So far I have emphasized the role of visualization in mathematics, and its power of persuasion. Here I will try to unite the visual and symbolic aspects of mathematics, and touch on the limitations of visualization.
Indeed, visualization works perfectly well in the geometric the- ory offinitereflection groups, but needs to be refined for the more general theory of infinite Coxeter groups. We take a brief look at this more general theory, which is of special interest for us at this point. As truly fundamental mathematical objects, Coxeter groups provide an example of a theory where the links between mathemat- ical teaching and learning and cognitive psychology lie exposed. Besides the power of geometric interpretation and visualization, the theory of Coxeter groups relies on manipulation of words in canonical generators (chains of consecutive reflections, in the case of reflection groups) and provides one of the best examples of the effectiveness of thelanguage metaphorin mathematics.
It is tempting to try to link the psychology of symbolic manip- ulation in mathematics with the Chomskian conjecture that hu- mans have an innate facility for parsing human language. Basi- cally, parsing is the recognition/identification of the structure of a string of symbols (phonemes, letters, etc.) We parse everything we read or hear. Here is an example from Steven Pinker’s book [215, pp. 203–205] where this thesis is vigorously promoted:
Remarkable is the rapidity of the motion of the wing of the hum- mingbird.
To make sense of the phrase, we have to mentally bracket sub- phrases, making something like
[Remarkable is [the rapidity of
[the motion of [the wing of
A sentence might have a different bracket pattern, just compare [Remarkable is [the rapidity of [the motion]]]
and
[[The rapidity [that [the motion] has]] is remarkable]. Some patterns are harder to deal with than others: for example,
[[The rapidity that [the motion that [the wing] has] has] is re- markable].
Some bracketings are close to incomprehensible, even though the sentence conveys the same message:
[[The rapidity that [the motion that [the wing that [the hum- mingbird] has] has] has] is remarkable]. [?] 1
Gregory Cherlin kindly offered a brainteaser from his childhood: Punctuate: Smith where Jones had had had had had had had had had had had the profes- sor’s approval.
Different human languages have different grammars, resulting in different parsing patterns. The grammar is not innate; Pinker emphasizes that what is innate is the human capacity to generate parsing rules. Generation of parsing patterns is a part of language learning (and young children are extremely efficient at it). It is also a part of the interiorization of mental objects of mathematics, espe- cially when theseobjectsare represented by strings ofsymbols.2
Cognitive scientists are very much attracted to case studies of “savants”, autistic persons with an ability to handle arithmetic or calendrical calculations disproportionate to their low general IQ. As Snyder and Mitchell formulated it [227],
. . . savant skills for integer arithmetic . . . arise from an abil- ity to access some mental process which is common to us all, but which is not readily accessible to normal individuals.
The parsing mechanisms of the human brain are the key to the understanding of low-level arithmetic and formula process- ing.
What are these “hidden” pro- cesses? In one of the extreme cases (mentioned by Butterworth [156]), a severely autistic young man was unable to understand speech, but could handle factors and primes in numbers. This suggests that certain mathematical actions are related not so much to language itself, but to the parsing facility, one of the components of the language system. An autistic person may have difficulty in handling language for rea- sons unrelated to his parsing ability; for example he may fail to recognize the source of speech communication as another person (or to understand the difference between what he knows and what the other person knows). But, in order to achieve such feats as “dou- bling 8 388 628 up to 24 times to obtain 140 737 488 355 328 in several seconds” [227, p. 589], an autistic person still has to be able to input into his brain the numbers given, inevitably, as strings of phonemes or digits.
3.1 Parsing 45
I propose a conjecture that the parsing mechanisms of the hu- man brain are the key to the understanding of low-level arithmetic and formula processing.
Moving several levels up the hierarchy of mathematical pro- cesses, we have a fascinating idea in the theory of automatic theo- rem proving:rippling, a formalization of a common way of mathe- matical reasoning where “formulae are manipulated in a way that increases their similarities by incrementally decreasing their dif- ferences” [320, p. 13]. This is facilitated by subdividing the formula into parts which have to be preserved and parts which have to be changed. Again, we see that in order to understand how humans use rippling in mathematical thinking (and whether they actually use it), we have to understand how our brain parses mathematical formulae.
To be on the cautious side, I am prepared to accept that pars- ing might be much more prominent in the input/output functions of the brain than in the internal processing of information. In a rare case of a savant with higher than normal general intellectual abili- ties, Daniel Tammet is able to vividly describe the way he perceives the world, language, and numbers. It is obvious from his words that number processing happens to be directly wired into the vi- sual module of his brain. For him, many numbers have a unique visual form.
“Different numbers have different colours, shapes and tex- tures . . . [The number] one is very bright and shining, like someone flashing a light into my face. Two is like a move- ment from right to left. Five is a clap of thunder or the sound of a wave against a rock. Six I find more difficult: it’s more like a hole or a chasm. When I multiply numbers, I see two shapes in a landscape. The space between the im- ages makes a third shape, like a jigsaw piece. And that third shape gradually crystallises: I see a fuzziness that becomes clearer and clearer.” [426]
He adds that the whole process takes place in flash, “like sparks flying off”.
Although Daniel Tammet suffers from Asperger’s syndrome (a form of autism) which to some degree inhibits his social skills— he has to remind himself that other people have thoughts entirely separate from his own and not to assume that they automatically know everything he knows—he has outstanding linguistic skills, speaks seven languages and learned Icelandic in a week. He can also reciteπto 22,514 decimal places. His case appears to confirm the thesis by Snyder and Mitchell; indeed, he has “an ability to access some mental process . . . which is not readily accessible to normal individuals”. This very access, however, requires parsing of the input.