topics: |  math | cognitive | arithmetic

Many fascinating aspects of how humans deal with arithhmetic.   

Alex Bellos was the Guardian correspondent in Brazil for many years, and it
shows up in the narratives about Brazil in this fascinating math book!
(He also has a book on the Brazilian fascination for soccer)

Has much to say about the Indian contributions to mathematics.  Here is a
figure on the evolution of the numerals across the world: 

	
			Evolution of modern numerals (p.122)

Vedic Mathematics

Also has a whole section on Tirthaji Maharaj, the Shankaracharya of Puri
who practised mathematics in his spare time, and wrote a bestselling work 
on Vedic Mathematics, except that the "mathematics" here was
entirely his own, and there was nothing "vedic" about it. 

In 1958, he visited the US and impressed his mathematical interlocutors
during a visit to Caltech.

	In 1958, when he was 82 years old, Tirthaji visited the United
	States, which caused much controversy back home because Hindu
	spiritual leaders are forbidden from travelling abroad, and it was
	the first time that a Shankaracharya had ever left India. His trip
	provoked great curiosity in the US. The West Coast would later become
	a focus for flower power, gurus and meditation, but back then no one
	had seen anyone like him. When Tirthaji arrived in California the Los
	Angeles Times called him ‘one of the most important – and least-known
	– personages in the world’.

	Tirthaji had a full schedule of talks and TV appearances. Though he
	spoke mostly about world peace, he devoted one lecture entirely to
	Vedic Mathematics. The venue was the California Institute of
	Technology...  p.128


Bellos goes to Puri (well after Tirthaji has passed away) and meets
interlocutors whose faith in Vedic Science far exceeds the evidence.

Nonetheless, he remains impressed with Vedic Mathematics:

	Vedic Mathematics stands up to scrutiny, even though the sutras are
	mostly so vague as to be meaningless and to accept the story of their
	origin in the Vedas requires the suspension of disbelief.

He goes on to give an explanation of the interesting urdhva-tiryaka rule for
multiplying two multi-digit numbers in your head.  Such algorithmic thinking
has long been a part of the Indian mathematical tradition, but sadly, the
sUtras of Tirthaji were inventions by an undoubtedly competent mathematician,
possibly Tirthaji himself, and posed into cryptic Sanskrit verse.

I think Bellos is impressed because he finds these ideas new, as I was when I
first encountered agreeing that the sUtras are "vedic".

Nothing "Vedic" about Vedic Mathematics

The well-known astrophysicist Jayant Narlikar has written about Tirthaji in 
his work on science, The Scientific Edge (2003):

	K.S. Shukla, a renowned scholar of ancient Indian mathematics...
	recalled meeting Swamiji [Tirthaji], showing him an authorized
	edition of Atharva Veda and pointing out that the sixteen sutras
	were not in any of its appendices (parishiShTas).  Swamiji is said
	to have replied that they occurred in his parishiShTa and in no
	other!  In short, Swamiji claimed the sutras to be Vedic on his own
	authority and no other. p.27 [incident narrated by SG Dani of TIFR]

---

Thus there is nothing "vedic" about the rules, except that they were
couched in terse sanskrit aphorisms similar to those used in the era of the
oral textual tradition.

Unfortunately these kind of hoaxes only make it harder for one to establish
the legitimate claims of Indian mathematical work.   What surprises me is
why a very learned religious man, and one who was clearly an idealist in
his youth -  he went to jail supporting a muslim cause - would do such a thing. 

But anyhow - onto excerptise...

Excerpts

Munduruku number system (ends in five)

The amazonian tribe, Munduruku, use only five words for numbers from 1 to
5.  The words from one to four - 

	one	pUg
	two	xep xep
	three	ebapug
	four	ebadipdip

have the same number of syllables as the count.  The word for five, 
pUg pogbi, means "a handful" and seems imprecise. 

---Bellos, p.15:

Still, I thought it odd that numbers larger than five did not crop up at all
in Amazonian daily life. I asked Pica (the linguist working with them) how
an Indian would say ‘six fish’. For example, just say that he or she was
preparing a meal for six people and he wanted to make sure everyone had a
fish each.

‘It is impossible,’ he said. ‘The sentence "I want fish for six people" does
not exist.’

What if you asked a Munduruku who had six children: ‘How many kids do you
have?’

Pica gave the same response: ‘He will say "I don’t know". It is impossible to
express.’

However, added Pica, the issue was a cultural one. It was not the case that
the Munduruku counted his first child, his second, his third, his fourth, his
fifth and then scratched his head because he could go no further. For the
Munduruku, the whole idea of counting children was ludicrous. The whole idea,
in fact, of counting anything was ludicrous.



experiment:  
	shown a circle with a varying number of dots in it, participants have
	to tell where they belong on a line between a circle with one dot,
	and another with ten. 

numbers as perceived by Munduruku are not equally spaced  -
five onwards, they appear much higher than four etc.   
This is similar to
children in kindergarten, who tend to overestimate numbers in the tens.

Logarithmic view of numbers - larger numbers get compressed.

[Robert Siegler and Julie Booth at CMU 2004]:
	kindergarten pupils (with an average age of 5.8 years), first-graders
	(6.9) and second-graders (7.8). The kindergarten pupil, with no
	formal maths education, maps out numbers logarithmically. By the
	first year at school, when the pupils are being introduced to number
	words and symbols, the curve is straightening. And by the second year
	at school, the numbers are at last evenly laid out along the line.

(see another number experiment below)

Animals that count

Animals that count and add - e.g. Clever Hans the horse

The lesson of Clever Hans was that when teaching animals to count, supreme
care must be taken to eliminate involuntary human prompting. For the maths
education of Ai, a chimpanzee brought to Japan from West Africa in the late
1970s, the chances of human cues were eliminated because she learned using a
touch-screen computer.

Ai is now 31 and lives at the Primate Research Institute in Inuyama, a small
tourist town in central Japan. Her forehead is high and balding, the hair on
her chin is white and she has the dark sunken eyes of ape middle age. She is
known there as a ‘student’, never a ‘research subject’. Every day Ai attends
classes where she is given tasks. She turns up at 9 a.m. on the dot after
spending the night outdoors with a group of other chimps on a giant tree-like
construction of wood, metal and rope. On the day I saw her she sat with her
head close to a computer, tapping sequences of digits on the screen when they
appeared. When she completed a task correctly an 8mm cube of apple whizzed
down a tube to her right. Ai caught it in her hand and scoffed it
instantly. Her mindless gaze, the nonchalant tapping of a flashing, beeping
computer and the mundanity of continual reward reminded me of an old lady
doing the slots.

When Ai was a child she became a great ape in both senses of the word by
becoming the first non-human to count with Arabic numerals. (These are the
symbols 1, 2, 3 and so on, that are used in almost all countries except,
ironically, in parts of the Arab world.) In order for her to do this
satisfactorily, Tetsuro Matsuzawa, director of the Primate Research
Institute, needed to teach her the two elements that comprise human
understanding of number:
quantity and order.

Chimps learn number sequences

Numbers express an amount, and they also express a position. These two
concepts are linked, but different. For example, in ‘five carrots’ five is
the quantity of carrots, or the cardinality.
When I count from 1 to 20 I am using the convenient feature that
numbers can be ordered in succession. I am not referring to 20 objects, just
reciting a sequence, or the number
ordinality.

We slip effortlessly between these two. To chimpanzees, however, the
interconnection is not obvious at all.

Matsuzawa first taught Ai that one red pencil refers to the symbol ‘1’ and
two red pencils to ‘2’. After 1 and 2, she learned 3 and then all the other
digits up to 9. When shown, say, the number 5 she could tap a square with
five objects, and when shown a square with five objects she could tap the
digit 5. Her education was reward-driven: whenever she got a computer task
correct, a tube by the computer dispensd a piece of food.

Once Ai had mastered the cardinality of the digits from 1 to 9, Matsuzawa
introduced tasks to teach her how they were ordered. His tests flashed digits
up on the screen and Ai had to tap them in ascending order. If the screen
showed 4 and 2, she had to touch 2 and then 4 to win her cube of apple. She
grasped this pretty quickly. Ai's competence in both the cardinality and
ordinality tasks meant that Matsuzawa could reasonably say that his student
had learned to count. The achievement made her a national hero in Japan and a
global icon for her species.

Matsuzawa then introduced the concept of zero. Ai picked up the cardinality
of the symbol 0 easily. Whenever a square appeared on the screen with nothing
in it, she would tap the digit.

Then Matsuzawa wanted to see if she was able to infer an understanding of the
ordinality of zero. Ai was shown a random sequence of screens with two
digits, just like when she was learning the ordinality of 1 to 9, although
now sometimes one of the digits was a 0. Where did she think zero's place was
in the ordering of numbers?

In the first session Ai placed 0 between 6 and 7. Matsuzawa calculated this
by averaging out which numbers she thought 0 came after and which numbers she
thought it came before. In subsequent sessions Ai's positioning of 0 went
under 6, then under 5, 4 and after a few hundred trials 0 was down to around
1. She remained confused, however, if 0 was more or less than 1. Even though
Ai had learned to manipulate numbers perfectly well, she lacked the depth of
human numerical understanding.

A habit she did learn, however, was showmanship. She is now a total pro,
tending to
perform better at her computer tasks in front of visitors, especially camera
crews.

How ants measure distance

At the University of Ulm, in Germany, academics put some Saharan desert ants
at the end of a tunnel and sent them down it foragng for food. Once they
reached the food, however, some of the ants had the bottom of their legs
clipped off and other ants were given stilts made from pig
bristles. (Apparently this is not as cruel as it sounds, since the legs of
desert ants are routinely frazzled off in the Saharan sun.) 

The ants with shorter legs undershot the journey home, while the ones with
longer legs overshot it, suggesting that instead of using their eyes, the
ants judged distance with an internal pedometer.

Chimpanzee visual memory : The feats of Ayumu

Chimpanzees may have limits to their mathematical proficiency, yet, while
studying this, Matsuzawa discovered that they have other cognitive abilities
that are vastly superior to ours.

In 2000 Ai gave birth to a son, Ayumu. On the day I visited the Primate
Research Institute, Ayumu was in class right next to his mum. He is smaller,
with pinker skin on his face and hands and blacker hair. Ayumu was sitting in
front of his own computer, tapping away at the screen when numbers flashed up
and avidly scoffing the apple cubes when he won them. He is a self-confident
lad, living up to his privileged status as son and heir of the dominant
female in the group.

Ayumu was never taught how to use the touch-screen displays, although as a
baby he would sit by his mother as she attended class every day. One day
Matsuzawa opened the classroom door only halfway, just enough for Ayumu to
come in but too narrow for Ai to join him. Ayumu went straight up to the
computer monitor. The staff watched him eagerly to see what he had
learned. He pressed the screen to start, and the digits 1 and 2
appeared. This was a simple ordering task. Ayumu clicked on 2. Wrong. He kept
on pressing 2. Wrong again. Then he tried to press 1 and 2 at the same
time. Wrong. Eventually he got it right: he pressed 1, then 2 and an apple
cube shot down into his palm. Before long, Ayumu was better at all the
computer tasks than his mum.

Far ahead of humans

A couple of years ago Matsuzawa introduced a new type of number task. On
pressing the start button, the numbers 1 to 5 were displayed in a random
pattern on the screen. After 0.65 seconds the numbers turned into small white
squares. The task was to tap the white squares in the correct order,
remembering which square had been which number.

Ayumu completed this task correctly about 80 percent of the time, which was
about the same amount as a sample group of Japanese children. Matsuzawa then
reduced the time that the numbers were visible, to 0.43 seconds, and while
Ayumu barely noticed the difference, the children's performances dropped
significantly, to a success rate of about 60 percent. When Matsuzawa reduced
the time that the numbers were visible again – to only 0.21 seconds, Ayumu
was still registering 80 percent, but the kids dropped to 40.

See this video to convince yourself that you simply cannot do what this
chimp is doing!!   


	in many ways, the human brain is deficient to animal brains:

---

This experiment revealed that Ayumu had an extraordinary photographic memory,
as do the other chimps in Inuyama, although none is as good as he
is. Matsuzawa has increased the number of digits in further experiments and
now Ayumu can remember the positioning of eight digits made visible for only
0.21 seconds. Matsuzawa reduced the time interval and Ayumu can now remember
the positioning of five digits visible for only 0.09 seconds – which is
barely enough time for a human to register the numbers, let alone remember
them.

read more about chimps - and how they can learn language (and also about
Ayumu)  in this excerptise of Sue Savage-Rumbaugh's 
Apes, Language, and the Human Mind (1998).

5-mo babies can count

In 1992, Karen Wynn, at the University of Arizona, sat a five-month-old baby
in front of a small stage, with the Sesame Street puppets Elmo and
Ernie. Elmo was placed on the stage. The screen came down. Then another Elmo
was placed behind the screen. The screen was taken away. Sometimes two Elmos
were revealed, sometimes an Elmo and an Ernie together and sometimes only one
Elmo or only one Ernie. The babies stared for longer when just one puppet was
revealed, rather than when two of the wrong puppets were revealed. In other
words, the arithmetical impossibility of 1 + 1 = 1 was much more disturbing
than the metamorphosis of Elmos into Ernies. Babies’ knowledge of
mathematical laws seems much more deeply rooted than their knowledge of
physical ones.

Munduruku numbers

Stanislas Dehaene and Pierre Pica devised experiments for the Munduruku in
the Amazon, one of which was very simple: he wanted to know just what they
understood by their number words. Back in the rainforest Pica assembled a
group of volunteers and showed them varying numbers of dots on a screen,
asking them to say aloud the number of dots they saw.

The Munduruku numbers are:

one	pUg
two	xep xep
three	ebapug
four	ebadipdip
five	pUg pogbi

When there was one dot on the screen, the Munduruku said pg.  When there were
two, the said xep xep. But beyond two they were not precise.  When three dots
showed up, ebapug was said only about 80 percent of the time. The reaction to
four dots was ebadipdip in only 70 percent of cases. When shown five dots, pg
pogbi was the answer managed only 28 percent of the time, with ebadipdip
being given instead in 15 percent of answers. In other words, for three and
above the Munduruku's number words were really just estimates. They were
ounting ‘one’, ‘two’, ‘threeish’, ‘fourish’ ‘fiveish’. 

Pica started to wonder whether pUg pogbi, which literally means ‘handful’,
even really qualified as a number. Maybe they could not count up to five,
but only to fourish?

Pica also noticed an interesting linguistic feature of their number words. He
pointed out to me that from one to four, the number of syllables of each word
is equal to the number itself. This observation really excited him. ‘It is as
if the syllables are an aural way of counting,’

Pica also tested the Munduruku's abilities to estimate large numbers. In one
test, illustrated overleaf, the subjects were shown a computer animation of
two sets of several dots falling into a can. They were then asked to say if
these two sets added together in the can – no longer visible for comparison –
amounted to more than a third set of dots that then appeared on the
screen. This tested whether they could calculate additions in an approximate
way. They could, performing just as well as a group of French adults given
the same task.