It is remarkable that Tom Lehrer should be able to revive the positivism debate with a song! Click here to read the hundreds of comments in YouTube. It's only apparently about abuse of mathematics in sociology. The deeper question is if any science is possible without mathematics. The Lehrers of this world say no, "but they all are soreheads"...
The question if sociology is a science or not, compared to natural science, using math as the measuring rod, should at least, if not only, be answered with reference to Aristotle's concept of 'scientific knowledge' in Analytica Posteriora. Why? Because Aristotle's concept is the original concept of scientific knowledge.
According to the Grand Master of Science, Aristotle, to have scientific knowledge is to be able to produce a valid syllogism the conclusion of which is universal scientific knowledge according to the schema: All M are P; All S are M: All S are P. In this schema the scientific knowledge is assumed to be carried by the form "All S are P".
According to Sir Karl Popper falsifiability is the key to scientific knowledge. Read his famous "The Logic of Scientific Discovery" ("Logic der Forschung", Vienna 1934) about this. You will see that his knowledge of Aristotles' theory of the syllogism was less than perfect. Aristotle had already taken account of the logic of falsifiability with the rule that a universal statement is falsifiable by its particular counterpart, "Some S are not P".
A simplified example from physics. All apples reasonably near Earth, (S), fall towards Earth (P), due to some specifiable causes (M): "All apples fall". This knowledge could be falsified exhibiting an apple that observably does not fall given such conditions and causes: "Some apples don't fall". Note that such an observation of an apple not falling would verify this particular statement. Note also that I here use the term "verify" in a simple and intuitively understandable sense, not in the over-philosophied sense attributed to it by Sir Popper.
Another example, this time from sociology. All human societies on Earth (S) depend, in an intuitively understandable and acceptable sense of the word "depend", on a mother tongue (P), due to some specifiable causes (M): "All human societies depend on mother tongues". Again this knowledge can be falsified logically exhibiting a human society that observably does not depend on a mother tongue given such conditions and causes: "Some human societies don't depend on mother tongues".
My argument presupposes a distinction between specific scientific knowledge in the form of conclusions of valid syllogisms, where math according to Aristotle's schema plays no role, and the gross, popular myth of what it means to be a "science". A science of something can only be the set of statements of scientific knowledge about that thing, and the members of this set must be the kind of conclusions referred to above. When in doubt, call the Aristotelian version of science "A-science" and all other versions "B-sciences".
With Gaileo the term "science" came to acquire the sense of "B-science", with mathematical descriptions of technological, physical experiments among the criteria, but seen from an Aristotelian point of view, Galileo's experiments were technological, not scientific. E.g., "Galileo proposed that a falling body would fall with a uniform acceleration, as long as the resistance of the medium through which it was falling remained negligible, or in the limiting case of its falling through a vacuum." (Citation from Wikipedia).
To observe and measure physically if bodies with different masses fall in the same way, no scientific theory is necessary. Galilean experiments are all about measurable observations and numeric descriptions. Note that I only say that his experiments were not scientific, and that therefore the corresponding mathematical descriptions were not scientific, always assuming Aristotle's definition of scientific knowledge and the viewpoint of A-science. The physical experiments were technological.
E.g., some physical experiments certainly made possible the description of some movement properties of some freely falling bodies. In each case the distance travelled by a body falling for time t approximated a number d that under ideal conditions in a medium that offers no resistance, assuming a so-called gravitational constant g, was 1/2gt^2.
But this equation is not scientific knowledge. It is a mathematical description. The scientific knowledge, assuming that it is deducible from antecedent universal premises, is expressed by the whole universal statement
All physical bodies fall according to the equation d = 1/2gt^2
in its logical or syllogistic context. The formulation of the mathematical equation, "d = 1/2gt^2" is only a part of the universal statement. It is a mathematical description which functions as an empirical point of reference for the universal theory "All physical bodies fall according to the equation..." This may be formulated through intuitive induction from some particular observations.
According to Aristotle, such a conclusion is logically indemonstrable unless another universal theory is assumed as its antecedent, but then the antecedent must be accepted as indemonstrable, or one gets enmeshed in an endless regress. Only when arriving at the position of the very conclusion of the syllogism, beginning with the first indemonstrable universal premise, does a mathematical formulation acquire its function as a description of the outcome of some technological experiment.
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