Pattern, language and algebra

I’ve spent a good deal of time exploring how mathematics can be seen in how the body lives – the mental magnitudes that are our experience of time and space, the presence of arithmetic reasoning in pre-verbal humans and nonverbal animals, cells in the brain that abstract visual attributes (like verticality), the algebraic forms in language, and probabilistic learning, to name just a few.

But I believe that the cognitive structures on which mathematics is built (and which mathematics reflects) are deep, and interwoven across the whole range of human experience. Perhaps our now highly specialized domains of research are inhibiting our ability to see the depth of these structures. I thought this, again, when a particular study on the neural architecture underlying particular language abilities was brought to my attention. The study, published in the Journal of Cognitive Neuroscience, investigated the presence of this architecture in the newborn brain.

Breaking the linguistic code requires the extraction of at least two types of information from the speech signal: the relations between linguistic units and their sequential position. Further, these different types of information need to be integrated into a coherent representation of language structure. The brain networks responsible for these abilities are well-known in adults, but not in young infants. Our results show that the neural architecture underlying these abilities is operational at birth.

The focus of the study was on the infants’ ability to discriminate patterns in spoken syllables, specifically ABB patterns like “mubaba” from ABC patterns like “mubage” The experiments were also designed to determine if the infants could distinguish ABB patterns from AAB patterns. The former is about identifying the repetition, while the latter about identifying the position of the repetition. Changes in the concentration of oxygenated hemoglobin and deoxygenated hemoglobin were used as indicators of neural activity. Results suggest that the newborn brain can distinguish both ABB sequences and AAB sequences from a sequence without repetition (an ABC sequence). And neural activity was most pronounced in the temporal areas of the left hemisphere. Findings also suggested that newborns are able to distinguish the initial vs. final position of the repetition, with this response being observed more in frontal regions.

All of this seems to say that newborns are sensitive to sequential position in speech and can integrate this information with other patterns. This identification of pattern to meaning, or the meaningfulness of position, certainly resembles something about mathematics, where the meaningfulness of pattern and position is everywhere.

The connection between pattern, language and algebra is more directly addressed in a more recent paper: Phonological reduplication in sign language (Frontiers in Psychology 6/2014). Here the role of algebraic rules in American Sign Language is considered, where words are formed by shape and movement.

This is the statement of how we are to understand rule:

The plural rule generates plural forms by copying the singular noun stem (Nstem) and appending the suffix s to its end (Nstem + s). This simple description entails several critical assumptions concerning mental architecture…First, it assumes that the mind encodes abstract categories (e.g., noun stem, Nstem), and such categories are distinct from their instances (e.g., dog, letter). Second, men- tal categories are potentially open-ended—they include not only familiar instances (e.g., the familiar nouns dog, cat) but also novel ones. Third, within such category, all instances—familiar or novel—are equal members of this class. Thus, mental categories form equivalence classes. Fourth, mental processes manipulate such abstract categories—in the present case, it is assumed that the plural rule copies the Nstem category. Doing so requires that rules operate on algebraic variables, akin to variables from algebraic numeric operations (e.g., X→X+1)1. Finally, because rule description appeals only to this abstract category, the rule will apply equally to any of its members, irrespective of whether any given member is familiar or novel, and regardless of its similarity to existing familiar items.

The hypothesis that the language system encodes algebraic rules is supported by a lot of data, but the paper does include a discussion of the alternative associationist architectures, or connectionist networks, where generalizations don’t depend on abstract classes but rather on specific instances that become associated (like an association between rog-rogs and dog-dogs). The authors argue, however, that algebraic rules provide the best computational explanation for experimental observations of both speakers and signers.

We also note that our evidence for rules does not negate the possibility that some aspects of linguistic knowledge are associative, or even iconic (Ormel et al., 2009; Thompson et al., 2009, 2010, 2012). While these alternative representations and computational mechanisms might be ultimately necessary to offer a full account of the language system, our present results suggest that they are not sufficient. At its core, signers’ phonological knowledge includes productive algebraic rules, akin to the ones previously documented in spoken language phonology.

All of this suggests the presence of deeply rooted algebraic tendencies that we wouldn’t find by looking for hardwired or primitive mathematical abilities. Yet it seems that abstraction and equivalence, in some algebraic sense, just happens as the body lives. The infant is ready to recognize and integrate patterns that will enable linguistic abilities and the signer seems to be operating on equivalence classes with gestures. This should encourage us to look at the formalization of algebraic ideas, and our subsequent investigation of them in mathematics, in a new way.  It’s as if we’re turning ourselves inside-out and successfully harnessing the productivity of abstraction and equivalence.  While these are not the only mathematical things the body does, the fairly specific focus of these studies suggests that abstraction and generalization as actions run deep and broad in our make-up.

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