One of the proponents of paraphyletic taxa, Richard Zander, published a paper in 2011 in which he introduced the issue as follows:
Structuralism (Overton 1975; Matthews 2001) was introduced by Ferdinand de Saussure (1857-1913) to linguistics (Balzer et al. 1987; Barry 2002), and spread as a postmodern "rejection of all things past" to architecture, art, anthropology, literary theory, psychology, psychoanalysis, mathematics, and, as is now evident, to systematics (Rieppel and Grande 1994: 249). It is intended to replace empiricism, which involves observation, data, and experiment leading to testable hypotheses, and empiricism rejects hidden causes and unnameable or unobservable entities. Structuralism emphasizes the importance of irreducible hidden structures that are essentially engraved in the ground of being, exist at all times as axiomatic, and are not properly dealt with by theories of changes or processes in nature (Piaget 1970; Gilbert et al. 1996; Brading and Landry 2006). Such ahistorical and synchronic structure is, in structuralism, the ultimate reality that contributes to the human-mediated appearance of what we observe. Although the word "structuralism" is sometimes used to refer to biological developmental design limitations on evolution, in this article it refers to apparently fundamental nesting patterns of similarity that imply evolutionary relationship.The point ultimately being that phylogenetic systematics is structuralist because it does not allow the hypothesis that one genus produces another, and because it assumes the existence of common ancestors in the past that Zander considers to be "hidden causes" and "unobservable entities". I, personally, consider that to be nonsense. As far as I can tell, phylogenetic systematists work with plenty of taxonomic, evolutionary and phylogenetic hypotheses, use evidence, do not reject the existence of the past, accept evolutionary change, and so on. For example, species diversify into more species and evolve new traits, and that is change. Arguing that it only counts if there are also shifts between supraspecific taxa at the same level seems silly. And hypothetical ancestors have to be assumed under any practice accepting the reality of common descent.
But note that structuralism is here described as something akin to a belief in Platonic ideal forms. An alternative definition is then mentioned at the end, and it is the one used by the first of the two bloggers I mentioned.
Here is biochemist Larry Moran:
the basic idea is that the form (structure) of modern organisms is a property of the laws of physics and chemistry and not something that evolution discovered. He would argue that if you replay the tape of life you will always get species that look pretty much like the species we see today because the basic forms (Baupläne) are the inevitable consequences of the underlying physics.So this is not about some underlying ideal forms, but structuralism is a view of evolution that considers the morphospace available to organisms to be extremely small and sharply restricted by what 'works' under the physical laws of our universe. Moran sits at the other end of the spectrum and has a view of evolution that considers morphospace to be wide open and the shapes taken by actual existing organisms to be mostly determined by chance and historical contingency.
To this replies PZ Myers:
no, that is not the structuralism I have studied. There is a grain of truth to it, in that structuralism does imply that there are physical/chemical constraints on form, but only the extremists would suggest that that means life on Mars would evolve to look like life on earth. (...) The core idea in structuralism is the importance of phenotypic interactions, rather than genetic control. Why do isolated cells tend to round up into spherical forms? It's not because there are genes instructing them to do so, but because that's a form that minimizes energy. Why do they flatten when they contact an adhesive surface? For the same reason. You don't need to even mention genes.Here structuralism describes the claim that not all morphology we see is explicitly coded for in genes. Instead some shapes just predictably happen because of the interactions of the organism with the physical environment - meaning that much less has to be specified in genes.
And oh dear, Wikipedia has got even more possible definitions. Following the link to the philosophy of science entry I do not, by the way, find a description that squares with Zander's as structuralism being dangerously non-empirical but instead one that claims "all aspects of reality are best understood in terms of empirical scientific constructs of entities and their relations" and that seems to be fairly irrelevant to the practice of classification anyway.
Surely most if not all phylogenetic systematists would not recognise themselves in Zander's definition of the word, and surely Moran's and Myer's definitions are orthogonal to questions of principles of classification. But it would be nice to sort this out a bit, if only so I can understand what exactly complete strangers are certain I believe.