Number a zero-dimensional datum R.W. Oldford University of Waterloo
Encoding–decoding What’s this? A “Quick Response” or “QR” code . . . first used in the Japanese automotive industry. It encodes something . . . what? . . . needs to be decoded . What do they mean by “quick response”?
Encoding–decoding What’s this? A “Universal Product Code” or “UPC” . . . here, UPC-A, a North American standard. It encodes something . . . what? . . . needs to be decoded .
Encoding–decoding UPC-A encodes numbers A standardized encoding of numbers as a bit pattern of equal width lines where black is 1, white is 0. Encodes 12 decimal digits as in table at left ◮ S and E are ‘Start’ and ‘End’ bit patterns 101 digit L pattern R pattern 0 ◮ M is the Middle bit pattern 01010 0001101 1110010 1 0011001 1100110 ◮ S , M , and E each have 2 bars 2 0010011 1101100 ◮ Each L and R is a 7 bit pattern for a 3 0111101 1000010 single decimal digit, last R is a ‘check 4 0100011 1011100 digit’ 5 0110001 1001110 ◮ 95 bits to produce the whole pattern 6 0101111 1010000 ( = 7 × 12 + 2 × 3 + 5 ). 7 0111011 1000100 8 0110111 1001000 9 0001011 1110100 So what’s the number? Hard! Designed for a machine to decode!
Encoding–decoding and are visual representations of the same number. They are visualizations of that number and, as such, encode the number in the visualization . Unfortunately, they are both designed to be “read”, that is decoded, by a machine . They are not designed to be decoded by a human; and we are very different from a machine (viz. being the product of evolution rather than design). We need an encoding designed to be decoded by a human.
Encoding–decoding OK, here’s one that is specifically designed for humans: So, what number is it? Is a better visualization than ? Is it worse? What would make it better? How about ∩ ∩ � ∩ ∩ ? Any better? Worse? The first is Khmer, the second ancient Egyptian.
Encoding-Decoding Comparing encodings QR UPC-A Khmer Ancient Egyptian ∩ ∩ � ∩ ∩ Which is easiest for us to decode? Why? What number is this? A cultural context: When decoded, it gives the answer to the ultimate question of life , the universe , and everything.
What is in a number? Some possible properties a number might have are ◮ a visual representation ◮ a picture that encodes the number and which can be decoded ◮ easily, efficiently, accurately ◮ a label, a unique identity (identify and distinguish between numbers) ◮ an ordering, in that it whether one number is smaller (or precedes) than another can be determined (e.g. order by height) ◮ magnitude, or size, is meaningful . . . ◮ numerosity is meaningful (i.e. how many? cardinality, or count) ◮ maybe only differences (intervals) between two numbers are meaningful (e.g. temperatures in degrees Fahrenheight or Celsius) ◮ ratios of two numbers are meaningful (e.g. “length”) Do the various encodings so far considered have any of these properties? How should an encoding be designed? Are some better than others? What numbers can be encoded? What numbers are natural to us as humans?
Where do numbers come from? Researchers have found that (e.g. see Nieder and Dehaene, 2009) “. . . basic numerical competence does not depend on language and education, but is rooted in biological primitives that can be explored in innumerate indigenous cultures, infants, and even animals.” ◮ many kinds of animals (e.g. pigeons, parrots, rats, dolphins, monkeys, chimpanzees) can distinguish numerosity (how many). E.g. rhesus monkeys have been shown ◮ to be able to be trained to order numbers (1-9) of things by magnitude – even novel numbers! See Brannon and Terrace, 1998. ◮ to use numerosity naturally (untrained) to select food sources having a greater number – 1 vs 2, 2 vs 3, 3 vs 4 and 5, but not 3 vs 8, or 4 vs any of 5, 6, or 8. See Hauser, Carey, and Hauser, 2000. ◮ human infants of only a few months age can discriminate, represent, and remember particular small numbers of items – See Starkey and Cooper, 1980. ◮ success similar to rhesus monkeys of Hauser, Carey, and Hauser, 2000. ◮ there is evolutionary value in numerical competence ◮ ability depends on culture, language, training ◮ the Pirahã culture contains only the counting words “one, two, many” – see Gordon 2004. ◮ perform poorly on numerical tasks for quantities beyond 3 ◮ use of fingers beyond 3, with the exception of 5, was a range; e.g. shift from 5 to 3 fingers to indicate 4 ◮ analog estimation might be being used beyond 3 ◮ small numbers, at least, seem natural
Where do numbers come from? Ancient Egyptian numerals Natural symbols. Circa 3,000 BC Ox yoke, coil of rope, papyrus or water lily, finger, frog, a god or maybe just a “wtf?” man
Where do numbers come from? Ancient Egyptian numerals Numerals in hieroglyphics from a modern Egyptian building
What’s in a number? Pictures provide meaning The ostensive definition of the number is the picture, its arrangement of elements is its pictorial form. The diagram is the meaning which is apprehended from its pictorial form. Ludwig Wittgenstein In early numerical systems, the meaning of the number is often given by its picture. Each one here itself provides an ostensive definition of 4 . They are what we mean by 4 . || |||| ❦ ❦ ❦ ❦ || Babylonian Egyptian Chinese Mayan Attic Greek
What’s in a number? Pictures provide meaning Even an irrational number can be given an ostensive definition. √ For example, here is the pictorial meaning of 2 : √ ◮ The length of the dark line is what is meant by 2 . √ ◮ The leftmost picture provides the meaning of 2 ◮ The rest of the sequence provides the reasoning and identifies (without words) ◮ The smallest square defines a unit ◮ A (sighted) alien from a sufficiently advanced civilization will understand this! Adapted from Meno’s Dialogue (by Plato) where Socrates leads a slave boy through questions about a word picture to realize that he (the slave boy) actually knows that a square of area 2 exists (sort of √ . . . actually 2 2 in the dialogue). Socrates Plato
What’s in a number? Pictures provide meaning Even an operation or a theorem can be given its meaning through a pictorial form. ❦ ❦ ❦ ❦ ◮ the meaning of 3 × 4 ❦ ❦ ❦ ❦ ◮ defines multiplication diagrammatically via a “perspicuous representation” – transparent, clear, evident . . . (Wittgenstein) ❦ ❦ ❦ ❦ Sense of use of “squaring x ” or “ x squared” for x 2 ◮ Definitions proof of the Theorem of Pythagoras transparent, clear, evident from the preservation of area
Where do numbers come from? A brain for numbers “Healthy human brains come equipped with several circuits that contribute to number processing” – we have a brain for numbers – Sandrini and Rusconi, 2009 . major lobes: Frontal Parietal lobe in orange, showing the intraparietal sulcus (IPS) in red. The principal in blue, Parietal lobe functions of the IPS include perceptual-motor coordination and visual attention. Here is in yellow, Temporal in also where numerical processing occurs as well as visuospatial working memory. green, Occipital in pink
Where do numbers come from? A brain for numbers ◮ symbolic numerical processing (recognition of numbers, comparisons of magnitude) seems to occur largely in one location in the brain, the intraparietal sulcus (or IPS) ◮ nature and nurture interact; our brain adapts its circuitry to training. ◮ because numbers are an essential part of our culture we spend much time training ourselves (esp. when young) to think about and with numbers. ◮ allows quick encoding and decoding of visual representations of number (e.g. Khmer decimal digits) ◮ studies show it matters little how the numbers are represented – by dots, by Hindu-Arabic numerals, or by words – See Ansari, 2007 ◮ in Western cultures, at least, there is evidence that people implicitly have a mental spatial representation of numbers that orders small to large numbers from left to right spatially ◮ some neurons (in the parietal cortex, esp. IPS) are tuned to be stimulated more by particular numbers. ◮ however presented (words, symbols, spatial arrangement of dots, or dots presented in time) ◮ time to compare relative magnitudes increases as the distance between numbers decreases ◮ does not appear to be a single, isolated piece of cortex that responds only to number ◮ “neuronal populations coding for number are highly distributed in the IPS and are intertwined and overlapping with representations of other quantitative parameters” (like line length) ◮ very large and very small numbers seem to be logarithmically compressed ◮ ranking numbers by magnitude seems to be different from cardinality and may involve overlapping circuitry within parieto-frontal areas See Nieder and Dehaene, 2009, Sandrini and Rusconi, 2009, and Rusconi et al, 2009 .
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