Wednesday, March 27, 2013

Big Data

Probably since forever philosophers and mathematicians have dreamed of mechanizing thought, of removing judgment from thinking. The newest aspirant in this millennial quest is Big Data, and not surprising there is an eponymous book (excerpted here) with the following provided summary:

This revelatory exploration of big data, which refers to our newfound ability to crunch vast amounts of information, analyze it instantly and draw profound and surprising conclusions from it, discusses how it will change our lives and what we can do to protect ourselves from its hazards.

Big Data (BD) is the new New Thing, the method by which diligence can substitute for thought. The idea actually has a certain charm as it reverberates with our sense of justice. Collecting data is hard work, but it is generally the kind of work for which effort is rewarded. Work hard and you will do well. Put in the hours and the data will pile up.  It’s an activity that rewards virtue.

In this it is entirely unlike coming up with a plausible analysis, aka thinking. This activity is totally unfair.  Lazy people can have excellent ideas. Sloth is no bar to insight and profligacy no guarantee of intellectual stagnation.  Here even the wicked, sloppy, and lazy can prosper. How unfair.

In a just world, virtue would be rewarded. In a just world hard work would guarantee enlightenment. We don’t live in a just world. Big Data is the unfounded belief that this can be remedied. The hard work of data gathering can substitute for the caprice of thought. It cannot be, and, unfortunately, believing it can is likely to deform scientific practice.  To see this, consider the following quote:

The era of big data challenges the way we live and interact with the world. Most strikingly, society will need to shed some of its obsession for causality in exchange for simple correlations: not knowing why but only what. This overturns centuries of established practices and challenges our most basic understanding of how to make decisions and comprehend reality (10).

And that is precisely the problem.  Big Data is part of an enterprise aimed at reforming scientific practice. Dump why aim for what.  However, contrary to the prevailing conception, without a model/theory it is not clear what it means to just “look for correlations.” Data do not speak for themselves. So gathering lots of data will not result in eloquent models that understand the whats that matter. Big data sets cannot pull themselves up by the bootstraps (nothing can pull itself up by the bootstraps!) thereby yielding useful models. So, without explicit thoughtful models that guide the enterprise, we will be saddled with implicit models that obscure (and trivialize) what we are doing (as noted here without good models it is even difficult to separate good data from bad).

None of this would be worth mentioning were it not for the mesmerizing powers of Big Data. We have seen this before (here, and here for example).  Big Data is the modern avatar to classical empiricist methodology. It’s appeal is its promises to provide insight without intellectual sweat. This time, however, Empiricism has found a slogan attached to a technology, Google being the all-powerful mantra. Not surprisingly, money-making slogans can be very enticing and Google intellectuals (e.g. Peter Norvig) can gain powerful platforms. And though I am quite sure that like all other (empiricist) attempts to circumvent thought, this too will ultimately fail, it’s demise may not come soon enough to prevent serious damage. So when you hear the siren calls of Big Data I suggest the following prophylactic procedure; repeat Kant’s dictum to yourself, viz. data without theory is blind, data without theory is blind, data without theory is blind…and hope it soon goes away.

Monday, March 25, 2013

Generative Grammar's Bad Press

David Pesetsky gave a talk at the most recent LSA in Boston that addressed the lack of sympathy for Generative Grammar in the popular press.  Indeed, it extends beyond that. Neither Science nor Nature are well know outlets of generative research.  At the end of the talk, I put in my two cents. I suggested that a large obstacle to generous treatment was the fact that the most influential and prolific living linguist (Guess who? A hint: initials are NC) was also one of the leading political dissidents in the West. Much as our media love to laud dissidents over there they have little taste for those that highlight the shortcomings of our own cultural, political and financial elite.  At any rate, suffice it to say that Chomsky's political writings have irritated a lot of people.  These people thus have an interest in poisoning this well and one strategy is to show that even in his area of professional competence his insights have proven false and his research overthrown.  This is an especially urgent task, it seems, for the liberal media like the New Yorker (see here). Glenn Greenwald has an excellent discussion of how character assassination is intended to work.  Sadly, similar BS is not absent even within linguistics.  I need not name names: we all know who the offenders are. So why is generative linguistics a hard sell? Well, part of the answer has nothing to do with the results of the field. What to do about it? In our small part of the world, we can refrain from character assassination and stick to debating the issues.  Ideas are fair game, personalities are not.

Animal Cognition and General Intelligence

There is a long tradition of attributing linguistic facility in humans to our greater general intelligence. This has always been a hard sell to anyone who has had a pet (a humbling experience; thanks Sampson) but it persists.  The conceit has been to substitute greater general intelligence for dedicated species specific capacities.  However, this tack has become harder the more we learn about animal cognition, given the increasing evidence that animals are pretty damn smart.  Here is another popular review from the front. Chimps, elephants and octopi have proven themselves to be very very clever. Nonetheless, not a word from any of them (though I have been trying to get Sampson to guest blog).  Smart as they are, anything remotely like our linguistic capacity is still unattested in even the smartest animals.  Think that maybe being smart and being linguistically facile are unrelated?

Wednesday, March 20, 2013

Guest Post: Jeffrey Watumull on Postal's Critique of Biolinguistics

Jeff Watumull send me an early version of the following and I immediately asked him if I could post it. It is a vigorous rebuttal of Postal's argument outlined in several of his more recent papers.  An earlier post on a similar topic generated a lot of interest and discussion. Jeff here argues (convincingly in my view) that Platonism and the biolinguistic program are perfectly compatible. If correct, and I will let you dear readers judge for yourselves, there is even less to Postal's Platonist critique of the biolinguistic program than I earlier conceded. Jeff argues that Postal's critique is a non-sequitur even on its Platonist own assumptions. The good news: if you find Platonism appealing you can still be a good biolinguist.  Whew! Thanks Jeff. Enjoy the piece. Oh yes, the post is on the long side. That's the price you pay for a comprehensive critique.

Biolinguistics and Platonism: Contradictory or Consilient?

Jeffrey Watumull

1 Introduction

In “The Incoherence of Chomsky’s ‘Biolinguistic’ Ontology” (Postal 2009), Postal attacks biolinguistics as “junk linguistics” (Postal 2009: 121) with an “awful” (Postal 2009: 114) ontology expounded in “gibberish” (Postal 2009: 118), the “persuasive fore of [which] has been achieved only via a mixture of intellectual and scholarly corruption” (Postal 2009: 104), whereas writings espousing Postal’s ontology “manifest substance and quality of argument at an incomparably higher intellectual level than [Chomsky’s]” (Postal 2009: 105).  As a proponent of biolinguistics, I am tempted to reply in kind to such invective, but to do so would be bad form and bad science.  A fallacy free and dispassionate—if disputatious—rebuttal is necessary and proper.

For Postal, language is a Platonic object, and therefore he concludes that the biolinguistic assumption of a physical basis for language is “absurd” (Postal 2009: 104).  To the contrary, I shall show Postal’s conclusion to be a non sequitur.

By engaging in this argument, I fully expect Postal to accuse me of having “chosen to defend something [i.e., biolinguistics] its own author [i.e., Chomsky] is unwilling to” (Postal 2009: 105), from which two conclusions necessarily follow in Postal’s mind: (i) I am a living testament to Chomsky’s “intellectual and scholarly corruption” of the youth; and (ii) “By exercising his undeniable right of silence here, Chomsky leaves unimpeded the inference that he has not attempted a refutation because he cannot” (Postal 2009: 105).  It goes without saying that I reject these conclusions and the premise from which they do not follow.  (Incidentally, (i) corrupting the young has noble precedents (e.g., a case from 399 BCE) and (ii) the argumentum a silentio is a classic fallacy.)

This is not an apologia for Chomsky.  Biolinguistics has no single author: it is a research program pursued by numerous individually-thinking scientists subordinate to no individual however foundational and influential.  Moreover the theoretical and empirical contributions of the diverse subprograms in which these scientists work are so numerous and important that none can be “dominant” (Postal 2009: 104): in the intersection of cognitive science, linguistics, and the formal sciences, the formal properties and functional architecture of linguistic cognition are being specified; evolutionary biology is investigating possible homologues/analogues of language in nonhuman animals; genetics is discovering some of the genes entering into the development and operation of the language faculty; neuroscience is mapping the physical substrate of linguistic processing; and this is but a subset of the biolinguistics program to “reinstate the concept of the biological basis of language capacities” (Lenneberg 1967: viii).

The subprogram I work in, call it mathematical biolinguistics, is so theoretically and empirically eclectic that I am naturally interested in its ontology.  It therefore cannot be “odd for [Postal’s] opposite in the present exchange to be anyone other than Chomsky” (Postal 2009: 105). 

In the next section I very briefly and very informally define the biolinguistics Postal impugns.  The third section is a rehearsal of Postal’s arguments for linguistic Platonism and ipso facto (so he assumes) against biolinguistics.  I proceed in the fourth section to analyze some of the flaws in these arguments, demonstrating that the ontologies of Platonism and biolinguistics—properly defined—are not mutually exclusive and contradictory, but in fact mutually reinforcing and consilient in a coherent and compelling philosophy of language.

I must add that my work and the ontology it assumes are not representative of all biolinguistic research.  Many would accept my thesis that, just as engineers have encoded abstract software into concrete hardware, evolution has encoded within the neurobiology of Homo sapiens sapiens a formal system (computable functions) generative of an infinite set of linguistic expressions, modulo my understanding of the formal system as a Platonic object.  Nor is mine the only coherent interpretation of biolinguistics.  So it must not be thought that someone with my philosophy is the only possible “opposite [to Postal] in the present exchange.”

2 Biolinguistics

Let the ontology of some research program be defined as “biolinguistic” if it assumes, investigates, and is informed by the biological basis of language—a definition subsuming many productive programs of research in the formal and natural sciences.  But so general a definition cannot adjudicate the case with Postal.  At issue here is the particular definition of biolinguistics that identifies language as I-language—i.e., a computational system (a function in intension) internal to the cognitive/neurobiological architecture of an individual of the species Homo sapiens sapiens—the properties of which are determined by the three factors that enter into the design of any biological system: genetics, external stimuli, and laws of nature.

That Chomsky invented the term I-language and has expatiated on the three factors does not render him the “author” (Postal 2009: 105) of biolinguistics—that would be a category error analogous to attributing “authorship” of evolutionary biology to Darwin given his invention of the term natural selection and expatiation on the factors entering into common descent with modification.  Biolinguistics and evolutionary biology are research programs to investigate objects and processes of nature.  Thus the only author of I-language is nature.  And thus anyone is free to recognize the ontology of biolinguistics as here defined.

3 Platonist Ontology

The incoherence of the biolinguistic ontology is claimed to derive from the fact that “there can be no such thing” (Postal 2009: 105) as biolinguistics, which assumes that “a mentally represented grammar and [the language-specific genetic endowment] UG are real objects, part of the physical world, where we understand mental states and representations to be physically encoded in some manner [in the brain].  Statements about particular grammars or about UG are true or false statements about steady states attained or the initial state (assumed fixed for the species), each of which is a definite real-world object, situated in space-time and entering into causal relations” (Chomsky 1983: 156-157).  To Postal, this ontology is as “absurd” as a “biomathematics” or a “biologic,” for “Were mathematics biological, brain research might resolve such questions as whether Goldbach’s Conjecture is true.  Were logic biological, one might seek grants to study the biological basis of the validity of Modus Ponens.  The ludicrous character of such potential research is a measure of the folly of the idea that these fields study biological things” (Postal 2009: 104, 105).
By analogy, Postal argues that the objects of linguistic inquiry are not physical (a fortiori not biological), but rather “like numbers, propositions, etc. are abstract objects, hence things not located in space and time, indeed not located anywhere.  They are also things which cannot be created or destroyed, which cannot cause or be caused.  [Natural languages] are collections of other abstract objects normally called sentences, each of which is a set” (Postal 2009: 105). 

In the paper under consideration, Postal does not expound this ontology (see Postal 2004); a “brief exposition of its essence” (Postal 2009: 106) suffices for his and my purposes.  Essential to the ontology—a form of linguistic Platonism—are the type/token distinction and discrete infinity.

3.1 Types/Tokens

“ES IST DER GEIST DER SICH DEN KORPER BAUT: [S]uch is the nine word inscription on a Harvard museum.  The count is nine because we count der both times; we are counting concrete physical objects, nine in a row.  When on the other hand statistics are compiled regarding students’ vocabularies, a firm line is drawn at repetitions; no cheating.  Such are two contrasting senses in which we use the word word.  A word in the second sense is not a physical object, not a dribble of ink or an incision in granite, but an abstract object.  In the second sense of the word word it is not two words der that turn up in the inscription, but one word der that gets inscribed twice.  Words in the first sense have come to be called tokens; words in the second sense are called types” (Quine 1987: 216-217).

The distinction applies to sentences: for instance, in the classic story by Dr. Seuss, there exist (by my quick count) six tokens of the one type I do not like green eggs and ham.  Postal defines sentence tokens and types as the objects of inquiry for biolinguistics and linguistic Platonism, respectively.  For biolinguistics, as Postal understands it, a sentence is nothing more than a “brain-internal token” (Postal 2009: 107)—a mental representation.  Such an object is defined by spatiotemporal (neurobiological) coordinates with causes (cognitive, chemical, etc.) and effects (e.g., in reasoning and communication).  For linguistic Platonism, as Postal understands it, this physical object is (if anything) a token of an abstract type, with only the latter being really real.  Empirically, “island constraints, conditions on parasitic gaps, binding issues, negative polarity items, etc.” obtain not of physical objects per se, but of abstractions: “Where is the French sentence Ça signifie quoi? — is it in France, the French Consulate in New York, President Sarkozy’s brain?  When did it begin, when will it end?  What is it made of physically?  What is its mass, its atomic structure?  Is it subject to gravity?  Such questions are nonsensical because they advance the false presumption that sentences are physical objects” (Postal 2009: 107).  For Postal this nonsense is nonfinite.

3.2 Discrete Infinity

“[T]he most elementary property of language—and an unusual one in the biological world—is that it is a system of discrete infinity consisting of hierarchically organized objects” (Chomsky 2008: 137).  “Any such system is based on a primitive operation that takes n objects already constructed, and constructs from them a new object: in the simplest case, the set of these n objects” (Chomsky 2005: 11).  Call [the operation] Merge.  Operating without bounds, Merge yields a discrete infinity of structured expressions” (Chomsky 2007: 5). 

Postal invokes the type/token distinction in his critique of this biolinguistic conception of discrete infinity.  He assumes that any object constructed by a physical system must be physical: “Consider a liver and its production of bile, a heart and its production of pulses of blood; all physical and obviously finite.  And so it must be with any cerebral physical production” (Postal 2009: 109).  Thus if language is a physical (neurobiological) system, then its productions (sentences) must be physical (neurobiological tokens).  But physical objects are by definition bounded by the finiteness of spatiotemporal and operational resources: “There is for Chomsky thus no coherent interpretation of the collection of brain-based expressions being infinite, since each would take time and energy to construct, [...] store, process, or whatever[...]; they have to be some kind of tokens” (Postal 2009: 109, 111).  More abstractly, a discretely (denumerably) infinite set is one with expressions (members) that can be related one-to-one with the expressions of one of its subsets (and with the natural numbers).  But if language is a neurobiological system, hence finite, then obviously it cannot contain or construct a set that can be related to the (countable) infinity of natural numbers: “every physical production takes time, energy, etc. and an infinite number of them requires that the physical universe be infinite and, internal to Chomsky’s assumptions, that the brain be” (Postal 2009: 111).  Reductio ad absurdum, supposedly.
If biolinguistics implies that expressions are bounded by the spatiotemporal and operational resources of neurobiology, then the (infinite) majority of expressions contained in the discrete infinity are generable only in principle: there exist infinitely many more possible sentences than can ever be generated in the physical universe.  So for the biolinguistic system to be defined as discretely infinite, it must be defined as an idealization: a system abstracted away from the contingent nature of the spatiotemporal and operational resources of neurobiology.  In other words, the biolinguistic system is discretely infinite only if abstracted from biology.  And this, Postal concludes, is the fundamental fallacy: 

If “the biological [Merge function] ‘ideally’ generates an infinite collection, most of the ‘expressions’ in the collection cannot be physical objects, not even ones in some future, and the [natural language] cannot be one either.  [A]lmost all sentences are too complex and too numerous [...] to be given a physical interpretation[...].  In effect, a distinction is made between real sentences and merely ‘possible’ ones, although this ‘possibility’ is unactualizable ever in the physical universe.  According to the biological view, [...] the supposedly ‘possible’ sentences are, absurdly, actually biologically impossible.  Thus internal to this ‘defense’ of Chomsky’s biolinguistic view, the overwhelming majority of sentences cannot be assigned any reality whatever internal to the supposed governing ontology.  This means the ontology can only claim [natural language] is infinite because, incoherently, it is counting things the ontology cannot recognize as real” (Postal 2009: 111).
If, however, tokens as physical objects can implement abstract types, then presumably a recursive rule—a finite type—could be tokenized as a procedure in the mind/brain.  This Postal concedes: although “nothing physical is a rule or recursive,” because recursive rules are Platonic, a “physical structure can encode rules” (Postal 2009: 110).  Presumably therefore Merge—the mentally-represented/neurobiologically-implemented recursive procedure posited in biolinguistics to generate discrete infinity—is a legitimate posit.  But Postal objects: “an interpretation of physical things as representing particular abstractions [is] something Chomsky’s explicit brain ontology has no place for” (Postal 2009: 110).  Furthermore, Merge is supposed to generate sets, and sets are Platonic abstractions, but as “an aspect of the spatiotemporal world, [Merge] cannot ‘generate’ an abstract object like a set” (Postal 2009: 114).  So Merge is either biological—not mathematical and hence incapable of generating a set (let alone an infinite one)—or it is mathematical—hence nonbiological but capable of generating discrete infinity.  In sum, language is either physical or it is Platonic, and only under the latter definition can it be predicated of that “most elementary property,” discrete infinity—or so Postal maintains.
4 Mathematical Biolinguistic Ontology

Let me affirm at the outset my commitment to mathematical Platonism, which informs my biolinguistic ontology in ways to be discussed.  More strongly than Chomsky, who does grant mathematical Platonism “a certain initial plausibility,” I am convinced of the existence of “a Platonic heaven [of] arithmetic and [...] set theory,” inter alia, that “the truths of arithmetic are what they are, independent of any facts of individual psychology, and we seem to discover these truths somewhat in the way that we discover facts about the physical world” (Chomsky 1986: 33).  It follows from this position that I must be committed to linguistic Platonism for any linguistic objects reducible to or properly characterized as mathematical objects.  And indeed in my theory of natural language (see Watumull 2012), the quiddities that define a system as linguistic are ultimately mathematical in nature.  (The “essence” of language, if you will, is mathematical—a proposition I shall not defend here, assuming it to be essentially correct, for at issue in this discussion is not whether the proposition is true, but whether it is consistent with a biolinguistic ontology if true.) 

4.1 Overlapping Magisteria

I and others (see, e.g., Hauser, Chomsky, Fitch 2002; Watumull, Hauser, Berwick 2013) posit a recursive function generative of structured sets of expressions as central to natural language; this function is defined in intension as internal to the mind/brain of an individual of the species Homo sapiens sapiens.  So conceived, I-language has mathematical and biological aspects. 

Nonsense!  Postal would spout: The ontologies of mathematics and biology are nonoverlapping magisteria!  Assuming mathematical Platonism, I concur that a mathematical object per se such as a recursive function (the type) is not physical.  However even Postal (2009: 110) concedes that such an object can be physically encoded (as a token).  The rules of arithmetic for instance are multiply realizable, from the analog abacus to the digital computer to the brain; mutatis mutandis for other functions, sets, etc.  And mutatis mutandis for abstract objects definable as mathematical at the proper level of analysis, such as a computer program:

            “You know that if your computer beats you at chess, it is really the program that has beaten you, not the   silicon atoms or the computer as such.  The abstract program is instantiated physically as a high-level  behaviour of vast numbers of atoms, but the explanation of why it has beaten you cannot be expressed without also referring to the program in its own right.  That program has also been instantiated, unchanged, in a long chain of different physical substrates, including neurons in the brains of the programmers and radio waves when you downloaded the program via wireless networking, and finally as states of long- and   short-term memory banks in your computer.  The specifics of that chain of instantiations may be relevant to     explaining how the program reached you, but it is irrelevant to why it beat you: there, the content of the     knowledge (in it, and in you) is the whole story.  That story is an explanation that refers ineluctably to abstractions; and therefore those abstractions exist, and really do affect physical objects in the way required by the explanation” (Deutsch 2011: 114-115).
(Though I shall not rehearse the argument here, I am convinced by Gold (2006) that “mathematical objects may be abstract, but they’re NOT [necessarily] acausal” because they can be essential to—ineliminable from—causal explanations.  The potential implications of this thesis for linguistic Platonism are not uninteresting.)

I take the multiple realizability of the chess program to evidence the reality of abstractions as well as anything can (and I assume Postal would agree): something “substrate neutral” (Dennett 1995) is held constant across multiple media.  That something I submit is a computable function; equivalently, that constant is a form of Turing machine (the mathematical abstraction representing the formal properties and functions definitional of—and hence universal to—any computational system). 

4.2 The Linguistic Turing Machine

Within mathematical biolinguistics, it has been argued that I-language is a form of Turing machine (see Watumull 2012; Watumull, Hauser, Berwick 2012), even by those Postal diagnoses as allergic to such abstractions: 

“[E]ven though we have a finite brain, that brain is really more like the control unit for an infinite computer.  That is, a finite automaton is limited strictly to its own memory capacity, and we are not.  We are like a Turing machine in the sense that although we have a finite control unit for a brain, nevertheless we can use indefinite amounts of memory that are given to us externally[, say on a “tape,”] to perform more and more complex computations[...].  We do not have to learn anything new to extend our capacities in this way” (Chomsky 2004: 41-42).

As Postal would observe, this “involves an interpretation of physical things as representing particular abstractions,” which he concedes is coherent in general because obviously “physical structure can encode rules” and other abstract objects (e.g., recursive functions) (Postal 2009: 110)—computer programs, I should say, are a case in point.

4.3 Idealization

Postal (2012: 18) has dismissed discussion of a linguistic Turing machine as “confus[ing] an ideal machine[...], an abstract object, with a machine, the human brain, every aspect of which is physical.”  I-language qua Turing machine is obviously an idealization, with its unbounded running time and access to unbounded memory, enabling unbounded computation.  And obviously “[unboundedness] denotes something physically counterfactual as far as brains and computers are concerned.  Similarly, the claim ‘we can go on indefinitely’ [...] is subordinated to the counterfactual ‘if we just have more and more time.’  Alas we do not, so we can’t go on indefinitely” (Postal 2012: 18).  Alas it is Postal who is confused.

4.3.1 Indefinite Computation
Postal’s first confusion is particular to the idealization of indefinite computation.  Consider arithmetic.  My brain (and presumably Postal’s) and my computer encode a program (call it ADD) that determines functions of the form fADD(X + Y) = Z (but not W) over an infinite range.  Analogously, my brain (and Postal’s) but not (yet) my computer encodes a program (call it MERGE) that determines functions of the form fMERGE(α, β) = {α, β}—with syntactic structures assigned definite semantic and phonological forms—over an infinite range.  These programs are of course limited in performance by spatiotemporal constraints, but the programs themselves—the functions in intension—retain their deterministic form even as physical resources vary (e.g., ADD determines that 2 + 2 = 4 independent of performance resources).  

Assuming a mathematical biolinguistic ontology, I-language is a cognitive-neurobiological token of an abstract type; it “generates” sets in the way axioms “generate” theorems.  As the mathematician Gregory Chaitin observes, “theorems are compressed into the axioms” so that “I think of axioms as a computer program for generating all theorems” (Chaitin 2005: 65).  Consider how a computer program explicitly representing the Euclidean axioms encodes only a finite number of bits; it does not—indeed cannot—encode the infinite number of bits that could be derived from the postulates, but it would be obtuse to deny that such an infinity is implicit (compressed) in the explicit axioms.  Likewise, zn+1 = zn2 + c defines the Mandelbrot set (as I-language defines the set of linguistic expressions) so that the infinite complexity of the latter really is implicitly represented in the finite simplicity of the former. 

So while it is true that physically we cannot perform indefinite computation, we are endowed physically with a competence that does define a set that could be generated by indefinite computation.  (A subtle spin on the notion of competence perhaps more palatable to Postal defines it as “the ability to handle arbitrary new cases when they arise” such that “infinite knowledge” defines an “open-ended response capability” (Tabor 2009: 162).)  Postal must concede the mathematical truth that linguistic competence, formalized as a function in intension, does indeed define an infinite set.  However, he could contest my could as introducing a hypothetical that guts biolinguistics of any biological substance, but that would be unwise. 

Language is a complex phenomenon: we can investigate its computational (mathematical) properties independent of its biological aspects just as legitimately as we can investigate its biological properties independent of its social aspects (with no pretense to be carving language at its ontological joints).  In each domain, laws—or, at minimum, robust generalizations—license counterfactuals (as is well understood in the philosophy of science).  In discussing indefinite computation, counterfactuals are licensed by the laws expounded in computability theory:

“[T]he question whether a function is effectively computable hinges solely on the behavior of that function in neighborhoods of infinity[...].  The class of effectively computable functions is obtained in the ideal case where all of the practical restrictions on running time and memory space are removed.  Thus the class is a theoretical upper bound on what can ever in any century be considered computable” (Enderton 1977: 530).

A theory of linguistic competence establishes an “upper bound,” or rather delineates the boundary conditions, on what can ever be considered a linguistic pattern (e.g., a grammatical sentence).  Some of those patterns extend into “neighborhoods of infinity” by the iteration of a recursive function.  Tautologically, those neighborhoods are physically inaccessible, but that is irrelevant.  What is important is the mathematical induction from finite to infinite: Merge applies to any two arguments to form a a set containing those two elements such that its application can only be bounded by stipulation.  In fact a recursive function such as Merge characterizes the “iterative conception of a set,” with sets of discrete objects “recursively generated at each stage,” such that “the way sets are inductively generated” is formally equivalent to “the way the natural numbers [...] are inductively generated” (Boolos 1971: 223). 

The natural numbers are subsumed in the computable numbers, “the real numbers whose expressions as a decimal are calculable by finite means” (Turing 1936: 230).  (The phrase “finite means” should strike a chord with many language scientists.)  It was by defining the computable numbers that Turing proved the coherency of a finitary procedure generative of an infinite set.

“For instance, there would be a machine to calculate the decimal expansion of π[...].  π being an infinite decimal, the work of the machine would never end, and it would need an unlimited amount of working space on its ‘tape’.  But it would arrive at every decimal place in some finite time, having used only a finite quantity of tape.  And everything about the process could be defined by a finite table[...].  This meant that [Turing] had a way of representing a number like π, an infinite decimal, by a finite table.  The same would be true of the square root of three, or the logarithm of seven—or any other number defined by some rule” (Hodges 1983: 100).

Though they have not been sufficiently explicitly acknowledged as such, Turing’s concepts are foundational to the biolinguistic program.  I-language is a way of representing an infinite set by a finite table (a function).  The set of linguistic expressions being infinite, “the work of the machine would never end,” but Postal must concede that nevertheless I-language “would arrive at every [sentence] in some finite time, having used only a finite quantity of tape.  And everything about the process could be defined by a finite table.”  This gives a rigorous sense to the linguistic notion “infinite use of finite means.” Generation and Explanation

But for all the foregoing, the finitude/infinitude distinction is not so fundamental given the fact that “[a] formal system can simply be defined to be any mechanical procedure for producing formulas” (Gödel 1934: 370).  The infinitude of the set of expressions generated is not as fundamental as the finitude of I-language (the generative function) for the following reason: it is only because the function is finite that it can enumerate the elements of the set (infinite or not); and such a compact function could be—and ex hypothesi is—neurobiologicall encoded.  Even assuming Postal’s ontology in which “[natural languages] are collections of [...] abstract objects” (Postal 2009: 105), membership in these collections is granted (and thereby constrained) by the finitary procedure, for not just any (abstract) object qualifies.  In order for an object to be classified as linguistic, it must be generated by I-language; in other words, to be a linguistic object is to be generated by I-language.  And thus I-language explains why a given natural language contains the member expressions it does. 
This notion of I-language as explanation generalizes to the notion of formal system as scientific theory:

“I think of a scientific theory as a binary computer program for calculating observations, which are also written in binary.  And you have a law of nature if there is compression, if the experimental data is compressed into a computer program that has a smaller number of bits than are in the data that it explains.  The greater the degree of compression, the better the law, the more you understand the data.  But if the experimental data cannot be compressed, if the smallest program for calculating it is just as large as it is [...], then the data is lawless, unstructured, patternless, not amenable to scientific study, incomprehensible. In a word, random, irreducible” (Chaitin 2005: 64).

This notion is particularly important, as Turing (1954: 592) observed, “[w]hen the number is infinite, or in some way not yet completed [...],” as it is for the discrete infinity (unboundedness) of language; “a list of answers will not suffice.  Some kind of rule or systematic procedure must be given.”  Otherwise the list is arbitrary and unconstrained.  So for linguistics, in reply to the question “Why does the infinite natural language L contain the expressions it does?” we answer “Because it is generated by the finite I-language f.”  Thus I-language can be conceived of as the theory explicative of linguistic data because it is the mechanism (Turing machine) generative thereof.

4.3.2 “the thing in itself”

Second, with respect to idealization generally, for mathematical biolinguistics to have defined I-language as a Turing machine is not to have confused the physical with the abstract, but rather to have abstracted away from the contingencies of the physical, and thereby discovered the mathematical constants that must of necessity be implemented for any system—here biological—to be linguistic (on my theory).  This abstraction from the physical is part and parcel of the methodology and, more importantly, the metaphysics of normal science, which proceeds by the “making of 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” (Weinberg 1976: 28).  The idealization is the way things really are.  Consider Euclidean objects: e.g., dimensionless points, breadthless lines, perfect circles, and the like.  These objects do not exist in the physical world.  The points, lines, and circles drawn by geometers are but imperfect approximations of abstract Forms—the objects in themselves—which constitute the ontology of geometry.  For instance, the theorem that a tangent to a circle intersects the circle at a single point is true only of the idealized objects; in any concrete representation, the intersection of the line with the circle cannot be a point in the technical sense as “that which has no part,” for there will always be some overlap.  As Plato understood (Republic VI: 510d), physical reality is an intransparent and inconstant surface deep beneath which exist the pellucid and perfect constants of reality, formal in nature:

            “[A]lthough [geometers] use visible figures and make claims about them, their thought isn’t directed to them but to the originals of which these figures are images.  They make their claims for the sake of the Square itself and the Diagonal itself, not the particular square and diagonal they draw; and so on in all cases.  These figures that they make and draw, of which shadows and reflections in water are images, they now in turn use as images, in seeking to behold those realities—the things in themselves—that one cannot comprehend except by means of thought.”

Analogously, any particular I-language (implemented in a particular mind/brain) is an imperfect representation of a form (or Form) of Turing machine.  But, Postal would object, the linguistic Turing machine is Platonic, hence nonbiological, and hence bio-linguistics is contradictory.  But, I should rebut, this objection is a non sequitur.
I am assuming (too strongly perhaps) that fundamentally a system is linguistic in virtue of mathematical (nonbiological) aspects.  Nevertheless, in our universe, the only implementations of these mathematical aspects yet discovered (or devised) are biological; indeed the existence of these mathematical systems is known to us only by their biological manifestations—i.e., in our linguistic brains and behaviors—which is reason enough to pursue bio-linguistics.  To borrow some rhetorical equipment, biology is the ladder we climb to the “Platonic heaven” of linguistic Forms, though it would be scientific suicide to throw the ladder away once up it.  That chance and necessity—biological evolution and mathematical Form—have converged to form I-language is an astonishing fact in need of scientific explanation.  It is a fact that one biological system (i.e., the human brain) has encoded within it and/or has access to Platonic objects.  (Postal must assume that our finite brains can access an infinite set of Platonic sentences.  The ontological status of the latter is not obvious to me, but obviously I am committed to the existence of the encoding within our brains of a finite Platonic function for unbounded computation.)  Surely a research program formulated to investigate this encoding/access is not perforce incoherent.

I do however deny any implication here that such complex cognition, “in some most mysterious manner, springs only from the organic chemistry, or perhaps the quantum mechanics, of processes that take place in carbon-based biological brains.  [I] have no patience with this parochial, bio-chauvinistic view[:] the key is not the stuff out of which brains are made, but the patterns that can come to exist inside the stuff of a brain” (Hofstadter 1999 [1979]: P-4, P-3).  Thus, as with chess patterns, it is not by necessity that linguistic patterns spring from the stuff of the brain; but the fact remains that they can and do.  And thus linguistics is just as much a biological science as it is a formal science.   

To reiterate, at present there exists no procedure other than human intuition to decide the set of linguistic patterns.  The neurobiology cannot answer the question whether some pattern is linguistic (e.g., whether some sentence is grammatical), but it encodes the procedure that enables the human to intuit the answer to such a question.  Analogously, neurobiological research would not establish the truth of Goldbach’s conjecture or the validity of reasoning by modus ponens, but rather would be unified with research in cognitive science to establish (discover) the rules and representations encoded neurobiologically that enable cognitive conjecture and reasoning.

4.4 Encoding Abstract Objects in Physical Systems

Postal believes that an “explicit brain ontology” as assumed in biolinguistics “has no place for” the encoding of an abstract object such as a Turing machine in a physical system such as the brain—but I see no grounds whatsoever for this belief.  Not only is this belief contradicted by Chomsky’s Turing machine analogy, but Postal himself quotes Chomsky discussing how in biolinguistics “we understand mental states and representations to be physically encoded in some manner” (1983: 156-157); and to physically encode something presumes a non-physical something to be so encoded.  For this reason “it is the mentalistic studies that will ultimately be of greatest value for the investigation of neurophysiological mechanisms, since they alone are concerned with determining abstractly the properties that such mechanisms must exhibit and the functions they must perform” (Chomsky 1965: 193).

It is in this sense of neurobiology encoding mathematical properties and functions that, “astonishingly” (Postal 2012: 23), we observe the obvious fact that “We don’t have sets in our heads.  So you have to know that when we develop a theory about our thinking, about our computation, internal processing and so on in terms of sets, that it’s going to have to be translated into some terms that are neurologically realizable.  [Y]ou talk about a generative grammar as being based on an operation of Merge that forms sets, and so on and so forth.  That’s something metaphorical, and the metaphor has to be spelled out someday” (Chomsky 2012: 91).  In other words, while the formal aspects of a Turing machine (e.g., Merge, sets, etc.) are, ex hypothesi, realized neurologically, it would be absurd (astonishing) to expect physical representation of our arbitrary notations (e.g., fMERGE(X, Y) = {X, Y}).  As Turing observed, in researching the similarities of minds and machines, “we should look [...] for mathematical analogies of function” (1950: 439)—similarities in software, not hardware.

Of course an ontological commitment to abstract properties and functions is not necessarily a commitment to Platonism (as Aristotle demonstrated and many in the biolinguistics program would argue), but it is certainly the default setting.  So it can be argued that I-language is just like Deutsch’s chess program: a multiply realizable computable function (or system of computable functions).  Indeed given that a Turing machine is a mathematical abstraction, I-language qua Turing machine is necessarily and properly defined as a physically (neurobiologically) encoded Platonic object.
5 Conclusion

I have argued that mathematical biolinguistics is based on the perfectly coherent concept of computation—as formulated by Turing—unifying mathematical Platonism and biolinguistics: evolution has encoded within the neurobiology of Homo sapiens sapiens a formal system (computable function(s)) generative of an infinite set of linguistic expressions (just as engineers have encoded within the hardware of computers finite functions generative of infinite output).  This thesis, I submit, is or would be accepted by the majority of researchers in biolinguistics, perhaps modulo the Platonism, for indeed it is not necessary to accept the reality of mathematical objects to accept the reality of physical computation.  However, I am a mathematical Platonist, and thus do recognize the reality of mathematical objects, and thus do argue I-language to be a concretization (an “embodiment” in the technical sense) of a mathematical abstraction (a Turing machine), which to my mind best explains the design of language.


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