Another Impossibility Argument

Things that are not possible don’t happen.  Sounds innocuous huh?  Perhaps, but that’s the basis of POS arguments.  If the evidence cannot dictate the outcome of acquisition, and nonetheless the outcome is not random, then there must be something other than the evidence guiding the process. Substitute ‘PLD’ for ‘evidence’ and ‘UG’ for ‘something other than the evidence’ and you get your standard POS argument. There have been and can be legitimate fights about how rich the data/PLD is and how rich UG is, but the form of the argument is dispositive and that’s why it’s so pretty and so useful.  A good argument form is worth its weight in theoretical gold. 

That this is what I believe is not news for those of you who have been following this blog.  And don’t worry, at least for today, I am not going to present another linguistic POS argument (although I am tempted to generate some new examples so that we can get off of the Yes/No question data).  Rather, what I want to do is publicize another application of the same argument form, the one deployed in Gallistel/King (G/K) in favor of the conclusion that connectionist architectures are biologically impossible.

The argument that they provide is quit simple: connectionism requires too many neurons to code various competences.  They dub the problematic fact over which connectionist models necessarily stumble and fall ‘the infinitude of the possible’ (IoP). The problem as they understand it is endemic; computations in a connectionist/neural net architecture (C/NN) cannot be “implemented by compact procedures.” This means that such a C/NN cannot “produce as an output an answer that the maker of the system did not hard wire into the look-up tables. (261)” In effect, C/NNs are fancy lists (aka look-up tables) where all possibilities are computed out rather than being implicit in more compact form in a generative procedure. And this leads to a fundamental problem: the brain is just not big enough to house the required C/NNs.  Big as brains are, they are still too small to explicitly code all the required possible realizable cognitive states.

G/K’s argument is all in service of arguing that neuroscientists must assume that brains are effectively Turing-von Neumann (TvN) machines with addressable, symbolic, read/write memories. In a nutshell:

 … a critical distinction between procedures implemented by means of look-up tables and … compact procedures …is that the specification of the physical structure of a look-up table requires more information than will ever be extracted by the use of that table.  By contrast, the information required to specify the structure of a mechanism that implements a compact procedure may be hundreds of orders of magnitude less than the information that can be extracted using that mechanism (xi).

What’s the argument? Like POS arguments, it starts with a rich description of various animal competences. The three that play staring roles in the book are dead reckoning in ants, bee dancing and food caching behavior in jays.  Those of you who like Animal Planet will love these sections. It is literally unbelievable what these bugs and birds can do.  Here is a glimpse of jay behavior as reported in G/K (c.f. 213-217).

In summer, when times are good, scrub jays collect and store food in different locations for later winter feasting. They cache this food in as many as 10,000 different locations. Doing this involves remembering what they hid, where they hid it, when they hid it, whether they emptied it, if the morsel was tasty, how quickly the morsel goes bad, and who was watching them when they hid it. This is a lot of information. Moreover, it is very specific information, sensitive to six different parameters.  Moreover, the values for these parameters are indeterminate and thus the number of possible memories these jays can produce and access is potentially unbounded.  Though there is an upper bound on the actual memories stored, the number of potentially storable memories is effectively unbounded (aka, infinite).  This is the big fact and it has a big implication. In order to store these memories the jays need some sort of template that roughly says ‘stored X at Y at time Z, X goes bad in W days, X is +/- retrieved, X is +/- tasty, storing was +/- observed.’ This template requires a brain/mind that can link variables, value variables, write to memory and retrieve from memory so as to store useful information and access it when necessary.  Note, we can treat this template as a large sentence frame, much like ‘X weighs Y pounds’ and like the latter there is no upper bound on the number of possible realizations of this frame (e.g. John weighs 200 pounds, Mary weighs 90 pounds, Trigger weighs 1000 pounds etc.).  These templates combined with the capacity to put actual food type/time/place/ etc. values for the variables constitute “compact procedures” for coding the relevant actual information required. Notice how “small” it is relative to the number of actual instances of such templates (finite specification versus unbounded number of instances).

If this description is correct (G/K review the evidence extensively), here is what is neuronally impossible: to list all the potential instantiations of this kind of proposition and simply choose the ones that are actual. Why?

… the infinitude of the possible looms. There are many possible locations, many possible kinds of food, many possible rates of decay, many possible delays between caching and recovery – and no restrictions on the possible combinations of these possibilities. No architecture with finite resources can cope with this infinitude by allocating resources in advance to every possible combination. (217)

However, current neuroscience takes read/write memory (a necessary feature of a system able to code the above information) to be neurobiologically implausible. Thus, current neuroscience only investigates systems (viz. C/NNs) that cannot in principle handle these kinds of behavioral data.  What’s the principled reason for this inadequacy? Computation in C/NNs is not implemented by compact procedures. Rather, C/NNs are effectively elaborate look-up tables and so cannot “output an answer that the maker of the system did not hard wire into one of its look-up tables.” (261).

That’s the impossibility argument. If we assume that brains mediate cognition then it cannot be the case that animal brains are C/NN devices.

I strongly recommend reading these sections of G/K’s book. There is terrific detailed discussion of how many neurons it would take to realize a C/NN capable of dead reckoning. By G/K’s estimates (261) it would take all the neurons in an ant’s brain (ants are terrific dead reckoners) to realize a more or less adequate system of dead reckoning.

This is fun stuff, but I am no expert in these matters and though I tend to trust Gallistel when he tells me something, I am in no position to independently verify his calculations regarding the number of neurons required to implement a reasonable C/NN device.[1]  However, G/K’s conclusion should resonate with generativists.  Grammars are compact procedures for coding what is essentially an infinite number of possible sentences and UG is (part of ) a compact procedure for coding what is (at the very least) a very large number of possible Gs.[2] Thus, whatever might be true of other animals, human brains clearly capable of language cannot be C/NNs.[3] Why do I mention this. For two reasons:

First, there is still a large cohort of neuroscientists, psychologists and computationalists who try to analyze linguistic phenomena in C/NN terms. They look at pictures of brains, see interconnected neurons, and conclude that our brains are C/NNs and that our linguistic competence must be analysed to fit these “data.” G/K argue that this is exactly the wrong conclusion to draw. Note that the IoP is trivially obvious in the domain of language, so the G/K argument is very potent here. And that is cause for celebration as these same net-enchanted types are also rather grubby empiricists.

G/K discuss the unholy affinities between associationism and C/NN infatuation (c.f. chapter 11) (the slogan “what fires together wires together” reveals all).  Putting a stake through the C/NN worldview also serves to weaken the empiricist-associationist learning conception of acquisition.[4] I doubt it will kill it. Nothing can it appears. But perhaps it can intellectually wound it yet again (though only if the G/K material is taken seriously, which I suspect it won’t be either because the net-nuts won’t get it or they will simply ignore it). So an attack on C/NNs is also a welcome attack on empiricist/associationist conceptions and that’s always a valuable public service.

Second, this is very good news for the minimalistically inclined. Here’s why.  Minimalists are counting on animal brains being similar enough to human brains for there to be features of the former that can be used to explain some of the properties of FL/UG.  Recall the conceit: take the cognitive capacities of our ancestors as given and ask what we need to add to get linguistically capable minds/brains.  However, were animal brains C/NNs and ours clearly are not (recall how easy it is to launch the IoP considerations in the domain of language) then it is very hard to see how something along these lines could be true.  Is it really plausible that the shift to language brought with it an entirely new kind of brain architecture?  The question answers itself.  So G/K’s conclusions regarding animal brain architectures is very good news.

Note, we can turn this argument around: if as minimalisms requires (and seems independently reasonable) human brains are continuous with non-human ones and if the IoP requires that brains have TvN architectures, then language in humans provides a strong prima facie argument for TvNs in animals.  For the reasons that G&K offer (not unlike those Chomsky originally deployed) human linguistic competence cannot supervene on C/NN systems. And as G/K note, this has profound implications for current work in neuroscience, viz. the bulk of the work is looking in the wrong places for answers that cannot possibly serve. Quite a conclusion, but that’s what a good impossibility argument delivers.

Let me end with one last observation: there is a tendency to think that neuroscientists hold the high intellectual ground and cognitive science/linguistics must accommodate itself to its conclusions. G/K demonstrate that this is nonsense.  Cognition supervenes on brains. If some kind of brain cannot support what we know are the cognitive facts, then this view of the brain must be wrong.  Anything else would be old fashion dualism. Hmm, wouldn’t it be amusing if to defend their actual practice neuroscientists had to endorse hard core Cartesian dualism? Yet, if G/K are right, that’s what they are effectively doing right now.

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[1] Note that the G/K argument is about physical implementation and so is additional (though related) to the arguments presented by Fodor and Pylyshyn or Marcus against connectionism. Not only does C/NN get the cognition wrong, it is also physically impossible given how many neurons it would take to implement a C/NN device to do even a half-assed job. 

[2] If UG is parametric with a finite number of parameters then the number of possible Gs will also be finite. However, it is virtually certain that the number of parameters (if there are parameters) is very high (see here for discussion) and so the space of possibilities is very very large. OF course, if there are no parameters, then there need be no upper bound on the number of possible Gs.

[3] Which does not mean that for some kinds of pattern recognition the brain cannot use C/NNs. The point is that these are not sufficient.

[4] The cognoscenti will notice that I am here carrying on my effort to change our terminology so that we use ‘learning’ to denote one species of acquisition, the species tightly tied to empiricism/associationism.