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Where Does Mathematics Come From? Part 1

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Sources: What Counts: How Every Brain is Hardwired for Math by Brian Butterworth The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene The Math Instinct by Keith Devlin Where Mathematics Comes From by George Lakoff and Rafael Núñez

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Possibilities The Garden of Eden Someone smart invented it and it spread; in other words, it is a cultural tool Everyone is essentially hardwired or genetically programmed to do some mathematics Some combination of the above

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The Number Module Brian Butterworth suggests that with rare exceptions we are all born with a Number Module that 1) categorizes the world in terms of numerosities – the number of things in a collection – and 2) allows us to make use of the cultural tools that extend the capabilities of this number module.

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The Number Module In his book, What Counts Butterworth provides several pieces of evidence for his theory, and this evidence is worth our looking at because it gives us insight into where mathematics comes from, at least in its simplest forms. We will also examine some of Dehaene’s ideas from The Number Sense

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Evidences for the Number Module Ability to deal with numerosity in children and animals Widespread ability to deal with numerosity, independent of education Identifiable structures in the brain Fast and automatic operations (comparing or identifying numerosities)

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Evidences from Animal Studies A chimpanzee named Sheba was trained by Sarah Boysen to understand the meanings of the digits 1 – 9. Sheba was able to perform arithmetic with them. In one experiment, a number of oranges were hidden in various places in Sheba’s cage – for example, two oranges under a chair, and 4 more in a box. Sheba’s task was to explore her cage, then come back and choose the digit that matched the total number of oranges. She succeeded from the very first trial. Then, rather than oranges, the experimenters hid the digit “2” and the digit “4.” Sheba explored the hiding places and then chose the digit “6.”

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Evidences from Animal Studies Two macaques named Able and Baker were able to distinguish the largest among up to five digits. When presented with a string of digits they would use a joystick to point to one of them. They then received that number of fruit candies. They soon learned to pick the largest number. For example, with the sequence “5 2 1 8 3” they would pick 8 – not infallibly, but much more often than chance alone would predict.

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Evidences from Animal Studies Rats can be trained to press Lever A a set number of times before pressing lever B. (Varied both number of presses and frequency) A raven name Jakob learned to open the lid of a box having the same number of dots on it as a card alongside two boxes. (2, 3, 4, 5, and 6 spots) A parrot was trained to say the number of objects on a tray. Lionesses can “count” the number of distinct roars from intruding lions and compare them with the number of friends she has with her.

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Evidences from Infant Studies Babies can discriminate between 2 and 3 objects a few days after birth. When babies are repeatedly shown cards with two objects on them, eventually they get bored and start spending less and less time looking at them. When a card with 3 objects is then shown, the baby spends more time, demonstrating that it can distinguish between 2 and 3 objects. These are called habituation experiments because they depend on the baby becoming habituated to certain stimuli and then acting surprised (by staring longer) at different stimuli. Infants can also distinguish between words of two and three syllables a few days after birth.

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Evidences from Infant Studies Four and five month old infants express surprise at “impossible” arithmetic, such as when two puppets are put behind a screen and the screen drops to reveal either one or three puppets instead. Thus it seems babies know 1+1 = 2, and similarly that 2 – 1 = 1. Slightly older infants know that 2 + 1 = 3 and 3 – 1 = 2. Babies as young as 6 months can discriminate between 2 and 3 actions, such as a puppet jumping 2 or 3 times.

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Evidences from Infant Studies In one experiment using six- through eight- month old babies, babies were presented two stimuli: a card with either two or three common objects on it, and a recording of a drum beating either two or three times. Babies spent more time looking at two-object cards when hearing two beats, and three-object cards when hearing three beats. Thus it seems they could coordinate the numerosity of objects in space and sounds over time.

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Evidences from Anthropology & Archeology Tallies Numeration Systems Counting Systems Counting Boards Number Words

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Tallies The radius bone of a wolf, discovered in Moravia, Czecholsovakia in 1937, and dated to 30,000 years ago, has fifty-five deep notches carved into it. Twenty-five notches of similar length, arranged in- groups of five, followed by a single notch twice as long which appears to terminate the series. Then starting from the next notch, also twice as long, a new set of notches runs up to thirty.

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Ishango Bone Ishango Bone, discovered in 1961 in central Africa. About 20,000 years old

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Ishango Bone

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Lartet Bone Discovered in Dodogne, France. About 30,000 years old. It has various markings that are neither decorative nor random (different sets are made with different tools, techniques, and stroke directions). Some suggest that the marks are meant to record different observations of the moon.

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Ancient, Pre-Historic and Pre- Contact Cultures Around 10 to 11 thousand years ago, the people of Mesopotamia used clay tokens to represent amounts of grain, oil, etc. for trade. These tokens were pressed into the surface of a clay “wallet” then sealed inside as a record of a successful trade contract. These impressions in clay eventually became stylized pictographs, and later, symbols representing numerosities.

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Clay Tokens

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Clay Envelope

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8 "gur-sag-gal" of barley, 16 "pounds" of wool and 16 "quarts" of oil

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Sumerian Cuneiform

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Babylonian Cuneiform

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Mayan Number System

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Egyptian Number System

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Various written systems were developed, some more advanced than others We’ll play around with some arithmetic in these systems eventually

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Other Systems of Counting Some systems have only 1, 2, and “many” Two-counting: –Examples from Australia, South America, South Africa, and Papua New Guinea –One, two, two-one, two-two, two-two-one, two-two-two, and so on.

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Examples from Papua New Guinea The counting systems in Papua New Guinea are best described in terms of the cycles (rather than the base) that they use. For example, if pairs were important to a language group, then the counting system might feature a 2-cycle, with six objects being thought of as three groups of two. Many systems would probably have a second cycle combining number words. The second cycles are commonly cycles of five so that, for example, the number 14 might be two fives and two twos.

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Counting systems based on composite units of 5 and 20 are also common in Papua New Guinea. The 800 different language groups have their own counting systems with a variety of basic number words. Commonly used number words are hand as 5, and person (10 fingers and 10 toes) as 20. A few groups have a hand as 4 (without the thumb) or as 6 (with the thumb as two knuckles).

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Other Systems of Counting (in Oceana & Papua New Guinea) 10-cycles, including some in which 7 is denoted by10-3, 8 by10-2, 9 by 10-1; in others, 6 is denoted by 2X3, 8 by 2X4, 7 by 2X3+1; 5-cycles, sometimes using groups of 10, 20, or 100 as well 3-, 4-, and 6- cycles with various other groupings.

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Body Counting

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1 little finger 2 ring finger 3 middle finger 4 fore finger 5 thumb 6 hollow between radius and wrist 7 forearm 8 inside of elbow joint 9 upper arm 10 point of shoulder 11 side of neck 12 ear 13 point on the head above the ear 14 muscle above the temple 15 crown of the head

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Counting Words Derived from Body Parts: The word for the number...is derived from a phrase meaning... 15Three fists 10Two hands 20Man complete 100Five men finished 9Hand and hand less one 2Raise a separate finger 6To cross over 6Take the thumb 9One in the belly 40A mattress

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Ainu Counting Words NumberMeaning of Ainu wordNumberMeaning of Ainu word 1Beginning-to-be402 X 20 4Much603 X 20 5Hand804 X 20 64 from 103010 from 2 X 20 73 from 105010 from 3 X 20 8Two steps down7010 from 4 X 20 9One step down9010 from 5 X 20 10Two sided (i.e. both hands)1005 X 20 20Whole (man)11010 from 6 X 20

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Counting Boards

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And so on… We’ll talk more about the development of our modern number system later.

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Subitizing Studies have shown that almost universally, humans have the ability to distinguish among 1, 2, and 3 items without counting. This is called subitizing. Most of us can subitize objects up to about four, and we can subitize sequences (or sounds, e.g.) up to about five or so. There is good evidence that this happens suddenly (subitize comes from the Latin word for sudden) and that it is an inborn, universal trait. This may be what the experiments with infants and animals are tapping into.

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Subitizing

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Naming Time vs Numerosity

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Subitizing It is interesting to note that most languages have a special status for the number words for “one,” “two,” and “three.” In languages with genders, these first three counting words were the only ones inflected to agree with gender (Old German, zwei, zwo, zween). In English, we have the first three ordinal words following a different pattern from the rest: first, second, third, fourth, fifth, sixth, nth. Finally, the words used for “2” and “second” often have the connotation of “another” and words for “three” often have the connotation of “a lot” or “beyond” (e.g. “tres” in French).

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Number Module Works well with small collections Allows for distinguishing and comparing numerosities. Also shows interesting effects: –Distance effects –Compaction of large numbers –Stroop Effect –Focus on quantity instead of numerical meaning

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Distance Effects Tell which number in each pair is larger: 2 and 9 3 and 4

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Distance Effects Tell whether each of the following pairs are the same

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4 and 4

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2 and 9

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4 and 5

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Compression of Large Numbers It is harder to distinguish between 93 and 97 than between 13 and 17

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Stroop Effect XOYXOY

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Green Red Green

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Numerical Stroop Effect Which member of each pair has the largest value? 2 9

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Which numeral of each pair is written largest? 2 9

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Attention to numerosity instead of numeral meaning Comparing 79 to 65 is easier than comparing 71 to 65, even though the “ten’s digits” should make both tasks equally easy.

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A Little About Brains... Double dissociation studies of individuals with certain brain injuries or impairments have shown: –The independence of language and number –The independence of number and memory –The independence of number and reasoning

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A Side Note Dehaene has suggested that the multiplication tables are stored as verbal information. If this is true, then memorizing them doesn’t necessarily make use of their numerical meaning and vice versa. Thus it is something like memorizing the following:

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The Devil’s Address Book AbbeyErnestlives onZoeErnestAvenue BrunoErnestlives onAbbeyZoeAvenue ChloeErnestlives onAbbeyErnestAvenue DallasErnestlives onBrunoZoeAvenue Ernest lives onBrunoErnestAvenue FrancisErnestlives onChloeZoeAvenue GilbertErnestlives onChloeErnestAvenue HenryErnestlives onDallasZoeAvenue KentErnestlives onDallasErnestAvenue

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A Little About Brains... Left-hemisphere injuries are more likely to lead to problems with numbers In fact, the left parietal lobe seems to be the most likely candidate for the location of number sense Interestingly, this is where finger manipulation skills are also centered.

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A Little About Brains... Gerstmann’s syndrome has four symptoms: –Finger agnosia –Acalculia –Left-right disorientation –Agraphia Butterworth concludes that finger counting provides a major means by which we move beyond the limitations of our Number Module.

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