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Ivan M. Havel CTS, Prague

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. 305477. 261043. 459421. 639833. 672143. 234967. 639739. 978797. 541763. 766109. THE GENESIS OF THE ELEMENTARY MENTAL CATEGORIES

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Ivan M. Havel CTS, Prague

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  1. SEEING NUMBERS Ivan M.Havel CTS, Prague

  2. Oliver Sacks, The twins In: The Man Who Mistook His Wife for a Hat. London 1985, pp.185–203. 305477 261043 459421 639833 672143 234967 639739 978797 541763 766109

  3. 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. • Ideas of causal dependence among events; of end and means; of • subject and attribute. • Judgments affirming, denying, doubting, supposing any of the • above ideas. • 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.

  4. NUMBERS, NUMBERS, NUMBERS(natural) COUNT number of something, cardinal number POČET ČÍSLO ČÍSLOVKA ČÍSLICE POČETNOST idea, abstract concept NUMBER figure, word, or group of figures denoting a number* "NUMBER" NUMERAL FIGURE basic numeral symbol, digit NUMEROSITY, ABUNDANCE great number of something * NUMERACY abilityto reason with (some) numbers ARITMETIC naive or formal theory of (all) numbers

  5. verbal representations of numbers transmission of functional information THE TRIPLE-CODE MODELschematic functional and anatomical architecture (Dehaene & Cohen, 1995) “NUMBER LINE” analogical quantity representation seeing Arabic numerals NUMBER SENSEanabilitytoquicklyunderstand,approximate, andmanipulatenumericalquantities(Dehaene)

  6. THE ART OF COUNTING

  7. SUBITIZING = telling number of objects at a glance (E. L. Kaufman, 1949) singleton pair triad quartet parallel preattentive processing serial processing

  8. 2.5 1 0.9 2 0.8 0.7 1.5 0.6 REACTION TIME (seconds) 0.5 1 PROPORTION OF ERRORS 0.4 0.3 0.5 0.2 0.1 0 0 1 2 3 4 5 6 7 8 SUBITIZING = telling number of objects at a glance (E. L. Kaufman, 1949) parallel preattentive processing serial processing NUMBER OF OBJECTS Adapted from Lakoff and Núñez (2000)

  9. 1 2.5 0.9 0.8 2 0.7 0.6 1.5 0.5 PROPORTION OF ERRORS 0.4 1 0.3 0.2 0.5 0.1 parallel preattentive processing serial processing 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 … 0 SUBITIZING vs. ORDINAL COUNTING SUBITIZING = telling number of objects at a glance (E. L. Kaufman, 1949) serial processing REACTION TIME (seconds) ? parallel processing NUMBER OF OBJECTS Adapted from Lakoff and Núñez (2000)

  10. 1,2 1,0 0,8 REACTION TIME (seconds) 0,6 0.4 0 51015 NUMBER OF DISTRACTORS VISUAL SEARCH PARADIGMS FOR FOCAL ATTENTION Modified from C. Koch: The Quest for Consciousness (2004) parallel processing target to be searched SERIAL SEARCH (algorithmic) phase transition ? PARALLEL SEARCH (Gestalt) target pops out serial processing

  11. REQUIRED COGNITIVE “SENSES” ForRequired 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 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. ~

  12. FROM SUBITIZING TO NUMERALS CROSS-CULTURAL CONVERGENCE

  13. NUMBER OF OBJECTS COUNTING STRATEGIES telling number at a glance (subitizing) actual counting (serial algorithm) DIFFERENT STRATEGIES count locally, guess globally manipulation with subsets ~

  14. 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.”(Sacks, ibid. p. 189) 3 x 37 = 111

  15. PYTHAGOREAN ARITHMETIC

  16. 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. Latincalculus "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

  17. SHAPES COUNTS FIGURATE NUMBERS 1 2 3 4 5 6 7 8 9 triangular numbers 1 3 6 10 15 21 28 36 square numbers 1 2 16 25 36 49 64 pentagonal numbers 1 5 12 22 35 51

  18. SHAPES COUNTS FIGURATE NUMBERS n+1 n hexagonal numbers 1 71937 61 91 oblong numbers 2 6 1220 30 42 rectilinear numbers 23 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 839953 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 14331439 1447 1451

  19. NON–COUNTABLES

  20. How many legs? NON–COUNTABLES Do we always need an exact count? NON–COUNTABLE EVEN LOCALLY LOCAL COUNTABILITYXGLOBAL NUMEROSITY • Numerosity of objects can be told at a glance. • Perception of non–countable collections is possible. • Is there a non–number arithmetic? LZE NAZÍRAT I NEPOČTY? EXISTUJÍ NEČÍSLA?

  21. How many legs? NON–COUNTS, PHONEY COUNTS Elephas multipodus Certainly more than 5 and less than 100. PROPOSAL: Use interval arithmetic. PROBLEM:Is the presumption of the existence of a "correct." number necessary?

  22. HOW MANY BLACK DOTS? Certainly more than 0 and less than 36. E. Lingelbach (1994)

  23. “WE SEE IT” PRIMALITY SANS ARITHMETIC

  24. Oliver Sacks, The twins In: The Man Who Mistook His Wife for a Hat. London 1985, pp.185–203. 305477 261043 459421 639833 672143 234967 639739 978797 541763 766109 • .THEY COULD NOT DO SIMPLE ARITHMETIC  playing with mental images?(Could they have any notion of “prime”? Rectilinear alignments of items?) • .EXTREME SENSE OF DETAIL perceiving large groups of tiny elements?(Spilled matches) • .PREDILECTION FOR PRIMES because primes boldly resist regular chopping? • .MORE TIME FOR LARGER (= LONGER) PRIMES (as–if) physical processing? • .NO RECORD ABOUT POSSIBLE RESTRICTIONS OF THE SET OF PRIMES(There are only two 6-digit Mersenne primes) • .THEY COMMUNICATED IN SPOKEN ENGLISH (decadic numerals)

  25. SIEVE OF ERATOSTHENES REQUIRED CAPACITIES – knowledge of numeral representations – arrangement of numerals by their size – multiplication – search – comparision – no prior knowledge about primes

  26. COMPOSITE NUMBER !

  27. etc. etc. PRIME NUMBER ! etc. etc.

  28. alternative start: stack the pile up in two columns + 1D row? returnPRIME! – + 2D rectangle ? – No need to know the number! No need of numeracy! returnCOMPOSITE! "corner slit"?(the last column lower) etc. – + shift the barrier move the top row onto the last column

  29. SHAPES OF LARGE NUMBERS

  30. COUNTS NUMBERS FIGURES NUMBER LINE 1 23 4 5 6 78 9 1 2 3 4 5 6 7 8 9 THE NUMBER LINE 45 51

  31. NUMERALSHAPE 9 5 0 4 2 5 8 6 3 6

  32. SHAPENUMERAL 6 9 5 0 4 2 5 8 6 3

  33. VELKÁ ČÍSLA NA JEDEN POHLED 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

  34. SUBITIZING LARGE NUMBERS mnemonic + eidetic memory 17 633 561 47 506 398 412 log n 8 432 158 746 863 334 529 674 971 302 465 SUBITIZING LARGE NUMBERS mnemonic + eidetic memory c d L(c) ≈ d.log n n = 6 950 425 863

  35. 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

  36. NUMBERS IN SAVANT’S HEAD Daniel Tammetautistic 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)

  37. 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 Katinka Regtiem SYNAESTHESIA (Both letters and numerals are symbols – frequent objects of synaestesia)

  38. SEEING MANY PRIMES AT THE SAME TIME

  39. VIEWING THE GLOBAL STRUCTURE OF PRIMES ULAM’S SPIRAL Stanislaw Ulam (1963) THE NUMBER LINE

  40. 93 97 96 95 94 65 64 57 90 63 62 61 60 59 58 89 66 67 88 66 87 68 67 86 69 68 85 70 69 71 84 70 83 72 71 82 73 72 73 VIEWING THE GLOBAL STRUCTURE OF PRIMES ULAM’S SPIRAL Stanislaw Ulam (1963) 137 139 99 101 100 98 91 92 181 102 131 103 56 179 104 55 149 105 54 106 53 127 151 107 52 108 51 50 109 173 110 74 75 76 77 78 79 80 81 111 163 157 167

  41. VIEWING THE GLOBAL STRUCTURE OF PRIMES ULAM’S SPIRAL Stanislaw Ulam (1963) velikost 200 × 200.

  42. 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 ~

  43. 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.

  44. 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.

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