Is Integer Arithmetic Fundamental to Mental Processing?: The mind's secret arithmetic


Allan W. Snyder* and D. John Mitchell
Centre for the Mind, Institute of Advanced Studies
The Australian National University, Canberra, Australia 0200

Unlike the ability to acquire our native language, we struggle to learn multiplication and division. It may then come as a surprise that the mental machinery for performing lightning fast integer arithmetic calculations could be within us all even though it can not be readily accessed, nor do we have any idea of its primary function. We are led to this provocative hypothesis by analysing the extraordinary skills of autistic savants. In our view such individuals have privileged access to lower levels of information not normally available through introspection.

keywords: savants, autistic savants, unconscious arithmetic, mental arithmetic, cognition, mind, consciousness, brain sciences, consciousness

* author for correspondence A.W. Snyder, FRS

1. Introduction

We are largely unaware of the ways in which our brains process information. For example, we are not conscious that shape is computed from object shading or that perspective is derived (among various ways) by the gradient of texture (Hemholtz 1910, Snyder and Barlow 1988). And why should we be? It is the "object" itself that is of ultimate interest rather than the manner in which we derive its label. Indeed, this specific example explains why it is so difficult to draw naturally occurring scenes, as has been elaborated elsewhere (Snyder and Barlow 1988, Snyder and Thomas 1997). Analogously, the foundations of our most fundamental beliefs, our mindsets, are not normally available for introspection. We are highly concept driven (Bartlett 1932; Snyder 1998). Presumably, this confers advantage by allowing us to operate automatically. Essentially, a world of unconscious information is sifted through, by mechanisms of which we are unaware, to arrive at our final judgements. We might therefore be in for some genuine surprises if we had access to the mental processes used to construct our mindsets.

To gain insight into this fundamental problem we turn to a rare group of individuals, savants with early infantile autism, because they appear significantly less concept driven than normal individuals. Furthermore, this group has been a subject of scrutiny, so we can borrow from comprehensive empirical studies and much theoretical discussion (Kanner 1943; Asperger 1944; Frith 1989; Treffert 1989; Baron-Cohen 1994) in order to build a framework from which predictions can be made about unconscious mental processes. This leads to our hypothesis that the mental machinery for performing lightning fast integer arithmetic (lengthy multiplication, division, factorization and prime identification) is within us all, although it can not normally be accessed, nor do we know what primary function it serves.

2. Savants - minds with privileged access to lower levels of information?

Building on the pioneering work of Kanner 1943, Asperger 1944, Frith 1989, O'Connor 1989 and Hermelin and O'Connor 1990, we surmise that children with early infantile autism give insight into a mind with limited mindsets, a mind that is not concept driven (Snyder and Thomas 1997; Snyder 1998). In our view such a mind can tap into lower level details not readily available to introspection by normal individuals. This is consistent with the constellation of traits associated with early infantile autism, especially those of savants (Hill 1978; Treffert 1989; O'Connor 1989; Howe 1989; Nettlebeck and Young in press) who seem to be aware of information in some raw or interim state prior to it being formed into the "ultimate picture". For example, it explains (Snyder and Thomas 1997) how it is that Nadia (Selfe 1977), a mentally retarded three and a half year old, can draw natural scenes like that of Fig. 1 from memory, with astonishing life-like perspective and to do so "spontaneously" without any training or without even passing through the usual scribble stage.

Now, it is a surprising fact that normal individuals cannot draw naturalistic scenes unless they are taught the tricks and schema to do so (Gombrich 1960). The reason why this is so unexpected is that our brains obviously possess all of the necessary visual information required to draw, but we are apparently unable to access it for the purpose of drawing. For example, our brain performs the calculations necessary to label three-dimensional objects. Yet the difficulties of drawing even a simple sphere are legion. We are not consciously aware of how our brains derive shape from shading, perspective from gradients of texture, size invariance with distance and so on.


Fig 1 Autistic child's drawing at about three and a half years (Selfe 1977).

Clearly, it is the object label or symbolic identification that is of ultimate importance and not the actual attributes processed by the brain to formulate the label (Snyder and Barlow 1988). Indeed, normal preschool children draw, not so much what they see, but rather from what can be called their mental schema. And, these schema, like those of Fig. 2, tend to be invariant across cultures. The horse is conventionally drawn side-on, head to the left, and in bold outline form as is typical of late preschool art. Somehow, the autistic savant Nadia can directly tap the way in which our brain derives perspective, whereas normal individuals can not. Yet these autistic savant artists often struggle to recognize familiar faces (Selfe 1977).

So we believe that artistic savants have direct access to "lower" levels of neural information prior to it being integrated into the holistic picture - the ultimate label. All of us possess this same "lower" level information, but we can not normally access it. Ramachandran (1998 p. 287) in his remarkable new book enriches this possibility by suggeting neurobiological mechanisms.

In our opinion, all savant skills can be explained as analogous to those of drawing (Snyder, 1998). Put simply, savants have privileged access to lower levels of "raw" information. Take the ability of perfect pitch as an example. Our mechanism for hearing consists of discrete frequency analysers which allow for the possibility of perfect pitch. But, surprisingly only one in 10,000 persons possess absolute pitch (Profita and Bidder 1988) and it is debatable whether or not absolute pitch can be taught (Takeuchi and Hulse 1993). So, in analogy to our discussion above on vision, it is the holistic information content that is important for hearing and not the component attributes from which this is derived (Miller 1989; Heaton, Hermelin and Pring 1998). Yet, all musical savants possess absolute pitch (Miller 1989). They apparently have access to lower levels of auditory information while we do not.

These and other savant skills (Treffert 1989; Rimland and Fein 1988), including the extraordinary ability to recall seemingly meaningless detail as opposed to recall of concepts, unusual sensory discrimination of smell and touch, and even time keeping abilities, reinforce our view that savants are able to tap something that is in us all, but which is not normally accessible. And this is consistent with numerous observations as captured by O'Connor's (1989, page 4) statements that their "gift springs so to speak from the ground, unbidden, apparently untrained and at the age of somewhere between 5 and 8 years of age. There is often no family history of the talent" and it "is apparently not improved by practice." Also, the talents are "chiefly in the direction of imitation and there is little capacity for originality or for creativity" (Treffert 1989, page 9).

Hand drawn HorseHand drawn Horse

Fig 2 Representative drawings of normal children, each at age four years and two months. (Emma and Teneal, Parents on Campus Preschool, Australian National University).

To our knowledge no young savant (when the skill first emerges) has ever given any insight into the methods used, nor can they learn or be taught. With maturity the occasionally offered insights are suspect, possibly being contaminated by expectations or the acquisition of concepts concerning the particular skill. Furthermore, savant skills often recede or are lost altogether with the onset of maturity (Selfe 1977; Treffert 1989; Barnes and Earnshaw 1995).

All of this suggests that the unusual skills of savants can be used as a diagnostic tool to probe information from lower level mechanisms which is not available to introspection of the normal mind. But, the savant has not revealed unknown or unexpected mechanisms in the case of drawing or perfect pitch. The physics of natural scenes already tells us how perspective must be computed by the brain and discrete frequency analysers are already known to be the primary auditory receptors. Nor, in this vein, should the savants' astonishing feats of recall for detail reveal anything new about mental processing, since much evidence supports the view that we all store an enormous amount of information, with only a minute subset available for recall (Treffert 1989, Penfield and Roberts 1966). Indeed, our recall like our drawing skills appear to be concept oriented (Bartlet 1932).

So the extraordinary drawing skills of savants, their astonishing recall of detail and their ability of perfect pitch do not reveal unexpected mental processing. We all have the same raw information but just can not directly access it, at least on call. But what does the existence of savant lightning calculators tell us about mental processing in the normal mind?

3. Savant lightning calculators.

Because normal children struggle to learn multiplication and division, it is surprising that some savants perform integer arithmetic calculations mentally at "lightning" speeds (Treffert 1989, Myers 1903, Hill 1978, Smith 1983, Sacks 1985, Hermelin and O'Connor 1990, Welling 1994, Sullivan 1992). They do so unconsciously, without any apparent training, typically without being able to report on their methods, and often at an age when the normal child is struggling with elementary arithmetic concepts (O'Connor 1989). Examples include multiplying, factoring, dividing and identifying primes of six (and more) digits in a matter of seconds as well as specifying the number of objects (more than one hundred) at a glance. For example, one savant (Hill 1978) could give the cube root of a six figure number in 5 seconds and he could double 8,388,628 twenty four times to obtain 140,737,488,355,328 in several seconds. Joseph (Sullivan 1992), the inspiration for the film "Rain Man" about an autistic savant, could spontaneously answer "what number times what number gives 1234567890" by stating "9 times 137,174,210". Sacks (1985) observed autistic twins who could exchange prime numbers in excess of eight figures, possibly even 20 figures, and who could "see" the number of many objects at a glance. When a box of 111 matches fell to the floor the twins cried out 111 and 37, 37, 37. Similar skills were reported as early as 1801 about a child named Dase, who was also "singularly devoid of mathematical insight" and of low general intelligence (Treffert 1989, Myers 1903).

4. Is integer arithmetic fundamental to mental processing?

If, as we believe, all savant skills have a common origin, then the skill for integer arithmetic, (like that for drawing, perfect pitch, and recall for meaningless detail), arises from an ability to access some mental process which is common to us all, but which is not readily accessible by normal individuals.

From this reasoning, we believe that everyone has the underlying facility for performing lightning fast integer arithmetic. This facility can not normally be tapped for the purpose of arithmetic nor do we have any idea of its primary function. Rather, we must learn arithmetic the way we learn to draw naturalistic scenes, by implementing tricks and algorithms (Gombrich 1960; Snyder and Barlow 1988; Snyder and Thomas 1997). Learning arithmetic is hard work for normal individuals (Dehaene 1997), whereas it seems effortless for mathematical savants. Why this should be is deeply mysterious.

As with drawing, tricks and algorithms can be learned for doing rapid arithmetic, but some savant lightning calculators vastly out-perform those who adopt these methods, both in speed (Hermelin and O'Connor 1990) and complexity (Sacks 1985; Waterhouse 1988). For example, in a pioneering empirical study, a mathematics graduate trained in the appropriate algorithms took 11.46 seconds to generate all the primes between integers 301 and 393 whereas a non-verbal autistic young man who had not previously confronted such a task took only 1.16 seconds (Hermelin and O'Connor 1990). Not only was the savant ten times faster, but he also made far fewer errors. Importantly, no practically realizable algorithm has yet been invented for rapidly identifying primes in excess of 8 figures as apparently performed by the autistic savant twins (Sacks 1985).

It would be interesting to compare the active (functional) brain images of autistic savant calculators with those of individuals who calculate via learned algorithms. We might anticipate significant differences between the two, possibly analogous to those between native and second language performance as recently observed by Hirsch's group (Kim, Relkin, Lee and Hirsch 1997). Our native language is acquired unconsciously, whereas second language acquisition is hard work. Accordingly, the arithmetic ability of autistic savants could be functionally like that of a native language whereas it is expected to be more like a second language in most of us.

5. What is required for arithmetic calculations?

Apart from learning the nomenclature or the symbolic representation of numbers, integer arithmetic is simply the ability to separate groups into an equal number of elements - that is to equipartition. For example, 12 elements can be represented as two equal groups of 6 elements or 4 equal groups of 3 elements. Equipartitioning may also be pertinent to another common skill of autistic savants - calendar calculating - where the day of the week is given upon being presented with any date, say 1000 years in the past or future (Sacks 1985; Hermelin and O'Connor 1986; Treffert 1989; Young, R.L. and Nettlebeck, T. 1994). We surmise from this that equipartitioning is fundamental to some yet unknown aspect of mental processing. It is intriguing to contemplate which aspect, analytical or perceptual.

The actual method of calculation, while intriguing, is not central to our thesis. Rather, our hypothesis rests on the very existence of an ability to do lightning calculations without training. Perhaps mathematical savants tap a mental process which spatially represents groups and patterns (Welling 1994) and equipartitions them analogous to the mathematical procedure of factorizing. This could also explain why primes to mathematical savants are the odd man out in groups of numbers and are reacted to as if they are very peculiar indeed (Hermelin and O'Connor 1990). Others have suggested the possibility of savants using modular arithmetic (Sacks 1985; Steward 1975).

Intergers 101 - 120

Fig 3 Generating primes: to determine whether it is has any factors other than 1 or itself. We need only consider prime factors less than the square root of the number. There are well known tricks for rapidly determining whether a number is divisible by 2, 3, 5 or 11, but in general it is necessary to divide by each prime. However if we are testing a sequence of consecutive numbers it is not necessary to test every number separately by dividing by each prime. Once you have determined that a number is divisible by 7 then you know every 7th number thereafter is divisible by 7. This gives us an alternative way to find primes, one known in antiquity, that does not explicitly involve division.

Above we have adapted this ancient method to find the primes in the range 101-120 (row 1). First we eliminate the even numbers and numbers ending in 5 (row 2). Dividing 101 by 3 leaves a remainder of 2, revealing 102 as the first multiple of 3. We then eliminate every third number starting with 102 (row 3). Finally , dividing 101 by 7 leaves a remainder of 3, revealing 105 as the first multiple of 7. We then eliminate every seventh number starting with 105 (row 4). Since we need only consider prime factors less than the square root of 121 equals 11, the remaining numbers are all prime.

Memory and algorithms (learned or induced) are known to play a crucial role in the techniques employed by lightning calculators from the normal population (Smith 1983; Dehaene 1997). But the beautiful work of Anderson, O'Connor and Hermelin (in press) has ruled out the role of memory for savant calculators. They also found that the performance profile of the savant calculator closely matched that of the control who was using the Eratosthenes algorithm for identifying primes as suggested earlier by Hermelin and O'Connor (1990). But it remains possible that other strategies for finding primes (see Fig. 3), not all of which need be arithmetic, could also have similar performance profiles. Whatever the case, this suggests that savant calculators have privileged access to some form of algorithmic mental processing.

6. Music and number

Some autistic savants have the ability to keep time for extended periods with accuracy to the second (Treffert 1989). Apparently, our internal clocks are more precise than might have been imagined. For example, when one autistic child was awakened he said, "It's 2.14 AM", then he went back to sleep (Rimland and Fein 1988, p 485). This ability to equipartition time could also contribute to the impressive musical skills of many autistic savants (Treffert 1989, Hill 1978; Miller 1989) and, when coupled with equipartitioning of space, could suggest a mechanism which interrelates music and mathematics.

There no doubt are other surprises that can be revealed by those "abnormal" minds which are somehow aware of interim mental processes and information not normally available through introspection. For example, is it possible that the senses of normal individuals are mixed and accessible for comparison in a way that individuals with synaesthesia (Luria 1987; Cytovic and Wood 1982) might suggest?

7. Discussion

As Howe (1989, p. 83) so aptly puts it, 'We experience only "the whole": it takes the evidence provided by unusual people in whom mental integration is incomplete owing to retardation, brain damage, or some other kind of mental "disturbance" to make us appreciate how smooth functioning of a person's total mental system depends on the parts, or sub-systems, that underlie it.' This philosophy underpins our present investigation. In particular, we believe that savants offer a window into "lower" level information used to construct our percepts and our judgements. From this we have argued that some mental processing exists in us all, for purposes yet unknown, which is recruited by autistic lightning calculators to perform precise integer arithmetic calculations, such as multiplication, division, factoring and identifying primes.

This highly quantitative numerical ability of autistic lightning calculators is in sharp contrast with the well known qualitative sense of numerosity displayed by human babies and even animals (Gallistel and Gelman 1992; Gallistel 1990; Wynn 1992; Wynn 1995; Dehaene 1997). For example, babies and animals can estimate the number of objects in a collection with an error that is proportional to the number itself. This turns out to be accurate for perceiving and estimating 1, 2 or 3 objects but is grossly inaccurate for judging large numbers. As Dehaene 1997 (p 119) concludes in his authoritative overview, "An innate sense of approximate numerical quantities may well be imbedded in our genes; but when faced with exact symbolic calculation we lack the resources".

Our paper is concerned with mathematical savants. Now, savants are far more prevalent in the autistic population than in any other group and also it is among this group where multiple savant skills most frequently occur (Rimland and Fein 1988; Triffert 1989). But, only a small fraction of the total autistic population are savants. And, this fraction tends to be predominately composed of those with early infantile autism (Treffert 1989), a condition first described by Kanner (1943). Our theoretical perspective is derived from savants in this category, although it could apply to savants in general. We believe that savants with early infantile autism have privileged access to lower levels of information and that they are impoverished in concept formation, compared with the general autistic population (Charman and Baron-Cohen 1993). With maturity certain concepts can be acquired, but often at the loss or reduction of their savant skills (Selfe 1977; Smith 1983).

It is worth mentioning, as have mathematicians of acclaim (Hadamard 1949), that savant lightning calculations are idiosyncratic and not representative of what would normally be considered a mathematical talent. Mathematicians are primarily concerned with the conceptual, whereas autistic savants have extreme difficulty with learning even the simplest mathematical concepts. In this regard it is interesting that Baron-Cohen and his colleagues (1997) have recently found that fathers and grandfathers of children with autism were more than twice as often in the field of engineering than were fathers and grandfathers of normal children. Similarly, they found that autism occurred significantly more often in families of students in the field of physics, engineering and mathematics (Baron-Cohen et al. 1998). Although these studies address the general autistic population and are not restricted to savants, they are nonetheless fascinating and deserve further investigation.

Prevalent explanations for savant skills: Mathematical savants have fascinated their investigators through the centuries (Smith 1985, Treffert 1989) so it should be of no surprise that various theories have been advanced to explain the phenomenon. These are discussed in depth elsewhere (Treffert 1989; Nettelbeck and Young in press). We critique the conceptual thrust of the most prominent views so that they can be contrasted with the perspective leading to our claim that integer arithmetic is fundamental to mental processing. Essentially there are two popular explanations for savant skills: one holds that obsessive focussed learning promotes savant skills just as it does for any expertise: the other postulates that genius and savants alike have highly developed domain specific neural structures (innate talent). In contrast, our view is that the mechanism for savant skills resides equally in us all but that (without some abnormality like autism) it can not normally be accessed for the skill in question.

Mathematical savants arise from obsessive learning: Many authors believe that the extraordinary feats of lightning calculators are a consequence of their passion and preoccupation for learning mathematics in much the same way it is for truly innovative mathematicians, see for example, e.g. Smith (1983) and Rimland and Fein (1988). Dehaene (1997, p. 164) especially presents a compelling discussion, concluding that, "A talent for calculation thus seems to arise more from precocious training, often accompanied by an exceptional or even pathological capacity to concentrate on the narrow domain of numbers, than from an innate gift". Howe (1989 p. 150) agrees, "The circumstances that give rise to a retarded savant's achievements are not entirely different from those in which a person of normal or above average intelligence chooses to specialize in a particular area of interest".

Savants have better brains for arithmetic: Another view holds that both genius and savant alike are endowed with exceptional domain specific neural structures (innate talent) which promote their specialized skills. This view also has a number of distinguished advocates. For example, O'Connor (1989, p. 19) says "But just as there are specialized centres mediating speech, so there may be centres for calculation, graphic skills or music. One can suffer deficits in these abilities so why not also have specific gifts!" Ramachandran (1998, p. 197) enriches the story further, speculating about savants, "that some specialized brain regions may have become enlarged at the expense of others", e.g. the angular gyrus for mathematical talent. Howe (1989, p. 153), also suggests that the savant artist Nadia and the man with the seemingly perfect memory, Shereskeskii, might fall into this category.

While the obsessive learning and the better brains theory for savant arithmetic may not be mutually exclusive (Hermelin and O'Connor 1990) and while they have their compelling aspects, we find them improbable for the following reasons: Those who have protracted experience with savants frequently report that the core ability behind the skill emerges "spontaneously" and does not improve qualitatively with time even though it might become better articulated (O'Connor 1989; Selfe 1977; Treffert 1989). This argues against obsessive learning. Furthermore, from the perspective of either theory, it would appear highly coincidental that such a peculiar subset of mathematics should be so compelling to a significant fraction of autistic savants across all cultures, and also that many of these same savants simultaneously have several savant skills (Rimland and Fein 1988) each of which are similarly peculiar and restricted. And why is there little or no invention or creative component in the skill? All of this mitigates against either obsessive learning or better brains being a plausible explanation for mathematical savants, as does the fact that savant skills can even arise after an accident or illness in otherwise normal individuals (Treffert 1989).

The mechanisms for savant mathematics reside equally in us all but can not normally be accessed: Now in contrast to the popular views discussed above, the unique aspect of our perspective is that the mechanism and information drawn on for savant mathematics resides equally in us all but it can not be recruited by us for mathematics. In other words, we believe that mathematical savants, like all autistic savants, arise from their privileged access to lower levels of raw information. Their skills are essentially a form of mimicry, and thus naturally lead to drawing, perfect pitch, time telling, astonishing recall, hyperlexia, echolalia, etc. Hence, the very same peculiar savant skills appear across different cultures. However, unlike drawing and perfect pitch, we do not know what lower levels are recruited for savant mathematical skills. But, we hypothesize that savant mathematics is propelled by some fundamental mechanism which equipartitions - possibly in both space and time. Why is it that savants have privileged access to lower levels of information? Perhaps it is promoted by a loss of those centres that control executive or integrative mechanisms as elaborated on by Treffert (1989) and also by Baron-Cohen (1995) in relation to the fact that individuals with autism lack a theory of mind. This in turn could leave savants less concept driven (Snyder and Thomas, 1997; Snyder 1998), or as Frith (1989) argues, lacking central coherence (Pring, Hermelin and Heavey 1995; Heaton, Hermelin and Pring 1998).

An intriguing question remains. Although we do not normally have access to the lower levels of information as do savants, is there nonetheless some artificial means to promote this access, say via induced altered states of consciousness? Possibly pertinent to this suggestion is the fact mentioned above that savant skills have been known to follow a severe physical illness, an operation, or a near drowning (Treffert 1989). This reinforces our belief that savant skills are innate in us all but are normally suppressed.

So in conclusion, we believe that the mental apparatus to perform "lightning fast" integer arithmetic calculations such as multiplication and division resides in us all, even though it is not normally accessible. The brain appears to perform something tantamount to arithmetic calculations (or analogously equipartitioning) for some unknown aspect of mental processing. The challenge now is to unravel which aspect.

We appreciate the critical insights of Mike Anderson, Kirsty Galloway McLean, Ted Nettlebeck, Mandy Thomas and Robyn Young.


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