does the above equation have without reference to gravity itself? It has “operational meaning”, but not meaning within itself.
And, as Rothbard explains, a mathematical expression will be modified upon receipt of additional empirical information, such as the discovery of general relativity for example, which requires a new set of symbols to explain more generally the effects we are seeing. We can never know the underlying causes in nature, only the effects. We can only refine our explanation of these effects, with increasing precision, but we can never get to the cause, because, nature is unmotivated. Think about the mainstream approach, and their attempt to treat human action like “natural” phenomenon; they are constantly using empirical data to compare to their equations, and their equations will be revised if needed to determine a more “general” equation. The cause (human action) is right in front of them but they choose to ignore the implications of this.
It is inappropriate to use mathematical logic in praxeology to describe human action because, for example, the statement “humans act” contains Rothbard’s “meaning” within itself. Reducing this to a mathematical form could be done, but I believe Rothbard is saying it would be pointless to do so since the statement has already been reduced to its most elemental form (Occam’s Razor comes into play) and the statement of “verbal logic” describes the cause itself with absolute certainty. In other words, when we say “humans act”, we are done. In the study of human action, there is no more precision to be gained that can get us closer to the “cause” or “root” of the matter, as general relativity did with Newton’s Fg=Gm1m2/r^2.
And today, quantum physics tries to unify the “very large” (general relativity) with the “very small” (the quantum), i.e., they are trying to come up with a more “general” theory to explain the “effects” observed in the universe. Mathematical methodology is superior for this type of analysis, but they will never get to the cause, for the reasons explained above.
My “boiled down” explanation was longer than I intended, but it did help me get these ideas out of my mind and into written form. Thanks for your patience.