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Alexei Samsonovich George Mason University, Fairfax, VA. Cognitive Constructor: An Intelligent Tutoring System Based on a Biologically Inspired Cognitive Architecture (BICA). Cognitive Constructor. (Part I of the talk). Computational consciousness.
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Alexei Samsonovich George Mason University, Fairfax, VA Cognitive Constructor: An Intelligent Tutoring System Based on a BiologicallyInspired Cognitive Architecture (BICA)
Cognitive Constructor (Part I of the talk)
Computational consciousness • can be defined as a fully functional computational equivalent of the human mind in its higher cognitive abilities, including : • Awareness of self and a sense of presence • Understanding of other minds • General mechanisms of voluntary action • Working, semantic, episodic and procedural memories • Commonsense knowledge and the general ability to learn • A system of values • Two approaches: piece-by-piece vs. grow
“Growing” approach:A universal learner considered here • is a cognitive agent (an electronic student) that can acquire arbitrary new knowledge from its own experience with the help of an instructor. • The agent should be able to use acquired knowledge in further learning, with no a priori limitations on its bootstrapped cognitive growth abilities. • The hypothesis is that there is a ‘critical mass’: i.e., a minimal, relatively small set of cognitive abilities that, when implemented in the agent, together enable virtually unlimited, self-sustained cognitive growth of the agent.
Implemented rapid prototype A B C Snapshots of GMU BICA dynamics.A: Indoor virtual environment in which the robot (red dot) performs spatial learning and an object search tasks. B: Dynamics of the cognitive map. C: A snapshot of semantic, working and input-output memories of the agent.
Architecture dynamics in higher-level modeling D E D: A theoretical snapshot of the architecture engaged in sensory perception with shared attention. E: a snapshot of a virtual environment and architecture components in a social interaction paradigm in virtual urban settings.
Idea of a ‘Cognitive Constructor’ • Cognitive science laboratory experiments tell a lot about learning, but are disconnected from education settings • Classroom experiments are not easy and limited • What about experiments in silico? • Virtually no limitations on the number and the range of paradigms, settings, duration, etc. • Can literally see what is going on inside the simulated mind during learning
Cognitive Constructor will be • A rapid prototype and a demo of BICA • A linguist and a conversational agent • An electronic student and a meta-learner • An automated programmer (for example) • A virtually embodied intelligent agent • A cognitive engineer
Engine Reward Episodic memory Cognitive map Working memory Pedagogical Agent (BICA) Designer Cognitive Constructor Input- output Procedural memory Semantic memory Student Science Microworld Cognitive Constructor Interface Instructor
Example 1: Learn to do automated programming Task: Write a code to implement an iterative algorithm (e.g., compute a polynomial) Schema 1: Expression Schema 2: The for loop linguistic linguistic symbolic symbolic code code
Example:Schema of the for loop expression variable variable value for initial step final expression variable Eff-assign val var repeat Con-prim var < final Eff call Eff-assign (var = var + step)
Example 2: “Cognitive construction” of a solution to a pulley problem 4
Example 3:Learning by reading • Challenge: • Acquire natural language capabilities plus general world knowledge, starting from a minimal “critical mass” • Insights: • Cannot acquire practical knowledge by passive reading • Need to understand words practically • Need embedding in a virtual microworld • Need to be guided by a human instructor • The system needs to be human-mind-compatible • Need BICA
Example: LBR This is an idea of a learning system that can acquire natural language capabilities plus general world knowledge, starting from a minimal innate “critical mass” of intelligence, in order to provide an interactive assistant for solving real-world challenges. • Self-aware BICA in a virtual microworld will enable the extraction of knowledge from text
LBR Programmatic Approach: Electronic corpora (curriculum) Syntactic & semantic parser GMU BICA Speech generator Verbal channel Instructor Working memory Episodic memory Simulation microworld with human interface Semantic memory To a formal reasoning & planning system Formal knowledge base It works as guided self-learning.
Conclusions for BICA / CC • To simulate student learning, we need computational equivalents of elements of the human mind • The task involves many dimensions: language, embodiment in a microworld, a Self concept, meta-learning and self-regulated learning, episodic memory • Identifying the ‘critical mass’ of bootstrapped learning is a fundamental scientific challenge • For more interesting topics, watch for the AAAI 2008 Fall Symposium on BICA!
Semantic Cognitive Mapping A practical byproduct of GMU BICA research (Part II of the talk)
A B C Semantic cognitive map:A metric system for meaning? Semantic cognitive map is an embedding of a set of concepts {A, B, C, …} into a continuous metric space, the geometry of which captures the semantics of relations among concepts. In the figure, A and B are synonyms, while A and C are antonyms.
Example of a semantic cognitive map • What is a semantic cognitive map? Here is an example. • Consider the space of colors as a cube with coordinates given by RGB components. Similar colors are co-located next to each other, while opposite colors can be found on opposite sides of the cube. • Consider a set of points inside the cube. Semantics of points is their color given by their coordinates. Therefore, spatial relations between points reflect their sematic relations. This is the definitive property of a Semantic Cognitive Map.
Semantic cognitive mapping method applied to words of natural language • Paradigm of the method: • Embed words as vectors in a D-dimensional space • Optimize the energy function H: • Do principal component (PC) analysis of the result, rotate all vectors to PC coordinates
Semantic cognitive maps of English Microsoft Word Thesaurus: 15,783 words D = 26 Princeton WordNet: 20,477 words
English sorted list The top of PC#1: “positive” clear, well, accept, praise, support, increase, good, bright, right, respect, cheerful, rise, happy, straight, pleasant, improve, continue, honest, clean, safe, encourage, positive, definite, sound, sensible, decent, real, advance, perfect, direct, boost, present, neat, warm, friendly, pure, tidy, kindness, grow, constant, agree, delight, attractive, certain, approve, warmth, genuine, start, fine, honor, upright, fair, alert, smart, correct, kindly, fit, steady, sure, strong, polite, honesty, obvious, kind, order, fresh, helpful, acknowledge, make, smooth, complete, acceptable, succeed, approval, love, openly, understanding, strengthen, generous, honestly, healthy, suitable, gain, open, welcome, interest, reliable, pick up, favor, careful, proper, agreeable, friendliness, clearly, valid, develop, original, secure, swell, rich, help, revive, consistent, please, trust, together, build up, great, attentive, confidence, true, wonderful, exact, sunny, carry on, clever, pleasing, expand, maintain, clarity, confident, value, stable, practical, sweet, trustworthy, sensitive, logical, brightness, adore, compliment, enhance, dependable, reasonable, consent, comfortable, bolster, certainty, expert, useful, lovely, sympathetic, agreement, cheerfulness, cheery, decided, precise, keep, join, joyful, legitimate, pleasure, growth, thorough, advantage, absolute, warmly, like, improvement, allow, respectable, permit, take up, prepared, all right, organized, faithful, raise, effective, adequate, lively, meet, mount, add, honorable, competent, credit, admire, joy, fix, assured, awake, delightful, cheer, positively, have, decency, caring, applaud, truthful, acceptance, politeness, virtue, perfectly, firm, grace, pristine, directly, begin, pleasantness, …
English sorted listThe bottom of PC#1: “negative” dull, decline, poor, stop, bad, uncertain, fail, sad, end, reject, neglect, unpleasant, drop, weak, ignore, vague, decrease, low, deny, miserable, wrong, lose, doubtful, distant, dislike, dirty, careless, weaken, criticize, dim, inadequate, gloomy, foul, down, upset, hate, obscure, insecure, unhappy, inappropriate, give up, unclear, discourage, separate, dubious, clash, destroy, false, reduce, withdraw, oppose, abandon, useless, small, break, cold, disregard, refuse, unreliable, bleak, irregular, depressed, badly, depression, untidy, unacceptable, dishonest, stupid, diminish, tired, doubt, dark, damage, nasty, desolate, cut, awkward, partial, disagreeable, uncertainty, scorn, ill, mess up, ugly, unsound, retreat, unwell, leave, unfit, evil, unsure, shabby, hopeless, depress, negative, rude, poorly, deflate, immoral, improper, minor, lacking, offensive, awful, hollow, dejected, uneven, shaky, repel, exclude, dip, loss, tasteless, conflict, corrupt, messy, appalling, deteriorate, unfriendly, disappear, veto, go down, vagueness, collapse, turn down, nobody, reserved, disgusting, undermine, horrible, dreadful, cloudy, separately, fall, indifference, melancholy, coldness, unsatisfactory, put off, wicked, hazy, rudeness, incompetent, wickedness, failure, chill, inept, bore, foolish, split, illogical, vile, recede, contradict, worsen, fake, senseless, clumsy, impossible, unprepared, treacherous, remote, unfair, unlikely, departure, break down, harmful, little, unhealthy, finish, misery, unsettled, break up, rundown, remove, inattentive, fade, invisible, confusion, flawed, despondent, forbid, condemn, narrow, cruel, inferior, reduction, indistinct, unfriendliness, limited, violate, crooked, run down, disreputable, sadness, confused, bar, question, waste, …
English sorted listThe top of PC#2: “exciting, tuff” stiff, hard, tough, harsh, strong, heavy, difficult, sharp, serious, loud, tense, strict, tight, severe, deep, intense, extreme, demanding, anxious, rough, inflexible, rigid, extraordinary, harshly, critical, violent, great, stiffness, elaborate, fierce, seriously, unusual, wild, arrogant, fiery, important, provoke, passionate, challenging, agitated, forceful, keen, excited, powerful, aggressive, strenuous, stubborn, angry, thick, taut, edgy, unyielding, pompous, exceptional, bias, firm, pungent, resistant, involved, severely, aggravate, complex, significant, intractable, obstinate, extremely, nervous, anxiously, agitate, hot, heavily, burning, raucous, noisy, enormous, busy, roughly, steep, exciting, furious, violently, grave, brash, stout, overpowering, arduous, weighty, huge, stern, excessive, rebellious, acute, fantastic, gross, inflexibility, upset, expensive, tricky, hectic, forced, thorny, enthusiastic, unbending, uncompromising, formidable, special, complicated, tempestuous, concerned, impatient, troubled, worried, heated, jumpy, fat, intensely, strident, remarkable, chunky, pointed, incredible, different, overwrought, wildly, overwhelming, out of the ordinary, conceited, testing, superior, peppery, annoyed, cutting, restless, exceptionally, ferocious, noisily, hefty, occupied, unnatural, flaming, piercing, obdurate, biting, strange, monstrous, unusually, oppressive, massive, beside yourself, acerbic, robust, fiercely, hit, taxing, excessively, desperately, prodigious, haughty, snobbish, virulent, formal, panic, crucial, sturdy, dramatic, grueling, extraordinarily, husky, concern, vital, extravagant, close, unruly, boisterous, peculiar, bulky, bizarre, self-important, rich, awkward, profound, raucously, major, pretentious, big, overbearing, heavy-duty, …
English sorted listThe bottom of PC#2: “calm, easy” calm, soft, relaxed, easy, mild, gentle, modest, quiet, simple, calmly, smooth, relax, ordinary, free, easygoing, calm down, cool, slightly, compliant, insignificant, empty, light, indifferent, weak, minor, peaceful, flexible, slight, tranquil, moderate, gently, normal, loose, soothe, lax, lenient, blunt, unconcerned, quietly, humble, tiny, serene, release, dry, run of the mill, still, understated, softness, meek, submissive, thin, apathetic, modestly, straightforward, normally, soothing, common, lightly, docile, natural, trivial, restrained, mildly, placid, impartiality, unremarkable, surrender, neutral, bland, laid-back, give in, tame, small, composed, softly, stillness, flexibility, mellow, temperate, pale, leisurely, ease, tolerant, dull, pacify, weakness, plain, calmness, pliant, supple, unimportant, settle down, yielding, relaxing, undemanding, hushed, serenity, quietness, slack, peaceable, peacefully, pliable, dispassionate, uninvolved, chill out, untroubled, innocent, appease, slender, minute, unassuming, reasonable, kindly, muted, unpretentious, lightweight, routine, pacific, tranquillity, slim, yield, nonchalant, placate, undisturbed, spare, unflustered, unexceptional, peace, acquiescent, pleasant, open, malleable, smile, economical, sedate, average, give up, delicately, laissez-faire, subservient, soften, idle, everyday, gentleness, ductile, unperturbed, low-key, blasé, biddable, sparse, elastic, leniently, alleviate, little, middling, unflappable, unimpressed, moderately, undistinguished, lightheartedly, lull, insignificantly, stress-free, permissive, defuse, available, commonplace, quiet down, limp, typically, hush, skinny, minimalist, humbly, relaxation, tolerance, unwind, well-behaved, faint, trim, regularly, coolly, accommodating, kind, usually, restful, orderly, serenely, at ease, effortless, mollify, boring, generally, at peace, uncomplicated, carefree, low key, slowly, humility, relieve...
English sorted listThe top of PC#3: “open, free” release, free, open, go, expose, fire, new, freedom, brief, start, loose, opening, independent, reveal, excess, leave, give, quick, let go, discharge, separate, undo, green, hasty, fast, untie, far, young, let loose, different, early, deliver, carelessly, extravagance, quickly, original, set free, first, disclose, single, precipitate, fresh, rapid, surrender, sell, make public, rapidly, set off, hastily, speed up, periphery, circulate, fly, liberate, relent, front, youthful, uncover, dismiss, far away, border, liberated, prompt, activist, bare, firstly, automatically, loosen, unintentionally, give in, edge, distribute, dig up, split, let off, depart, unwrap, accidentally, exposed, far-flung, unlock, at liberty, surplus, publish, hurried, take off, unfasten, divulge, outward, dilettante, unveil, outside, unusual, carelessness, unthinkingly, unconventional, recklessness, yield, unearth, preliminary, unchain, unfetter, spendthrift, send out, initial, run off, publicize, starting, introductory, send, impulsive, share, betray, let out, unbind, give up, released, fleeting, margin, wastefulness, surface, beyond, merrily, innocently, unconstrained, liberation, swift, unrestricted, fringe, before, disinter, modern, speedy, supply, spread about, exude, immediately, go off, flying, temporary, outskirts, profligacy, space, exhume, briefly, rash, landlord, unmarried, hurry, rashly, forward, inexperienced, leave go of, move, emancipate, flexible, give notice to, branch out, promptly, dig out, unhook, at large, unleash, individual, external, far afield, a long way, give out, unconsciously, sudden, reckless, momentary, unused, available, uninhibited, overkill, too much of a good thing, heavy-handedness, speedily, hit the road, accelerate, disengage, begin, casually, improvidence, fringes, outer edge, suburbs, outer reaches, preparatory, uncovered, unconfined, make tracks, get a move on, bring to light, uncontrolled, lend, allow to go, ahead, deliverance
English sorted listThe bottom of PC#3: “close, repress” close, detain, restraint, restrain, confine, capture, conceal, hold back, fasten, stay, compact, hold, final, slow, center, restricted, control, local, hold up, swallow, slowly, cover, keep, wait, deliberately, join, arrest, carefully, late, curb, holdup, eat, suppress, conclude, protracted, fixed, restrict, lock, imprisoned, lock up, seal, lingering, catch, last, freeze, confined, hide, bury, old, put away, conventional, linger, finalize, wrap up, caution, prolonged, closing, prudence, cautious, slow down, cover up, inside, seriousness, delay, finish, later, withhold, tie up, hitch, protect, finally, bind, contract, secure, back, jail, keep secret, moderation, catch-22, restriction, interior, on purpose, guarded, wariness, rear, keep under wraps, long, domestic, firm up, knowingly, unite, inward, stretched, trap, permanent, cushion, shut, conquer, shut in, fetter, keep in custody, set, close up, restrained, conservative, cautiously, put behind bars, parsimonious, shield, finale, keep quiet, detention, suppression, bring under control, bolt, block, captive, tenant, long-drawn-out, conclusion, impede, send down, settle, keep under control, put in jail, take, lockup, concluding, expanded, stretched out, lengthened, time-consuming, traditional, modify, associate, delayed, tame, at length, finish off, jailed, interned, behind bars, wary, hold down, lastly, experienced, considered, clip, follow, self-denial, caged, incarcerated, locked up, ordinary, covered, result, wrap, lengthy, repress, careful, captivity, lock in, dull, sober, parsimony, finishing, keep down, take in for questioning, constrict, take captive, plant, flat, push to, put down the lid, thrifty, slow up, cage, remain, land, get under somebody's feet, get in somebody's way, make somebody late, borrow, clasp, spun out, temperance, after, economical, the same, take somebody on, sign up, guzzle, within, secret, fence in, polish off, retain, lasting, put up the shutters, cocoon.
German sorted list (auto-translated):The top of PC #1: “positive” honestly, well, healthy, to recover, suitably , good, kindly, help good, promote, healthy, honest, honest, to please, friendly, success have, suitable, suitable, persist, kind, honest, good, kind, promote, to love, kind, succeed, recognize, quite, promote, decently, wages, to strengthen, pleasing, surely, certainly, clearly, honour, industrious up, suitable, pleasantly, wages, supported, industrious, healthy, wages, right, industriously, pleasing, entrance, right, cheerfully, attractively, pleasing, put, love, cheerful, to appreciate, praise, cheerful, freshly, decent, pleasant, develop, clear, strong, pleasant, friendly, decent, clear, courageously its, considered, occur, decent, lucky, agreements, yield, friendly, pleasing, put up, agreements, permitted, solemnly, recognize, friendly, yields, success, resemblances, beautifully, attentively, added, agreement , wages, wages, luck, confirmed, promoting acknowledgments, on the dam its, yields , resemblances, yield, yield, pleasant, encouraged, recognition, recognize , permitted, exciting, encourage, clear, accomodating, win respected, on the legs bring, right, exciting, remain, carefully, pleased, lively, recognized, help, to recognized, cheerful, beautiful, joy, prosper, support, fixed, assure, resemblance, to be connected, lucky, energize, use, completely, beautiful, attractive, better, strongly, justified, fixed, clearly, exciting, solemn, lucky, confessed, help, wise, attractive, fixed, freshness, promote, promoted, certainly, admire, core, promoted, live, to blow, solemn, conscientiously, into the hand take, cleanly, perfectly, are awake, eight to give, arrive, affection, kind, lively, help, to please, forward, careful, lively, desire, correctly, come occur, recognize, help, straight, finished, acknowledgment, luck, luck, careful, attentive, exactly, help, agreements, honour, good, praise, steadily, healthier
German sorted list (auto-translated):The top of PC #2: “exciting, tuff” mad, madly, defy themselves mad, over-elevated, self-assured, critical, self-assuredly, haste, grosssprecherisch, up showers, critical, practice-substantial, critically, grossspurig, practice-substantially, hectically, grosssprecherische, hectic, herrisch, grossspurige, grosssprecherischen, hectic, arrogant, grossspurigen, herrische, arrogantly, self-assured, practice-substantial, excited, applied, anmassend, arrogant, fastidiously, exert, sudden, anmassende, extremely, blown up, rebel, immodestly, herrischen, applied, suddenly, accelerated, extraordinarily, hurry, look up-exciting, zornige, arrogant, instructions, excessive, immodest, excite, affectedly, heavy, anmassenden, waste, oppose, more absurd, unusually, excessively, arrogantly, applied, strike, affected, sudden, exaggerate, to the load put, immodest, unusual, fastidious, affected, blown up, unusually, fastidious, excessive, suddenly, rush, unusual, moved, dare, hectic, zornig, luxuriös, Getue, immoderately, much, anger, provoke, blown up, erpicht, worry, defiance to offer, mad, excited, arrogant, raising, greedy, movement, protzig, unrest, importunately, hastily, excited, greedily, remarkably, importunate, extremely, precede, stress, magnificently, to take itself out, ereifern, rave themselves, heated up, Rabatz make, look up-exciting, into which shoes push, dicktuerisch, entrüsten, raising, proudly, protzige, luxuriösen, zornigen, rear up, luxuriöse, open, fast, immoderate, spanningness, surprisingly, accumulation, forgotten, cut themselves open, importantly, sudden, excite themselves, and ago, prevail, race, greedy, heat up, request, immoderate, thoughtless, attack, resist, look up-exciting, dicktuerische, extraordinary, passionately, heavily, work, protzigen, attack, klotzig, hero, many, radically, attacked, chests, delirium, spanningnesses, revolt themselves, saucy, extraordinary, hasty, importunate, noise,…
German sorted list (auto-translated)The top of PC#3: “open, free” Released, handed over, distributed, wasted, give, give, refunded, give , solve, deliver, deliveries, release prayer, hand over, hand over, waste, show, release, paid, given, spent, give, release, give, release, which give liberty, gifts, release, give, loslassest, give, donate, spend, left, given, out-divided, separately, released, spend, hand over, give, hand over, deliver, abandon, leave, presented, release, release, spend, wasted, released, return, released, freely, revealed, gave, reject, released, exposed, hand over released , spent, spend, spend, release, forgets, delivered, provided, offers, issues, handed over, sensitive, recompensed, issued, liberating, returned, to come leave, distributed, release, hand over, return, sacrifices, returned, release, answer, donate, loosens, returns, release, distribute, dismisses, solved, abandoned, supplies, abandon, relinquishments, distribute, less, distribute, wasted, abandons, distribute, donate, donates, refund, separate, return, surrendered, presented, spent, mürbe, gave, handing over, separate, done without, deliver, give, spend, release, out-divides, undressed , sacrifices, delivered, returns, solves, relieved, abandons, returns , giving, offered, apart from, sends, relieves, delivers, wastes , out-divided, solves, more flexible separates, gives, loosely, spends, gives, pays, relieves, pays, out-divided, betrayed, separates, solved, susceptible, separates, released, releases, turns, now, released, expenditure-divided, solved, sensitive, releases, separates, pays, sacrifices, returned, freely moves, paid, separated, freilassest, away-given, revealing liberties, sensitively, carried, separates, delivered, pimpelige, soft, distribute, spend, scattered, out-divided, out-divided, sold, distributed, be responsible for, cut open, away-given, return, loosely, …
PC-by-PC correlation analysis:English vs. English, English vs. French
PC-by-PC correlation analysis:English vs. English, English vs. Spanish
PC-by-PC correlation analysis:English vs. Russian, English vs. ANEW
English Words have similar locations on different maps English English English English French
Application of semantic cognitive mapping: “Semantic twisting”
Application of semantic cognitive mapping:Document search / classification / validation 36,000 English words, 500 Medline abstracts allocated in 10-D space
Results: the first two sentences ofthe most “positive” abstract • “The development of noninvasive methods capable of affording ever higher resolution images is an ongoing major objective of scientific investigation for the benefit of both clinical medicine and neuroscience. Since its development in the early 1970s, magnetic resonance imaging (MRI) has remained a technology of choice for medical imaging development because of the wide range of potential clinical applications.”
Results: the first two sentences of the most “negative” abstract • “7-Nor-20-oxopregn-5-en-3 beta-yl acetate was converted into (20R)-5 beta,6 beta-epoxy-7-nor-5 beta-pregnane-3 beta,20-diyl diacetate in three steps. Stereospecific migration of the 6 alpha-hydride ion led to a 6-oxo derivative with a 5 alpha-configuration.”
Results: the first two sentences of the most “calming” abstract • “Tacrolimus is used widely for immuno-suppression following transplantation. It has rarely been linked to the development of peripheral neuropathy.”
Results:The most “exciting” abstract • “Drosophila courtship is a complex behavior. A new study shows that glia modulate neurotransmission to influence male preference, but the authors should have resisted the temptation to describe their results in tabloid language.” • { if you do not believe, here is the reference: Levine, J. D. (2008). Glia and romance. Nature Neuroscience, 11(1):8-10. }
Results:The most “positive” sonnet • So is it not with me as with that Muse, • Stirr'd by a painted beauty to his verse, • Who heaven itself for ornament doth use • And every fair with his fair doth rehearse, • Making a couplement of proud compare' • With sun and moon, with earth and sea's rich gems, • With April's first-born flowers, and all things rare, • That heaven's air in this huge rondure hems. • O! let me, true in love, but truly write, • And then believe me, my love is as fair • As any mother's child, though not so bright • As those gold candles fix'd in heaven's air: • Let them say more that like of hearsay well; • I will not praise that purpose not to sell.
Results:The most “negative” sonnet • Poor soul, the centre of my sinful earth, • My sinful earth these rebel powers array, • Why dost thou pine within and suffer dearth, • Painting thy outward walls so costly gay? • Why so large cost, having so short a lease, • Dost thou upon thy fading mansion spend? • Shall worms, inheritors of this excess, • Eat up thy charge? Is this thy body's end? • Then soul, live thou upon thy servant's loss, • And let that pine to aggravate thy store; • Buy terms divine in selling hours of dross; • Within be fed, without be rich no more: • So shall thou feed on Death, that feeds on men, • And Death once dead, there's no more dying then.
The same method was applied to45 jokes of/about famous scientists • The most “positive” joke: • Berthelot was so proud of his Legion d'Honneur that he had a lab coat made with a hole in it, through which you could see and admire the medal • The most “exciting” joke: • Evariste Galois was not only a mathematical genius but also a dedicated revolutionary. Ironically, he proved that many problems cannot be solved by radicals.