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␈↓ α∧␈↓␈↓ u1


␈↓ 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and

␈↓ α∧␈↓αDeath␈↓. by Steve J. Heims. M.I.T. Press.  547 pp. $19.95.

␈↓ α∧␈↓␈↓ 

by John McCarthy 

␈↓ α∧␈↓␈↓ αTProfessor␈α
Kline␈α
recounts␈α
a␈α
series␈α
of␈α
"shocks",␈α
"disasters"␈α
and␈α
"shattering"␈αexperiences␈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∞firing␈α
the␈α
rocket␈α
in␈α
the␈α
prescribed␈α
direction␈α∞for␈α
the

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

␈↓ α∧␈↓␈↓ αTThe␈α
ancient␈αGreeks␈α
were␈α"shocked"␈α
to␈α
discover␈αthat␈α
the␈αside␈α
and␈α
diagonal␈αof␈α
a␈αsquare␈α
could

␈↓ α∧␈↓not␈αbe␈α
integer␈αmultiples␈α
of␈αa␈α
common␈αlength.␈α This␈α
spoiled␈αtheir␈α
plan␈αto␈α
found␈αall␈αmathematics␈α
on

␈↓ α∧␈↓that␈αof␈αwhole␈αnumbers.␈α
 Nineteenth␈αcentury␈αmathematics␈αwas␈α
"shattered"␈αby␈αthe␈αdiscovery␈αof␈α
non-

␈↓ α∧␈↓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-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␈α
about␈αthe

␈↓ α∧␈↓world (as Kant had said).

␈↓ α∧␈↓␈↓ αTOnce␈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∧␈↓first␈αinformal␈αand␈αthen␈αincreasingly␈αaxiomatized,␈αculminating␈αin␈αformalisms␈αso␈αwell␈αdescribed␈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␈αsurprising␈αparadox␈αof␈αthe␈αthe␈αset␈αof␈αall␈αsets␈αthat␈αare␈αnot␈αmembers␈αof␈αthemselves.␈α (Is␈αit␈αa

␈↓ 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␈αto␈α
prove␈αmathematics␈α
consistent␈αby␈α
showing␈αthat␈α
starting␈αfrom␈α
the␈αaxioms␈αand␈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 be carried out, and this was another surprise.
␈↓ α∧␈↓␈↓ u2


␈↓ α∧␈↓␈↓ αTHowever,␈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 different approaches.

␈↓ α∧␈↓␈↓ αTProfessor␈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⊗confidence␈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␈αto␈α
be␈α
found␈α
in␈α
set␈αtheory,␈α
and␈α
our␈α
confidence␈α
in␈αthis␈α
is

␈↓ α∧␈↓greater than our confidence in any part of physics.

␈↓ α∧␈↓␈↓ αTHeims has a thesis and presents the lives of two mathematicians as an illustration.

␈↓ α∧␈↓␈↓ αTJohn␈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∧␈↓␈↓ αTvon␈α
Neumann's␈αenormous␈α
popularity␈αand␈α
reputation␈α
also␈αcame␈α
from␈αhis␈α
willingness␈αto␈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

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

␈↓ α∧␈↓␈↓ αTNorbert␈αWiener␈α(1894-1964)␈αwas␈αchild␈αprodigy␈αintensely␈αeducated␈αby␈αhis␈αfather,␈αa␈αprofessor

␈↓ 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∧␈↓␈↓ αTWiener␈α
stories␈αare␈α
often␈α
about␈αhis␈α
constant␈αsolicitation␈α
of␈α
assurance␈αthat␈α
his␈αcontributions␈α
to
␈↓ α∧␈↓␈↓ u3


␈↓ α∧␈↓mathematics␈α∞were␈α
outstanding␈α∞-␈α∞which␈α
they␈α∞were.␈α
 Although␈α∞he␈α∞was␈α
inclined␈α∞to␈α∞pontificate␈α
and

␈↓ 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∧␈↓␈↓ αTIn␈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␈αof␈αfeedback,␈αwherein␈αthe␈αoutput␈αof␈αa␈αprocess␈αwas␈αcompared␈αwith␈αa␈αgoal␈αand␈αthe␈αdifference

␈↓ 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
field.␈α∞ von␈α
Neumann

␈↓ α∧␈↓began␈αto␈αconstruct␈αa␈α"general␈αlogical␈αtheory␈αof␈αautomata"␈αand␈αproduced␈αsome␈αfragments␈αincluding

␈↓ α∧␈↓a␈α∩way␈α∪of␈α∩constructing␈α∪reliable␈α∩computers␈α∪from␈α∩unreliable␈α∪components␈α∩and␈α∪a␈α∩theory␈α∪of␈α∩self-

␈↓ α∧␈↓reproducing machines.

␈↓ α∧␈↓␈↓ αTWhile␈α∂Heims␈α∂doesn't␈α∂attempt␈α∂to␈α∂evaluate␈α∂the␈α∂subsequent␈α∂influence␈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∩first␈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␈αmental␈αprocesses␈αsuch␈αas␈αlearning␈αand␈αheuristic␈αsearch

␈↓ 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

␈↓ α∧␈↓science and the philosophy of mind.

␈↓ α∧␈↓␈↓ αTWhen␈α
the␈α
first␈α
conference␈αon␈α
artificial␈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∧␈↓␈↓ αTBoth␈α⊃men␈α∩were␈α⊃interested␈α⊃in␈α∩human␈α⊃affairs.␈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␈αand␈αconflict.␈α Both␈αtheories␈αare␈αstill␈αbeing␈αapplied␈αand␈αextended.␈α He␈αwas␈αalarmed␈αby

␈↓ α∧␈↓Soviet␈αexpansionism␈αafter␈αWorld␈αWar␈αII␈αand␈αadvocated␈αa␈αstrong␈αU.S.␈αmilitary␈αposition␈αincluding

␈↓ α∧␈↓the development of the hydrogen bomb to which he also made technological contributions.
␈↓ α∧␈↓␈↓ u4


␈↓ α∧␈↓␈↓ αTWiener␈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∪significant␈α∩problems

␈↓ α∧␈↓require␈α∞more␈α∂than␈α∞just␈α∂the␈α∞ideas␈α∞of␈α∂feedback␈α∞and␈α∂filtering.␈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∧␈↓␈↓ αTHeims'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␈αis␈αascribed␈αsolely␈αto␈αa␈αdesire␈αto␈αintimidate␈αthe␈αSoviet␈αUnion,␈αand␈αEisenhower's␈α1955␈αatoms-

␈↓ α∧␈↓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␈α⊂Japanese␈α⊂babies.␈α⊃ Like␈α⊂the

␈↓ α∧␈↓curate's egg, parts of the book are good.