1. Axiomatisation and the formal sciences

The introductory chapter of LFS presents a rough characterisation of what the formal sciences are. For Tomalin, the phrase ‘the formal sciences’

This crucially assumes that logical consequences in the formal sciences are deductive consequences, not semantic consequences. Thus Tomalin immediately signals what will become a central theme of the book, that the kind of proofs that are characteristic of formal sciences are derivations. This excludes some tools of logic and mathematics that are typically taken to fall within formal and axiomatic methods. As Tomalin tells the story, what was really central to the birth of TGG was the idea of a formal theory, which he identifies with Hilbert's proof theory (Beweistheorie). So model theory appears to be excluded from the methods of the formal sciences – a very radical restriction, excluding from the formal sciences a huge amount of pure and applied mathematics logic, and large areas of formal linguistics. (As we shall see, the exclusion of semantics was not in fact characteristic of the methods used to rigorise the calculus.)

Tomalin's view of what 19th-century mathematicians would have understood by ‘rigorisation’ is the one expressed by Grabiner (Reference Grabiner1981: 5, quoted in LFS: 26f.):Footnote ²

Newton, Leibniz and others had accepted certain results of the calculus on the grounds that they were predictively accurate. Cauchy wanted to prove these already accepted results from explicitly stated axioms and, crucially, explicit definitions of concepts such as ‘limit’, ‘convergence’, ‘continuity’, ‘derivative’, etc. The idea was not only to put the calculus on an epistemologically secure foundation but also to see the connection between the defined concepts and the result proved. Thus the proofs were not mere symbol manipulations; as Grabiner says, ‘the derivation of a result by manipulating symbols was not a proof of the result’. Both the definitions and the axioms were supplied with meaning through a reduction to algebra, which was regarded as more secure than the calculus was in the 18th century.

Tomalin does not separate out the various components making up Cauchy's axiomatic-deductive methods, but they could be stated as practical maxims. Our proposed phrasing of the maxims attributable to Cauchy (we will suggest other maxims later) would be as follows:

1. (A) State the fundamental assumptions of the pre-formal theory as axioms.

2. (B) Define all non-primitive concepts explicitly.

3. (C) In proofs of theorems, use only definitions, axioms and previously established theorems.

4. (D) Reduce the axioms, definitions, and theorems to a better-understood and less controversial theory.

The use of (A) and (B) in mathematics dates back at least to Euclid's Elements, so they are certainly not novel in 19th-century mathematics. Even Newton called his laws of motion ‘axioms’, so (A) does not distinguish Cauchy's rigorous methods from Newton's empirically justified, prediction-oriented calculus. But in (B) we see a dramatic difference between Euclid and Cauchy, on the one hand, and Newton and Leibniz, on the other: Cauchy wanted to replace Newton's inchoate intuitive notion of ‘fluxion’ and Leibniz's use of ‘momentary differences’ by explicitly defined concepts.

And even more importantly, (C) and (D) are key components of Cauchy's contribution to the development of axiomatic-deductive methods. The use of reductive methods like (D) is typical when theories are axiomatised for epistemological reasons. For Cauchy, (D) was adopted in part to exhibit the relationships between concepts, fundamental assumptions, and theorems. But there are also other epistemological reasons for adopting (D). One is epistemological foundationalism: the view that theories must be made secure against rational doubt. The idea that mathematical and logical theories can achieve foundationalist goals by means of (D) came under attack in the last half of the 20th century, and the need for foundationalist projects in mathematics has been rejected in recent decades (see Shapiro Reference Shapiro1991 for an excellent discussion). Tomalin, however, appears to assume that epistemological foundationalism is not just a motivation for using the axiomatic method, but part of it: he fails to distinguish between foundations and foundationalism (see, for example, pages 28, 38–45).

Tomalin does not compare the goals of different mathematical or logical programmes that developed axiomatic methods and advocated their use. Indeed, he writes as if there is a unique, univocal axiomatic-deductive method regardless of goals. This, we think, is a mistake: not every maxim is adopted by every theorist, because different maxims are useful in achieving different goals.

The set-theoretic paradoxes that were discovered in the second half of the 19th century (LFS: 29–32) provoked at least three philosophical movements which Tomalin surveys: Logicism, Formalism, and Intuitionism. It is the first two that loom largest in LFS.

Logicism is the programme in the philosophy of mathematics that was pursued by Frege, and by Whitehead and Russell; Tomalin describes it as the project of using logic ‘as a basis for arithmetic’ (33), though in fact there are various different formulations (see Rayo Reference Rayo and Stewart2005 for a discussion).

Hilbert's Formalism is a different philosophical programme. Most philosophers of mathematics would characterise it as the programme of showing that the whole of mathematics is consistent and complete. Hilbert's proof theory was not the whole programme; rather, it was the most famous strategy for working on the programme.

Tomalin claims that ‘perhaps the main achievement of this book has been to associate TGG with both Formalism and Logicism’ (186). The trouble is, Tomalin equivocates on ‘Formalism’, correctly pointing out a common misconception, but then apparently falling into it himself. To make this clear we need to explain a bit about Hilbert's programme.

A crucial step in Hilbert's strategy for showing the consistency of mathematics (see Hilbert Reference Hilbert and van Heijenoort1927) was to convert ordinary arithmetical propositions into formulas by a fully automatic procedure referred to by Tomalin as formalisation (40). Hilbert's insight here is that the propositions of mathematics can be mechanically paired with syntactically defined formulas. This conversion step is a formalisation in the sense that it mechanically produces formulas that serve as meaningless representatives of the propositions of ordinary arithmetic. Hilbert calls them copies (Abbilder) of those propositions.

A formula is then said to be provable if it is the last line of a derivation in a sense that should be very familiar to those who know generative grammar: a derivation is a sequence of formulas each of which either is an axiom, or is a substitution instance of an axiom, or is obtained from the axioms and substitution instances by an inference rule. Consistency was then to be proved by showing that some mathematical contradiction like 0≠0 is not provable. (That shows that the whole system is consistent, because in an inconsistent system everything is provable. Incidentally, the linguist acquainted with basic results on generative capacity will see here the seeds of Gödel's celebrated later result: to assume we have a mechanical method for showing unprovability of a formula is to assume that the set of all provable formulas is a decidable set; the problem is that it might not be.)

Inference rules function rather like rewriting rules in a generative grammar (see section 5 below), and are formal in the sense that they rely purely on form: they do not depend on any interpretation of the formulas. Hilbert thus employs string manipulation techniques on meaningless copies of the meaningful propositions of arithmetic, as a way of building consistency proofs. But it does not follow from this that mathematics itself is a game played with meaningless symbols: the propositions of mathematics are not meaningless (Hilbert's strategy merely employs meaningless Abbilder as surrogates for them).

Tomalin seems to understand all this; he spends a page (44) warning that it is a ‘common misconception’ that Hilbert thought mathematics was a game played with meaningless symbols – a misconception that was partly due to people ignoring the ‘distinction between the formalisation process and the metamathematical process’.

Note carefully that Tomalin uses ‘formalisation’ for the step of converting mathematical propositions into formulas; ‘formal’ for systems and proofs that make no reference to meaning; and ‘Formalism’ for the larger enterprise of proving mathematics consistent.

But only five pages later, in the context of pointing ahead to his subsequent discussion of Hilbert's influence on Carnap and Quine, Tomalin (49) changes his terminological policy, and conflates Formalism with being formal. He notes correctly the ‘interest in the syntax of formal languages’ displayed in Quine's Mathematical Logic (1940), but incorrectly claims ‘his presentation is conspicuously influenced by Formalism’. The example he cites involves a passage where Quine

In Tomalin's view, it is not the programme of showing all of mathematics to be consistent that has ‘conspicuously influenced’ Quine in electing to work with strings over a fixed finite alphabet; nor is it what Tomalin earlier called the formalisation step; it is merely being formal.

This is not a mere quibble. Which of the three – the formalisation step, the project of Formalism, or being formal – does Tomalin claim to have linked to the origins of TGG?

Tomalin mentions some other axiomatisations of mathematical and logical theories that all satisfy at least maxims (A)–(C): John Young's Fundamental Concepts of Algebra and Geometry (1911), W. V. O. Quine's Mathematical Logic (1940), and Alonso Church's Introduction to Mathematical Logic (1944), for example.Footnote ³ In the context of discussing Church, he hints (49) at what he understands the axiomatic-deductive method amounted to:

The first three ‘topics’ that Tomalin mentions are not topics at all, but just standard categories employed in the syntax and semantics of typical formal languages. What Tomalin might have in mind is that the following maxim is often adopted in the most formal presentations of mathematical and logical theories:

1. (E) Use a well-defined invented formal language to state the axioms.

But the last three ‘topics’ are simply distinct types of logic, some of which had properties that were well known by the mid-1940s. Possibly Tomalin means to suggest another maxim, (F):

1. (F) Identify the logic that is used in deriving the consequences of the (non-logical) theory.

(This would be a special case of Suppes' observation (1956: 156) that in mathematics ‘a statement of what theories are assumed is needed’.)

The motivation for adopting (E) is often taken to be to rid axiomatisations of the vagueness and ambiguity of natural language. The motivation for adopting (F) is that without it there is no explicit objective standard of what follows from the axioms of the theory.

LFS began by claiming that the formal sciences ‘can be viewed as being unified by the same basic scientific method’ (3). It should be clear that (A)–(F) do not characterise any single method. They are a family of maxims, from which different researchers adopt different subsets for particular purposes. There is no unified ‘axiomatic-deductive method’: which of the maxims a theorist chooses to respect will partly depend on the goals of the axiomatisation project, and goals are diverse.

2. The mathematicising of linguistics

Tomalin's long chapter headed ‘Mathematical linguistics’ is not about mathematical linguistics. It is a survey, in 53 pages, of historical material rich enough to make a book on its own: the extent of the influence of mathematical logic, recursive function theory, and logical empiricism on the mathematicisation of linguistics in the period 1926 to 1953. It is necessarily somewhat superficial, but could serve as a useful guide to sources for the future work on the use of formal methods in linguistics and how they influenced linguists' conceptions of good linguistic theorising.

The chapter argues that ‘the’ axiomatic method as developed in mathematics and logic was applied to presenting linguistic theories during the quarter of a century before Chomsky (Reference Chomsky1955 [1975]). So part of what makes the chapter important given the goals of LFS is its role in showing that there was tacit agreement among a group of American linguists (in particular those most influenced by Bloomfield) about the general form linguistic theories should take.

Tomalin begins by discussing Bloomfield's paper ‘A set of postulates for the science of language’ (1926), which attempted to set out an entire approach to linguistic theorising in axiomatic form. He quotes one or two examples of Bloomfield's axioms and definitions, but does not discuss the content of the paper in detail. This is an interesting signal that LFS itself takes the curious view that the form of linguistic theories is worth discussing quite independently of their content. This is curious because evaluation of the appropriateness of axiomatising must surely depend in part on what is to be axiomatised.

We do not agree at all with Tomalin's comparison between Bloomfield (Reference Bloomfield1926) and Cauchy's rigorisation of the calculus. Tomalin writes (55):

But the mere fact that both Cauchy and Bloomfield (like Euclid) respected maxims (A) and (B) does not make it reasonable to draw a favorable comparison between Bloomfield's short paper and Cauchy's major project. The two projects are utterly different in important ways. Maxim (C) is irrelevant to Bloomfield (Reference Bloomfield1926), because he did not prove, or try to prove, anything. Nor did Bloomfield follow maxim (D): he did not reduce any linguistic theory or grammar to a different theory that he favored at the time – say, Weiss's (Reference Weiss1925) axiomatisation of behaviourist psychology (which, as Tomalin notes, was a model for Bloomfield Reference Bloomfield1926).

Bloomfield (Reference Bloomfield1926) laid out the assumptions of his general approach to linguistic theorising, and urged that other linguists should do so as well. He wanted technical vocabulary to be explicitly defined in order to avoid certain kinds of error, and to show which specialised linguistic terms were undefinable and primitive and which could be defined in terms of the primitives (1926:26). He warned that otherwise linguists were in danger of falling prey to making obscure speculations about languages, as did Humboldt, or engaging in fruitless psychological disputes like the ones that had gone on between Paul, Wundt, and Delbrück.

Tomalin performs a very useful service in reminding linguists (or perhaps in many cases informing them for the first time) of the almost forgotten work of Harwood (Reference Harwood1955), which suggested stating grammars for generating finite sets of strings in two parts: (i) a set of strings taken to be axioms, and (ii) a set of inference rules for deriving further strings from them. Chomsky (Reference Chomsky1957) does cite this work, but only to reject it as inadequate for not containing any provision for recursion. Harwood's motivation was apparently to consider methods for measuring economy of description of finite corpora. That may explain why Harwood seems content with a description that entails a finite bound on sentence length.

Tomalin's consideration of recursive devices (66) attempts to link Dedekind's use of recursive functions, Peano's inductive definition of the successor function, and Bar-Hillel's definition of ‘sentence’ that uses ‘sentence’ in the definiens (Bar-Hillel calls the latter ‘recursive in disguise’). He then explores the influence on non-TGG linguistics of the Lvov-Warsaw school of logic founded by Twardowski.Footnote ⁴ The logicians who emerged from this extraordinarily important school included Łukasiewicz, Lesniewski, Ajdukiewicz, and Tarski. Tomalin is especially (and rightly) interested in the influence of Ajdukiewicz's definitions of logical calculi on Bar-Hillel's conception of syntax, which gave rise to modern categorial grammar. In our view, however, Bar-Hillel's work was more important to the history of linguistics than Tomalin explicitly recognises: it is not just all the work in categorial or type-logical syntax and semantics that largely sprang from it (Bach, Carpenter, Dowty, Jacobson, Lambek, Montague, Moortgat, Partee, Steedman, Szabolcsi, …), but also to some significant degree early HPSG, and Chomskyan minimalism, and the work of Keenan & Stabler (Reference Keenan and Edward2003).

As Tomalin sees it, Bar-Hillel attempted to combine two things. One was the methods Harris (Reference Harris1951) developed to attempt to read off the structure of languages from corpora. The other was Ajdukiewicz's notation and approach to defining logical calculi. Tomalin holds that Bar-Hillel's adaptation of Ajdukiewicz-style formation rules ‘implies that the type of formal languages used in symbolic logic (particularly the techniques developed by Ajdukiewicz) are closely related to natural languages and that therefore methods developed to analyse the former can readily be adapted in order to provide analyses of the latter’ (73). Tomalin regards it as significant that categories can be combined to form well-formed strings by a ‘purely mechanical process’ (72). So Bar-Hillel's mechanical combinatory procedure could be regarded as analogous to Hilbert's formalisation, but it is not formal in the sense of lacking a semantics.

Tomalin fails to note that, although Bar-Hillel never uses the word ‘generate’, he was unquestionably proposing a generative grammar, several years before Chomsky. Bar-Hillel's proposal was for a bottom-up generative grammar, in the sense that well-formed expressions are built up from smaller categorised parts (as in the post-1990 minimalist program) rather than being derived top-down from a start symbol as in pre-1958 TGG. This is important for Tomalin's project. As he notes later in the book (164), the ‘particular concepts and techniques that eventually came to be associated with TGG were already spreading throughout the linguistics community, albeit in an imprecisely articulated form, by the mid-1950s.’ Bar-Hillel's work, rooted in the work of the Lvov-Warsaw school, should have been mentioned in the same context.

3. Logical empiricism

Tomalin is the first historian of linguistics to seriously examine the relevance to TGG of Carnap's constructional system theory, and Nelson Goodman's revision of it. Carnap's project encapsulated the logical empiricist research programme. It was set out in Carnap's Der logische Aufbau der Welt: Versuch einer Konstitutionstheorie der Begriffe (1928) (hereafter Aufbau). In The Structure of Appearance (1951; henceforth TSA), Goodman significantly modified Carnap's project.

Tomalin aims to articulate the direct line of intellectual influence from Carnap to Goodman and Quine, and from Goodman and Quine to Chomsky. He claims that the Aufbau ‘exerted a lasting influence over syntactic theory in the 1950s’ (LFS: 73). This is certainly right, in the sense that Aufbau influenced conceptions of what a scientific theory is, quite generally, from the 1930s to at least the 1960s.

It is well known that Chomsky took courses from Goodman at the University of Pennsylvania; but LFS, to its credit, actually discusses the content of Goodman's TSA (a revision of his dissertation), and Carnap's Aufbau. What Tomalin aims to show is that Goodman's revision of Carnap's logical empiricist project for rationally reconstructing all of science was adopted in Chomsky (Reference Chomsky1951) and (1955 [1975]).

However, LFS does not discuss in any detail the aims of the logical empiricist program as articulated in the Aufbau. What matters to Tomalin is that the Aufbau endorsed maxims (A)–(D) in its proposal for constructing (or really, reconstructing) all of science. Hilbert had earlier proposed extending his program of Formalism to physics this way (see Corry Reference Corry2004). But what the logical positivists wanted to do was to reconstruct all of science, psychological and sociological theories included; it was not part of their project to offer an explicit proof of consistency.

The Aufbau aimed to provide a metatheoretical framework that would (i) unify all scientific theories, (ii) secure the foundation of science by showing that it can be reconstructed on a purely experiential base, and (iii) show that all philosophical problems are pseudo-problems. Importantly, the Aufbau did not rationally reconstruct any particular scientific theory. It merely set out a programme, and addressed metatheoretical problems that might impede it. This is the way Carnap describes the project of reconstructing science:

The fundamental experiential basis was an ‘autopsychological basis’ that ‘consists only of conscious experiences’ (1967: 102f.). These conscious phenomenal experiences are ‘the given’: a theory-neutral indubitable foundation in terms of which all concepts and objects of science were to be explicitly defined. The empiricism of ‘logical empiricism’ is located in this indubitable autopsychological base. As Tomalin notes, this reductive empiricist base was intended to be the secure epistemological foundation of science, and the idea that empirical knowledge needs such a base is the foundationalist dogma of empiricism that Quine (Reference Quine1953) inveighs against.

Tomalin seems to think that the Aufbau aimed ‘to consider questions of knowledge acquisition’ (74). This is something of a misunderstanding. The goal of the project described in the Aufbau was to justify the claims that scientific knowledge is unified, based only on experience, epistemologically secure, and free of metaphysics. The acquisition of scientific knowledge can be described historically or psychologically, but it is hard to understand how a programme for reconstructing all of science on an experiential basis could answer any question about the way people acquire scientific knowledge. For surely, even working scientists acquire much of their scientific knowledge from the testimony and publications of other scientists, not merely from their own phenomenal experiences.

Tomalin tells us that Quine's familiarity with the Aufbau began in the early 1930s: it motivated him to visit the Vienna Circle in 1932 and 1933, and ultimately to visit Carnap, who was lecturing in Prague at the time. (We would add that Quine also visited Warsaw during this year; Dresner (Reference Dresner1999) argues that through the influence of Carnap, Ajdukiewicz had a lasting impact on Quine's philosophy of language.) He reports that Quine first ‘came into regular contact’ (78) with Goodman when Carnap was lecturing at Harvard on logical syntax in 1935. And Tomalin emphasises that Quine and Goodman shared an interest in the Aufbau.

What is more important, because of its later direct influence on Chomsky (Reference Chomsky1955 [1975]), is that Tomalin discusses – perhaps for the first time for an audience of linguists – the content of TSA, Goodman's revision of Carnap's programme in the Aufbau. Moreover, Tomalin emphasises that by 1943 Goodman had developed a deep interest in economy and simplicity as criteria of adequacy for constructional systems that would influence early TGG.

The discussion of economy in LFS includes the following striking passage that is very closely paraphrased in Chomsky (Reference Chomsky1955 [1975]: 114, n. 2, quoted in LFS: 116). Goodman writes:

This idea of economy was, as Tomalin shows, adopted in early TGG as an adequacy condition on being one of the genuine primitives of an axiomatic deductive system, the primitives being all those concepts (or terms denoting them) that cannot be explicitly defined in the system without circularity. Highly formalised mathematical and logical theories aim for this kind of economy, i.e., they require that the primitives be independent of each other. If the goal of using the axiomatic-deductive method is to make the structure of the subject matter perspicuous, then reflecting on the relations between the primitive concepts of the pre-formal theory (and their relations to each other as stated in the axioms) might be thought to display a structure of the subject matter of the theory. But, we should note, there will be other structures that are attributable to the subject matter under reaxiomatisations of the theory.

And we also note that Goodman speaks of the network of interrelationships: he seems to be claiming in the above passage that basal economy might serve to constrain the theory up to the unique structure of a scientific subject matter!

Although Tomalin does not mention this point, the above passage from Goodman might be understood to express Hilbert's idea that the axioms of a mathematical theory give an implicit definition of the primitives of the theory. For example, in his correspondence with Frege, Hilbert wrote that in geometry terms like ‘point’ and ‘line’ should not be explicitly defined in intuitively spatial terms, as Frege thought, but rather that the set of axioms should be taken to implicitly define them, thus identifying the relations that exhibit the structure of the subject matter. In philosophy, this kind of definition is called an ‘implicit functional definition’, and it has been used by philosophers to state functionalist definitions of mind as an alternative to behaviourism. (The connections to syntax should be clear: a phrase structure grammar gives an implicit functional definition of the categories it employs.) This suggests that it might be worth investigating the genealogy and uses of implicit definitions from Hilbert through Carnap and Goodman to Chomsky. Instead, what Tomalin himself takes from the above quotation is this:

This is a seriously uncharitable reading of Goodman. It is more plausible that Goodman is introducing a further standard part of the axiomatic-deductive method (see Suppes Reference Suppes1956: 156). It could be expressed in a further maxim:

1. (G) Ensure that all primitive terms are independent of each other.

Tomalin's aim in the following section, entitled ‘Formal linguistic theory’ (88–106), is to establish a stream of influence flowing from Hilbert through Carnap's logical syntax to Bloomfieldian linguistics. The first step is to show Hilbert influencing Carnap. But there is a major flaw in Tomalin's exposition. In order to illustrate the ‘indebtedness to Formalism’ in Carnap's book he quotes Carnap's The Logical Structure of Language saying that a system is formal when

But once again, Tomalin is merely referring to being formal, though he calls it ‘Formalism’. It is true that Hilbert has been mistaken for a ‘formalist’ (in the game-playing sense – the ‘common misconception’ referred to earlier), and also that Carnap was one of those who misread Hilbert in this way; but Corry (Reference Corry2004) argues convincingly that it is indeed a misreading (as Tomalin seemed to accept on page 44). Tomalin is thus trying to trace Hilbertian influence through the work of someone who misunderstood Hilbert. If Tomalin is going to catalogue the genuine influence of Hilbert's actual views on Carnap and early TGG, then these threads of misunderstanding need to be sorted out.

The next step in Tomalin's argument is to establish the influence of Carnap on Bloomfield. Tomalin regards Bloomfield as a Carnapian ‘formalist’. But why? Solely because ‘Bloomfield repeatedly expressed skepticism about the role of meaning in linguistic theory’ (95). It is true that Bloomfield located the study of meaning in psychology, not linguistics; and, notoriously, by the time he wrote Language (1933) he regarded behaviourism as the most promising research program in psychology. But Bloomfield's view that linguistics is about form in natural languages and not about meaning is present in the postulates that he published two years before the first (German) edition of the Aufbau. Bloomfield (Reference Bloomfield1926: 27) states that the language of a speech community is ‘the totality of utterances that can be made in that speech community’, and that ‘an utterance is made up wholly of forms’. If utterances are constituted by their non-semantic properties, and linguistics studies utterances, then it is no part of linguistics to investigate meanings. Bloomfield's exclusion of meaning from the subject matter of linguistics is thus entirely independent of Carnap (Reference Carnap1928).

Bloomfield's view of linguistics has nothing to do with either formalisation, or Formalism, or being formal, as these were earlier characterised. Bloomfield proposed axioms and definitions, but they were fully interpreted: when he said ‘utterance’ he meant to refer to utterances. He was not interested in proofs of consistency; he proposed no mechanical procedures for obtaining such proofs; and he did not take either grammatical descriptions or general linguistic theories to be uninterpreted symbol systems.

4. Goodman, Chomsky, and generative grammars

The fourth chapter opens with a brief personal and intellectual biography of Noam Chomsky up to the mid-1950s. After that it attempts to ‘identify the presence of a particular influence’ on Chomsky, and to ‘trace its development as his research gradually matured during the 1950s’ (108). (This sounds like development over a whole academic career, but keep in mind that Tomalin is talking about largely unpublished work by a young man between the ages of 22 and 28!) The influences discussed include reminders of what was established in chapter 3, which makes the chapter rather repetitious.

Tomalin understands Chomsky's MA thesis, The Morphophonemics of Modern Hebrew (1951, henceforth MMH), as aiming to combine Harris' goal of developing a discovery procedure for grammars from a corpus with Goodman's constructional system theory and conception of simplicity. There is a long discussion of Chomsky's use of the concept of simplicity in MMH, and of his more substantive view of simplicity in his massive 1955–1956 work The Logical Structure of Linguistic Theory (Chomsky Reference Chomsky1955 [1975], LSLT). In effect, Tomalin thinks that Chomsky took Goodman's program in TSA and applied it to Hebrew morphophonemics. (This would mean that Goodman's influence on Chomsky predated the publication of TSA, perhaps through material presented in classes.)

In section 4.4, ‘Constructive nominalist syntax’ (121–125), Tomalin discusses Chomsky's first published paper, ‘Systems of syntactic analysis’ (1953), which definitely does show some influence of Goodman's technical work. But we see again the equivocation about Formalism: ‘The Formalist emphasis of Chomsky's paper is clear’, he asserts, because he aims to keep syntactic analysis ‘purely formal’ (122), following the early 1950s ‘trend in syntactic theory to develop methods of analysis that do not require access to semantic information’ (121). This remark embodies a confusion (the same one that we just saw with reference to Bloomfield). The semantic information that the analysis needs no access to is the semantics of the natural language being analysed, not the semantics of the language in which the analysis is couched. Chomsky's interest in a mechanical procedure for going from a corpus to a description is perhaps reminiscent of what Tomalin calls ‘formalisation’; but it is not the project of proving mathematics consistent that is influencing linguistics here.

We finally arrive at a discussion of early TGG in Tomalin's fifth chapter, which revisits (i) Chomsky's much-discussed rejection of discovery procedures and advocacy of evaluation measures; (ii) the idea that the ‘constructional levels’ of the Aufbau are hierarchical and so is the general theory of linguistic structure in LSLT; and (iii) the differences between transformational rules, recursive rules, and formal syntax. (The frequent returns to previously discussed topics for the purpose of examining their influence on early TGG reveals again that LFS is organised in a less than optimal way.)

Tomalin misses an important opportunity in his treatment of Hockett (Reference Hockett1955) on pages 144f., in a section where he rehearses Chomsky's already much-repeated arguments against finite-state automata as an adequate model for English. To set the stage for this, Tomalin notes that in the introduction to A Manual of Phonology Hockett sketches ‘a Grammatical Headquarters (GHQ), which is responsible for generating the sentences that are spoken’ (LFS: 144). This takes the form of a stochastic finite-state generator, not just another structuralist analytical method. What Hockett proposes is in effect that a probabilistic generative grammar is an actual component of the human language user. The sketch given is crude; the illustrative example, which generates a tiny finite language, is just an expository toy, and many important differences separate Hockett's GHQ from later conceptions of the language faculty. Nonetheless, Tomalin completely misses this evidence that Hockett (whose book Chomsky actually reviewed in International Journal of American Linguistics in 1955) was an advocate of both generative grammars and their neuropsychological reality (see Hockett Reference Hockett1948, also missed by Tomalin, where the latter point is extremely clear).

Another opportunity is missed in the discussion of Chomsky's arguments against stochastic models of languages – the only new topic introduced in this chapter. LFS cites the true claim from Chomsky (Reference Chomsky1957) that Colorless green ideas sleep furiously and Furiously sleep ideas green colorless had the same frequency in English before 1957 (namely, zero). But he adds the false claim that ‘[t]herefore, [Chomsky] is obliged to conclude that frequency reveals nothing about grammaticality’ (148). This is not true. Chomsky does assert that ‘in any statistical model for grammaticalness, these sentences will be ruled out on identical grounds as equally “remote” from English’ (1957: 16), but he is wrong. He is assuming that the probability of a type of event must be regarded as zero if it has not occurred so far. That is the result that one gets from using the technique now known as maximum likelihood estimation (MLE). Chomsky was not obliged to adopt MLE. A more suitable technique had been developed during the Second World War by A. M. Turing and I. J. Good (see Sampson Reference Sampson2001, chapter 7, for an elementary exposition), and although it took a while to become known, Good had published on it by 1953.

Chomsky was simply not very interested in applying statistical methods to linguistic material, and knew little about them. In his disdain for such work he was followed by most linguists for the next forty years. But when Pereira (Reference Pereira2000) finally applied Good-Turing estimation (‘smoothing’) to the question of how different the probabilities of the two famous word sequences are from normal English text, he found that the first (the grammatical one) had a probability 200,000 times that of the second.

In a section devoted to the topic of discovery procedures and grammar evaluation (149–155), Tomalin addresses Chomsky's early bias toward, and later bias against, logical empiricism and the influence of Goodman's thinking. Here we encounter a significant mistake in philosophical interpretation: Tomalin erroneously takes Goodman to be a logical empiricist. Tomalin refers to ‘the kind of logical empiricism advocated by Goodman’ (150) and writes that in the mid-1950s Chomsky abandoned the ‘hard-line logical empiricism espoused by Goodman, and (seemingly) championed by Chomsky’ (151). One can certainly read parts of LSLT as following the path of Carnapian (reductive) logical empiricism, and Syntactic Structures as rejecting it. But it is a mistake to charge Goodman with being a reductive empiricist. He was an outspoken critic of reductive empiricism, and of foundationalism, just as Quine was. As Geoffrey Hellman put it (1977: xxiii):

Tomalin counts his extended discussion of the influence Goodman and Quine had on Chomsky as one of the significant contributions that LFS makes to the history of linguistics. He is certainly right that the influence of Carnap, Goodman, and Quine on early TGG had previously been given no serious attention. But perhaps because Tomalin is venturing into literature that is unfamiliar to him, he gets Goodman wrong.

The question of whether Chomsky followed Quine and Goodman in rejecting foundationalism is worth considering on at least two levels: with reference to the epistemology of linguistics, and with reference to the epistemological question of how we can ‘know’ a language. There is some support for thinking that Chomsky retained his earlier foundationalism. However, to show this we first have to point out a further mistake of Tomalin's in interpreting Goodman.

Tomalin speculates that Chomsky's mistrust of inductive generalisation was based on Goodman's arguments in Fact, Fiction and Forecast (1954, henceforth FFF). He thinks that Goodman sees induction as a research method, and that the standard Humean problem of induction as discussed in FFF was taken as an ‘alarming condemnation of empiricism as a practical philosophy’ (153). But in fact the point of Goodman's discussion of Hume's (old) problem of induction in FFF is to argue that principles of induction (e.g., the principle that the future will resemble the past) are justified in just the same way that deductive rules of inference are justified. Goodman sees inductive and deductive rules of inference as equally justified. His discussion concerns the justification of rules of inference generally, not research methods, or empiricism (reductive or not), and there is little reason to think it contains the seeds of Chomsky's doubts. If it did, it would follow that justification of deductive inference was no better off. One may speculate that Chomsky's mistrust of conclusions reached by inductive inference stemmed from his foundationalism, since even good inductive arguments do not establish their conclusions with the same certainty as good deductive arguments. But his doubts concerning induction are not based on Goodman's arguments.

5. Transformational generative grammars and the work of Post

Tomalin moves on to discuss TGG specifically, first describing some aspects of the theory developed in LSLT, and then discussing transformations and recursion. It is regrettable that he does not make use of or cite the very important and thought-provoking assessment of LSLT in Sampson (Reference Sampson1979); it would have been useful to see Tomalin engage with the contrarian argument given there.

In the remainder of the discussion of TGG we find the most significant citational failure in LFS: its failure to note the importance of the work of the mathematical logician Emil Leon Post for the mathematical foundations of TGG. Post is cited very briefly seven times in LFS, always in order to mention (in a rather repetitive way) his work on the mathematics of mechanically enumerable sets of positive integers as summarised informally for a mathematical audience in Post (Reference Post1944). There Post uses the device of finitely describable idealized procedures (e.g., ‘write down 2, 4, 8, 16, 32, … , and continue thus forever’) as a way of specifying infinite sets whose membership could be computationally enumerated. In the terminology Post uses, these procedures generate the sets. And to read Tomalin, one might think that the only relevant thing about Post is that Chomsky (Reference Chomsky1959: 137n.) credits him with having used the technical term ‘generates’ to denote the relation between an enumeration procedure and the enumerated set (see LFS: 64, 169f.).

That citation of Post (Reference Post1944) as the source for ‘generates’ appears to be the only actual bibliographical citation of Post that Chomsky ever made. The passing mentions of Post in Chomsky (Reference Chomsky, Ernest Nagel and Alfred1962: 539) and (1965: 9) are not accompanied by citations. Chomsky nowhere mentions Post or references his work in LSLT, though he does cite Rosenbloom (Reference Rosenbloom1950), which was strongly influenced by Post (see especially Rosenbloom's chapter IV). This may mean that Chomsky got his understanding of Post mainly via Rosenbloom. The fact is that Chomsky never cites the papers in which Post makes the contributions that are most relevant to TGG: the ones applying his formalisation of deduction to general combinatorial problems about strings.

Post's 1920 doctoral dissertation, cut by a third to produce the published version in Post (Reference Post1921), was devoted to providing a fully mathematicised treatment of the deductive system of propositional logic employed in Whitehead & Russell (1910–Reference Whitehead and Bertrand1913). Part of this involved devising a kind of rule system for manipulating uninterpreted symbol strings, one which would be general enough to embrace any imaginable deductive consequence relation (see 1921: 276). This involved the formalisation step in that it mechanically mapped logical propositions to meaningless strings of symbols; it was formal in the sense that no semantic properties of propositions were relevant to what the system did with their corresponding strings; and it was a kind of Formalism in that it was aimed at establishing consistency proofs, not for mathematics but for Whitehead & Russell's propositional calculus. It would thus have been essential material for Tomalin, given his aim of linking Formalism to the origins of TGG, but he overlooks the whole of Post's main body of work.

Taking up his work twenty years later (after a long period of illness), Post (Reference Post1943) defined a ‘canonical production system’ over a finite symbol alphabet A as a set of initially given strings defined over A (the ‘primitive assertions’) and a set of rules (he called them ‘productions’) for deriving further strings. A rule, very much like a (generalised) transformation in early TGG, consists of a structural description and a structural change. The structural description is a set of patterns of the form g ₀P ₁g ₁P ₂g ₂ … P _kg _k against which symbol strings are to be matched. The gs are given strings that must be matched by copies of themselves, and the Ps are free variables over strings that can be matched by any substring. A set of such patterns ‘produces’ a further string, defined by an analog of the structural change of a transformation: another sequence of strings and variables in which all of the free variables are copied from those in the structural description. A canonical production system is said to generate the set of all and only those strings that can be produced through iterative use of its rules.

What these papers reveal is that it was Post who invented rewriting systems. Linguists appear to have overlooked this point, but theoretical computer scientists have not; see the very clear discussions in Brainerd & Landweber (Reference Brainerd and Lawrence1974: 159ff.) and Kozen (Reference Kozen1997: 256ff.).

Post also did the first work on generative power. The extremely general canonical systems of Post (Reference Post1943) were equivalent to Church's lambda calculus and to Turing machines. Post defined a normal production system as a very restricted kind of canonical system, with only one initial string, only one pattern in a structural description, only one variable in a pattern, and rules limited to the very spare form ‘g ₀P ₁ produces P ₁g ₁’ – more simply, ‘xZ→Zy’ where x and y are specific strings and Z is a variable over strings. Each such rule removes the designated x from the beginning of the string and adds y at the end. Post proved by a succession of reductions that normal systems can define any set of strings over an alphabet A that canonical systems can define, provided only that certain extra symbols not in A are permitted to appear in the strings manipulated by the rules. These extra symbols are what linguists now call non-terminals, or syntactic category symbols. (The free string variables are different: they appear only in rules. The extra non-terminal symbols appear in intermediate strings in derivations, though not in the strings ultimately counted as belonging to the language.)

The differences between Post's systems and the TGGs of Chomsky (Reference Chomsky1957) lie mainly in the additional devices that Chomsky assumed, like rule ordering and obligatory application; but these turn out neither to restrict nor to enhance generative power. Post had, in effect, already proved that restricting transformations to the form ‘xZ→Zy’ did not reduce the generative power of a grammar (and thus that ‘Is string w generated by grammar G?’ is not in general decidable – a partial anticipation of the later result of Peters & Ritchie concerning the 1965 version of TGG).

There is another simplified form of rule which Post also showed did not reduce generative power: a system in which all the rules are of the form ‘ZxW→ZyW’. Post (Reference Post1947) called this a rule of ‘semi-Thue’ type. The Norwegian mathematician Axel Thue (Reference Thue1914) had introduced bidirectional rules of the form ‘ZxW↔ZyW’, meaning that ZxW and ZyW may be substituted for each other (equivalent to a pair of rules containing ZxW→ZyW and ZyW→ZxW), and had posed the question of finding a general method for solving problems like ‘Can the rules convert string v into string w?’ for arbitrary v and w. Post (Reference Post1947) settled the question by proving that there was no such method. In fact he proved a stronger claim: the problem is undecidable even for unidirectional replacement rules (like ZxW→ZyW).

The semi-Thue type of rule is of particular importance, because Chomsky (Reference Chomsky, Ernest Nagel and Alfred1962: 539) directly acknowledges that ‘[a] rewriting rule is a special case of a production in the sense of Post; a rule of the form ZXW→ZYW’. The type-0 rules in the classification of Chomsky (Reference Chomsky1959) are simply semi-Thue rules.

Thus, more than ten years before Syntactic Structures, Post had defined rewriting systems of a very general sort (canonical systems) and had shown that neither limiting them to normal systems (xZ→Zy) nor limiting them to type-0 rewriting rules (ZxW→ZyW) altered their Turing-equivalence.

In short, Post's role in developing the methods used in early TGG has been underestimated. This was an important topic for Tomalin's project, but because of his reliance on Chomsky for bibliographical references, he never came upon the relevant papers.Footnote ⁵