### Math and Language

I've always believed that there is a strong similarity between mathematics and language. It seems that there's a recent study which flies directly in the face of this idea. Here's the abstract:

"A central question in cognitive neuroscience concerns

^{ }the extent to which language enables other higher cognitive

^{ }functions. In the case of mathematics, the resources of the

^{ }language faculty, both lexical and syntactic, have been claimed

^{ }to be important for exact calculation, and some functional brain

^{ }imaging studies have shown that calculation is associated with

^{ }activation of a network of left-hemisphere language regions,

^{ }such as the angular gyrus and the banks of the intraparietal

^{ }sulcus. We investigate the integrity of mathematical calculations

^{ }in three men with large left-hemisphere perisylvian lesions.

^{ }Despite severe grammatical impairment and some difficulty in

^{ }processing phonological and orthographic number words, all basic

^{ }computational procedures were intact across patients. All three

^{ }patients solved mathematical problems involving recursiveness

^{ }and structure-dependent operations (for example, in generating

^{ }solutions to bracket equations). To our knowledge, these results

^{ }demonstrate for the first time the remarkable independence of

^{ }mathematical calculations from language grammar in the mature

^{ }cognitive system."

Here's a link to the original journal article.

Here's a link to a review in Nature.

Anyway, in an effort to hold my beliefs in light of prevailing wisdom, I argue that the capacity for arithmetic is not the kind of "grammar" that prevails in mathematics.

## 1 Comments:

it looks like they didn't just test simple calculation...they also incorporated tasks that would require a mental pushdown automaton to compute (although i might be wrong, i got a D in my algorithms class).

also, even in the absence of such higher order tasks, i find it interesting that the subjects do not have to 'read' the numbers in order to calculate simple arithmetic.

it would also be interesting to see if the subjects could learn a new non-natural language grammar -- such as a lisp or forth derivative that replaced 'wordy' syntax with more 'symbolic' syntax. i vaguely remember standardized test questions where they would introduce new binary operators & associated notations then asks questions on their utilization. perhaps these types of questions could be posed to the subjects.

personally, i find the functionotopic (if you would excuse the

portmanteu) organization of the brain and its associated emergent behavior to be one of its most interesting aspects. i would be interested to see if there is a grammotopic organization that is akin to the tonotopic and visuotopic organization present in the sensory systems...the last paragraph seems to imply that there is a qualitative difference in the fashion in which the brain processes natural language and more formal grammars: "Number words may be important in childrenâ€™s acquisition of numerical concepts and their digital, orthographic, phonological, and sensory representations (9, 41). Similarly, language grammar might provide a 'bootstrapping' template to facilitate the use of other hierarchical and generative systems, such as mathematics. However, once these resources are in place, mathematics can be sustained without the grammatical and lexical resources of the language faculty. As in the case of the relation between grammar and performance on 'theory-of-mind' reasoning tasks (42), grammar may thus be seen as a co-opted system that can support the expression of mathematical reasoning, but the possession of grammar neither guarantees nor jeopardizes successful performance on calculation problems."

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