Once again, this post got away from me, so I am dividing it into two parts.
As I mentioned in a recent previous post, I have just finished re-reading Language & Mind (L&M) and have been struck, once again, about how relevant much of the discussion is to current concerns. One topic, however, that does not get much play today, but is quite well developed in L&M is it’s discussion of Descartes’ very expansive conceptions of linguistic creativity and how it relates to the development of the generative program. The discussion is surprisingly complex and I would like to review its main themes here. This will reiterate some points made in earlier posts (here, here) but I hope it also deepens the discussion a bit.
Human linguistic creativity is front and center in L&M as it constitutes the central fact animating Chomsky’s proposal for Transformational Generative Grammar (TGG). The argument is that a TGG competence theory is a necessary part of any account of the obvious fact that humans regularly use language in novel ways. Here’s L&M (11-12):
…the normal use of language is innovative, in the sense of much of what we say in the course of normal use is entirely new, not a repetition of anything that we have heard before and not even similar in pattern - in any useful sense of the terms “similar” and “pattern” – to sentences or discourse that we have heard in the past. This is a truism, but an important one, often overlooked and not infrequently denied in the behaviorist period of linguistics…when it was almost universally claimed that a person’s knowledge of language is representable as a stored set of patterns, overlearned through constant repetition and detailed training, with innovation being at most a matter of “analogy.” The fact surely is, however, that the number of sentences in one’s native language that one will immediately understand with no feeling of difficulty or strangeness is astronomical; and that the number of patterns underlying our normal use of language and corresponding to meaningful and easily comprehensible sentences in our language is order of magnitudes greater than the number of seconds in a lifetime. It is in this sense the normal use of language is innovative.
There are several points worth highlighting in the above quote. First, note that normal use is “not even similar in pattern” to what we have heard before. In other words, linguistic competence is not an instance of pattern matching or recognition in any interesting sense of “pattern” or “matching.” Native speaker use extends both to novel sentences and to novel sentence patterns effortlessly. Why is this important?
IMO, one of the pitfalls of much work critical of GG is the assimilation of linguistic competence to a species of pattern matching. The idea is that a set of templates (i.e. in L&M terms: “a stored set of patterns”) combined with a large vocabulary can easily generate a large set of possible sentences in the sense of templates saturated by lexical items that fit.  Note, that such templates can be hierarchically organized and so display one of the properties of natural language Gs (i.e. hierarchical structures). Moreover, if the patterns are extractable from a subset of the relevant data then these patterns/templates can be used to project novel sentences. However, what the pattern matching conception of projection misses is that the patterns we find in Gs are not finite and the reason for this is that we can embed patterns within patterns within patterns within…you get the point. We can call the outputs of recursive rules “patterns” but this is misleading for once one sees that the patterns are endless, then Gs are not well conceived of as collections of patterns but collections of rules that generate patterns. And once one sees this, then the linguistic problem is (i) to describe these rules and their interactions and (ii) to further explain how these rules are acquired (i.e. not how the patterns are acquired).
The shift in perspective from patterns (and patternings in the data (see note 5)) to generative procedures and the (often very abstract) objects that they manipulate changes what the acquisition problem amounts to. One important implication of this shift of perspective is that scouring strings for patterns in the data (as many statistical learning systems like to do) is a waste of time because these systems are looking for the wrong things (at least in syntax). They are looking for patterns whereas they should be looking for rules. As the output of the “learning” has to be systems of rules, not systems of patterns, and as rules are, at best, implicit in patterns, not explicitly manifest by them, theories that don’t focus on rules are going to be of little linguistic interest.
Let me make this point another way: unboundedness implies novelty, but novelty can exist without unboundedness. The creativity issue relates to the accommodation of novel structures. This can occur even in small finite domains (e.g. loan words in phonology might be an example). Creativity implies projection/induction, which must specify a dimension of generalization along which inputs can be generalized so as to apply to instances beyond the input. This, btw, is universally acknowledged by anyone working on learning. Unboundedness makes projection a no-brainer. However, it also has a second important implication. It requires that the generalizations being made involve recursive rules. The unboundedness we find in syntax cannot be satisfied via pattern matching. It requires a specification of rules that can be repeatedly applied to create novel patterns. Thus, it is important to keep the issue of unboundedness separate from that of projection. What makes the unboundedness of syntax so important is that it requires that we move beyond the pattern-template-categorization conception of cognition.
Dare I add (more accurately, can I resist adding) that pattern matching is the flavor of choice for the Empricistically (E) inclined. Why? Well, as noted, everyone agrees that induction must allow generalization beyond the input data. Thus even Es endorse this for Es recognize that cognition involves projection beyond the input (i.e. “learning”). The question is the nature of this induction. Es like to think that learning is a function from input to patterns abstracted from the input, the input patterns being perceptually available in their patternings, albeit sometimes noisily. In other words, learning amounts to abstracting a finite set of patterns from the perceptual input and then creating new instances of those patterns by subbing novel atoms (e.g. lexical items) into the abstracted patterns. E research programs amount to finding ways to induce/abstract patterns/templates from the perceptual patternings in the data. The various statistical techniques Es explore are in service of finding these patterns in the (standardly, very noisy) input. Unboundedness implies that this kind of induction is, at best, incomplete. Or, more accurately, the observation that the number of patterns is unbounded implies that learning must involve more than pattern detection/abstraction. In domains where the number of patterns is effectively infinite, learning is a function from inputs to rules that generate patterns, not to patterns themselves. See link in note 6 for more discussion.
An aside: Most connectionist learners (and deep learners) are pattern matchers and, in light of the above, are simply “learning” the wrong things. No matter how many “patterns” the intermediate layers converge on from the (mega) data they are exposed to they will not settle on enough given that the number of patterns that human native speakers are competent in is effectively unbounded. Unless the intermediate layers acquire rules that can be recursively applied they have not acquired the right kinds of things and thus all of this modeling is irrelevant no matter how much of the data any given model covers.
Another aside: this point was made explicitly in the quote above but to no avail. As L&M notes critically (11): “it was almost universally claimed that a person’s knowledge of language is representable as a stored set of patterns, overlearned through constant repetition and detailed training.” Add some statistical massaging and a few neural nets and things have not changed much. The name of the inductive game in the E world is to look for perceptual available patterns in the signal, abstract them and use them to accommodate novelty. The unboundedness of linguistic patterns that L&M highlights implies that this learning strategy won’t suffice the language case, and this is a very important observation.
Ok, back to L&M
Second, the quote above notes that there is no useful sense of “analogy” that can get one from the specific patterns one might abstract from the perceptual data to the unbounded number of patterns with which native speakers display competence. In other words, “analogy” is not the secret sauce that gets one from input to rules So, when you hear someone talk about analogical processes reach for your favorite anti-BS device. If “analogy” is offered as part of any explanation of an inferential capacity you can be absolutely sure that no account is actually being offered. Simply put, unless the dimensions of analogy are explicitly specified the story being proffered is nothing but wind (in both the Ecclesiastes and the scatological sense of the term).
Third, the kind of infinity human linguistic creativity displays has a special character: it is a discrete infinity. L&M observes that human language (unlike animal communication systems) does not consist of a “fixed, finite number of linguistic dimensions, each of which is associated with a particular nonlinguistic dimension in such a way that selection of a point along the linguistic dimension determines and signals selection of a point along the associated nonlinguistic dimension” (69). So, for example, higher pitch or chirp being associated with greater intention to aggressively defend territory or the way that “readings of a speedometer can be said, with an obvious idealization, to be infinite in variety” (12).
L&M notes that these sorts of systems can be infinite, in the sense of containing “an indefinitely large range of potential signals.” However, in such cases the variation is “continuous” while human linguistic expression exploits “discrete” structures that can be used to “express indefinitely many new thoughts, intentions, feelings, and so on.” ‘New thoughts’ in the previous quote clearly meaning new kinds of thoughts (e.g. the signals are not all how fast the car is moving). As L&M makes clear, the difference between these two kinds of systems is “not one of “more” or “less,” but rather of an entirely different principle of organization,” one that does not work by “selecting a point along some linguistic dimension that signals a corresponding point along an associate nonlinguistic dimension.” (69-70).
In sum, human linguistic creativity implicates something like a TGG that pairs discrete hierarchical structures relevant to meanings with discrete hierarchical structures relevant to sounds and does so recursively. Anything that doesn’t do at least this is going to be linguistically irrelevant as it ignores the observable truism that humans are, as matter of course, capable of using an unbounded number of linguistic expressions effortlessly. Theories that fail to address this obvious fact are not wrong. They are irrelevant.
Is hierarchical recursion all that there is to linguistic creativity? No!! Chomsky makes a point of this in the preface to the enlarged edition of L&M. Linguistic creativity is NOT identical to the “recursive property in generative grammars” as interesting as such Gs evidently are (L&M: viii). To repeat, recursion is a necessary feature of any account aiming to account for linguistic creativity, BUT the Cartesian conception of linguistic creativity consists of far more than what even the most explanatorily adequate theory of grammar specifies. What more?
 This is not unique to the linguistic cognition. Lots of work in cog sci seems to identify higher cognition with categorization and pattern matching. One of the most important contributions of modern linguistics to cog sci has been to demonstrate that there is much more to cognition than this. In fact, the hard problems have less to do with pattern recognition than with pattern generation via rules of various sorts. See notes 5 and 6 for more off handed remarks of deep interest.
 I suspect that some partisans of Construction Grammar fall victim to the same misapprehension.
 Many cog-neuro types confuse hierarchy with recursion. A recent prominent example is in Frankland and Greene’s work on theta roles. See here for some discussion. Suffice it to say, that one can have hierarchy without recursion, and recursion without hierarchy in the derived objects that are generated. What makes linguistic objects distinctive is that they are the products of recursive processes that deliver hierarchically structured objects.
 Note that unbounded implies novelty, but novelty can exist without unboundedness. The creativity issue relates to easy handling of novel structures. This can occur even in small finite domains. Creativity implies projection, which must specify a dimension of generalization along which inputs can be extended to apply to instances beyond the input. Unboundedness makes projection a no-brainer. It further implies that the generalization involves recursive rules. Unboundedness cannot be pattern matching. It requires a specification of rules that can be repeatedly applied to create novel patterns. Thus, it is important to keep the issue of unboundedness separate from that of projection. What makes the unboundedness of syntax so important is that it requires that we move beyond the pattern-template-categorization conception of cognition.
 It is arguable that some rules are more manifest in the data that others are and so are more accessible to inductive procedures. Chomsky makes this distinction in L&M, contrasting surface structures which contains “formal properties that are explicit in the signal” to deep structure and transformations for which there is very little to no such information in the signal (L&M:19). For another discussion of this distinction see (here).
 Thus the hope of unearthing phrases via differential intra-phrase versus inter-phrase transition probabilities.
 We really should distinguish between ‘learning’ and ‘acquisition.’ We should reserve the first term for the pattern recognition variety and adopt the second for the induction to rules variety. Problems of the second type call for different tools/approaches than those in the first and calling both ‘learning’ merely obscures this fact and confuses matters.
 Although this is a sermon for another time, it is important to understand what a good model does: it characterizes the underlying mechanism. Good models model mechanism, not data. Data provides evidence for mechanism, and unless it does so, it is of little scientific interest. Thus, if a model identifies the wrong mechanism not matter how apparently successful in covering data, then it is the wrong model. Period. That’s one of the reasons connectionist models are of little interest, at least when it comes to syntactic matters.
I should add, that analogous creativity concerns drive Gallistel’s arguments against connectionist brain models. He notes that many animals display an effectively infinite variety of behaviors in specific domains (caching behavior in birds or dead reckoning in ants) and that these cannot be handled by connectionist devices that simply track the patterns attested. If Gallistel is right (and you know that I think he is) then the failure to appreciate the logic of infinity makes many current models of mind and brain beside the point.
 Note that unbounded implies novelty, but novelty can exist without unboundedness. The creativity issue relates to easy handling of novel structures. This can occur even in small sets. Creativity implies projection which must specify a dimension of generalization along which inputs can be extended to apply to instances beyond the input. Unboundedness makes projection a no-brainer. It further implies that the generalization is due to recursive rules that require more than establishing a fixed number of patterns that can be repeatedly filled to create novel instances of that pattern.