Number a zero-dimensional datum R.W. Oldford University of - - PowerPoint PPT Presentation
Number a zero-dimensional datum R.W. Oldford University of Waterloo Encodingdecoding Whats this? A Quick Response or QR code . . . first used in the Japanese automotive industry. It encodes something . . . what? . . . needs
a zero-dimensional datum R.W. Oldford
University of Waterloo
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”?
What’s this?
A “Universal Product Code” or “UPC” . . . here, UPC-A, a North American standard.
It encodes something . . . what? . . . needs to be decoded.
UPC-A encodes numbers
A standardized encoding of numbers as a bit pattern of equal width lines where black is 1, white is 0. digit L pattern R pattern 0001101 1110010 1 0011001 1100110 2 0010011 1101100 3 0111101 1000010 4 0100011 1011100 5 0110001 1001110 6 0101111 1010000 7 0111011 1000100 8 0110111 1001000 9 0001011 1110100 Encodes 12 decimal digits as in table at left ◮ S and E are ‘Start’ and ‘End’ bit patterns 101 ◮ M is the Middle bit pattern 01010 ◮ S,M, and E each have 2 bars ◮ Each L and R is a 7 bit pattern for a single decimal digit, last R is a ‘check digit’ ◮ 95 bits to produce the whole pattern (= 7 × 12 + 2 × 3 + 5).
So what’s the number? Hard! Designed for a machine to decode!
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
different from a machine (viz. being the product of evolution rather than design). We need an encoding designed to be decoded by a human.
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.
Comparing encodings QR UPC-A Khmer Ancient Egyptian
Why? What number is this?
A cultural context: When decoded, it gives the answer to the ultimate question of life , the universe , and everything.
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?
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
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
Ancient Egyptian numerals Numerals in hieroglyphics from a modern Egyptian building
Pictures provide meaning The ostensive definition of the number is the picture, its arrangement of elements is its pictorial form.
Ludwig Wittgenstein
The diagram is the meaning which is apprehended from its pictorial form.
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
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!
Socrates Plato 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).
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 x2 Definitions proof of the Theorem of Pythagoras transparent, clear, evident from the preservation of area
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 in blue, Parietal lobe in yellow, Temporal in green, Occipital in pink Parietal lobe in orange, showing the intraparietal sulcus (IPS) in red. The principal functions of the IPS include perceptual-motor coordination and visual attention. Here is also where numerical processing occurs as well as visuospatial working memory.
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,
◮ 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
◮
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 .
So . . . what are natural numbers?
In mathematics, we take the natural numbers to be N, that is the set of “counting numbers” {1, 2, 3, 4, . . .} an ‘infinite’ set. Or perhaps we take N0 the same set supplemented by 0 to be the natural numbers. But fractions were also ‘natural’ to early numerate societies and to our brains; provided they were not too small. ◮ ancient (natural) systems found little need for very large numbers or exceedingly small numbers. ◮ 0, negative numbers, rationals, irrationals, real, imaginary, . . . , come later ◮ in terms of comprehension, very large and very small numbers are logarithmically compressed Our modern world deals routinely in very large numbers and in very small; unfortunately beyond a certain size (large or small) these numbers cannot be easily
(small) numbers beyond some threshold appear to be the same.
Imagining big numbers Imagine the magnitude of the following numbers . . . imagine seeing that many things:
◮ 3 . . . ◮ 5 . . . ◮ 7 . . . ◮ 12 . . . ◮ 20 . . . ◮ 50 . . .
What visual images come to mind for each?
What if they are a lot bigger?
Imagining 9,000,000
Imagining 9,000,000
Imagining 9,000,000
Imagining 9,000,000
9,000,000 American children with no health coverage in 2007. Each one represented by a child’s block.
Imagining 9,000,000
9,000,000 American children with no health coverage in 2007.
Every day modern big numbers Imagine each of the following magnitudes:
◮ 9,960 . . . e.g. the average number of pieces of junk mail that are printed, shipped, delivered, and
disposed of in the US every three seconds
◮ 20,500 . . . e.g. the average number of tuna fished from the world’s oceans every fifteen minutes ◮ 183,000 . . . e.g. the estimated number of birds that die in the United States every day from
exposure to agricultural pesticides.
◮ 270,000 . . . e.g. the estimated number of sharks of all species killed around the world every day
for their fins
◮ 2,400,000 . . . e.g. the estimated number of pounds of plastic pollution that enter the world’s
◮ 9,000,000 . . . e.g. the number of American children with no health coverage in 2007 ◮ 170,000,000,000 . . . e.g. the number of US tax dollars given to bail out insurance giant AIG
in 2009 See Chris Jordan’s “running the numbers” at http://www.chrisjordan.com/gallery/rtn/ for other excellent examples.
Imagining small numbers Imagine the magnitude of the following fractions . . . imagine seeing it:
◮ 1 3 . . . ◮ 1 6 . . . ◮ 1 7 . . . ◮ 5 12 . . . ◮ 3 20 . . . ◮ 1 50 . . .
What are the pros and cons of these various visual fractions?
Imagining small numbers Communicating risk: visualizing a fraction (e.g. of deaths) via icons, say 3 out
Imagining small numbers Visualizing a fraction (e.g. of deaths) via icons: 15 out of 1000 people Works well provided the fraction is not too small. See Visual Fractions – simple
Imagining very small numbers What if the fractions are a lot smaller? Imagine the magnitude being:
◮ 34 100,000 . . . e.g. length in metres of a single pixel on a 17-inch monitor with a resolution of 1024 × 768 ◮ 1 10,000 . . . e.g. the average diameter of a human hair in metres ◮ 155 100,000,000. . . e.g. the wavelength in metres of light used in optical fibre ◮ 12 100,000,000. . . e.g. the diameter in metres of the human immunodeficiency virus, or HIV ◮ 1 13,983,816. . . e.g. the probability of winning the grand prize in Lotto 6/49
Conveying the randomness associated with a probability is an additional challenge. E.g.
1 16,777,215 an animation: Where’s Wally? (the UK name for Where’s
Waldo?)
Imagining very small numbers
Alternatively a short narrative based on shared experience might be used to convey the very small magnitude. The hidden nature conveys the uncertainty. The hidden nature conveys the uncertainty. Works . . . provided the experience is truly shared.
Changing the scale – by inversion
Sometimes, large numbers are more understandable than small numbers (and vice versa) so that inverting the number can lead to a more meaningful number. Here we take a small, difficult to understand, number and convert it to a large but hopefully more meaningful number.
If p is the small probability of some event (e.g. winning a lottery), we can convert this (via a negative binomial) to an expected length of time to win (assuming so many tickets purchased at each draw, and so many draws per time period). That is, calculate the expected number of draws before the first win, then translate that into a time period based on frequency of lottery draw (i.e. weekly, monthly, . . . ). Note that time periods (e.g. ‘years’) ground the numbers in reality. Associating known historical events might help give meaning to the length of time, and so ground the magnitude. The visual lengths of the timelines further ground the relative sizes.
Changing the scale – by inversion
Note also that a narrative, possibly humourous, helps engage.
Changing the scale – move probabilities to expectation
◮ Switch to expected losses/returns ◮ “On average” is fairly natural to most people ◮ Expectation reduces randomness to a constant
Changing the scale – move probabilities to expectation
Changing the scale – micromorts
Rescaling by changing units can work . . . provided the new unit is itself meaningful.
To communicate risk of death, Ronald Howard (1970s) suggested a micromort as a meaningful unit of measure for the probability of death. 1 micromort = one-in-a-million chance of death
Pr(Death) = 1 1, 000, 000 = 1 micromort To ground this look at background daily risk. In 2008 England and Wales, out of 54 million people only 18,000 died from external causes (accidents, murders, suicides, . . . ). This is a daily average of
54, 000, 000
1 365 = 18, 000 54 × 365
1 1, 000, 000 ≈ 1 micromort. So we deal with about 1 micromort of deadly risk every day without worry.
Changing the scale – micromorts
Some examples: ◮
the risk of dying from general anesthetic (not the surgery, the anesthetic) is estimated to be about 1 in 100,000 or 10 micromorts.
◮
We expect therefore that one death will occur in every 100,000 operations; each operation is 10 micromorts
See Understanding uncertainty: Small but lethal by David Spiegelhalter and Mike Pearson.
And could convert this in turn to other units ◮
How many micromorts do you “spend” on a given activity?
◮
How many do you spend to travel a particular distance?
◮
How far can you travel (or often can you do some activity) on a hundred micromorts? Some animations by David Spiegelhalter and Mike Pearson.
Changing the scale – microlives
Micromorts are units of probability (of death). As we did the lotteries we could switch to units of expectation, in this case expected lifetime. We introduce the microlife.
1 microlife is 30 minutes of expected lifetime.
In Canada today, a 20 year old male has a life expectancy of 80.8 years, or 60.8 years to go (Canadian females the same age have life expectancy of 84.6 years). This gives our 20 year old male approximately 1,065,946 microlives left (females have an extra 66,621). Chronic risks can then be measured in the number of microlives they remove (from the more than a million available). To ground this look at background daily risk. For example, here are some things that would cost a 30 year old man 1 microlife:
◮
smoke 2 cigarettes
◮
drink 2 pints of strong beer
◮
each day of being 5 kg .overweight See David Spiegelhalter’s Understanding uncertainty: Microlives
Very large numbers and very small numbers do not come up much in simple societies. And so are, in some sense, ‘unnatural’. Once numbers grow beyond a certain magnitude they all seem the same . . . Similarly, exceedingly small values become indistinguishable beyond a certain point. Both very large and very small numbers are difficult to visualize – in both senses of the word. The uncertainty associated with probability and randomly generated values increases the difficulty.