1. Introduction
In 1998, the author came upon an idea on how one could plot graphically on computer screens
point information above 2-dimensions. In order to do this he postulated that points should be
divisible. This violated Eulcid's definition of the indivisibility of a pont. Nevertheless, the idea
gelled into the invention of the Hgram graph for which a copyright was lodged with the Library of
Congress, U.S.A including several published papers [3 to 5]. Research on Hgram graphs continued for a period of nine years until
he retired from academic teaching. He was highly excited when he came across two years
later Mongotmery's article entitled "The Mind's Eye" in the Discover (May 1991) issue. In the
development of the Hgram graphs, he was not awared that considerable progress has been
made in brain probing by Allman and Kaas in which the main conclusion is that visual image
from the external world is mapped point-by-point onto the visual cortex like the working of a
pin-hole camera. The geometry of the image is practially preserved in this mapping. The part which
particularly interested the author was their discovery that the brain have multiple maps for the
same point. He immediately made the connections between his Hgram graphs and these multiple
maps in the brain. He postulated that in order to see beyond 2-dimensions, points must be split
into subpoints with different attributes. The brain solves this by creating multiple regions, each
sensitive to a different attribute. For example one region might be sensitive to motions, another
to a specific colour, another to a specific pattern and so on. Thus one could build up
multiparametric information on a point via these multiple maps just like the way sublayers are
used to represent subpoints in the Hgram graph [4,5].
This month, the author encounters an article in the July '97 issue of Discover in which Robert
Kunzig related the work of the French neuropsychologist Dahene on how the brain processes
numbers. Because of my past interest on multidimensional graphical visualisaton and my
present interest in sequence algebra, it gives me the opportunity to contribute some thoughts on
how the brain handles numbers. There is a big difference between mapping of visual images
and mapping of numbers. An image is mapped into the visual cortex with little geometrical
distortions. However, Dahene has not found any such map related to numbers. Neither has he
found the number line although he and Cohen have narrowed the region to the inferior parietal
cortex. It is not clear what functions are performed within this area in the processing of
numbers. The author postulates that there must exist in the brain special generators which can
fire the "number line" as a "timing line". He speculates that we count numbers by time intervals.
This paper cites examples to support his theory.
2. Number Generators In The Brain
Neurologists already know for a long time that neurons can fire impulses at fixed frequencies but
as they are not mathematicians, they do not associate this action with mathematical functions,
especially generating functions. A mathematical function is one which gives a single output
value for a single input value. Depending on the complexity of the relation, the processing could
be as simple as 1+1 or as complicated as finding the 20th Fermat's number. Obviously the
difference between these two task is that the former takes very little time whereas the latter takes
an inordinately long time. There are things which the brain is not designed to do well whereas a
digital computer might succeed. Why? Because the brain is built to solve survival problems.
Computing a large Fermat's number is not in the agenda of survial.
The fact that a neuron can fire an impulse train of fixed frequency must implies that there exist
an organic analogue of an impulse generator which takes the place of a mathematical function.
Imagine a mapping whereby when a subject receives the command "6", a neuron cluster will fire
six impulses at equal intervals and likewise for other integer numbers. This can easily be
simulated by a sequence algebraic generating function called Nat(z) [6]. Equation (1) shows the
firing of eleven impulses by Nat(z) by series expansion.
................................1.........1........1........1........1........1.........1........1...........1
Nat(z) := 1 + 1/z + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + O(---) .............(1).
..................................2........3........4.........5........6........7.........8.........9...........10
................................z.........z.........z.........z.........z........z.........z..........z...........z
The can be liken to the firing of impulses by a neuron as shown below:
----------|----------|----------|----------|----------|------
The author postulates that there might be neurons which could fire a fixed number of impulses
by intention. If this is the case then it is easy to associate the firing of one impulse with the
integer 1 and two with the integer 2 and so on. This is the case of numbers mapped into the time
domain and not a spatial domain. However, it is still a type of mapping. It is known that human
baby, and for that matter all higher animals, could count small numbers from birth. As we grow
older, we begin to use other parts of the brain to reinforce our arithmetic system, such as
language, patterns and so on. For example, at one time we are made to memorizing the
multiplication table by rotes. Here we are not using the number line any more. Furthermore, it is
found that a person with brain lesion to specific areas, finds it difficult to do precise counting and
interpolations. For example one might response that 1+1 = 3. Another might find difficulty to
ascertaining whether 5 falls between 3 and 7. If one assumes that damage to the brain could
cause damage to some generators then such observations could be explained using sequence
algebra [6 to 8].
For example, suppose in a healthy individual, the generator is Nat(z). However for
someone with brain lesions, we could simulate, just for illustration, damage to
Nat(z) by a new generator z^3/(z^3-1) which can be expanded as follows:
........................1..........1.........1............1
XNat(z) := 1 + ---- + ---- + ---- + O(---) ....................................(2).
..........................3..........6.........9...........12
.........................z..........z..........z...........z
Now the subject can only fire multiples of 3 time intervals and some multiples of 2. He can still
recognise numbers in the set such as 1, 3, 6, 9 and so on but he has no clue at all what you are
talking about when numbers such as 2, 4, 5, 7, .. are presented to him. He cannot place such
numbers in his "number line" .
As to why human babies can only count small numbers has also an explanation from the
generator model. Young children sense of time is very short. If you ask a child to wait for you
whilst you go to the loo, 5 minutes is an inordinately long time to him. Most probably, young
children fire timing impulses at a higher rate than adults. Most probably the timing line is
extended as we grow up. Young babies are only precoccupied with things immediate to them,
like hunger .. cry for milk. Notice a baby becomes very impatient within seconds if the first cry
is not quickly responded by the mother. Their time line is very short. This explains why their
number line is also very short.
What about some of these calculating prodigies? Are they born or trained? If the above
generator model is correct, they are trained. Through concentrated exercises, one could extend
the range of expansion of the generator and do marvellous arithmetic with them. This is
evidenced by what sequence algebraists have done with generators. The only difference is that
sequence algebraists rely solely on digital computer to do the calculations. Why are calculating
prodigies mostly autistic people. A generating function is a very limited domain and since it is
endowed from birth, autistic people are on par with other new borns. These are the only people
who are willing to concentrate on using this limited resource. Great mathematicians are more
likely to be distracted because they have other more interesting explorations to do. By working a
tremendous amount of time with the generators, they are able to attain an expert level above
average persons. This could only occur if there is an ingredient of work versus reward and this
already exists to fire up their passion with the encouragement of parents. Autistic people never
graduate into great mathematicians probably this way of doing arithmetic is very resource
intensive. Normal people tend to migrate to link numbers with languages and use other parts of
the brain for arithmetic.
The question is whether all children should be stimulated to develop the generator components?
I would think it is a waste of time since we now have handheld calculators and other training
methods. That this is highly inefficient could be explained using the equation of divisibles as
shown in equation (3) [6].
...................................................................1
.........................................Comp(z) := ------------- ........................................(3).
...............................................................i....i
.............................................................z..(z - 1)
Finite generation of seven composite numbers starting from 4 is shown in equation (4).
....................1........2........2........1........2........4.......2..........1
Comp(z) := ---- + ---- + ---- + ---- + --- + --- + --- + O(---) .....................(4).
.......................4.......6........8........9.......10.......12....14..........15
.....................z........z........z.........z.........z........z.......z...........z
Presumably few are trained to write down a series of composite numbers by immitating the
principle of the equation of divisibles given by equation (3). And yet, if the brain counts numbers
by time intervals, it would be possible to train one to write down composites numbers and with
difficulties prime numbers this way. And yet, the process will get increasingly difficult as we
extend the range of composites or primes as we need to generate impulse trains with
increasingly large intervals. If you are a baby, you will not be able to cope. Even with adults,
you will soon loose track because you cannot generate intervals precisely with large intervals.
Thus doing arithmetic by counting numbers along a line has limitations whether it is a time line or
a number line.
Dahene's concept of a number line has another flaw. Evolution will never retain body parts
which are not repeatedly used. Long idleness will cause unused parts to atrophy. Imagine a
number line with a millioin integers assigned to the neuron clusters. How many of these
numbers will ever be used throughout one's life? Those not used will atrophy. On the other
hand, a neuron impulse generator model is highly economical. It certainly does not tie up a lot of
neuron clusters. As to why the number line is different from the mapping of visual images onto
the visual cortex, we have a parallel here in the Computer Monitor. All the pixels are repeatedly
refreshed up to 60 times per second. In other words they are all actively firing away whenever
a person open his eyes.
3. Going over past experimental evidence.
This section is based on meagre information available from the article in Discover [1].
(i) The Case Of Monsieur N: (reported by Larent Cohen, neurologist)
2+2 = 3 or 5 but never 9
N has no precise knowledge of the meaning of "9".
A dozen is either 6 or 10.
Can't tell whether a number is odd or even.
My diagnosis:
Perhaps in this case only the generator for Odd(z) is working as Nat(z) and Even(z) had been
damaged. He has some inkling of even numbers from the language part of the brain but he is
unable to generate a pulse train with even intervals. When asked what is a dozen, his answer are
in even numbers but rather imprecise. He seemed to fair better with odd numbers such as in
approximating 2 as between 3 and 5. This might be because he could generate 3 and 5 as odd
numbers but he is hazy about the even numbers except that his language association enables
him to approximate the interpolations. He can't distinguish between even and odd because
Even(z) generator is not functional. So he cannot do comparisons.
(ii) Robert Moyer and Thomas Landauer (Standford psychologists):
The smaller the number difference between two single digit integers, the longer it takes a person
to choose the large of two numbers.
My diagnosis:
Assume that a healthy person fires his Nat(z) to compare maginitude of numbers by time interval
counts. He can only fire one number at a time. Maybe there are multiple generators by which
he could fire both numbers. Time interval estimates are not very precise. If the time interval
difference is large, it is easy to pick the longer time interval but if these are close, he might have
to repeat the firing and use polling to decide which is the larger.
(iii) The concept of number line by Dahene
Probably both left and right brain have number lines since Monseur N has a large part of the left
brain destroyed. Joself Gerstmann tracked the region where the number line could occur to the
inferior parietal cortex of the left hemisphere on the side of the brain, above and behind the ear.
Its function is not clearcut. Whether the right or the left parietal cortex is involved is dependent
on whether someone is right- or left-handed. Dahene mentioned that whilst we know which area
is involved, there is absolutely no evidence of the neural code in that area. He speculated that
there is probably a number line hard-wired into our brains, each number corresponding to a
dedicated cluster of neurons.
My comments:
Probably Dahene was wrong. It is an uneconomical way of representing numbers. There is
probably no physical number line but clusters of cells which functions like pulse generators with
fixed intervals. Then counting numbers is reduced to counting time intervals. We don't see
time. Therefore why should we see numbers. Is this the reason why no one has detected a line
with neurons firing away when we say "6" or "7" to the test subject? I think so. Other evidences
mentioned above suggest also time interval counting rather than number mappings. A number
line with millions of assigned neuron clusters must be very uneconomical since most clusters will
never fire throughout a person's life.
(iv) Counting in lower animals
Even rats, pigeons and chimps can count quite well. Rats can learn to press a lever 4 times or 16
times to get food. This must be related to interval counts, not symbolic number processing.
Pigoens can learn to peck at a target 45 time rather than 50 times. Chips will pick a tray with 7
chocolates over one with 6 although they too suffer from the distance effect when trying to
choose the larger of two numbers [1]. Thus elementary ability to perceive and manipulate
number is part of our evolutionary heritage - we are born with it.
My comments:
The ability to count time intervals is absolute essential to survival. For example, if a monkey
jumps 90 feet from one tree top to another, he has got to know before landing the estimated time
for doing various things like grasping a branch at the landing site, how long a branch takes to flex
and so on. Young babies could count probably only a few numbers say up to 4. That is why
they are so impatient when hungry. 4 seconds is eternity to them which all mothers are awared
of. However if you are fiftyish and ask a young child of 3 or 4 years old to guess your age he is
likely to speculate: "a hundred?". All animals can count using time interval counts. The
interception of a gazelle's escape path by a cheetah has to do with time measurement. The
number line as such does not exist although one could say that a time line is also a number line.
But the time line is not a visual image. Hence the number line does not exist, it is only
conceptual.
(v) Child Prodigies Or Idiot Savant
A few of these calculating prodigies have also been great mathematicians - Carl Friedrich Gauss
for example but this is rare. As we grow up, our ability to generate impulses with increaing time
range will improve. So with concentrated training, it is possible to produce calculating genius. If
the premise that we compute by using time intervals is correct then one should expect that
autistic calculating prodigies should have a more precise sense of time than average children.
But I have no facilities to test this hypothesis. One could devise quite elaborate arithmetics
using time interval counting. Idiot savants are those who spend a lot of time on this confined
resource and develop it to an expert level. Good mathematicians have too much distractions to
do this for long.
(vi) Finding the number line by brain probing
Just because Allman and Kaas succeeded with visual images in the visual cortex, it does not
follow that the number line could be probed using the same procedures. You can probably
detect firing clusters but it is connected with the firing in short burst with finite number of
impulses. The time taken could be very short, say less than a millisecond. The signal might be
swamped by a lot of background noise of longer durations. So detecting these would be quite a
feat. I don't think present instrumentation is sophisticated enough to detect these machine gun
bursts.
(vii) Damaged Brains can self-repair over time
The brain is highly redundant. There must be other sites with these number generating clusters.
So I expect the brain could find an alternative site given the time to reconnect. If from early
days, a person is trained on mental calculations using both hemispheres, he might recover from
partial damage faster than those who are not thus trained. The firing of neurons with fixed
number of impulses probably can be intentional and therefore trainable. Some people might be
able to fire accurately small burst like expert machine gunners. But then we await much refined
measurement techniques to detect these.
4. Sequence Algebra To Simulate Number Defective Brains
Computer simulations might overcome some measurement difficulties. For example, we would
like to predict the arithmetic performance of a patient with a deficient generator of the z^3/(z^3-1) type
which means that he cannot generate the numbers 1,3,6,9 and so on. Please read previously
published papers in section 8 of this URL site concerning the generation of number sequences.
All you need is an algebraic package to do the simulation.
5. Summary
In this paper, the author forwards the thesis that the number line is not a space line which could
be identified by brain probing such as being done by neurologists in image mappings to the
visual cortex. Numbers are translated into time intervals and the ability to count time intervals
is inborn. Young children or animals are not born to visualise numbers along a line, rather to
count simple time intervals. Nature is very pragmatic. Any endowment handed down through
evolution are always connected with survival. For example, colour visions of some mammals
enable them to identify potential food sources more clearly at night against fuzzy backgrounds.
Forward vision is another. So why should we have a number line when ability to count time
intervals is inborn which also serves survival. The whole basis of computer technology is based
on precise timing generated by the crystal clock and not on storing every integers into the
memory for processing. We think the same reasoning applies to the time line against the number
line.
5 References
1. Kunzig R. : A Head for Numbers, Discover July '97, pp108 to 115.
2. Montgomery G. : The Mind's Eye, Discover May '91, pp 51 to 56.
3. Huen Y.K. (1988): An introduction to the HGRAM (an n-Dimensional Graph Paper).
Copyright application, July 1988, USA Library of Congress Copyright Registration No. Txu
354026.
4. Huen Y.K.: (1991): The HGRAM graphical format - its geometrical interpretation and its
applications, Int. J. Math. Educ. Sci.Technol.,
22,No.3, 403-418.
5. Liyaw A.P., Kiu C.M., Pok Y.M. and Huen Y.K.(1997): Hgram patterns of Routh stability
zones in linear systems, INT.J.MATH.EDUC.SCI.TECHNOL, 1997,VOL.28,NO.2,225-241.
6. Huen Y.K.
7. Huen Y.K.
8. Huen Y.K.
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