Speaking of which, it has been many years since I really had much contact with arithmetic. Even in high school, math competitions did not emphasize the actual operations of addition, subtraction, multiplication, and division, let alone during my entire university years. So much so that yesterday, when I suddenly had to handle a large number of digits in a number-guessing game, I abruptly discovered that the abacus in my mind had rusted over; moving each bead felt sluggish, and worse still, many of the beads had become badly loose, so that with one slip of the hand I would lose the exact count. After a few rounds my brain began to descend into utter confusion. In the middle of the night I thought of it again and played through Hundred Beads several times, only to find that I had not gotten it right once, and that my speed had also dropped dramatically. For the first fifty or so digits it was still all right, but by around seventy, with almost every new digit I had to feel for the counting beads; otherwise, if I was not careful, the beads would scatter and be impossible to hold together…
I have mentioned this technique of bead-based mental arithmetic many times on my blog, so I won’t do any more popularizing here. But perhaps people who have not learned bead-based mental arithmetic may not easily understand what I just said. In fact, “beads” is not a metaphorical usage at all, but precisely my inner image of numbers. I don’t know what numbers feel like to other people, but for me, real numbers appear in my mind in the form of counting beads.
In connection with my recent reflections on sensation and cognition, I have been thinking more about this inner image of numbers, because for me numbers are the simplest and most typical kind of multi-sensory composite image.
In fact, most concrete nouns evoke in ordinary people’s minds composite impressions of multiple senses. Take the word “apple,” for example: it calls up the auditory impression of the sound of the word “apple,” the visual impression of memories of an apple’s general shape and color, the tactile impression of an apple’s texture, and of course also impressions of smell and taste, and so on. Whenever any one of these impressions is stimulated again, the other impressions associated with it are always triggered along with it in a tangled mix. And certain impressions thus evoked further connect with other present stimuli, or else further evoke other past impressions; this alternating emergence of one impression after another is what constitutes the process called thinking or association.
Here there is no sharp boundary between consciousness and the unconscious, between thinking and association, between rational inference and emotional fluctuation. Even the clearest and most abstract thought is nothing more than the mutual prompting and alternating emergence of sensory impressions. It is just that the more organized a line of thought is, the more likely it is to have some clear main thread, or rather some stable sensory platform, to sort out the mixed and jumping impressions of sensation, arranging them more orderly and making them possible to express clearly.
In ordinary thinking, auditory impressions are probably a platform for organizing sensation. This is not to say that in the process of thinking one always relies primarily on the mobilization of auditory impressions; it is only to say that auditory impressions relating to words, as one cross-section, can present the whole process of thinking in a comparatively clear and orderly way. In fact, what is probably mobilized more in the process of thinking are visual and tactile impressions. Yet when the process of thinking is ultimately conceptualized and can be expressed in audible words, this is rather like a cross-section: it reflects the flow and result of thought, but if one thinks that what is grasped through this cross-section is the flow and result of thought as a whole, then one is mistaken. Behind the speakable surface lie even more unspeakable realities. Although surface and interior are indeed reliably connected, one must know that often similar surfaces conceal utterly different inner natures, while wildly different surfaces may simply be two aspects of a similar inner nature. Human words are always limited; they can only ever express one cross-section of the world. Moreover, richness of connotation and clarity of organization are often in conflict: ordinary language, as the broadest platform, contains a relatively high degree of richness, whereas abstracted and reduced logical language has narrowed almost into a line, and thus can achieve clear levels and orderly arrangement, yet it is far removed from richness.
But all this richness and certainty, and the like, are being discussed from the standpoint of the auditory sense. In fact, the method by which one can carve out an expressible cross-section from a mixed presentation of impressions is not necessarily always centered on hearing. For example, mathematical calculation—especially geometric deduction—places particular emphasis on visual impressions. The abstract symbols of mathematics carry no further auditory or tactile content, and their visual content is monochrome as well. The process of thought is silently worked out on the flat surface of draft paper, and the result is then organized into clear and concise steps of argument. The more formalized a science is, the less one needs to mobilize other, richer senses in understanding such deductions; one need only call upon monochrome vision.
Is there perhaps also some process of thinking that is neither dominated by hearing nor dominated by sight, but instead takes place in a cross-section dominated by touch? In fact, although touch is the most indispensable of the senses, precisely because it is everywhere and permeates everything, precisely because it is too rich, it is difficult to abstract it away and form a clear thread or platform. One typical tactile platform that comes to my mind is bead-based mental arithmetic.
For me, numbers on the one hand of course always evoke the image of their Arabic numerals, as well as their Chinese pronunciation. This is also their initial form when they present themselves to my mind from the outside. Yet these two forms are never the dominant impression; in my mind, numbers cannot be handled by relying on these two forms alone. It is like seeing “柒百零捌加廿五” and having to translate it into an impression like “708+25” before one can calculate; otherwise, although you know it is some arithmetic proposition, its image remains cold and unprocessable to you. For me, however, the image of numbers is not something to be translated into “708+25” or the like; rather, only when they are translated into the arrangement of counting beads on an abacus do they become intimate and calculable numbers.
So when I hear or see certain numbers presented to me, I cannot help but mentally move them onto the abacus in my mind before I can consider myself to have recognized them. To put it extravagantly, the question of which is larger, 4 or 6, can only be compared when they have been translated into beads—just as for ordinary people, “肆” and “陆” can only be compared in size after being translated into “4” and “6.”
Of course, this process of translating into beads happens instantaneously. Basically, when I hear the sound “si,” the fingers of my right hand immediately conjure up the corresponding sensation. And using tactile impressions of the fingers on this sensory platform to perform calculations is reliable and efficient (otherwise, why would anyone learn bead-based mental arithmetic?). Even for my already rusted abacus, it can still be guaranteed that two-digit addition and subtraction are faster than calling out the numbers, and three-digit addition and subtraction are much faster than using a calculator.
As for multiplication and division, because the multiplication tables still have to be introduced after all, hearing becomes somewhat more involved again. In addition, in addition and subtraction, as well as in the storage of number memory, the position of touch is almost absolute.
What, after all, is the significance of saying all this? I want to say that a cross-section that organizes sensory impressions with touch as the dominant sense is possible. As for what significance this possibility itself may have, let me leave that for later…
June 22, 2009
Latest Comments
- Vidya
2009-07-09 14:11:42
When I was in high school, I analyzed my psychological process when doing addition and subtraction. But it was not very economical.
For example, 5+6: two 5s are 10, and in my mind 6 is an image of 5 with a little extra head showing. Then it becomes 11.
Sometimes with multiplication, 7+8, I see 8 as 1 more than 7, then two 7s are 14 and plus 1 makes 15.
I see 9 as an image of negative one. And numbers within 5 are directly images of quantities (for example, beads). And so on. - Gu: The key is that these “images” of yours are all visual rather than tactile, right?
- Vidya
2009-07-13 19:33:41
Hehe, they are visual. It seems that tactile sensation really does require long-term practice in abacus calculation.
Translated from the Chinese original with AI assistance. The original text is authoritative.
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