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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
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.
THANK YOU FOR YOUR ATTENTION