/LMAR=0/XLINE=3/USET=20␈↓ α,␈↓␈↓ ⊂F1


␈↓ α,␈↓␈↓αMathematics: The Loss of Certainty␈↓. by Morris Kline. Oxford.  366 pp. $19.95.

␈↓ α,␈↓␈↓αJohn␈α∃von␈α∃Neumann␈α∃and␈α∃Norbert␈α∃Wiener:␈α∃From␈α∃Mathematics␈α∃to␈α∃the␈α∃Technologies␈α∃of␈α∃Life

␈↓ α,␈↓αand Death␈↓. by Steve J. Heims. M.I.T. Press.  547 pp. $19.95.

␈↓ α,␈↓␈↓ 
mby John McCarthy 

␈↓ α,␈↓␈↓ α|Professor␈α≥Kline␈α≥recounts␈α≥a␈α≥series␈α≥of␈α≥"shocks",␈α≥"disasters"␈α≥and␈α≥"shattering"␈α≥experiences

␈↓ α,␈↓leading␈α∀to␈α∀a␈α∀"loss␈α∀of␈α∀certainty"␈α∀in␈α∀mathematics.␈α∀ However,␈α∀he␈α∀doesn't␈α∀mean␈α∀that␈α∀the␈α∀astronaut

␈↓ α,␈↓should␈α⊃mistrust␈α⊃the␈α⊃computations␈α⊃that␈α⊃tell␈α⊃him␈α⊃that␈α⊃≡ring␈α⊃the␈α⊃rocket␈α⊃in␈α⊃the␈α⊃prescribed␈α⊃direction

␈↓ α,␈↓for the prescribed number of seconds will get him to the moon.

␈↓ α,␈↓␈↓ α|The␈α↔ancient␈α↔Greeks␈α↔were␈α↔"shocked"␈α↔to␈α↔discover␈α↔that␈α↔the␈α↔side␈α↔and␈α↔diagonal␈α↔of␈α↔a␈α↔square

␈↓ α,␈↓could␈α→not␈α→be␈α→integer␈α→multiples␈α→of␈α→a␈α→common␈α→length.␈α→ This␈α→spoiled␈α→their␈α→plan␈α→to␈α→found␈α→all

␈↓ α,␈↓mathematics␈α∂on␈α∂that␈α∂of␈α∂whole␈α∂numbers.␈α∂ Nineteenth␈α∂century␈α∂mathematics␈α∂was␈α∂"shattered"␈α∂by␈α∂the

␈↓ α,␈↓discovery␈α⊂of␈α⊂non-Euclidean␈α⊂geometry␈α⊂(violating␈α⊂Euclid's␈α⊂axiom␈α⊂that␈α⊂there␈α⊂is␈α⊂exactly␈α⊂one␈α⊂parallel

␈↓ α,␈↓to␈α⊂a␈α⊂line␈α⊂through␈α⊂an␈α⊂external␈α⊂point),␈α⊂which␈α⊂showed␈α⊂that␈α⊂Euclidean␈α⊂geometry␈α⊂isn't␈α⊂based␈α⊂on␈α⊂self-

␈↓ α,␈↓evident␈α∀axioms␈α∀about␈α∀physical␈α∀space␈α∀(as␈α∀most␈α∀people␈α∀believed).␈α∀ Nor␈α∀is␈α∀it␈α∀a␈α∀necessary␈α∀way␈α∀of

␈↓ α,␈↓thinking about the world (as Kant had said).

␈↓ α,␈↓␈↓ α|Once␈α∪detached␈α∪from␈α∪physics,␈α∪mathematics␈α∪developed␈α∪on␈α∪the␈α∪basis␈α∪of␈α∪the␈α∪theory␈α∪of␈α∪sets,␈α∪at

␈↓ α,␈↓≡rst␈α∀informal␈α∀and␈α∀then␈α∀increasingly␈α∀axiomatized,␈α∀culminating␈α∀in␈α∀formalisms␈α∀so␈α∀well␈α∀described

␈↓ α,␈↓that␈α⊗proofs␈α⊗can␈α⊗be␈α⊗checked␈α⊗by␈α⊗computer.␈α⊗ However,␈α⊗Gottlob␈α⊗Frege's␈α⊗plausible␈α⊗axioms␈α⊗led␈α⊗to

␈↓ α,␈↓Bertrand␈α≠Russell's␈α≠surprising␈α≠paradox␈α≠of␈α≠the␈α≠the␈α≠set␈α≠of␈α≠all␈α≠sets␈α≠that␈α≠are␈α≠not␈α≠members␈α≠of

␈↓ α,␈↓themselves.␈α~ (Is␈α~it␈α~a␈α~member␈α~of␈α~itself?).␈α~ L.E.J.␈α~Brouwer␈α~reacted␈α~with␈α~a␈α~doctrine␈α~that␈α~only

␈↓ α,␈↓constructive␈α'mathematical␈α'objects␈α'should␈α'be␈α'allowed␈α'(making␈α'for␈α'a␈α'picky␈α'and␈α'ugly

␈↓ α,␈↓mathematics),␈α∂whereas␈α∂David␈α∂Hilbert␈α∂proposed␈α∂to␈α∂prove␈α∂mathematics␈α∂consistent␈α∂by␈α∂showing␈α∂that

␈↓ α,␈↓starting␈α∃from␈α∃the␈α∃axioms␈α∃and␈α∃following␈α∃the␈α∃rules␈α∃could␈α∃never␈α∃lead␈α∃to␈α∃contradiction.␈α∃ In␈α∃1931

␈↓ α,␈↓Kurt␈α≤G␈↓
:␈↓odel␈α≤showed␈α≤that␈α≤Hilbert's␈α≤program␈α≤cannot␈α≤be␈α≤carried␈α≤out,␈α≤and␈α≤this␈α≤was␈α≤another

␈↓ α,␈↓surprise.
␈↓ α,␈↓␈↓ ⊂F2


␈↓ α,␈↓␈↓ α|However,␈α~Hilbert's␈α~program␈α~and␈α~Tarski's␈α~work␈α~led␈α~to␈α~metamathematics,␈α~which␈α~studies

␈↓ α,␈↓mathematical␈α∪theories␈α∪as␈α∪mathematical␈α∪objects.␈α∪ This␈α∪replaced␈α∪many␈α∪of␈α∪the␈α∪disputes␈α∪about␈α∪the

␈↓ α,␈↓foundations of mathematics by the peaceful study of the structure of the di≥erent approaches.

␈↓ α,␈↓␈↓ α|Professor␈α%Kline's␈α%presentation␈α%of␈α%these␈α%and␈α%other␈α%surprises␈α%as␈α%shocks␈α%that␈α%made

␈↓ α,␈↓mathematicians␈α≡lose␈α≡con≡dence␈α≡in␈α≥the␈α≡certainty␈α≡and␈α≡in␈α≡the␈α≡future␈α≡of␈α≡mathematics␈α≡seems

␈↓ α,␈↓overdrawn.␈α∪ While␈α∪the␈α∪consistency␈α∪of␈α∪even␈α∪arithmetic␈α∪cannot␈α∪be␈α∪proved,␈α∪most␈α∪mathematicians

␈↓ α,␈↓seem␈α⊗to␈α⊗believe␈α⊗(with␈α⊗G␈↓
:␈↓odel)␈α⊗that␈α⊗mathematical␈α⊗truth␈α⊗exists␈α⊗and␈α⊗that␈α⊗present␈α⊗mathematics␈α⊗is

␈↓ α,␈↓true.␈α⊃ No␈α⊃mathematician␈α⊃expects␈α⊃an␈α⊃inconsistency␈α⊃to␈α⊃be␈α⊃found␈α⊃in␈α⊃set␈α⊃theory,␈α⊃and␈α⊃our␈α⊃con≡dence

␈↓ α,␈↓in this is greater than our con≡dence in any part of physics.

␈↓ α,␈↓␈↓ α|Heims has a thesis and presents the lives of two mathematicians as an illustration.

␈↓ α,␈↓␈↓ α|John␈α~von␈α~Neumann␈α~(1903-1957)␈α~was␈α~perhaps␈α~the␈α~brightest␈α~of␈α~a␈α~remarkable␈α~group␈α~of

␈↓ α,␈↓Hungarian␈α&born␈α&mathematicians␈α&and␈α&physicists.␈α& Heims␈α&describes␈α&his␈α&contributions␈α&to

␈↓ α,␈↓mathematical␈α∃logic,␈α∃to␈α∃the␈α∃mathematical␈α∃foundations␈α∃of␈α∃quantum␈α∃mechanics,␈α∃to␈α∃the␈α∃theory␈α∃of

␈↓ α,␈↓games,␈α⊗to␈α⊗the␈α⊗development␈α⊗of␈α⊗computers,␈α⊗to␈α⊗the␈α⊗development␈α⊗of␈α⊗atomic␈α⊗bombs␈α⊗and␈α⊗peaceful

␈↓ α,␈↓nuclear energy, and to the relation of brain and computer.

␈↓ α,␈↓␈↓ α|von␈α→Neumann's␈α→enormous␈α→popularity␈α→and␈α_reputation␈α→also␈α→came␈α→from␈α→his␈α→willingness␈α→to

␈↓ α,␈↓listen␈α⊃to␈α⊃other␈α⊃scientists␈α⊃and␈α⊃his␈α⊃ability␈α⊃to␈α⊃clarify␈α⊃their␈α⊃ideas␈α⊃and␈α⊃often␈α⊃solve␈α⊃the␈α⊃problems␈α⊃they

␈↓ α,␈↓were␈α⊃posing.␈α⊃ No-one␈α⊃else␈α⊃then␈α⊃or␈α⊃since␈α⊃had␈α⊃anything␈α⊃like␈α⊃his␈α⊃reputation␈α⊃for␈α⊃this,␈α⊃but␈α⊃he␈α⊃might

␈↓ α,␈↓have made greater original contributions had he been less helpful to others.

␈↓ α,␈↓␈↓ α|Norbert␈α⊂Wiener␈α⊂(1894-1964)␈α⊂was␈α⊂child␈α⊂prodigy␈α⊂intensely␈α⊂educated␈α⊂by␈α⊂his␈α⊂father,␈α⊂a␈α⊂professor

␈↓ α,␈↓of␈α_languages␈α_at␈α_Harvard.␈α_ He␈α_received␈α_his␈α_PhD␈α_at␈α_18␈α_and␈α_immediately␈α_began␈α_a␈α_career␈α_of

␈↓ α,␈↓contributions␈α⊂to␈α⊂many␈α⊂branches␈α⊂of␈α⊂mathematics.␈α⊂ After␈α⊂World␈α⊂War␈α⊂II,␈α⊂he␈α⊂proposed␈α⊂a␈α⊂science␈α⊂of

␈↓ α,␈↓"cybernetics, the theory of feedback and control in animal and machine".

␈↓ α,␈↓␈↓ α|Wiener␈α∂stories␈α∂are␈α∂often␈α∂about␈α∂his␈α∂constant␈α∂solicitation␈α∂of␈α∂assurance␈α∂that␈α∂his␈α∂contributions␈α∂to

␈↓ α,␈↓mathematics␈α⊃were␈α⊃outstanding␈α⊃-␈α⊃which␈α⊃they␈α⊃were.␈α⊃ Although␈α⊃he␈α⊃was␈α⊃inclined␈α⊃to␈α⊃ponti≡cate␈α⊃and
␈↓ α,␈↓␈↓ ⊂F3


␈↓ α,␈↓had␈α⊂a␈α⊂higher␈α⊂opinion␈α⊂of␈α⊂the␈α⊂importance␈α⊂of␈α⊂some␈α⊂of␈α⊂his␈α⊂contributions␈α⊂than␈α⊂many␈α⊂others␈α⊂did,␈α⊂his

␈↓ α,␈↓two volumes of autobiography are usually very objective, especially about his earlier life.

␈↓ α,␈↓␈↓ α|In␈α≥the␈α≥late␈α≥forties,␈α≥Wiener␈α≥and␈α≥von␈α≥Neumann␈α≥shared␈α≥an␈α≥interest␈α≥in␈α≥the␈α≥relation␈α≥of

␈↓ α,␈↓computers␈α_to␈α_the␈α_brain,␈α_met␈α_often,␈α_and␈α_jointly␈α_organized␈α_meetings.␈α_ Wiener's␈α_approach␈α_was

␈↓ α,␈↓through␈α∂the␈α∂notion␈α∂of␈α∂feedback,␈α∂wherein␈α∂the␈α∂output␈α∂of␈α∂a␈α∂process␈α∂was␈α∂compared␈α∂with␈α∂a␈α∂goal␈α∂and

␈↓ α,␈↓the␈α∪di≥erence␈α∪used␈α∪to␈α∪control␈α∪the␈α∪process.␈α∪ He␈α∪coined␈α∪the␈α∪term␈α∪cybernetics␈α∪for␈α∪the␈α∪whole␈α∪≡eld.

␈↓ α,␈↓von␈α∃Neumann␈α∃began␈α∃to␈α∃construct␈α∃a␈α∃"general␈α∃logical␈α∃theory␈α∃of␈α∃automata"␈α∃and␈α∃produced␈α∃some

␈↓ α,␈↓fragments␈α∂including␈α∂a␈α∂way␈α∂of␈α∂constructing␈α∂reliable␈α∂computers␈α∂from␈α∂unreliable␈α∂components␈α∂and␈α∂a

␈↓ α,␈↓theory of self-reproducing machines.

␈↓ α,␈↓␈↓ α|While␈α⊗Heims␈α⊗doesn't␈α⊗attempt␈α⊗to␈α⊗evaluate␈α⊗the␈α⊗subsequent␈α⊗in∨uence␈α⊗of␈α⊗the␈α⊗work␈α⊗of␈α⊗either

␈↓ α,␈↓Wiener␈α∪or␈α∪von␈α∪Neumann,␈α∪neither␈α∪cybernetics␈α∪nor␈α∪the␈α∪general␈α∪theory␈α∪of␈α∪automata␈α∪has␈α∪been␈α∪as

␈↓ α,␈↓successful␈α_as␈α_the␈α_approach␈α_≡rst␈α_proposed␈α_about␈α_1950␈α_by␈α_the␈α_British␈α_logician␈α_and␈α_computer

␈↓ α,␈↓scientist␈α∀Alan␈α∀Turing.␈α∀ Turing␈α∀proposed␈α∀that␈α∀mental␈α∀processes␈α∀be␈α∀studied␈α∀by␈α∀programming␈α∀a

␈↓ α,␈↓computer␈α⊗to␈α⊗carry␈α⊗them␈α⊗out␈α⊗rather␈α⊗than␈α⊗by␈α⊗building␈α⊗machines␈α⊗that␈α⊗imitate␈α⊗the␈α⊗brain␈α⊗at␈α⊗the

␈↓ α,␈↓physiological␈α→level.␈α→ Programming␈α→concrete␈α→mental␈α→processes␈α→such␈α→as␈α→learning␈α→and␈α→heuristic

␈↓ α,␈↓search␈α∀in␈α∀connection␈α∀with␈α∀problem␈α∀solving␈α∀programs␈α∀has␈α∀proved␈α∀more␈α∀fruitful␈α∀in␈α∀psychology,

␈↓ α,␈↓computer science and the philosophy of mind.

␈↓ α,␈↓␈↓ α|When␈α∪the␈α∪≡rst␈α∪conference␈α∪on␈α∪arti≡cial␈α∪intelligence␈α∪was␈α∪organized␈α∪for␈α∪the␈α∪summer␈α∪of␈α∪1956,

␈↓ α,␈↓everyone had great hopes for a contribution from von Neumann, but he was already too sick.

␈↓ α,␈↓␈↓ α|Both␈α→men␈α→were␈α→interested␈α→in␈α→human␈α→a≥airs.␈α→ von␈α→Neumann␈α→developed␈α→a␈α→mathematical

␈↓ α,␈↓theory␈α⊂of␈α⊂an␈α⊂expanding␈α⊂economy␈α⊂in␈α⊂the␈α⊂1920s␈α⊂and␈α⊂a␈α⊂theory␈α⊂of␈α⊂games␈α⊂in␈α⊂the␈α⊂1940s␈α⊂for␈α⊂studying

␈↓ α,␈↓competition␈α∩and␈α∩con∨ict.␈α∩ Both␈α∩theories␈α∩are␈α∩still␈α∩being␈α∩applied␈α∩and␈α∩extended.␈α∩ He␈α∩was␈α∩alarmed

␈↓ α,␈↓by␈α_Soviet␈α_expansionism␈α_after␈α_World␈α_War␈α_II␈α_and␈α_advocated␈α_a␈α_strong␈α_U.S.␈α_military␈α_position

␈↓ α,␈↓including␈α≡the␈α≡development␈α≡of␈α≡the␈α≡hydrogen␈α≡bomb␈α≡to␈α≡which␈α≡he␈α≡also␈α≡made␈α≡technological

␈↓ α,␈↓contributions.
␈↓ α,␈↓␈↓ ⊂F4


␈↓ α,␈↓␈↓ α|Wiener␈α≥proposed␈α≥that␈α≥there␈α≥be␈α≥a␈α≥cybernetic␈α≥theory␈α≥of␈α≥human␈α≥biology␈α≥and␈α≥sociology

␈↓ α,␈↓emphasizing␈α⊗both␈α⊗random␈α⊗processes␈α⊗and␈α⊗stabilization␈α⊗by␈α⊗feedback.␈α⊗ These␈α⊗attempts␈α⊗achieved

␈↓ α,␈↓considerable␈α↔acclaim,␈α↔but␈α↔(I␈α↔think)␈α↔few␈α↔results␈α↔of␈α↔lasting␈α↔value,␈α↔because␈α↔signi≡cant␈α↔problems

␈↓ α,␈↓require␈α∩more␈α∩than␈α∩just␈α∩the␈α∩ideas␈α∩of␈α∩feedback␈α∩and␈α∩≡ltering.␈α∩ Perhaps␈α∩because␈α∩he␈α∩didn't␈α∩see␈α∩the

␈↓ α,␈↓problems␈α∃of␈α∃pattern␈α∃matching␈α∃and␈α∃heuristics,␈α∃he␈α∃expected␈α∃automatic␈α∃factories␈α∃to␈α∃replace␈α∃most

␈↓ α,␈↓manual␈α∂labor␈α∂before␈α∂1970.␈α∂ He␈α∂worried␈α∂about␈α∂the␈α∂expected␈α∂unemployment␈α∂but␈α∂had␈α∂few␈α∂concrete

␈↓ α,␈↓proposals.␈α∃ His␈α∃attitude␈α∃to␈α∃defense␈α∃was␈α∃the␈α∃opposite␈α∃of␈α∃von␈α∃Neumann's;␈α∃he␈α∃opposed␈α∃work␈α∃on

␈↓ α,␈↓defense␈α∂problems␈α∂after␈α∂the␈α∂end␈α∂of␈α∂World␈α∂War␈α∂II,␈α∂sometimes␈α∂holding␈α∂that␈α∂a␈α∂scientist␈α∂should␈α∂keep

␈↓ α,␈↓secret work that he thought could be used for military purposes.

␈↓ α,␈↓␈↓ α|Heims's␈α⊃thesis␈α⊃is␈α⊃that␈α⊃Wiener␈α⊃was␈α⊃moral␈α⊃and␈α⊃von␈α⊃Neumann␈α⊃was␈α⊃immoral␈α⊃in␈α⊃their␈α⊃attitudes

␈↓ α,␈↓toward␈α⊃the␈α⊃uses␈α⊃of␈α⊃science,␈α⊃especially␈α⊃military␈α⊃applications,␈α⊃but␈α⊃also␈α⊃industrial.␈α⊃ Aspects␈α⊃of␈α⊃their

␈↓ α,␈↓family␈α≠backgrounds,␈α≠early␈α≠work,␈α≠and␈α≠personal␈α≠lives␈α≠are␈α≠interpreted␈α≠as␈α≠precursors␈α≠of␈α≠their

␈↓ α,␈↓postwar␈α∀positions.␈α∀ The␈α∀"critical␈α∀science"␈α∀style␈α∀he␈α∀adopts␈α∀involves␈α∀loaded␈α∀adjectives␈α∀and␈α∀other

␈↓ α,␈↓unfairness␈α∀and␈α∀often␈α∀assumes␈α∀what␈α∀he␈α∀has␈α∀undertaken␈α∀to␈α∀prove.␈α∀ Thus␈α∀Truman's␈α∀decision␈α∀to

␈↓ α,␈↓use␈α≤the␈α≤atomic␈α≤bomb␈α≤is␈α≤ascribed␈α≤solely␈α≤to␈α≤a␈α≤desire␈α≤to␈α≤intimidate␈α≤the␈α≤Soviet␈α≤Union,␈α≤and

␈↓ α,␈↓Eisenhower's␈α!1955␈α!atoms-for-peace␈α!proposals␈α!are␈α!described␈α!as␈α!a␈α!"benign␈α!veneer".␈α! Both

␈↓ α,␈↓propositions␈α≠are␈α≠unsupported␈α≠by␈α≠argument.␈α≠ The␈α≠series␈α≠of␈α≠photographs␈α≠ends␈α≠with␈α≠two␈α≠of

␈↓ α,␈↓deformed Japanese babies.  Like the curate's egg, parts of the book are good.
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