Mental imagery in Mathema.cs and Computer Science Alain Finkel, LSV, ENS Cachan & CNRS ‐ France
The reality of mental imagery
Test of mental rota.on of Vandenberg
Test of mental rota.on
Test of mental rota.on
Test of mental rota.on
Test of mental rota.on
Mental imagery and percep.on Theorem: mental imagery ≡ percep.on Proof: • Perky, 1910 • Kosslyn, 1978, Mental speed is linear with the distance • Mellet, 1995, Observing the brain during MI and P. – Two ways: Where ? What/Who ? • virtual percep.on may replace mental imagery
Perky, 1910
Kosslyn, 1978
Mellet, 1995
Plan 1. The reality of mental imagery 2. Examples in mathema.cs and CS 3. Coming back to the story 4. My conclusions
algebraic iden..es (a + b) 2 = a 2 + 2ab + b 2
(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3
sum of first integers 1 + 2 + 3 + … + n = n(n+1)/2 • proof by recurrence; but no intui.on • audi.ve proof, playing on the formula: S = 1 + 2 + 3 + …+ (n ‐ 1) + n S = n + (n ‐ 1) + …+ 3 + 2 + 1 2S = (n+1)+(n+1)+…+(n+1) = n(n + 1)
visual proof
sum of odd integers 1 + 3 + 5 +… + (2n ‐ 1) = n 2
Rolle theorem Rolle's Theorem Suppose f con+nuous on [a,b], derivable on ]a,b[ with f (a) = f (b). Then there exists c ∈ ]a,b[ s.t. f‘(c)= 0
Rolle theorem
the point…
Func.ons… • a func.on may be: – a rela.on f sa.sfying the formula : (Verbal) • For all x, there exists at most an y s.t. f(x)=y • For all x,y,z, if (x,y) and (x,z) are in f then y=z – A graph s.t. every node has at most a successor (Visual) – a process which takes inputs, works and produces outputs (Visual‐Verbal‐sequen.al) – a algorithm compu.ng (Verbal+movement/.me) – a formula/equa.on which defines (verbal) – examples of func.ons: y=3x 2 +8, f(r)=1 et f(p/q)=0), g(n)=0 si T n stops (in n steps) on n.
An old one: the complex plane • Complex numbers were discovered or invented in the 16th century • they remained mysterious for 2 centuries • in the 18th century, Argand found the complex plane: mental representa.ons of the complex numbers • makes the complex numbers much easier to understand
the quadra.c equa.on • ax 2 + bx + c = 0 • Well‐known formula • Geometric interpreta.on
Ordering – 3 formula (verbal): • reflexive • symetric • an.‐symetric • transi.ve – A graph without circuit (visual)
Graphs and automata Graph: – rela.on (edges) between a set E of ver.ces. – adjacency matrix – picture of edges linking a finite number of ver.ces – Runs in a labyrinth – Give examples (metro,…) Finite automaton: – Labelled graph – 5‐uple (E,A, δ ,e 0 ,F) – Words recognized by an automaton – Labyrinth with rooms, named one‐way corridors, entrance and exit rooms – A machine whichs produces outputs
Regular/recognizable/ra.onal langage • Recognized by an automaton: visual‐dynamic • Generated by a grammar : verbal‐sequen.al • Described by a regular expression: verbal
Pumping lemma Lemma: Let L be the set of words recognized by a given finite automaton. There exists k L ∈ N such that any word w ∈ L, of length larger than k L , can be factored w = xuy , with u non empty and xu n y ∈ L for all n ∈ N. k L is bounded by the number k of ver.ces of the automaton Simple proof using the visual/kinesthesic representa.on: Aier at most k steps, any walk must go twice through the same vertex… The loop can be iterated
Plan 1. The reality of mental imagery 2. Examples in mathema.cs and CS 3. Coming back to the story 4. My conclusions
Mental images from 2000 years (in french ) Avant JC: Platon, Aristote, Epictète… • • 1600: Descartes 1700‐1800: Locke, Hume, Berkeley, Kant,… • • 1890‐1950: Husserl, Heidegger, Merleau‐Ponty, Proust, Freud 1900‐1930: Tichener, Binet (psycho. Introspec.ve) : profils visuels, audi.fs, moteurs • • 1910: Perky (conflit entre percep.on et évoca.on) 1930: Pavlov, Skinner, Watson (comportementalisme) • • 1933: Séman.que générale (Korzibsky) 1936: Turing, thèse de Church • 1940‐60: Ecole de Piaget • • 1956: Naissance des sciences cogni.ves, Miller (7+2), Galanter, Pribram 1956‐60: Naissance des thérapies cogni.ves: Beck et Ellis (USA) ‐ • importance des croyances ‐ pas de lien avec la PC avant 1980 1970‐1980: Paivio, Kosslyn, Pinker, Denis (double codage V/A) • • 1985: Gardner (intelligences mul.ples) 1986: Baddeley (Calepin visuo‐spa.al et boucle phonologique, apen.on) • • 1995: Damasio, Goleman, Berthoz 2000: Dehaene • 2009: •
Aristote « Jamais l’âme ne pense sans image »
Descartes • «L’imagina.on, à elle seule, est incapable de créer la science…toutefois…, nous devons, dans certains cas, recourir à elle. D’abord, en la fixant sur l’objet que nous voulons considérer, nous l’empêcherons de s’égarer et de nous gêner; ensuite et surtout, elle peut nous servir à éveiller en nous certaines idées » • Essaie de construire les mathéma.ques sans les images réputées moins fiables que les idées (mots).
Einstein « les mots et le langage, écrits ou parlés, ne semblent pas jouer le moindre rôle dans le mécanisme de ma pensée »
Renew of scien.fic mental imagery in 1960
Dual coding: verbal and imaging • Paivio, Smythe et Yuille (1968): – concrets words are easier to memorise. • Atwood (1971): – « visual » sentence sensible at visual interference – Abstract sentence sensible at audi.ve interférence. • Warrington et Shallice (1979): – read beper the word pyramide aier listening the word Egypte than aier having seen the picture of a pyramide.
Mental imagery Shepard, Metzler (1971): mental rota.on linear with the angle • Kosslyn (1975): 200 ms for dog + elephant / dog + fly • Kosslyn, Ball et Reiser (1978): mental explora.on linear with the distance • • Scien.fic ques.on: Coding = – unique amodal and proposi.onnal for all MRs ? Zenon Pylyshyn Or – plurimodal for each MRs ? Stephen M. Kosslyn Research with PET, brain vision,… •
Plan 1. The reality of mental imagery 2. Examples in mathema.cs and CS 3. Coming back to the story 4. My conclusions
Alan Turing • Design TM par.ally with introspec.on. • How does he made in his mind a mul.plica.on ? • Create states and ac.ons, data and programs, finally not very different… • States and ac.ons are a paradigm occuring in different domains: Leibniz, Laplace, Comte,…
My framework: Computa.onal theory of mind • Cogni.ve states and ac.ons more or less connected with brain states/ac.ons. • Cogni.ve behavior: sequence of states and ac.ons. • Representa.ons = objects (large states + ac.ons) • Mental imagery: ability to manipulate cogni.ve states and representa.ons. • cogni.ve automata
What is a mental representa.on? Repe..on is necessary… • An image, a movement, a sound, linked to a mathema.cal concept or proof • Several types of representa.ons, adapted to various people – Visual, visio spa.al: mental images or designs – Audi.ve, verbal, phonologic: a sound, a poetry, a sentence,... – Kinesthe.c: a movement, feeling, emo.on,…
More precisely but s.ll informal
Mental Représentation Concrete Conceptual Action Verbal Logic Symbolic Auditive Kinesthesic Visual
Visual Content space Time Structure Nombre d’objets Couleurs Durée Localisation Sujet dans l’image ? Mouvement Intensité Luminosité Concret/symbolique Contraste Brillance Transparence Texture Ombre Cadre Dimension 2/3 D Taille de l’image
Auditive Content space Time Structure Concret/symbolique Volume Durée Localisation Rythme Tonalité Hauteur Pauses Timbre Cadence Tempo
Kinesthesic Content space Time Structure Tactile Température Localisation Proprioceptif Vibration Durée Odeur Pression Mouvement Goût Intensité Humidité Texture
The difficulty of scien.fic defini.ons • Mental states: thoughts, beliefs, desires, percep.ons, images,… • No complete scien.fic defini.ons of MI, MR, MS. Anyway, we have to work with ! • Theory is difficult, not mature, not finished ! As complexity theory, as all theories…
What is the use of a mental representa.on? • Very important to get an intui.on, to understand, to remember • Good representa.ons are hard to find • One representa.on is only useful for a part of students; one needs several representa.ons of various types. • Good representa.ons oien need some effort to be used • Representa.ons do NOT replace algorithms and methods; they help to learn and use them
« Observa.ons » • Understand ≠ memorise • Understand a definition = build a well-adapted representation, manipulate it (Create, test, complete our representations) • well-adapted = multiples codings: verbal, visual, auditive, feeling, movement, kinesthesic, parameters of attention. • Understand a proof ≠ follow the proof line after line + build a well-adapted MR • Prove = create the proof • Teach= give few MR
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