Monday, September 19, 2011

The debate over the "Basic Metaphor of Infinity"

Volume 31, issue 6 of Behavioral and Brain Sciences (from 2008) is a special issue on the psychology of mathematics. The contribution to the issue by Lance J. Rips, Amber Bloomfield, and Jennifer Asmuth contains a criticism of, among other things, George Lakoff and Rafael Núñez' theory of number acquisition. There are a number of replies as well as a counterreply in the same issue.

Competing Inferences and Wrong Learning
The problem Rips et al. have with Lakoff and Núñez' theory is that it hypothesizes a cognitive apparatus that, although it does indeed predict the phenomena in question, could equally be used to predict the opposite patterns. So, Lakoff and Núñez can explain how children learn that addition is commutative or that there are infinitely many natural numbers; but their theory could explain the opposite just as well.

For instance, with respect to the "closure" (as they call it) of addition, they ask,
given everyday limits on the disposition of objects, why don't people acquire the opposite “nonclosure” property – that collections of objects cannot always be collected together – and project it to numbers? (p. 636)
In the same vein, they object to Lakoff and Núñez' idea of the "Basic Metaphor of Infinity":
Although there may be a potential metaphorical mapping from iterated physical processes to infinite sets of numbers, it is at least as easy to imagine other mappings from iterated processes to finite sets. Why would people follow the first type of inference rather than the second? (p. 636)
Thus, the Basic Metaphor of Infinity cannot save abstract arithmetic, since it's open to the exact same problem as arithmetic was in the first place.

In other words, real-world experience is not unambiguous---it doesn't crystallize into one single, coherent pattern with no competing alternatives. There is no reason why one pattern should end up winning out all alternatives.

The Generalizations of the Brain
Lakoff replies:
This is true of direct experience in the world, but not of the neural circuitry learned on the basis of repeated successful “small” cases of object collection, taking steps in a given direction, and so on. Our theory holds despite such physical limitations on large collections in the world. (p. 658)
I am not entirely sure how to read this objection. Apparently, neither do Rips et al. because in their counterreply, they state that they "don't see how transposing the problem from mind to brain helps solve it."

But here's a reading: Maybe Lakoff is saying that we don't have any basic experience (n < 4) of sets that we couldn't add more objects to? So because there's no experience of failure within that limited domain, we conclude that the process can go on indefinitely?

Categorizing Experience: Small and Big Counterexamples
There's problem with this idea, though; it seems to entail that any process, if it's long enough, will be conceptualized as infinite. We should then learn that, for instance, counting a pile of coins will be an unending process.

Conversely, suppose we do learn about the bounds of very long but finite processes (such as counting the leaves on a tree). Then Lakoff needs to explain why we do not generalize real-world limitations on indefinite processes (such as walking along a road until we get tired). Either we learn from actual physical bounds, or we don't. Lakoff seems to want it both ways, depending on what he finds in actual mathematics in the particular case.

His best bet here, I guess, is to insist on the unity of a process like counting---any counterexample to one instance of a counting process will automatically teach us something about all counting processes. This will allow him to fold the infinity of the number line into the infinity of possible future experiences with counting. Or, put differently, fold the infinity within the model into the infinity of models.

So, for instance, we do experience examples of bounds on counting within the sensible horizon. We can count three coins and be done. But if the process is indefinite, we do not encounter such example. We never walk 3 steps and then feel that we have exhausted the possibilities of taking further steps. We might be physically blocked, but that's more like being interrupted during counting than it's like encountering a limit.

This would do the trick, as far as I can see. But of course, it would come at the price of presupposing a perfectly functioning ability to recognize different activities as instances of the same process as well as the ability to separate contingent from essential obstacles.

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