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␈↓ α∧␈↓␈↓ u1


␈↓ α∧␈↓α␈↓ ∧6A PROGRAMME FOR META-EPISTEMOLOGY

␈↓ α∧␈↓␈↓ αTMeta-epistemology␈α
will␈α∞be␈α
to␈α∞epistemology␈α
as␈α
metamathematics␈α∞is␈α
to␈α∞mathematics.␈α
 Taking
␈↓ α∧␈↓epistemology␈α
as␈α
the␈α
theory␈α
of␈α
knowledge␈αand␈α
how␈α
it␈α
is␈α
obtained,␈α
meta-epistemology␈αstudies␈α
systems
␈↓ α∧␈↓that␈α∂seek␈α⊂knowledge␈α∂in␈α⊂a␈α∂world.␈α∂ An␈α⊂epistemological␈α∂system␈α⊂is␈α∂then␈α∂a␈α⊂world␈α∂and␈α⊂a␈α∂knowledge
␈↓ α∧␈↓seeker in that world.

␈↓ α∧␈↓␈↓ αTOur␈αultimate␈α
goal␈αis␈α
to␈αstudy␈α
epistemological␈αsystems␈α
technically.␈α Thus␈α
we␈αmay␈α
consider␈αa
␈↓ α∧␈↓knowledge␈α⊃seeking␈α⊂computer␈α⊃program␈α⊂connected␈α⊃to␈α⊂a␈α⊃system␈α⊂of␈α⊃finite␈α⊂state␈α⊃automata␈α⊃and␈α⊂ask
␈↓ α∧␈↓whether␈α
a␈α
given␈α
knowledge␈αseeking␈α
strategy␈α
can␈α
learn␈α
certain␈αfacts␈α
about␈α
the␈α
"external␈αworld".␈α
 To
␈↓ α∧␈↓the␈αextent␈αthat␈αepistemological␈αquestions␈αcan␈αbe␈αput␈αin␈αsuch␈αa␈αtechnical␈αform,␈αtheir␈αanswers␈αare␈αto
␈↓ α∧␈↓be␈α∂obtained␈α∂by␈α∞mathematics.␈α∂ The␈α∂philosophical␈α∞issues␈α∂will␈α∂then␈α∞concern␈α∂whether␈α∂the␈α∞technical
␈↓ α∧␈↓form␈αof␈αthe␈αquestion␈αproperly␈αrepresents␈αthe␈αoriginal␈αphilosophical␈αproblem.␈α If␈αthe␈αexperience␈αof
␈↓ α∧␈↓metamathematics␈α⊃is␈α⊃a␈α∩portent,␈α⊃the␈α⊃mathematical␈α⊃answers␈α∩won't␈α⊃be␈α⊃accepted␈α∩as␈α⊃philosophically
␈↓ α∧␈↓conclusive␈α∀but␈α∀will␈α∀nevertheless␈α∀transform␈α∀the␈α∀way␈α∀in␈α∀which␈α∀philosophers␈α∀think␈α∀about␈α∪the
␈↓ α∧␈↓problems.␈α I␈αhave␈α
in␈αmind␈αGoedel's␈αincompleteness␈α
theorems,␈αGentzen's␈αresults␈αon␈α
the␈αconsistency
␈↓ α∧␈↓of arithmetic and the result that intuitionist and classical arithmetic are equiconsistent.

␈↓ α∧␈↓␈↓ αTI␈αmust␈αconfess␈αthat␈αI␈αcannot␈αyet␈αgive␈αtechnical␈αforms␈αof␈αepistemological␈αquestions␈αthat␈αseem
␈↓ α∧␈↓ripe for mathematical attack.  However, I have some ideas that may be helpful.

␈↓ α∧␈↓1.␈αAvoiding␈αheuristic␈αproblems.␈α Work␈αin␈αartificial␈αintelligence␈αsuggests␈αthat␈αfind␈αout␈αthe␈α
facts␈αof
␈↓ α∧␈↓the␈α∂world␈α∞has␈α∂two␈α∞components.␈α∂ One␈α∂component␈α∞is␈α∂a␈α∞general␈α∂framework␈α∞or␈α∂language␈α∂in␈α∞which
␈↓ α∧␈↓hypotheses␈αand␈αfacts␈α
can␈αbe␈αexpressed,␈α
and␈αthe␈αother␈α
is␈αa␈αstrategy␈α
of␈αsearch␈αand␈α
experiment.␈α In
␈↓ α∧␈↓artificial intelligence, the first is called epistemology, and the second is called heuristics.

␈↓ α∧␈↓␈↓ αTIt␈α∞is␈α∞our␈α
hope␈α∞that␈α∞the␈α
heuristic␈α∞problem␈α∞can␈α
be␈α∞avoided␈α∞in␈α
studying␈α∞epistemology␈α∞in␈α
the
␈↓ α∧␈↓following␈α∂way.␈α∂ Instead␈α∂of␈α∂imagining␈α∂a␈α∂computer␈α∂program␈α∂that␈α∂seeks␈α∂knowledge,␈α∂we␈α∂imagine␈α∂a
␈↓ α∧␈↓program␈α
that␈αis␈α
led␈αtowards␈α
the␈α
truth␈αby␈α
a␈αSocratic␈α
tutor.␈α
 Its␈αtutor␈α
supplies␈αno␈α
knowledge␈α
to␈αbe
␈↓ α∧␈↓accepted␈α∞on␈α∞the␈α∞tutor's␈α∞authority,␈α∞but␈α∞merely␈α∞proposes␈α∞observations,␈α∞experiments␈α∞and␈α∞inferences.
␈↓ α∧␈↓The␈α⊃student␈α⊃program␈α⊃accepts␈α⊃or␈α⊃rejects␈α⊃these␈α⊃steps␈α⊃according␈α⊃to␈α⊃its␈α⊃built-in␈α⊃epistemology,␈α⊃i.e.
␈↓ α∧␈↓knowledge,␈αrules␈αof␈αinference,␈αand␈αrules␈αof␈αreasonable␈αconjecture.␈α We␈αrequire␈αthat␈αthe␈αstudent␈αbe
␈↓ α∧␈↓consistent␈αin␈αthat␈α
no␈αSocrates␈αcan␈α
lead␈αit␈αinto␈αa␈α
contradiction.␈α (Later␈αwe␈α
shall␈αhave␈αto␈α
relax␈αthis
␈↓ α∧␈↓stricture␈αa␈α
little).␈α If␈αthe␈α
Socratic␈αtutor␈α
can␈αlead␈αit␈α
to␈αdiscover␈α
a␈αcertain␈αfact␈α
about␈αits␈α
world,␈αthen
␈↓ α∧␈↓we␈αshall␈αsay␈αthat␈α
its␈αepistemology␈αis␈αadequate␈α
for␈αobtaining␈αthis␈αfact.␈α
 Since␈αwe,␈αthe␈αcreators␈αof␈α
the
␈↓ α∧␈↓epistemological␈α∞system,␈α
also␈α∞created␈α
the␈α∞world␈α∞in␈α
which␈α∞knowledge␈α
is␈α∞sought,␈α
we␈α∞know␈α∞what␈α
the
␈↓ α∧␈↓facts are, so it is a definite question as to whether the program has discovered them.

␈↓ α∧␈↓␈↓ αTWe␈αmust␈αhope␈αthat␈αavoiding␈αheuristics␈αis␈αsuccessful,␈αbecause␈αstudying␈αsystems␈αthat␈αcombine
␈↓ α∧␈↓epistemology and heuristics will be very difficult.

␈↓ α∧␈↓␈↓ αTConsider␈α
first␈αKant's␈α
idea␈αthat␈α
our␈αability␈α
to␈α
understand␈αthe␈α
world␈αdepends␈α
on␈αhaving␈α
some
␈↓ α∧␈↓innate␈α
ideas.␈α Many␈α
of␈αhis␈α
original␈αproposals␈α
as␈α
to␈αwhat␈α
modes␈αof␈α
perceiving␈αthe␈α
world␈αare␈α
innate,
␈↓ α∧␈↓e.g.␈αEuclidean␈αgeometry,␈αhaven't␈αsurvived␈αintact,␈αand␈αothers␈αare␈αinsufficiently␈αdefinitely␈αdescribed,
␈↓ α∧␈↓but␈α
something␈α
along␈α
this␈α∞line␈α
is␈α
still␈α
defended.␈α
 Here␈α∞is␈α
an␈α
attempt␈α
at␈α
a␈α∞corresponding␈α
technical
␈↓ α∧␈↓question.

␈↓ α∧␈↓␈↓ αTLet␈αthe␈α
world␈αbe␈αa␈α
system␈αwith␈αa␈α
state␈αthat␈α
varies␈αas␈αa␈α
function␈αof␈αa␈α
real␈αvariable␈α
␈↓↓t␈↓␈αcalled
␈↓ α∧␈↓time.␈α⊂ Let␈α⊂the␈α⊂"student"␈α⊂be␈α⊂computer␈α⊃program␈α⊂with␈α⊂sense␈α⊂and␈α⊂motor␈α⊂organs␈α⊂connected␈α⊃to␈α⊂this
␈↓ α∧␈↓␈↓ u2


␈↓ α∧␈↓"outside␈αworld".␈α We␈αsuppose␈αthat␈αthe␈αway␈αthe␈αstate␈αof␈αthe␈αoutside␈αworld␈αchanges␈αas␈αa␈αfunction␈αof
␈↓ α∧␈↓its␈α∞previous␈α∞history␈α∞and␈α∞its␈α∞motor␈α∞inputs␈α∂is␈α∞described␈α∞by␈α∞a␈α∞collection␈α∞of␈α∞sentences␈α∞of␈α∂set␈α∞theory.
␈↓ α∧␈↓Suppose␈α∂further␈α∂that␈α∞the␈α∂particular␈α∂"outside␈α∂world"␈α∞is␈α∂an␈α∂arbitrary␈α∂member␈α∞of␈α∂a␈α∂certain␈α∂set␈α∞of
␈↓ α∧␈↓"possible outside worlds" - this set also being defined by a sentence of set theory.

␈↓ α∧␈↓␈↓ αTNow␈α
we␈α
can␈α
try␈α∞to␈α
embody␈α
different␈α
epistemological␈α∞strategies␈α
in␈α
the␈α
student␈α∞program.␈α
 A
␈↓ α∧␈↓rather␈α⊂strong␈α⊂empiricist␈α⊂strategy␈α⊂would␈α⊂allow␈α∂only␈α⊂hypotheses␈α⊂that␈α⊂were␈α⊂expressed␈α⊂in␈α⊂terms␈α∂of
␈↓ α∧␈↓sensory-motor␈αrelations␈αin␈αsome␈αrestricted␈αlanguage.␈α A␈αKantian␈αprogram␈αwould␈αallow␈αhypotheses
␈↓ α∧␈↓that␈α∂conformed␈α⊂to␈α∂some␈α∂preferred␈α⊂␈↓↓synthetic␈α∂a␈α⊂priori␈↓␈α∂ideas.␈α∂ It␈α⊂seems␈α∂to␈α∂me␈α⊂that␈α∂a␈α⊂set␈α∂theoretic
␈↓ α∧␈↓epistemology␈αcorresponding␈αto␈αmodern␈αscientific␈αpractice␈αwould␈αallow␈αthe␈αtutor␈αto␈αpropose␈αnew␈αset
␈↓ α∧␈↓theoretic␈α∂constants␈α∂without␈α∂having␈α∂to␈α∂define␈α∂them␈α∂and␈α∂propose␈α∂sentences␈α∂of␈α∂set␈α∂theory␈α⊂as␈α∂laws
␈↓ α∧␈↓connecting␈αthem.␈α
 Depending␈αon␈αwhat␈α
was␈αproposed␈α
and␈αwhat␈αthe␈α
world␈αis␈α
like,␈αthe␈αstudent␈α
might
␈↓ α∧␈↓be␈α∞able␈α
to␈α∞decide␈α
on␈α∞these␈α
proposed␈α∞laws,␈α
but␈α∞no␈α
a␈α∞priori␈α
requirement␈α∞would␈α
be␈α∞made␈α∞that␈α
the
␈↓ α∧␈↓proposed laws be decidable experimentally.

␈↓ α∧␈↓␈↓ αTWe␈α∞should␈α∞compare␈α∞the␈α∂efficacy␈α∞of␈α∞many␈α∞specific␈α∂innate␈α∞modes␈α∞of␈α∞perception,␈α∂e.g.␈α∞spatial
␈↓ α∧␈↓and␈α
temporal,␈α
with␈α∞the␈α
general␈α
mode␈α
that␈α∞says␈α
that␈α
the␈α
world␈α∞is␈α
a␈α
causal␈α∞system␈α
mathematically
␈↓ α∧␈↓describable,␈αe.g.␈αby␈αsentences␈αof␈αZermelo-Fraenkel␈αset␈αtheory.␈α The␈αlatter␈αpresumably␈αhas␈αa␈αcertain
␈↓ α∧␈↓universality.␈α Any␈αspecific␈α
mode␈αof␈αperception,␈α
e.g.␈αthe␈αtendency␈α
to␈αinterpret␈αsensations␈α
as␈αarising
␈↓ α∧␈↓from␈α
persistent␈α
objects␈α
located␈α
in␈α
three-dimensional␈α
space,␈α
can␈α
be␈α
formulated␈α
as␈α
a␈α
hypothesis␈α
in
␈↓ α∧␈↓set theory.

␈↓ α∧␈↓␈↓ αTSystems␈α∞of␈α
the␈α∞type␈α
described␈α∞above␈α∞separate␈α
the␈α∞knower␈α
from␈α∞the␈α
outside␈α∞world.␈α∞ At␈α
the
␈↓ α∧␈↓cost␈α
of␈α
elaboration␈α∞that␈α
may␈α
make␈α
theorems␈α∞harder␈α
to␈α
formulate␈α
and␈α∞prove,␈α
we␈α
can␈α∞remove␈α
the
␈↓ α∧␈↓separation.␈α∂ Consider␈α∂a␈α∂two␈α∞or␈α∂three␈α∂dimensional␈α∂cellular␈α∞automaton␈α∂system.␈α∂ In␈α∂such␈α∂a␈α∞system,
␈↓ α∧␈↓each␈α
cell␈α
is␈α
a␈α∞finite␈α
automaton␈α
whose␈α
state␈α
at␈α∞time␈α
␈↓↓t+1␈↓␈α
depends␈α
on␈α
its␈α∞state␈α
and␈α
the␈α
states␈α∞of␈α
its
␈↓ α∧␈↓immediate␈αneighbors␈αat␈αtime␈α␈↓↓t.␈↓␈αA␈αgood␈αexample␈αis␈αConway's␈αLife␈αautomaton␈αthat␈αuses␈αan␈αinfinite
␈↓ α∧␈↓two␈αdimensional␈αarray␈αof␈αtwo␈αstate␈αautomata␈α-␈αone␈αlocated␈αat␈αeach␈αpoint␈αof␈αthe␈αplane␈αwith␈αinteger
␈↓ α∧␈↓co-ordinates.␈α
 The␈αstate␈α
of␈αa␈α
cell␈αat␈α
time␈α␈↓↓t+1␈↓␈α
is␈α0␈α
if␈αthe␈α
number␈αof␈α
its␈αeight␈α
neigbors␈αin␈α
state␈αone␈α
is
␈↓ α∧␈↓less␈αthan␈α
2␈αor␈α
more␈αthan␈α3.␈α
 If␈α2␈α
neighbors␈αare␈αin␈α
state␈α1,␈α
it␈αretains␈αits␈α
state␈αand␈α
if␈α3␈αneighbors␈α
are
␈↓ α∧␈↓in␈α∞state␈α
1,␈α∞it␈α∞goes␈α
to␈α∞state␈α∞1.␈α
 It␈α∞has␈α
been␈α∞shown␈α∞that␈α
the␈α∞Life␈α∞cellular␈α
automaton␈α∞is␈α∞a␈α
universal
␈↓ α∧␈↓computer␈α∂and␈α∂constructor.␈α⊂ In␈α∂particular,␈α∂self-reproducing␈α∂arrays␈α⊂of␈α∂Life␈α∂cells␈α∂that␈α⊂can␈α∂execute
␈↓ α∧␈↓arbitrary␈α∪computer␈α∀programs␈α∪are␈α∪possible.␈α∀ We␈α∪can␈α∪imagine␈α∀equipping␈α∪such␈α∪an␈α∀array␈α∪with
␈↓ α∧␈↓different␈α
epistemological␈α
programs␈α
and␈α
ask␈α
what␈α
programs␈α
would␈α
do␈α
formulate␈α
and␈α
confirm␈α
the
␈↓ α∧␈↓hypothesis␈αthat␈αthe␈αfundamental␈αphysics␈αof␈αtheir␈αworld␈αwas␈αthe␈αLife␈αcellular␈αautomaton.␈α If␈αthere
␈↓ α∧␈↓were␈αmany␈αof␈αthem,␈αwhat␈αhypotheses␈αwould␈αthey␈αformulate␈αabout␈αother␈αminds,␈αconsciousness,␈αetc?
␈↓ α∧␈↓Would␈αthey␈αdiscriminate␈αproperly␈αbetween␈αtheir␈αprograms␈α(minds)␈αand␈αthe␈αconfigurations␈αof␈αcells
␈↓ α∧␈↓(brains)␈α⊗constituting␈α⊗the␈α↔computer␈α⊗executing␈α⊗the␈α⊗program?␈α↔ Indeed,␈α⊗is␈α⊗this␈α↔the␈α⊗interesting
␈↓ α∧␈↓distinction?