On conceptual naturalness
Appeal to general considerations of conceptual naturalness such as simplicity, economy, or non-redundancy is not unique to generative grammar. It has been employed fruitfully in the more developed natural sciences—in particular, theoretical physics. The discussion of physics that follows attempts to elucidate this notion in a way that, ultimately, should illuminate its role in contemporary theoretical linguistics.
Consider, for example, Einstein’s principle that all physical laws must be Lorentz invariant. As Putnam (1962) notes: ‘This is a rather vague principle, since it involves the general notion of a physical law. Yet in spite of its vagueness, or perhaps because of its vagueness, scientists have found it an extremely useful leading principle.’ This is because they have ‘no difficulty in recognizing laws’: a law of nature will be an equation relating ‘real magnitudes’ that has ‘certain characteristics of simplicity and plausibility’ In other words, determining whether Einstein’s principle may be applied to any particular case will involve ‘general considerations of conceptual naturalness’.
In a different area, Bohr’s quantum mechanical ‘Correspondence Principle’ (circa 1913) is arguably rooted in such considerations. It states that, in the classical limit, the results obtained from quantum mechanics should converge with those obtained from classical mechanics. According to some physicists, the research work carried out during the years 1919–1925 that finally led to quantum mechanics may be described as systematic guessing guided by the Correspondence Principle. This is then a case where considerations of conceptual naturalness appear to have played a direct role in the progress of science.
The appeal to conceptual naturalness manifests itself also in the quest for mathematical beauty, which motivates many a theoretical physicist as Dirac notes:
Theoretical physicists accept the need for mathematical beauty as an act of faith. There is no compelling reason for it, but it has proved a very profitable objective in the past. For example, the main reason why the theory of relativity is so universally accepted is its mathematical beauty. (Dirac, 1968)
In the natural sciences, while hypothesis formation may be guided by appeals to conceptual naturalness, any given hypothesis will carry weight only to the extent that it can be subjected to the inexorable test of experiment. This is the essence of the scientific method, which governs physics and linguistics alike. But there is no chosen method for elaborating the scientific hypotheses themselves. The scientific method is not concerned with that, nor could it be, for it is not possible to set up explicit rules or criteria in this area. This does not mean that ‘anything goes’. But it does mean that there is a lot of diversity in the ways scientists deal with problems and arrive at solutions.
Dirac discusses this diversity of methods in theoretical physics:
One can distinguish between two main procedures for a theoretical physicist. One of them is to work from the experimental basis. For this, one must keep in close touch with the experimental physicists. One reads about all the results they obtain and tries to fit them into a comprehensive and satisfying scheme.
The other procedure is to work from the mathematical basis. One examines and criticizes the existing theory. One tries to pinpoint the faults in it and then tries to remove them. The difficulty here is to remove the faults without destroying the very great successes of the existing theory.
There are these two general procedures, but of course the distinction between them is not hard-and-fast. There are all grades of procedures between the extremes. (Dirac, 1968)
Dirac designates the two types of procedures as ‘experimental’ and ‘mathematical’, respectively. He then proceeds to give several examples of the mathematical procedure:
Maxwell’s investigation of an inconsistency in the electromagnetic equations of his time led to his introducing the displacement current, which led to the theory of electromagnetic waves Einstein noticed a difficulty in the theory of an atom in equilibrium in blackbody radiation and was led to introduce stimulated emission, which has led to the modern lasers, [this is Einstein, 1917; RF&JRV] But the supreme example is Einstein’s discovery of his law of gravitation, which came from the need to reconcile Newtonian gravitation with special relativity’. (Dirac, 1968)
Dirac’s notions also apply to the founding work in quantum mechanics between 1913 and 1925. The following description is striking:
Whether one follows the experimental or the mathematical procedure depends largely on the subject of study, but not entirely so. It also depends on the man. This is illustrated by the discovery of quantum mechanics.
Two men are involved, Heisenberg and Schrödinger. Heisenberg was working from the experimental basis, using the results of spectroscopy, which by 1925 had accumulated an enormous amount of data. Much of this was not useful, but some was, for example the relative intensities of the lines of a multiplet. It was Heisenberg’s genius that he was able to pick out the important things from the great wealth of information and arrange them in a natural scheme. He was thus led to matrices.
Schrödinger’s approach was quite different. He worked from the mathematical basis. He was not well informed about the latest spectroscopic results, like Heisenberg was, but had the idea at the back of his mind that spectral frequencies should be fixed by eigenvalue equations, something like those that fix the frequencies of systems of vibrating springs. He had this idea for a long time, and was eventually able to find the right equation, in an indirect way. (Dirac, 1968)
The ‘mathematical procedure’ typically arises in what Husserl has called the ‘Galilean style of science’, in recognition of its origins in the work of Galileo. Weinberg (1976) characterizes this style as follows:
…we have all been making abstract mathematical models of the universe to which at least the physicists give a higher degree of reality than they accord the ordinary world of sensation.
More generally, one can define Galilean science as the search for mathematical patterns in nature. As Chomsky notes, implementing the Galilean style entails a ‘readiness to tolerate unexplained phenomena or even as yet unexplained counterevidence to theoretical constructions that have achieved a certain degree of explanatory depth in some limited domain, much as Galileo did not abandon his enterprise because he was unable to give a coherent explanation for the fact that objects do not fly off the earth’s surface’ (1980, 9–10).
A significant feature of the Generative Revolution in linguistics has been the development of a Galilean style in that field. And, to a great extent, the recent developments within MP must be viewed in this light—specifically, as Dirac’s mathematical procedure (method) at work within linguistics. Dirac has identified two main methods within the mathematical procedure itself: one is to remove inconsistencies, the other, to unite theories that were previously disjoint (see Dirac, 1968). In linguistics, the inconsistencies primarily concern overlapping grammatical conditions, as discussed earlier, which conflict with the basic assumption that CHL has an optimal design. Note further that this assumption itself relates directly to the quest for mathematical beauty, which informs the Galilean style.
One aspect of Dirac’s mathematical procedure as applied in linguistics involves the effort to extend and deepen the mathematical formalism used to express syntactic concepts and syntactic principles. We will refer to this facet of the Minimalist endeavor as the ‘Generative Program’ for the study of language (GP) because it originates in Chomsky’s foundational work in the fifties and sixties and has been essential to the development of the Galilean style in linguistics. However, it should be obvious that linguistics and physics are at very different stages of mathematical maturation. From this perspective, it is useful to distinguish the ‘Galilean character’ of an area, i.e., how much of the subject matter can be analyzed mathematically, from what one could call its ‘Pythagorean character’, how much of mathematics is put to use in the Galilean treatment. Linguistics and physics have the same Galilean character, although they obviously differ in Pythagorean character.
The difference in mathematical status between physics and linguistics partly reflects the more general difference between physics and biology—especially from the perspective that generative grammar is ultimately a branch of theoretical biology, more specifically, of theoretical developmental biology. In biology, the genetic code rather than mathematics has been the tool of choice for explaining life.
This, however, appears to be a historical accident, not the result of some principled difference between biology and the physical sciences. Mathematics has a central explanatory role to play in biology, as discussed in Stewart (1998), whose title, Life’s other secret, is intended as contrapuntal to ‘life’s first secret’, which is the genetic code:
The mathematical control of the growing organism is the other secret— the second secret, if you will—of life. Without it, we will never solve the deeper mysteries of the living world—for life is a partnership between genes and mathematics, and we must take proper account of the role of both partners. This cognizance of both secrets has run like a shining thread through the history of the biological sciences—but it has attracted the mavericks, not the mainstream scientist. Instead of thinking the way most biologists think, these mavericks have been taking a much different approach to biology by thinking the way most physical scientists and mathematicians think. This difference in working philosophy is the main reason why understanding of the deeper aspects of life has been left to the mavericks. (Stewart, 1998: xi)
The main message of d’Arcy Thompson, one of the great mavericks in biology, is that ‘life is founded on mathematical patterns of the physical world.’18 Thus one role of theoretical biology is to identify such mathematical patterns and elucidate the way they function in organisms:
The role of mathematics [in biology] is to analyze the implications of models—not ‘nature red in truth and complexity’, as Tennyson did not quite say, but nature stripped down to its essence. Mathematics pursues the necessary consequences of certain structural features. If a planet can be considered a uniform sphere, what would its gravitational attraction be like?… If the movement of cells in some circumstances is controlled by physical forces and does not greatly depend on complicated internal features such as mitochondria, what will the cells do? From this point of view, the role of mathematics is not to explain biology in detail, but to help us separate out which properties of life are consequences of the deep mathematical patterns of the inorganic universe, and which are the result of more or less arbitrary initial conditions programmed into lengthy sequences of DNA code. (Stewart, 1998:243–244)
It is worth noting at this point that Chomsky was aware that both approaches, separately or jointly, might account for the human language faculty In criticizing the empiricist view of language acquisition in the first chapter of Chomsky 1965 (written in 1958–1959, as mentioned in Huybregts and van Riemsdijk 1982), he notes:
…there is surely no reason today for taking seriously the position that attributes a complex human achievement entirely to months (or at most years) of experience, rather than to millions of years of evolution or to principles of neural organization that may be even more deeply grounded in physical law… (p. 59)
However, twenty years later, Chomsky is openly skeptical of a purely genetic approach to evolution.
It does seem very hard to believe that the specific character of organisms can be accounted for purely in terms of random mutation and selectional controls. I would imagine that biology of 100 years from now is going to deal with evolution of organisms the way it now deals with evolution of amino acids, assuming that there is just a fairly small space of physically possible systems that can realize complicated structures. (Huybregts and van Riemsdijk, 1982:23)
From this point of view, the more promising approach is ‘d’Arcy Thompson’s attempt to show that many properties of organisms, like symmetry, for example, do not really have anything to do with a specific selection but just with the ways in which things can exist in the physical world’ (Huybregts and van Riemsdijk, 1982:23).
The mathematical perspective informs the Generative Program (GP), in effect, ‘the study of language’s other secret’. Thus Chomsky’s mathematical work defines a central facet of GP, beginning with his construction of the foundations of modern generative grammar in Chomsky (1951) and (1975a).
Because the MP is a particular implementation of GP, the notion of ‘perfection’ often invoked within MP is ultimately a mathematical notion, calling for a higher level of mathematical formalization in syntax. The Minimalist conjecture that CHL is a ‘perfect system’ is a tentative claim about the form and the complexity of each computation. The claim is (i) that each computation can be represented as an abstract mathematical structure completely defined by interface (output) conditions and ii) that this structure is an extremum in some mathematical space. A natural metric for the comparison of computations is their complexity as measured by their length. Note that, if the only constraints on CHL are those that follow from legibility conditions at the interfaces, then it is unavoidable that some notion of computational cost should be part of the definition of ‘effective’ computations, since, within such a system, it is always possible to combine a computation with a ‘vacuous one’ (i.e., one that has a null effect). The unidirectionality of movement (if it is a fact) would then be a particular design feature aimed at reducing the likelihood of vacuous steps.
Considerations of economy have a long standing legitimacy in the physical sciences. It was in physics that an economy principle of any depth was first advanced. This was the principle of least time, discovered by Fermat circa 1650. That principle states that, out of all possible paths that it might take to get from one point to another, light takes the path which requires the shortest time. Fermat’s principle is a particular instance of the general physical principle of ‘least action’. Another important economy principle of physics is ‘the idea that the inorganic world is fundamentally lazy: it generally behaves in whatever manner requires the least energy’ (Stewart, 1998:16). That idea was for Thompson (1942) a central principle underpinning the mathematics of growth and form found in living organisms.
Comparing Fermat’s principle with Snell’s theory of light, Feynman notes that such economy principles have a special philosophical character distinct from causal explanations of phenomena.
With Snell’s theory we can ‘understand’ light. Light goes along, it sees a surface, it bends because it does something at the surface. The idea of causality, that it goes from one point to another, and another, and so on, is easy to understand. But the principle of least time is a completely different philosophical principle about the way nature works. Instead of saying it is a causal thing, that when we do one thing, something else happens, and so on, it says this: we set up the situation, and light decides which is the shortest time, or the extreme one, and chooses that path. (Feynman et al., 1963:26–7)
Feynman’s observation extends to all economy considerations developed in the natural sciences. Economy principles fall under what 17th and 18th philosophers called ‘final causes’, as opposed to ‘efficient causes’. Efficient causes are essentially mechanistic in nature like those invoked in a Newtonian account of the dynamics of a point particle, for example, or Snell’s account of refraction as described by Feynman above. Final causes involve a deeper level of understanding, as Feynman notes:
Now in the further development of science, we want more than just a formula. First we have an observation, then we have numbers that we measure, then we have a law which summarizes all the numbers. But the real glory of science is that we can find a way of thinking such that the law is evident. (Feynman et al., 1963:26–3)
Thus, the distinction between efficient and final causes is locally one of levels of analysis and globally one of levels of explanation.
The notion level’ (of analysis, of explanation) is evidently crucial. The natural sciences provide instances where successful explanatory theories that had been developed at a certain level were later unified with theories at some other level. This is the case for classical thermodynamics, which is deducible from statistical mechanics (hence a reduction). Also the unification of structural chemistry with physics was made possible by the development of quantum mechanics, which provided a common foundation (see Chomsky 1995a, and Smith, 1999 for discussion). However, the explanatory import of a theoretical principle at some given level L is in general relatively independent of the possibility of unifying L with other levels. A case in point is that of the principle of least action’ mentioned above (the general principle subsuming Fermat’s principle of least time), which is reducible to other principles in every area where it applies (see Jourdain, 1913 and Lanczos, 1970 for discussion). Thus, it applies in classical mechanics, where it is known as ‘Hamilton’s principle’. And, indeed, Hamilton’s principle is an alternative formulation of classical mechanics, equivalent to the Newtonian formulation. As it turns out, though, the Hamiltonian formulation has desirable features not found within the Newtonian formulation. For example, the Hamiltonian formalism can be generalized to all types of coordinates and, furthermore, is more convenient than Newton’s equations when the system is complex. But the real importance of the Hamiltonian formalism arises from the fact, both, that it can be generalized to classical electricity and magnetism (with an appropriate Lagrangian) and that it constitutes the point of departure for the quantization of physical systems (see the discussion in Cohen-Tannoudji et al., 1996:1476–1491, for example).
There may be deep reasons for this remarkable generality. The following excerpt from Toffoli (1999) is intriguing in that respect:
We are taught to regard with awe the variational principles of mechanics [such as Hamilton’s principle RF-JRV]. There is something miraculous about them, and something timeless too: the storms of relativity and quantum mechanics have come and gone, but Hamilton’s principle of least action still shines among our most precious jewels.
But perhaps the reason that these principles have survived such physical upheavals is that after all they are not strictly physical principles! To me, they appear to be the expression, in a physical context, of general facts about computation, much as the second law of thermodynamics is the expression, in the same context, of general facts about information. More specifically, just as entropy measures, on a log scale, the number of possible microscopic states consistent with a given macroscopic description, so I argue that action measures, again on a log scale, the number of possible microscopic laws consistent with a given macroscopic behavior. If entropy measures in how many different states you could be in detail and still be substantially the same, then action measures how many different recipes you could follow in detail and still behave substantially the same. (Toffoli, 1999:349–350)
If this is on the right track, the computational significance of the Hamiltonian formalism supersedes any deduction of it in any particular subdomain.
The computational nature of economy considerations provides a link between physics and linguistics, at least metaphorically. Whether it is stronger than that will have to be determined by a future neuroscience that can validate the physical approach to complex mental structures as suggested by Chomsky extending the views of d’Arcy Thompson. In any event, economy considerations contribute substantially to what constitutes the ‘perfection’ of the computational system in both domains. Whether these considerations for each domain turn out to be related or the same remains an empirical question for the future.
In linguistics, there are several ways the ‘perfection’ of CHL could be manifested in terms of economy conditions. Shortness of derivation is only one symptom of perfection. Another manifestation, possibly equivalent in some cases, would be the existence of symmetries across levels of analysis, given that such symmetries enhance the economy of computations.
To illustrate, consider the following well-known contrast in anaphoric interpretation for the paradigm in (2):
While Mary in (2a) may be construed as anaphoric with she, this is not a possible construal for (2b). Exactly how we account for this depends crucially on what representations are available. Prior to the Minimalist Program these anaphoric representations would be given in terms of co-indexing generated by a rule of Index NP (see Freidin and Lasnik, 1981 for discussion). Thus the construals under discussion would be given as (3a) and (3b) respectively, where the viable construal of (2b) is given as (3c).
However, given the Inclusiveness Condition (4), which we take to be central to the Minimalist Program, indices are not legitimate elements of representations.
Therefore the construals of (2) indicated in (3) will have to be represented another way. We shall assume that a definite pronoun is a definite description with a silent NP component (cf. Postal, 1966 and Brody 1982). Specifically, we posit the following underlying representation for a pronoun:
The NP component of the pronoun determines its interpretation: two different interpretations of a pronoun reflect two distinct underlying representations of that pronoun. For example, the sentence in (2a) is represented as in (8) when she is construed as anaphoric with Mary, but as in (9) when she is construed as referring to Clea.
We propose to relate the interpretive contrast in (2) to symmetries in the representations of the structures involved.
The defining property of a pronominal element like she in (2) is that its PF representation is invariant under substitution of its NP component. Call this the pronominal symmetry:
No matter what representation is assigned to NP, the PF representation of pro remains constant. We formalize this as in (11):
Thus all elements in the range of pro share the same PF representation.
Now, there is a general principle in grammar that items in a structure are not interpreted in isolation, but always with respect to some larger domain. Technically, grammar constructs an interpretation of the head and of the specifier of x only at the level of some constituent properly containing x. Call this the Generalized Phase Conjecture, in reference to the analysis proposed in Chomsky (1999):
Chomsky (1999) considers a notion of phase that seems appropriate for the interpretation of expressions involving displacements. We conjecture that a different notion of phase exists for the assessment of anaphoric relations. Specifically, the phase for a pronominal expression pro is its c-command domain.
Considering now the paradigm in (2), let us call Pshe the phase for the pronoun she. For the form in (2a), Pshe is the embedded TP [she solved the problem]. The pronominal symmetry associated with she carries over to Pshe: the PF representation of the phase of she is invariant under substitution of NP in the representation of she, quite obviously. We assume this to be a general requirement for phases, stated as (13):
Given the PPC, the PF invariance of a pronoun pro, which constitutes the pronominal symmetry, must in general carry over to the phase of pro Ppro. We evaluate satisfaction of the PPC by extending the notion of ‘range’ to Ppro:
Accordingly the range of pro in (15a) establishes the set of parallel structures in (15b):
Then:
In this way (2a) satisfies the PPG.
Consider next whether ii) satisfies the PPC. In that structure, the phase Pshe is the matrix TP containing in addition to the pronoun a second DP which could, but need not, relate to the interpretation of the pronoun. In the case where she in (2b) is interpreted as Mary, the corresponding representation is that in (17) (with the phi-features [3rd person, singular, feminine]):
The set of parallel structures established by the range of pro in this case includes one structure in which a pair of expressions are anaphorically linked, to wit the structure in (17). This conformation is subject to the Parallelism Principle34 as formulated in (19):
Given the definition of the range of pro, case (ii) doesn’t apply to the set of parallel structures. The application of case (i) amounts to revising the definition of the range as follows:
If we apply this definition to the structure in (17), then, the range of Pshe relative to she includes such structures as those in (21):
Because the set in (21) is not PF invariant, the PPC is violated.35 In this way, the construal (3b) of (2b) is excluded. Note that, in the case of the construal in (3c), the range of Pshe may not include the structure in (17)—the structure where she is anaphoric with Mary—by (ii) and (iii) of (20). It is easy to check that (2b), under construal (3c), satisfies the PPC: no matter the value of pro within the admissible range, the pronoun and its phase will both remain PF invariant. In essence, Principle C reflects a conflict between Parallelism and Phasal Coherence: in the case of a structure such as (17), there is no coherent definition of ‘range of a phase’ that can satisfy both principles.
To summarize, Principle C follows from the interaction of the Principle of Phasal Coherence, related to QR, with the Parallelism Principle.37 This account immediately extends to the contrast in (21) if the chunk him in himself is treated as a pronoun falling under the analysis above:
The above account also extends to the following paradigm from French:
(23a–b) show the standard contrast for disjoint reference under c-command as in (2). In surprising contrast, (23c) allows the coreferential interpretation between the pronominal matrix subject and the complement subject via the pronoun in the relative clause. The same anaphoric behavior obtains when pronominal matrix subject is replaced by an R-expression, as in (23d). However, disjoint reference obtains again if the pronoun in the relative clause in (23c) is replaced by an R-expression, as illustrated in (23e). Note that (23f) results from transposing the matrix and complement subjects in (23d) and thus this pair is exactly parallel to (23a—b). This analysis extends to the pair (23e,g). The example in (23h) is grammatical as expected, in contrast to (23f).
The paradigm in (23) shows that the constituent [le juriste que DP est] ‘the jurist that DP is’ has the same anaphoric behavior as the DP in it. For the purpose of applying Principle C, it behaves as a pronoun when DP is a pronoun and as a name otherwise. DP in turn behaves as if it occupied the position of the head modified by the relative clause. Noting that the predicate juriste and its subject DP within the restrictive relative clause construction [le juriste que DP est] share the same set of phi-features assume that the notion of symmetry extends to such pairs of constituents:
Consider in this light the form [le juriste que pro (NP) est], with pro(NP) a pronoun. The PF of the pair (juriste, pro(NP)} remains invariant under the substitution of NP for NP in the pair ((juriste, pro(NP)). By the above extension, pro(NP) establishes a range not only for its own phase, but also for the phase of the raised predicate le juriste, since pro(NP) and juriste share the same phi-features. The PPC gives rise to the contrast between (23g) and (23h). Note that (24) entails that, in a similar fashion, the notion of symmetry may be extended to the pair (Mary, she) in the structure in (3a), since the DP Mary and the pronoun she share the same phi-features. However, in that case, if an NP different from Mary is substituted for Mary within the pair (Mary, the PF of the pair is altered (we assume that the substitution takes place across the board). No PF invariance obtains and the PPC is then not relevant to the pair (Mary, she).
To the extent that the kind of analysis proposed above is viable, it provides a modest illustration of what is being referred to as the ‘perfection’ of the grammatical system. The possibility then arises that the abstract analytical principles involved in the formal definition of a computation turn out to have exactly the right empirical consequences. This is an exciting prospect, which, combined with that of potentially rich mathematical developments, is stirring imaginations. The authors of this note understand the excitement, and share in it. Uriagereka’s Rhyme and reason is a particular expression of that entirely natural and legitimate reaction. In essential respects, linguistics is no different from other fields in natural sciences at comparable stages of development.
