There’s a long contested divide between “hard” and “soft” sciences. For the most part, this divide was always artificial and has become increasingly obviously baseless thanks largely to the interdisciplinary nature of modern fields in the sciences (and other academic fields). For example, some neuroscientists spent their undergraduate years and some of their graduate career taking courses in e.g., biological physics, electromagnetism, differential equations, systems modelling, and other topics associated with “hard” sciences. Others spent their undergraduate years taking courses in psychology, struggled through the one or two requisite courses in biological psychology, got as far in mathematics as a graduate “multivariate statistics” course that was much more elementary than undergraduate courses in statistics and probability taken by mathematics majors, and are used to reading particular papers in journals that (like many journals in the social & behavioral sciences) are designed to provide as much information as is possible without describing the actual research methods. Such journals usually include an abstract, a long introduction which describes past research and the motivation for the current research, followed by the methods section and a technical “version” of the results, and then back to a much more readable style with a non-technical version of the results (“discussion”) often accompanied by an even more simplified presentation of the findings in a “conclusion” section. That way, most researchers can read the abstract and conclusion, the researchers that are building upon the study can read these as well as the introduction and discussion sections, and everybody can skip the technical “mathy” stuff in the research methods and results sections (which are often highly formulaic anyway; for example, fMRI studies frequently include information about the fMRI machine at a level of technological complexity far beyond almost all readers, but don’t include information like the actual data they ran some statistical analysis/analyses on and other much more relevant information).

As another example, most statisticians don’t have degrees in statistics, but rather in fields such as psychology, engineering, economics, etc. In fact, much of modern statistics developed from the work of eugenicists, behavioral scientists, and a worker at a Guinness Brewery (William S. Gosset).

That said, to pretend that one can get away with the kind of “garbage in, garbage out” statistical methods in the social & psychological sciences if one’s field is machine learning or involves statistical mechanics, or that those in the so-called “hard sciences” can publish findings about possibly non-existent and certainly constructed concepts (e.g., intelligence, religiosity, and personality traits in general) that relies on measurements which don’t actually qualify as measurements according to measurement theory as is done constantly in the so-called “soft sciences”, is a fantasy. Unfortunately, the effect of the increasingly interdisciplinary nature of the sciences on the always fuzzy and at least fairly non-existent line between hard and soft sciences has mostly resulted in those who have little background in STEM entering fields which should require them too. I decided a quick, grossly simplified, and rather distorted contrast between two research papers could serve as a foundation to explore the far more nuanced nature of what to me isn’t a matter of “hard” vs. “soft” science but a more general problem: a vast number of researchers using technology and mathematics they don’t adequately understand because the universities they attended opted not to teach them (and usually actively dissuaded them from learning by requiring them to take courses in statistics that actually teach them to associate particular research questions with particular statistical tests and how to enter data into SPSS in order to run those tests without understanding the underlying logic/mathematics).

Here’s the abstract from a paper (“The Importance of Perceived Care and Connectedness with Friends and Parents for Adolescent Social Anxiety”) published in a leading sociology journal:

“Nonclinical social anxiety in adolescence can be highly problematic, as it likely affects current and especially new social interactions. Relationships with significant others, such as close friends, mothers, and fathers, could aid socially anxious adolescents’ participation in social situations, thereby helping reduce feelings of social anxiety. We examined whether making friends as well as high friendship quality help reduce social anxiety over time, and whether friends’, mothers’, and fathers’ care interact in reducing social anxiety. Using longitudinal data from 2,194 participants in a social network (48% girls; Mage = 13.58) followed for 3 years, we estimated friendship selection and influence processes via a continuous time-modeling approach using SIENA. We controlled for the effects of depressive symptoms, self-esteem, gender, age, and family structure. Our findings suggest that perceived care by friends mediated the effect of making friends on social anxiety. Perceptions of mother and father, as well as friend care and connectedness, respectively, did not interact in decreasing social anxiety. Nonetheless, care and connectedness with mothers, fathers, and friends jointly predicted decreases in social anxiety. Caring relationships with friends and parents each play a role in mutually protecting early adolescents against increasing in social anxiety over time.”

Here’s the abstract from a recent paper (“Emergence of Massless Dirac Fermions in Graphene’s Hofstadter Butterfly at Switches of the Quantum Hall Phase Connectivity”) in a leading physics journal:

“The fractal spectrum of magnetic minibands (Hofstadter butterfly), induced by the moiré superlattice of graphene on a hexagonal crystal substrate, is known to exhibit gapped Dirac cones. We show that the gap can be closed by slightly misaligning the substrate, producing a hierarchy of conical singularities (Dirac points) in the band structure at rational values Φ=(p/q)(h/e) of the magnetic flux per supercell. Each Dirac point signals a switch of the topological quantum number in the connected component of the quantum Hall phase diagram. Model calculations reveal the scale-invariant conductivity σ=2qe 2 /πh and Klein tunneling associated with massless Dirac fermions at these connectivity switches.”

And from a leading mathematics journal we have “Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set”:

“A fundamental theme in holomorphic dynamics is that the local geometry of parameter space (e.g. the Mandelbrot set) near a parameter reflects the geometry of the Julia set, hence ultimately the dynamical properties, of the corresponding dynamical system. We establish a new instance of this phenomenon in terms of entropy.

Indeed, we prove an “entropy formula” relating the entropy of a polynomial restricted to its Hubbard tree to the Hausdorff dimension of the set of rays landing on the corresponding vein in the Mandelbrot set. The results contribute to the recent program of W. Thurston of understanding the geometry of the Mandelbrot set via the core entropy.”

What do they all have in common? They deal directly or indirectly with connectedness. However, connectedness in the 2nd two are intimately connected, and both are wholly disconnected to the connectedness of the first.