Transcript Prezentace aplikace PowerPoint

SEEING NUMBERS
Ivan M.Havel
CTS, Prague
Oliver Sacks, The twins
In: The Man Who Mistook His Wife for a Hat. London 1985, pp.185–203.
261043
639833
234967
978797
766109
305477
459421
672143
639739
541763
THE GENESIS OF THE ELEMENTARY MENTAL CATEGORIES
William James (1890)
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; of
subject 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.
NUMBERS, NUMBERS, NUMBERS
(natural)
"NUMBER"
COUNT
number of something, cardinal number
NUMBER
idea, abstract concept
NUMERAL
figure, word, or group of figures
denoting a number*
FIGURE
basic numeral symbol, digit
NUMEROSITY, ABUNDANCE great number of something
*
NUMERACY
ability to reason with (some) numbers
ARITMETIC
naive or formal theory of (all) numbers
THE TRIPLE-CODE MODEL
schematic functional and anatomical architecture
(Dehaene & Cohen, 1995)
“ NUMBER LINE ”
analogical quantity representation
verbal representations
of numbers
transmission of
seeing Arabic numerals
NUMBER SENSE an ability to quickly understand, approximate,
and manipulate numerical quantities
(Dehaene)
THE ART OF COUNTING
SUBITIZING
= telling number of objects at a glance (E. L. Kaufman, 1949)
singleton
pair
parallel preattentive processing
triad
quartet
serial processing
SUBITIZING
= telling number of objects at a glance (E. L. Kaufman, 1949)
2.5
1
0.8
2
0.7
0.6
1.5
0.5
1
0.4
0.3
0.5
0.2
parallel 0.1
preattentive processing
REACTION TIME (seconds)
PROPORTION OF ERRORS
0.9
serial processing
0
0
1
2
3
4
5
6
7
8
NUMBER OF OBJECTS
Adapted from Lakoff and Núñez (2000)
SUBITIZINGSUBITIZING
vs. ORDINAL COUNTING
= telling number of objects at a glance (E. L. Kaufman, 1949)
2.5
1
0.8
2
0.7
0.6
1.5
0.5
1
0.4
?
0.3
0.2
0.5
parallel processing
parallel 0.1
preattentive processing
REACTION TIME (seconds)
PROPORTION OF ERRORS
0.9
serial processing
0
0
1
2
3
4
5
6
7
8
9
10
NUMBER OF OBJECTS
Adapted from Lakoff and Núñez (2000)
11
VISUAL SEARCH PARADIGMS
FOR FOCAL ATTENTION
Modified from C. Koch: The Quest for Consciousness (2004)
parallel processing
1,0
phase transition ?
REACTION TIME (seconds)
1,2
target to be searched
0,8
0,6
0.4
target pops out
0
5
10
NUMBER OF DISTRACTORS
15
serial processing
REQUIRED COGNITIVE “SENSES”
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
Alan Turing (1936):
 SENSE OF SYMBOLIC REPRESENTATION
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.
FROM SUBITIZING TO NUMERALS
CROSS-CULTURAL CONVERGENCE
COUNTING STRATEGIES
telling number at a glance (subitizing)
actual counting
(serial algorithm)
NUMBER OF OBJECTS
count locally, guess globally manipulation with subsets
~
SUBITIZING + MULTIPLYING
=
x
3 x 4 = 12
Sacks’ twins:
“111,” they both cried simultaneously...
then they murmured “37”, “37”, “37”
“We didn’t count,” they said. “We saw the 111.”
3 x 37 = 111
(Sacks, ibid. p. 189)
PYTHAGOREAN ARITHMETIC
LAYERS OF PEBBLES
PYTHAGOREAN PROTO–ARITHMETIC
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
3
4
5
6
7
8
9
SHAPES
COUNTS
FIGURATE NUMBERS
triangular numbers
1
3
6
10
15
21
28
36
square numbers
1
2
16
25
36
49
64
pentagonal numbers
1
1
2
5
3
4
12
5
22
6
7
35
8
9
51
SHAPES
COUNTS
FIGURATE NUMBERS
hexagonal numbers
1
7
19
37
61
91
30
42
n +1
oblong numbers
n
2
6
12
20
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
NON–COUNTABLES
NON–COUNTABLES
Do we always need an exact count ?
How many legs?
NON–COUNTABLE EVEN LOCALLY
LOCAL COUNTABILITY X GLOBAL NUMEROSITY
• Numerosity of objects can be told at a glance.
• Perception of non–countable collections is possible.
• Is there a non–number arithmetic ?
NON–COUNTS, PHONEY COUNTS
Elephas multipodus
How many legs?
Certainly more than 5 and less than 100.
PROPOSAL: Use interval arithmetic.
PROBLEM:
Is the presumption of the existence of a "correct."
number necessary?
HOW MANY BLACK DOTS ?
Certainly more than 0 and less than 36.
E. Lingelbach (1994)
“WE SEE IT”
PRIMALITY SANS ARITHMETIC
Oliver Sacks, The twins
In: The Man Who Mistook His Wife for a Hat. London 1985, pp.185–203.
261043
639833
234967
978797
766109
305477
459421
672143
639739
541763
(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 ERATOSTHENES
REQUIRED CAPACITIES
– knowledge of numeral representations
– arrangement of numerals by their size
– multiplication
– search
– comparision
– no prior knowledge about primes
COMPOSITE NUMBER !
etc.
etc. NUMBER !
PRIME
etc.
etc.
alternative
start:
stack the pile up in two columns
1D row ?
+
2D rectangle ?
+
–
No need to know the number !
No need of numeracy !
–
return PRIME !
return COMPOSITE !
"corner slit" ?
etc.
(the last column lower)
+
move the top row
onto the last column
–
shift the barrier
SHAPES OF LARGE NUMBERS
COUNTS
NUMBERS
1
2
6
1
2
FIGURES
3
7
3
4
5
NUMBER LINE
4
5
8
6
9
7
8
9
THE NUMBER LINE
45
51
NUMERAL  SHAPE
6 950 4 25 8 63
SHAPE  NUMERAL
6 9 50 4 25 8 6 3
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
SUBITIZING LARGE NUMBERS
mnemonic + eidetic memory
c
d
n = 6 950 425 863
L(c) ≈ d . log n
17 633 561
47 506 398 412
863 334 529 674
971 302 465
log n
8 432 158 746
COGNITIVE SENSE OF CONTINUOUS SHAPES
ZDENĚK SÝKORA (Prague)
Line No. 56, 1988
Line No. 100, 1992
Phase No. 31, 1989
XXX
XXX
Line No. 50, 1988
NUMBERS IN SAVANT’S HEAD
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)
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. 31
SYNAESTHESIA
(Both letters and numerals are symbols – frequent objects of synaestesia)
Katinka Regtiem
...
SEEING MANY PRIMES
AT THE SAME TIME
VIEWING THE GLOBAL STRUCTURE OF PRIMES
ULAM’S SPIRAL
Stanislaw Ulam (1963)
THE NUMBER LINE
VIEWING THE GLOBAL STRUCTURE OF PRIMES
ULAM’S SPIRAL
Stanislaw Ulam (1963)
137
139
149
151
101 100 99
98 97 96
95
94
93
102 65
63
60
59
58 57 90
64
62 61
92
103 66
56
89
104 67
55
88
105 68
68
54
87
106 69
69
53
86
107 70
70
52
85
108 71
51
84
109 72
50 83
110 73 74
75
76
77 78
79
80
81 82
111
157
91
163
167
181
131
179
127
173
VIEWING THE GLOBAL STRUCTURE OF PRIMES
ULAM’S SPIRAL
Stanislaw Ulam (1963)
velikost 200 × 200.
VIEWING THE GLOBAL STRUCTURE OF PRIMES
INVERSE APPROCH: LET PRIMES GO FIRST !
primes
2
3
5
7
11 13
17 19 23 29 31 37 41 43 47 53 59
...
THE PRIME LINE
composites
THE NUMBER LINE
~
VIEWING THE GLOBAL STRUCTURE OF PRIMES
INVERSE APPROCH: LET PRIMES GO FIRST !
...
primes
THE PRIME LINE
composites
THE NUMBER LINE
Stanislas Dehaene: Numbers are represented as distributions
of activation on the mental number line.
THANK YOU
FOR
YOUR
ATTENTION
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.