Ivan M. Havel

CTS, Prague

SEEING NUMBERS

459421 672143639739

639833

261043

766109

234967 54176

3

305477

978797

Oliver Sacks, The twins

In: The Man Who Mistook His Wife for a Hat. London 1985, pp.185–203.

1. Elementary sorts of sensation, and feelings of personal activity;

2. Emotions; desires; instincts; ideas of worth; æsthetic ideas;

3. Ideas of time and space and number

4. Ideas of difference and resemblance, and of their degrees.

5. Ideas of causal dependence among events; of end and means;

ofsubject and attribute.

6. Judgments affirming, denying, doubting, supposing any of the above ideas.

7. Judgments that the former judgments logically involve,

exclude, or are indifferent to, each other.

We may postulate that all these forms of thought have a natural

origin.

THE GENESIS OF THE ELEMENTARY MENTAL CATEGORIES

William James (1890)

NUMBER

COUNT

NUMERAL

FIGURE

ability to reason with (some) numbers

NUMERACY

"NUMBER"

NUMBERS, NUMBERS, NUMBERS(natural)

NUMEROSITY, ABUNDANCE NUMEROSITY, ABUNDANCE

idea, abstract concept

number of something, cardinal number

figure, word, or group of figures denoting a number*

basic numeral symbol, digit

POČET

ČÍSLO

ČÍSLOVKA

ČÍSLICE

POČETNOST

*

naive or formal theory of (all) numbers

ARITMETIC

great number of something

THE TRIPLE-CODE MODELschematic functional and anatomical architecture

(Dehaene & Cohen, 1995)

seeing Arabic numerals

“ NUMBER LINE ”

verbal representations of numbers

NUMBER SENSE an ability to quickly understand,

approximate, and manipulate numerical quantities (Dehaene)

transmission offunctiona

l

information

analogical quantity representation

THE ART OF COUNTING

SUBITIZING= telling number of objects at a glance (E. L. Kaufman, 1949)

serial processing

singleton pairtriad

quartet

parallel preattentive processing

SUBITIZING

1 2 3 4 5 6 7 8

NUMBER OF OBJECTS

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

PR

OPO

RTIO

N O

F ER

RO

RS

2.5

2

1.5

1

0.5

0

REA

CTIO

N T

IME (

seco

nds)

= telling number of objects at a glance (E. L. Kaufman, 1949)

serial processing

Adapted from Lakoff and Núñez (2000)

parallel preattentive processing

SUBITIZING

parallel processing

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

PR

OPO

RTIO

N O

F ER

RO

RS

?

NUMBER OF OBJECTS

= telling number of objects at a glance (E. L. Kaufman, 1949)

Adapted from Lakoff and Núñez (2000)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 …

2.5

2

1.5

1

0.5

0

REA

CTIO

N T

IME (

seco

nds)

seria

l pro

cess

ing

parallel preattentive processing serial processing

SUBITIZING vs. ORDINAL COUNTING

VISUAL SEARCH PARADIGMS FOR FOCAL ATTENTION

Modified from C. Koch: The Quest for Consciousness (2004)

SERIAL S

EARCH

(algorithmic)

target pops out

REA

CTIO

N T

IME (

seco

nds)

1,2

1,0

0,8

0,6

0.4

target to be searched

phase

tra

nsi

tion ?

parallel processing

serial processing

PARALLEL SEARCH (Gestalt)

0 5 10 15

0 5 10 15 NUMBER OF DISTRACTORSNUMBER OF DISTRACTORS

For Required

SUBITIZING SENSE OF SAMENESS and DIFFERENCE

COMPARISON OF COUNTS SENSE OF NUMEROSITY

ACTUAL COUNTING SENSE OF ORDER

NUMBER LINE ABSTRACT CONCEPT OF NUMBER

ARITHMETIC OPERATIONS ADVANCED NUMERACY

USING NUMERALS SENSE OF SYMBOLIC REPRESENTATION

REQUIRED COGNITIVE “SENSES”

Alan Turing (1936):

The behavior of the [human] computer at any moment is determined by the symbols which he is observing, and his "state of mind" at that moment.

CROSS-CULTURAL CONVERGENCE

FROM SUBITIZING TO NUMERALS

COUNTING STRATEGIES

count locally, guess globally

telling number at a glance (subitizing)

NUMBER OF OBJECTS

actual counting (serial algorithm)

DIFFERENT STRATEGIES

manipulation with subsets~

3 x 4 = 12

3 x 37 = 111

“111,” they both cried simultaneously...then they murmured “37”, “37”, “37”

“We didn’t count,” they said. “We saw the 111.”

(Sacks, ibid. p. 189)

x=

Sacks’ twins:

SUBITIZING + MULTIPLYING

http://z.about.com/d/gosouthamerica/1/0/v/J/15208482typicalcell.jpg

PYTHAGOREAN ARITHMETIC

LAYERS OF PEBBLESPYTHAGOREAN PROTO–ARITHMETIC

1 2 3 4 5 6 7 8 9

Πυθαγόρας (582–500 B.C.)

Mathematics education in ancient Babylon, Egypt, Greece and Rome used limestone pebbles in visual patterns to reveal the fundamental relationships among numbers.

Latin calculus "reckoning, account," originally "pebble used as a reckoning counter," diminutive of calx (gen. calcis) "limestone" (cf. calcium).

Greek pséfois "pebble", hence "reconing" pséfois logizesthai (Hérodotos) or en pséfó legein (Aristotle).

1 2 16 25 36 49 64

1 5 12 22 35 51

1 3 6 10 15 21 28 36

1 2 3 4 5 6 7 8 9

triangular numbers

square numbers

pentagonal numbers

SHAPES COUNTS FIGURATE NUMBERS

n

n + 11 7 19 37 61 91

hexagonal numbers

2 6 12 20 30 42

oblong numbers

rectilinear numbers

2 3 5 7 11 . . . 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 641

643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 953 967

971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451

SHAPES COUNTS FIGURATE NUMBERS

NON–COUNTABLES

NON–COUNTABLES

How many legs?

Do we always need an exact count ?

LOCAL COUNTABILITY X GLOBAL NUMEROSITY

NON–COUNTABLE EVEN LOCALLY

• Numerosity of objects can be told at a glance.

• Perception of non–countable collections is

possible.

• Is there a non–number arithmetic ?

How many legs?

Certainly more than 5 and less than 100.

Elephas multipodus

NON–COUNTS, PHONEY COUNTS

PROBLEM: Is the presumption of the existence of a "correct." number necessary?

PROPOSAL: Use interval arithmetic.

HOW MANY BLACK DOTS ?

E. Lingelbach (1994)

Certainly more than 0 and less than 36.

“WE SEE IT”

PRIMALITY SANS ARITHMETIC

459421 672143639739

639833

261043

766109

234967 54176

3

305477

978797

Oliver Sacks, The twins

In: The Man Who Mistook His Wife for a Hat. London 1985, pp.185–203.

(1) .THEY COULD NOT DO SIMPLE ARITHMETIC playing with mental images ?

(Could they have any notion of “prime” ? Rectilinear alignments of items?)

(2) .EXTREME SENSE OF DETAIL perceiving large groups of tiny elements ?(Spilled matches)

(3) .PREDILECTION FOR PRIMES because primes boldly resist regular chopping ?

(4) .MORE TIME FOR LARGER (= LONGER) PRIMES (as–if) physical processing ?

(5) .NO RECORD ABOUT POSSIBLE RESTRICTIONS OF THE SET OF PRIMES(There are only two 6-digit Mersenne primes)

(6) .THEY COMMUNICATED IN SPOKEN ENGLISH (decadic numerals)

SIEVE OF ERATOSTHENESREQUIRED CAPACITIES

– knowledge of numeral representations

– arrangement of numerals by their size

– multiplication

– search

– comparision

– no prior knowledge about primes

COMPOSITE NUMBER !

PRIME NUMBER !etc.

etc.etc.

etc.

2D rectangle ?

return PRIME !

return COMPOSITE !

"corner slit" ?(the last column lower)

shift the barrier move the top row onto the last column

+–

+–

+ –

alternative start:

etc.

No need to know the number !

No need of numeracy !

1D row ?

stack the pile up in two columns

SHAPES OF LARGE NUMBERS

COUNTS NUMBERS FIGURES NUMBER LINE

1 2 3 4 5

6 7 8 9

45 51

1 2 3 4 5 6 7 8 9

THE NUMBER LINE

360596 524 8

NUMERAL SHAPE

360596 524 8

SHAPE NUMERAL

SUBITIZING LARGE NUMBERS mnemonic + eidetic memory

6 950 425 863 17 633 561 47 506 398 412

8 432 158 746 863 334 529 674 971 302 465

VELKÁ ČÍSLA NA JEDEN POHLED

6 950 425 863n =

d L(c) ≈ d . log n

c

17 633 561 47 506 398 412

8 432 158 746 863 334 529 674 971 302 465

log n

SUBITIZING LARGE NUMBERS mnemonic + eidetic memory

Line No. 56, 1988 Line No. 100, 1992 Phase No. 31, 1989

XXX XXX Line No. 50, 1988

COGNITIVE SENSE OF CONTINUOUS SHAPESZDENĚK SÝKORA (Prague)

Daniel Tammet autistic savant :

I have always thought of abstract information—

numbers for example—in visual, dynamic form.

Numbers assume complex, multi-dimensional shapes

in my head that I manipulate to form the solution to

sums, or compare when determining whether they are

prime or not.

(Interview for Scientific American, January 8, 2009)

NUMBERS IN SAVANT ’S HEAD

VISUAL MNEMONICS(direct seeing numerals)

Solomon Shereshevsky (1886 - 1958):

„Даже цифры напоминают мне образы...“

1 = a proud, well-built man (гордый стройный человек);

2 = a high-spirited woman (женщина веселая);

3 = a gloomy person (угрюмый человек);

6 = a man with a swollen foot;

7 = a man with a moustache;

8 = a very stout woman - a sack within a sack....

87: “As for the number 87, what I see is a fat woman and a man twirling his moustache.”

Luria, A.R. (1968/1987). The Mind of a Mnemonist, p. 31SYNAESTH ESIA(Both letters and numerals are symbols – frequent objects of synaestesia)

Kati

nka

Reg

tiem

SEEING MANY PRIMES AT THE SAME TIME

VIEWING THE GLOBAL STRUCTURE OF PRIMES ULAM’S SPIRAL

Stanislaw Ulam (1963)

THE NUMBER LINE

73

72

71

70

69

68

67

66

50

51

52

53

54

55

56

74 75 76 77 78 79 80 81

5761 60 59 5863 6265 64

82

83

84

85

86

87

88

89

90

110

109

108

107

106

105

104

103

102

92 919397 96 95 949899100101

111

127

137139

131

149

151

157 163 167

173

179

181

73

72

71

70

69

68

67

66

VIEWING THE GLOBAL STRUCTURE OF PRIMES ULAM’S SPIRAL

Stanislaw Ulam (1963)

velikost 200 × 200.

VIEWING THE GLOBAL STRUCTURE OF PRIMES ULAM’S SPIRAL

Stanislaw Ulam (1963)

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59

INVERSE APPROCH: LET PRIMES GO FIRST !

~

VIEWING THE GLOBAL STRUCTURE OF PRIMES

composites

primes

THE PRIME LINE

THE NUMBER LINE

. .

.

INVERSE APPROCH: LET PRIMES GO FIRST !

VIEWING THE GLOBAL STRUCTURE OF PRIMES

composites

primes

THE NUMBER LINE

. .

. THE PRIME LINE

Stanislas Dehaene: Numbers are represented as distributions of activation on the mental number line.

fulltext: www.cts.cuni.cz/new/data/Repd76e8c7a.pdf

Havel, I. M.: Seeing Numbers. In: Witnessed Years: Essays in Honour of Petr Hájek, P. Cintula et al. (eds.), Colledge Publications, London 2009, pp. 71–86.

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