The Dynamics of Duality : A Fresh Look at the Philosophy of Duality (Mathematical Logic and Its Applications)

The

_Dynamics

of

_Duality:

A Fresh Look

at

the

_Philosophy

of

_Duality

Yoshihiro

_Maruyama

The

Hakubi Centre and Graduate School of

_{Letters, Kyoto University}

Mathematical,

Physical,

and Life

Sciences

_{Division, University}

of

Oxford

“‘

[\mathrm{T}]\mathrm{h}\mathrm{e}

development

of

_philosophy

since the Renaissance has

_by

and

_large

_gone from

_right

to left

_Particularly

in

_physics,

this

_development

has reached a

peak

in our own

time,

in

that,

to a

large

_extent,

the

possibility

of

knowledge

of the

_{objectivisable}

statesof affairs is

_denied,

and it is asserted that we must

be content to

_predict

results of observations. This is

_really

the end of all theo‐ retical science inthe usual sense”’

_(Kurt

Gödel,

The modern

_development

_of

the

foundations of

mathematics in the

_light

_{of philosophy,}

lecture never

delivered)1

1

_Unity,

_Disunity,

and Pluralistic

_Unity

Since the modernist

_killing

of natural

_{philosophy seeking}

auniversal

_conception

of the cosmos as aunited

whole,

our

_system

of

knowledge

hasbeen

optimised

for the sake of each

particular domain,

and has

_accordingly

been

_massively

_fragmented

and disenchanted

_(in

termsof Weber’s

_theory

of

_modernity).

_Today

welack aunifiedviewof the

world, living

inthe _age of

_disunity

surrounded

_{by myriads}

of uncertainties and

contingencies,

which invade both science and

_society

as

exemplified by

different sciences of

chance,

including

quantum theory

and artificial

_intelligence

on the basis of statistical machine

learning,

and

_by

the

_{distinctively}

after‐modern features of risk

society

(in

terms of the theories of

late/reflexive/liquid modermity),

respectively

(this

is not

_necessarily

_negative

like

_quite

some

people

_respect

diversity

morethan

uniformity;

_{contemporary art,}

for

example,

_gets

alotof

_inspirations

from uncertainties and

contingencies

outthere innatureand

_society).

This _{modern and late‐modern process of disunification and contextualisation in favour of} scientific and othersorts of

_pluralism

rather than monism has gone hand‐in‐hand with:

lAccordingto Gòdel, scepticism, materialism andpositivism standontheleft side,andmetaphysics, idealismand

theologyontheright. Gödel’sphilosophyofphysicsis _{rarely addressed,}the onlyexception beinghis workon_general

relativity,andyetthismanuscript suggestshispositionisnevernaive_{realism/platonism.} Heindeedpaysprudentattention towhat he calls the leftspiritof thetime,somesortof antirealismoragnosticism,andseriouslyattemptstomake foundations of mathematics coherewith it_throughKantian_epistemologyand Husserlianphenomenology. Itisanendeavourtoreconcile therightand theleft,andinthepresentarticleweshall addressexactlythesameissuewith the idea of_{duality (between} the_rightand the_{left). Apart}from_that,thisquotealso sounds likeChomsky’scriticismof statistical datascience,and the leftwardpropensityGödelwasanxious aboutseemstohave survived until thepresent,whetherunfortunatelyor_fortunately.

\bullet the late‐modern shift of

analytic philosophy

from the Vienna circle to the Stanford

school,

the former

_advocating

the

_unity

of

_science,

as seen in their series of

pub‐

lications entitled International

_{Encyclopedia of Unified}

_Science,

and the latter the

disunity

of

_science,

as seen intheir

_publications

such as The

Plurality

of

Science

by

Suppes

[52]

and The

_Disunity

_of

Science

_by

Galison et al.

_[23];

\bullet and _atthe sametimewith the

post‐modern

shift of continental

philosophy

as_repre‐

sented

_by

the so‐called “end of

_grand

narratives” thesis

_by

_(way

_notorious)

_Lyotard

[35],

which was,

arguably,

a

global phenomenon

as_maybeobservedevenintheana‐

lytic philosophy

of science movefrom the

global

general

philosophy

of science tothe local

_philosophy

of

_special

sciences

_(from

_singular

“science” to

_plural

_{“sciences”).}

\bullet Note that this

change

furtherwenthand‐in‐hand with the demise of

foundationalism,

the

_unity

of science’s nearest kin and a

_particular

kind of

grand

narrative as had been endorsed

_by

_quite

a

_majority

of

_past

philosophers longing

for the so‐called

Archimedean

_vantage

_point

_according

to

_Rorty

in his

_Philosophy

and the Mirror

_of

Nature

_[49],

who also

_emphasised

the

_priority

of the

_contingent

overthe

absolute,

and

thereby

tooka

_step

towards

post‐analytic philosophy

(note

further that the mirror of

naturewasthrown _awayinartas

well;

Klee_{says art}doesnot

_reproduce

the

_visible).

From another

_angle,

the rise of

_{analytic philosophy}

_{itself was, in} a _way, a

_symptom

of

disunity

orthe end of

grand

narratives in

philosophy

as a

whole, mostly

annihilating

the

tradition of

_systematic

_philosophy

in the

_Anglophone

world that had still remained even after the death of natural

_{philosophy. Today}

science

_(including

_humanities)

isfundamen‐

tally pluralist

in various

_respects,

and the

_unity

ofscience isalmost

_empirically

refuted as

it were, not

_just

innatural sciences but in other sciences aswell. To name but a

few,

the

atmosphere

of

_logic

has shifted from the absolute

_logic

of the

_single

worldtorelative

_logics

ofdifferent

_domains,

orfrom the

Russell‐Wittgenstein’s logic

as

_representing

thestructure

of the worldtothe

_{Carnap’s logic}

as

endorsing

theso‐called

_principle

of

tolerance,

andnow

logical pluralism

(see,

e.g.,

[2])

is

mostly

considered

something

obvious in

practice,

ifnot

in

_theory.

Pure mathematics has shifted from the

_{Bourbaki‐style conceptual}

modernist mathematics

_juggling

with

_generic

structures to the more down‐to‐earth

computational

post‐modernist

mathematics with nuanced

_perspectives

on

particular

structures as exem‐

plified by,

say,

quantum

groups and low‐dimensional

topology

(which

also utilise

category

theory

in substantial

_manners).

The

_{post‐modern}

mathematics is similar to the 19th

century

mathematics,

and

_yet

it _{still builds upon the}

_conceptual

methods of modern

mathematics,

whichwere neveravailable atthat time and allowed for solutionsto several

long‐standing problems

as even the

general public

have heard of in

popular

scienceread‐

Berman

_[7]

and

_{others, though}

it is

_basically

a

sociological

concept

andnever

applied

to

mathematics and other formal sciences before. Therecent

_propensity

for \mathrm{t}

‘retrieving

real‐ ism”

_[17]

inboth

_analytic

and continental

_traditions,

as

exemplified

by

structural realism

[30]

and

_speculative

realism

_[41],

wouldcount as a case of re‐enchantment in

philosophy

as

well;

note that

philosophy

before the realist turn had been

disenchanted,

from

things

to ideas

_(the

_Copernican

_turn),

and to words

_(the

_linguistic

_turn),

_according

to

_Rorty

[50].

The movein

philosophy

from

things,

to

ideas,

to

words,

and backto

things

may be

compared

with themove inmathematics from

things,

to

infinities,

to

symbols,

and back

to

_things.

This is afiner

_analysis

of disenchantment and re‐enchantment in

philosophy

and

_mathematics;

the same, more nuanced

analysis

could be elaborated for other fields aswell.

All this tells us that the notorious idea of the end of

grand

narratives was indeed

generally

truein science as well. Yet if the

post‐modernist terminology

is

inappropriate

I would instead

_rely

_{upon the}

_concept

of “unitism” as

opposed

to that of

“disunitism“,

which

_actually

better illustrates the

_plot

of the

_{story here,}

that

_is,

the transition from

unity,

to

_disunity,

and to

_something

_doing

_justice

to both. Here I would characterise unitism and disunitism

_by

the presence of

global

meaning

and absence of

it,

respectively;

remember that

_disunitism,

or _{course, may still}

keep

some local

meanings.

The end of

grand

narrative,

then,

maybeunderstood asthemove from unitismto disunitism about

knowledge:

the

_global

_unity

of the

_kingdom

of

_{knowledge collapses}

_(through

the “unexit” affair

_perhaps)

whilst

_{globally meaningful knowledge}

_{(resp. Truth)}

cutsdownto

_separate

pieces

ofmore local

knowledge

(resp.

truths,

which are not

_yet

_{“post‐truth”),}

and the

multiply

divided

_kingdom

of disunified

_knowledge

starts tothrive

_thereafter,

followed

_by

the

_repeated,

_unending

formation ofmore and morelocalised

knowledge

communities in

which

_agents

are

loosely

interlaced with each other

by

their

epistemic

family

resemblance

and in which

_{knowledge production}

_gets

moreandmore

localised,

and

only

makessense moreandmore

locally.

Put another way, there is afundamental

discrepancy

between the

following

twoviews onthe terminusof

reason/science/civilisation,

_i.e.,

thefinal

endpoint

orutmost limit towhich the

_dynamics

of

_{reason/knowledge/society}

_eventually

leadsus:

\bullet the modernist view that the world

gradually

_{converges into} oneand the sameideal

limit as in the

_Hegelian

and Kantian or neo‐Kantian

thoughts

[20,

21,

22] (for

in‐

stance because the convergence or

patching

condition of

reason/knowledge/society

is

_satisfied}

to the

_physicist

this means

physics finally

leads to the so‐called

_theory

of

_everything,

_yet

it isstill unclear whether it

happens

atthe end of the

_day

or

not);

\bullet the

post‐modernist

viewthat the world

diverges

into diverse local limits

(for

instance

limit or

global

element

patching

local elements

together;

to be

precise

there is

_yet

another _case,

_i.e.,

there is no limit

whatsoever;

a

_toy

example

of this case would be

that

_{philosophical}

debates or real‐life

quarrels usually

lead to no

_{converging point,}

people eternally

arguing against

each

other;

it is

hopeless,

but

_might

be

_reality);

\bullet the mathematician would

image

what each case would look like in terms of so‐

called filter convergenceor sheaf condition

(the

latter can

actually

model

_quantum

contextuality

as done

by

Isham‐Butterfield and

Abramsky

et al

_although

every

option

is

_{mathematically possible, nonetheless,}

there is

_usually

a

unique

limit,

i.e.,

if space involved is

compact

Hausdorff

_(see

anytextbookon ultrafilter convergence;

note however that ultrafilters may not exist without indeterministic

principles;

to

put

it in

_logical

_terms,

_complete

or

maximally

consistenttheories _{may not}

exist).

The

_“kingdom

of

_knowledge”

is not

_merely

a

fancy metaphor:

division in

society

and

division in

_knowledge

_(and

also division in

_reason)

are not

_separate

_phenomena,

but

they

are _consequencesof the same

disposition

inhuman

history,

which I characterise

by

disunitism here. Whilst

_Lyotard

focused his attention to

_knowledge

in his end of

_grand

narratives

_thesis,

it is

_part

of theevenwider

tendency

of

meaning

lossor

meaning

bleach

from the

_{unitism/disunitism}

perspective.

And this

_completes

the first half of the shift. Thereare ofcoursecountermovements

against

disunitism,

and even

_proponents

of dis‐

unity

are not

_totally

content with the

_disunity

of science in some sense. The

concept

of

re‐enchantment

_by

Berman

_[7]

is

_actually

an ideal for the future as well as an account

of the after‐modern intellectual

_{tendency. Although}

we have come to

_naturally

_sepa‐ rate

_objects

and

_subjects,

fact and

_value,

andso

fourth,

in the

mechanistically

divided

world after the scientific revolution in the

_Renaissance,

and

_although

holistic

_meaning

has

thereby

been lost in our

conception

of the

universe, nevertheless,

Berman

[7]

argues that

they ought

tobe reunited in ordertoovercomethe

problems

of

_contemporary

society

and

torecoverits holistic

coherency

inthe_process of the re‐enchantment of the world. Yet I

would ratherarguethat the mechanisticviewis

just

dualtotheholistic view of the

world,

and so

they

are

structurally equivalent

to each other inacertain sense. It is even math‐

ematically

true that the Newtonian mechanistic and Leibnizian holistic views of space

are indeed

dually equivalent

to each other. The same holds for the

theory

of

meaning

inthe

_{analytic tradition,}

and the truth‐conditional and verification‐conditional semantics

maybeseen as

equivalent

as Ihave

argued

elsewhere. This instantiatesthe basic idea of

duality

discussed

_through

the

_following

sections. What is at issue

_{here, however,}

is the

unity

versus

disunity

debate. And _my

major point

about it is that

duality

allows us to

reinstate

_unity

without

_breaking

_disunity.

Letus

unpack

the

_meaning

of this in thecaseof

the holistic _one, and what

_duality

says, ofcourse, is not that

they

are

equal. Although

Hegelian

dialectics would urge us to find

_yet

another view to reconcile the two views in

conflict,

however,

duality

refuses todo so, and proposes to see them as

_representing

the samestructure.

_{Duality keeps pluralism}

inthat anyof thetwo views is not

denied,

and

thatthere is no

global,

third view

encompassing

the two

_views,

and

_yet

it attains

_unity

in that there is one and thesame structureshared

_by

them. What this

_pluralistic

unity

meansfurther shall be articulated in the

following

sections.2

2

Dualism

The

major

concernof the

_present

article lies in

elucidating

the

dynamics

or

“logic”’

of

duality

via both

_{philosophical}

and mathematical

_{analyses. Duality}

has been the funda‐ mentalwayof human

thinking

as

broadly

seeninhumanintellectual

history;

for

example,

Cartesian dualism and

_Hegelian

dialectics instantiate

_(of

course

different)

forms of

duality

inabroadersense. Inthe

following

Ishall illustrate

philosophy

asa

conceptual

enterprise

pursing

the ideaof

_duality,

arguing

thatanumber of fundamental

problems

in

philosophy,

including

the

_{long‐standing}

realismversus antirealism

issue,

_mayberesolved

by

virtue of

duality.

The

_unity

of realism and antirealism has been discussed in my

preceding

works

[36, 38]

aswell. Thissortof idea constitutes the

philosophical

strand of the

present

article,

and

_yet

there is

_another,

mathematical strand to shed

_light

on the mechanism of dual‐

ity.

In

_light

of

_duality

theories

_developed

sofar

through

the

duality

theorists’ dedicated collaborative

_endeavour,

I wouldsay, we are now

ready

to

unpack

the

general

mechanism

of the way how

duality

emerges,

changes,

and

breaks; indeed,

a succinct answer tothe

question

shall be

given

later. In the

_following

Iam

_going

to

_give

a

conceptual

account of

dualism, duality,

and

_disduality

in the

_{abstract; disduality}

is

_basically

meant to be the absence of

_duality,

or

duality‐breaking.

Dualism and

duality

are related to each

_other,

and

_yet

_surely

different. How

_they

are different isnot that

obvious, though.

Developments

of

_philosophy

have centred around a tension between realism and an‐ tirealism

_(or

in Gödel’s tems between the

_“right”

and the

_“left”;

see the

quotation

and

footnote

_above).

And the dualistic tension may be illustrated

by asking

the nature of

a

_variety

of fundamental

_concepts.

What is

Meaning?

The realist asserts it consists in

the

_{correspondence}

of

_language

to

_reality,

whilst the antirealist contends it lies in the

autonomous

_system

or internalstructure of

language

or

linguistic

practice

(cf.

the

early

Wittgenstein

[57]

vs. the later

Wittgenstein

[58];

Davidson

[14]

vs. Dummett

[18]).

What

is Truth? To the

_realist,

it is the

_{correspondence}

of assertions tofactsor statesof

affairs;

2Inmy 2016SynthesepaperI haveactuallyformulated the idea ofpluralisticunityinahttledifferent,moreabstract manner. Theconceptionofpluralisticunityheremaybeseen as aninstance ofthat.

to the

_antirealist,

it has no outward

reference,

constituted

by

the internal

coherency

of

assertions or

by

somesort of instrumental

pragmatics

(cf.

Russellvs.

Bradley

[8]).

What

is

_Being?

To the

_realist,

itis

_persistent

_substance;

tothe

_antirealist,

it emerges withinan

evolving

process,

cognition, structure,

network,

environment,

orcontext

_(cf.

Aristotle

_[1]

vs.

Cassirer/Heidegger/Whitehead [10,

_26,

56 What is

_{Intelligence?}

To the

_realist,

it ismore than behavioural

simulation,

characterised

by

the

intentionality

of

mind;

tothe

antirealist,

it is

_fully

conferred

_by

_copycatting

as inthe

Turing

testor the ChineseRoom

(cf.

Searle

_[51]

vs.

Turing

[53]).

Whatis

Space?

To the

realist,

it is acollection of

points

with no

_extension;

to the

_antirealist,

it is astructure of

regions, relations, properties,

or

information

_(cf.

Newtonvs. Leibniz

[54]; Cantor/Russell

vs.

Husserl/Whitehead

[3]).

To

cast these instances of dualism in more

general

_terms,

dualism may be conceived of as

arising

between the

_epistemic

and the

_ontic,

orbetween the formal and the

conceptual

in Lawvere’s terms

_[32],

as inthe

figure

below:

The best known

_{dualism, presumably,}

would be the Cartesian dualism between mind and

matter

_(or

mind and

_body),

inwhich the ontic realm ofmatter and the

_epistemic

realm of mind are

separated.

The Kantian dualism between

thing‐in‐itself

and _appearancecan

readily

beseen as a caseof the

_{ontic‐epistemic}

dualism.

Cassirer,

the

logical

Neo‐Kantian

of the

_{Marburg School,}

asserted the

_priority

of thefunctional overthe substantival

[10],

having

builta

purely functional,

_genetic

view of

knowledge,

whichwas

mainly

concerned

with modern science at an

early

_stage

of his

thought

asin Substance and Function

[10],

and

_yet

_eventually

evolved to _encompass

_{everything including}

humanities in his mature

Philosophy of Symbolic

Form

_[11].

It is an

all‐encompassing

magnificent Philosophy

of

Culture

_[34],

indeed

_{subsuming myth,}

_art,

_language,

humanities,

and both

_empirical

and

exact sciences. Cassirernow counts as a_{precursor of what} is called Structural Realism

[19,

30,

31].

Yet hisfunctionalist

_philosophy

would better be characterisedas

Higher‐Order

acertain

sense),

just

as in

_category

theory. Cassirer, however,

doesnot

support

ordinary

abstractionism from the concrete at

_all,

and that he rather _puts

_strong

_emphasis

onthe

generation

of the abstract via

_{conceptual symbolic}

formation withno

_prior

reference to

reality,

whichitselfcomes out of the

symbolic

construction. The

symbolic

generation

of

reality

had beenacentral theme in his

philosophy

asawhole. His

_genetic

vieweven

payed

due attention to _{the process of how}structures are

generated,

_just

as in

_{type theory.}

In

light

of

_this,

Cassirer’s

_philosophy

may be

regarded

as a

conceptual underpinning

of the

enterprise

of

_category

_theory,

and his

_dichotomy

between substance and function asthat

of

_{categorical duality.}

Cassirer

_actually

started his career with workon Leibniz and his

relationalism

_[9],

at which weshall have a

_glance

inthe

_following.

Mathematially,

the

_{ontic‐epistemic}

_duality

is best

_recognised

in the nature of space aforementioned. There were two

conceptions

of space at the dawn of mathematical sci‐

ence: the Newtonian realist

_conception

of absolute _space and the Leibnizian antirealist

conception

of relational _space

_[54].

Hundreds _{of years}

later,

Whitehead

_[56]

_recognised

a

similar tensionbetween

_point‐free

_{space and}

_point‐set

space

(as

part

of his

inquiry

into Process and

_Reality),

_advocating

the latter

_point‐free

_conception,

which may also be found inthe

_{phenomenology}

of Brentano and Husserlas well

(see

[3]).

Onthe one

hand,

points

are

recognition‐transcendent

entities

_(just

like

_prime

_{ideals/filters,}

_maximally

consistent

theories,

or what Hilbert called ideal

elements;

recall that the

algebraic

geometer

indeed

identify

points

with

prime

ideals, which,

in

_{general, only}

exist with the

_help

of the axiom of choiceor someindeterministic

principle

like

that).

Onthe

other,

regions

(or

any other aforementioned

_entities)

are more

recognition‐friendly

and

epistemically

better

grounded

(just

as

point‐free topology

can be

developed constructively

or even

predicatively

in the

formal

_topology

_style).

In

_general,

realism and antirealism are

grounded

_{upon the ontic}

and the

_epistemic,

_{respectively.}

As shown in the

_figure,

_quite

some

major

philosophers,

whether in the

analytic

or

continental

_tradition,

have wondered about versions of the

ontic‐epistemic

dualism. The fundamental

_problem

of suchadualism isto accountfor how thetwodifferent realmscan

interact with each

_other;

in the

_particular

case of the Cartesian

dualism,

it boils down

to

_explicating

how the mind can know about the material world when

they

are

totally

(and

so

causally)

separated.

Howcan

they causally

interact at allwhen

_they

are

causally

separated?

It _appears

_impossible;

this is the

_typical

_waythe

_philosopher

_gets

troubled

_by

dualism in

_accounting

for the

_{ontic‐epistemic}

interaction.

Philosophy

of mathematics faces an instance of the interaction

problem

as well. If the

realm of mathematical

_objects

and the realm of human existenceare

totally separated

(in

particular causally

separated),

howcanhuman

beings

get

epistemic

access tomathemat‐ ical entities? If mathematical

_objects

exist in a Platonic

_universe,

as in Gödel’s realist

philosophy

for

_example,

it seems

quite

hard to account for how it is

_possible

to have a

causal connection between humans in the

_ordinary

universe and mathematical entities in the Platonic universe when the two universes are

causally

disconnected

(this

sort of

problem

isknown asthe Benacerraf’s dilemma

[5]).

Yet if mathematical

objects

exist in

the

_mind,

as in Brouwer’s antirealist

philosophy

(he

counts as an antirealist at least in

the sense of Dummett

[18]),

theaccount of interaction between the ontic and the

epis‐

temic is much

_easier,

since the ontic

is, just

by

assumption,

reduced to the

_epistemic

in this _case;

_by

_contrast,

_then,

it

_gets

harder to account for the existence and

_objectiv‐

ity

of mathematical

_entities,

_especially

transfinite ones. For

instance,

how can humans

mentally

construct

_{far‐reaching}

transfinite entities and does

_everyone’s

mentalconstruc‐

tion

_{really yield}

the same results for sure?

(To

remedy

the existence

problem,

Brouwer

actually

endorsed

_arguably

non‐constructive

_principles.

Otherwise a continuum can be

countable as in recursive mathematics in the Russian

tradition;

recall that the number

of

_computable

reals are

only countably

many.)

Summing

_up, realist

ontology

makes it

difficult to account for the

_possibility

of

_epistemic

access to mathematical

_objects;

con‐

versely,

antirealist

_epistemology

_{yields ontological}

difficulties in

_{constructively}

_justifying

their existence and

_objectivity.

A moral drawn from the above discussion is that there is a trade‐off between real‐

ismand antirealism:

_{straightforward}

realist

_ontology

leadstoinvolved

_epistemology

with the

_urgent

issue of

_epistemic

access to entities unable to exist in our

ordinary, tangi‐

ble

_universe;

and

_{straightforward}

antirealist

_epistemology

toinvolved

_ontology

with the

compelling

problem

of

_securing

their existence and

_objectivity.

In

_general,

realism

gets

troubled

_by

our

accessibility

to abstract

_entities;

antirealismfacesa

challenge

ofwarrant‐

ing

their existence and

_objectivity.

Put

_simply,

an easier

_ontology

of

_something

often makes its

_epistemology

more

difficult,

and viceversa.

_Something

seemsreversed between

realist/ontological

and

_{antirealist/epistemological}

worldviews. And this is where the idea of

_duality

between realism and antirealismcomesinto the

play.

3

_Duality

Everything,

from Truth and

_Meaning

to

_Being

and

_Mind,

has dual facetsas aforemen‐

tioned.

_{Conceptually, duality}

theory,

in

_turn,

is an

_attempt

to unitethem

_together

with the ultimate aim of

_showing

that

_they

are the two sides ofone and thesame coin. Put another way,

duality

allows two

_things

_opposed

in dualism to be reconciled and united

as

just

two_{different appearances of}one and the same fundamental

reality;

in this_sense,

duality

is a sort of monism established on the

_top

of dualism

(cf.

Hegelian

dialectics asin Lawvere’s

philosophy

of

mathematics).

In Dummett’s

philosophy

on the

theory

of

meaning,

for

_example,

he makes a

binary

_opposition

between realism and antirealism

(cf.

Platonism and

_{Intuitionism/Constructivism);}

what isatstake there is

_basically

the

_legit‐

imacy

of

_{recognition‐transcendent}

truth

_conditions,

which is allowed in

realism,

but not

in antirealism. As in _{my recent} works

_[36,

38

_however,

the realist and antirealist con‐

ceptions

of

meaning

may be reconciled and unitedas

sharing

the samesort of

structure,

evenif

they

are

literally opposed.

In viewof Dummett’s constitution

thesis, according

to which thecontent of

_metaphysical

_{(anti)realism}

is constituted

_by

semantic

_{(anti)realism}

[18, 42],

this would

_arguably

count as aunification of

metaphysics

as well as the

theory

of

_meaning.

_Duality

thus conceived is aconstructive canon to deconstruct dualism as it were.

Whilst

_{having posed}

theCartesiandualismas

aforementioned,

Descartesalso

developed

analytic

geometry,

which is ina sense a_{precursor of}

duality

between

algebra

and

_geometry,

even

though

he

might

nothave beenaware that

_systems

of

equations

are dualto _spaces

of theirzeroloci

(logically

paraphrasing,

this amounts tothe fact that

systems

of axioms aredual to _{spaces of their}

models;

and such

correspondence

between

logic

and

algebraic

geometry

can be made

precise

in

duality

theory).

Notwithstanding

that Galois

theory

may be seen as an instance of

duality,

Galois himself wouldnot have been aware of the

essentially categorical duality underpinning

his

_theory,

either. It

_{would, then,}

be Riemann whofirst discovered

_duality

between

_geometry

and

_algebra

ina

mathematically

substantial

form;

indeed he showed how toreconstruct

_(what

are now

called)

Riemann surfaces from

function

_fields,

and vice versa,

thereby establishing

the

(dual)

equivalence

between them.

Even earlier than

_Riemann,

_however,

Dedekind‐WeUer and Kronecker

_{(mathematically)}

gave a

purely

algebraic

conception

of _space, a sort of precursor of what is now called

point‐free

geometry

(philosophically,

it would date backtoLeibnizas

aforementioned).

The

_history

of

_duality

inmathematical

_{form, thus,}

goes backtothe late 19th

century

(it

ought

to be noted here that

_duality

in mathematical form

_basically

means

categorically

representable duality,

and so

duality

in

projective geometry,

for

example,

doescount as an

origin

of

duality

in this _sense; it would not be

categorically

representable,

at least

to my

knowledge).

Duality

then flourished in the

_early

20th

_century,

as

exemplified

by Hilbert’s, Stone’s, Gelfand’s,

and

Pontryagin’s

dualities

_(Hilbert’s

Nullstellensatz is

essentially

adual

equivalence

between

finitely

generated

reduced k

‐algebras

and varieties overk foran

algebraically

closed field k

).

The

discovery

of dualitieswasthereafterfollowed

by applications

tofunctional

_analysis,

_{general topology,}

and universal

_algebra

ontheone

hand,

and to

_algebraic

_{geometry, representation}

_theory,

and number

_theory

onthe other

(interestingly,

the

_{Pontryagin duality plays}

avital rôle in number

theory

as

exemplified

by

André Weil’s Basic Number

_Theory

_[55]).

And it was

eventually accompanied by

one to

_identify

a universal form of

duality

(before

_{category theory}

it was

only vaguely

understood what

_exactly

different dualities have in common, andno one was ableto

_spell

out what

_presicely

_duality

_is).

_Today

there are a

_great

_variety

of dualities found across

quite

different kinds of science as in the

_{following figure}

_(and

even in

_engineering

such as

_{optimisation,}

linear

_programming,

control

theory,

and electrical circuit

theory;

duality

thus goes far

beyond

pure

mathematics,

and it may sometimes be of

genuine

practical

use as in those

_engineering

_theories):

These dualities are diverse at first

_sight,

and

yet

tightly

intertwined with each other in their

_conceptual

structures. _{To pursue links between different dualities is indeed}one of

the

_principal

aims of

_{duality theory}

in

_category

_{theory. Having}

a look at the above

pic‐

ture of

_dualities,

it is

_particularly

notable that the

_{physics duality}

between states and observables

_is,

in a _{way, akin} to the informational

_duality

between

_systems

and observ‐ able

_{properties/behaviours.}

Howcould we,

then,

shed

light

on structural

analogies

and

disanalogies

between diverse dualities? What is the

generic

structure or architecture of

duality

in the first

_place?

Such

questions

propel

the

_{investigation}

of

_{categorical duality}

theory.

The

_{duality‐theoretical correspondence}

between

_logic

and

_algebraic

_geometry

il‐ lustrates what Ulam calls an

analogy

between

analogies

in

mathematics;

note that the Stone

_duality

isconcerned with

_equivalence

between

_syntax

and

_semantics,

and it is actu‐

ally

a

strengthened

version of the Gödel’s

completeness

theorem

(to

reinforce this

point,

it is named Gödel‐Stone in the

picture;

technically,

the

_injectivity

ofan evaluation map

in the Stone

_{duality exactly}

amounts to

_{completeness,}

whereas the

surjectivity,

though

computer

science

_duality

above

_is,

in itsmathematical

_substance,

aform of Stone

duality,

and even the

physics

duality

between states and observables maybe formulated in the

Stone

_{duality style. Quite}

some

_part

of the

duality

_picture

above, therefore,

boils down

to Stone

_duality,

the

_{bird’s‐eye}

view of which shall be

_given

below. As evident in the above

_picture

of

_{categorical dualities,}

_category

_{theory today}

has found

_{widespread appli‐}

cations in diverse

_disciplines

of science

_{beyond mathematics;}

it would now be more like foundations of science in

_general

than foundations of mathematics in

_particular.3

Duality

is even crucialfor Hilbert’s programme, as

Coquand

etal.

_[13]

assert:

A

_partial

realisation _{of Hilbert’s programme has}

_recently

_proved

successful in commutative

_algebra

Oneof the

_key

tools is

_{Joyal’s point‐free}

versionof the Zariski

_spectrum

as a distributive lattice

In

_[13]

_they

contrive aconstructive version ofGrothendieck’s schemes

by replacing

their

basespaces with

point‐free

ones

through

theStone

duality

for distributive lattices. From a

categorical

point

of

view,

wecould _saythat the

_spectrum

functor

Spec

:Al

\mathrm{g}^{} \rightarrow \mathrm{S}\mathrm{p}\mathrm{a}

froman

algebraic

_category

Alg

toa

topological

category

Spa

amounts tothe introduction of ideal elements in Hilbert’s_sense, and its

_adjoint

functor the elimination of them. Dual‐

ity,

therefore,

has contributedto_{Hilbert’s programme and constructivism. The}

_point‐free

Tychonoff

theorem is

constructive;

this is classic. Yet the state‐of‐the‐art goes far

beyond

it, encompassing

not

_just

_{general topology}

but also somemainstream mathematics such as Grothendieck’s scheme

theory.

In

_light

of rich

_duality

theories

_developed

so

far,

we

give

succinct answerstothethree

questions

onthe mechanismof

duality

(for

detail,

see_my DPhil thesis

[39]):

\bullet How does

duality

emerge?

It is when the dual

aspects

ofa

single

entity

arein “har‐

mony”’

with each

_other;

what I call the

_harmony

condition

_explicates

this

_harmony.

Dual

_adjunctions

_{emerge when}

_algebraic

structures areharmonious with

topological

structures,

according

to

_(the

_harmony

condition

_of)

the

_{duality theory}

via

_categor‐

ical

_topology

and

_algebra.

In dual

_adjunctions

between

_algebras

and spaces, the

harmony

condition

_basically

means that the

algebraic

operations

induced on the

spectra

of

_algebras

are continuous. Dual

equivalences

are determined

by

the ratio

of

_existing

term functions over all

functions, according

to

(my

understanding

of)

natural

_{duality theory.}

'_This

is thethingcategorytheoryhas aimedat since its_{early days.} Granted thatquitesomecategorytheorists had moreorlessfoundationalistdoctrines, nevertheless,itwouldbeappropriatetothink ofcategorytheoryaslocal relative foundations rather thanglobalabsolutefoundations,whichiswhatsettheory isaboutinthenatureof theuniverseor the cumulativehierarchyofsets,justasbasechangeisafundamental idea ofcategorytheory. Settheorycansupporta multiverseviewasshownin recent_{developments,}andyet categorytheory intrinsicallydoesso,Iwouldsay. Notealso that

thepracticeof mathematicsisconcerned with different combinations of set‐theoretical andcategory‐theoretical ideas,and the_binaryoppositionbetweenset_theoryand_{categorytheory}are_{not very}constructiveorfruitfulin_{practice, apart}from

\bullet Howdoes

duality

mutate? Dual structures

_get

simplified

astermfunctions

_increase;

this is what natural

_duality

_theory

tells us. As a

limiting

_case, if

_existing

term

functionsare all functions

(

\mathrm{i}.\mathrm{e}., functional

completeness

in

logical

terms),

thendual

spaces are Stone

(aka. Boolean)

_spaces

(this

is the

primal duality theorem;

extra

structures on_space are

indispensable

in the

quasi‐primal duality

theorem).

Ifcon‐ tinuous functions coincide with term

functions,

then dual structures are coherent

spaces

(this

could be called continuous functional

completeness,

which entailsStone

duality

with

_respect

to coherent

_spaces).

\bullet How does

duality

break? It is caused

by

either an excess of the ontic or an ex‐ cess of the

_epistemic,

as shall be discussed below. There are some

impossibility

theorems known in non‐commutative

_algebra

_[6],

which exhibits an excess of the

epistemic.

Thiscanhowever beremedied

by

meansof sheaf

theory.

Theidea ofnon‐

commutative

_{duality theory}

via sheaf

_theory

is

_simple:

we take the commutative coreofanon‐commutative

algebra,

dualise

_it,

and

_equip

the dual space with asheaf

structure to account for the non‐commutative

_part.

The same methods works for a

broad

_variety

of non‐commutative

_{algebras, including}

_operator

_algebras,

_quantales,

and substructural

_logics.

In substructural

_logics,

the method is further extended in such a _way that in

general

we take the structural core of a substructural

logic,

dualise

_it,

and endowasheafstructurewith ittotakecareof thesubstructural

_part.

This _{process may} be

_{expressed by}

means of the

general

_concept

of Grothendieck

situations.

_(For

_detail,

seemy DPhil thesis

[39].)

Toelucidate how

_{duality changes}

in

_logical

contextsin

_particular,

for

_example,

when_you

weaken/strengthen

your

logic

or extend it with

_operators,

let us also

_present

a

bird’s‐eye

view ofdifferent

_logical

dualities in a

rough

and

_yet

intuitivemanner.

Stone‐type

dualities

basically

tellus that the

algebras

of

_propositions

are dual to the spacesof models in the

following

fashion:

\bullet Classical

logic

_isdual _tozero‐dimensional Hausdorff_spaces.

‐

Propositions

areclosed_opens,for which the law of excluded middle

(LEM)

holds,

sincethe union ofaclosed open and its

complement,

which is closed_open

again,

is

_equal

to _{the entire space.}

\bullet Intuitionistic

logic

is dual to certain non‐Hausdorff_spaces, that

_{is, compact}

sober

spaces such that its

compact

opens form a

basis,

and the interiors of their boolean

combinations are

compact.4

4Thisdefinition ofHeytingspacescameoutofmyjointwork with K._Sato;Lurie’s_{Higher Topos Theory[33]}gives yet

‐

Propositions

are

_compact

_opens. The

topological

meaning

of LEM is zero‐

dimensionality.

In

_general

itdoesnothold because the

_complement

ofa

compact

open is not

necessarily

compact

open.

\bullet Modal

logic

isdual _toVietoris

coalgebras

over

topological

_spaces.

‐ Modal

operators

amount to

_Kripke

relations or Vietoris

hyperspaces.

This is

what is called

_{Abramsky‐Kupke‐Kurz‐Venema duality}

inthe

_thesis,

_relating

to

powerdomain

constructions in domain

_theory

aswell.

Note that the existence of unit ensures that duals spaces are

_compact

(all

elements ofa

finitary algebra

concerned

_yield

_compact

_subspaces,

and_so,if there isaunit

element,

the entire space is

compact);

otherwise

_they

are

only locally

_compact.

Thesameholds for the

Gelfand

_duality

aswell. There _are, of course,evenmore

logical

_systems

_youcanthinkof: \bullet First‐order

logic

_may be dualised

by

two

approaches: topological

groupoids

(\mathrm{i}.\mathrm{e}.,

spaces of models with

automorphisms)

and

indexed/fibrational

topological

spaces

(i.e.,

duals of Lawvere

_{hyperdoctrines).}

‐ The latter

approach

extends to

_{higher‐order logic,}

thus

_giving

duals of

_triposes

or

higher‐order hyperdoctrines.

It

just

topologically

dualise the

propositional

value

_category

ofa

hyperdoctrine

or

tripos.

\bullet

Infinitary logic

forcesustotakenoteven

locally

_compact

_spacesinto

_account,

just

like

the

_duality

for frames

_{(aka. locales).}

And the

_resulting

_duality

is adual

adjunction

in

_general,

rather thanadual

equivalence.

‐

There maynot be

_enough

models or

_points

to

_separate

_{non‐equivalent proposi‐}

tions. There isno need for the axiom of choice thanksto

_infinitary

operations,

i.e.,

no need toreduce infinitaries on the

topological

side into finitaries on the

algebraic.

Note that all the otherdualities

_require

the axiomof choiceto warrant

the existence of

_enough

_points.

\bullet

Many‐valued logics

are diverse. It

depends

what sortof dual structure appears. It

is,

e.g., rational

polyhedra

for Lukasiewicz

logic.

For other

logics,

dual structures often include

_multi‐ary

relations on _spaces asinnatural

_{duality theory.}

‐ Dualities for

many‐valued logics

are

_mostly

subsumed under the framework of

dualities induced

_by

Janusian

_(aka.

_{schizophrenic)}

_objects

$\Omega$, or Chu

duality

theory

on value

objects

$\Omega$,whichmaybe

multiple‐valued.

(For

detail,

see

[39].)

You can combine some of

these,

and

thereby

obtain more

complex

dualities for more

are

_usually

required,

and

_yet

there is no

general

method to

_generate

them so far. The

structureof

_duality

combinations and

_coherency

conditions thus

_required

would be worth further elucidation. Note that this is a

rough

_picture

of dualities in

logic,

and there are some inaccuracies and omissions. Notice also that not all of these dualities are induced

by

Janusian

_(aka.

_{schizophrenic)}

objects, including

those for intuitionistic and modal

logics,

in which

_implication

and

_{modality, respectively,}

are not

_pointwise

_operations

on their

_spectra

_(for

mode detailed accounts see

[39]).

4

_Disduality

The absence of

_duality,

what is named

_disduality

in the

_present

_article,

is

_just

as in‐

teresting

on its own

right

as the presence of

duality.

According

to the discussionso

far,

duality

is about the

_{relationships}

between the

_epistemic

and the ontic. What is disdual‐

ity

then? In a

nutshell, disduality

is about an excess of the

epistemic

or the

ontic;

the

duality correspondence collapses

when either of the ontic and the

epistemic

is excessive. To articulate what is

_really

meant

here,

let us focus _{upon two} casesof

disduality

in the

following:

one is caused

_{by incompleteness}

and the other

_by

non‐commutativity

as in

quantum

theory.

The former shall

_give

a case of the excessof the

ontic,

and the latter a case of theexcess of the

_epistemic.

As mentioned

_{above, completeness}

maybe seen as aform of

duality

between theories

and models. What Gödel’s first

_{incompleteness}

theorem tells us is that there are not

enough

formal theories to characterise the truths of intended

_model(s)

_concerned,

or to

put

it

_differently,

there are some models which are unable to be axiomatised via formal

theories,

where theories are, of course, assumed to be

_finitary

_(or

_recursively

axiomatis‐

able)

and

_stronger

than the Robinson arithemetic

_(the

technicalstatementof this is that theset of

_{stronger‐than‐Robinson}

truths is not

_recursively

_enumerable).

If you allow for

infinitary theories,

you can nonetheless obtain a

complete

characterisation,

for

example,

of arithmetical

_truths,

and

_yet

this is not

_acceptable

from an

epistemological

point

of

view,

such asHilbert’sfinitism. This is a caseof

disduality

duetothe excessof the ontic.

Wenowturn to the otherkind of

_disduality.

Let us have a look at a case of the excess of the

epistemic.

There is some sort of

incompleteness

in

_quantum

_algebra.

The Gelfand

_duality

tellsus there is adual

equiva‐

lence between

_{(possibly nonunital)}

commutative

_{C’‐algebras}

and

_locally

compact

Haus‐ dorff spaces. There have been different

attempts

to

_generalise

it so as to include non‐

commutative

_algebras,

in

_{particular algebras}

of observables in

_quantum

_theory,

and

_yet,

as

long

as the duals of non‐commutative

algebras

are

purely topological,

this is

actually

the

_quantum

realm of

_{non‐commutativity.}

This is indeed a case of

disduality

due tothe excess of the

_epistemic:

there are _{too many} non‐commutative

algebras, compared

tothe

available amount of

_topological

spaces. The

disduality

may be remedied to extend the

notionof_spaceso asto

_include,

for

_example,

sheaves of

_algebras

inadditionto

_topological

spacesperse

(just

as Grothendieckindeed did in his scheme‐theoretical

duality);

in such a _case,

however,

both sides of

duality

_get

moreor less

algebraic

(the

same _may be said

about the Tannaka

_duality

for noncommutative

_compact

groups,inwhichcase duals are

categories

of

representations,

and so

fairly

algebraic).

There is another

_thought

on the notion of

disduality.

No canonical

agreement

exists on what

duality

meansinthefirst

place

even _among

_category

theorists aswell as_among

philosophers.

For

_example,

some _say

duality

is dual

equivalence,

whilst others say it is

dual

_adjunction

in

_general.

A weaker notion of

_duality

could count asa kind of

(weaker)

disduality.

In that case we can seehow far dual

adjunctions

are

from,

and

yet

how

they

(technically

always

but

_practically

_sometimes)

transform

into,

dual

_{equivalences.}

The different between dual

_equivalences

and dual

_adjunctions

do matterfroma

philosophical

point

of view.

_Think,

for

_example,

of

_physics,

which_maybe seen as

_pursuing

the

duality

between

_reality

and observation

_(recall

the state‐observable

duality

above).

If there is a

perfect

balance between

_reality

and observation that means there is a dual

equivalence

between them. Yet if there is more

reality

than can be reconstructed from observation

then it is adual

adjunction

which is not a dual

equivalence.

Likewise ifthere are more

observationalor

_epistemic

differences than

reality

can

metaphysically

accommodate then

it

_{is, again,}

a dual

adjunction

which is not a dual

equivalence.

This is not

just

about

physics,

and there are, for

example,

subtle theories of different balances between states

and observation in theoretical

_computer

science. What this sort ofstorytells us is that

there can still be some sort of weaker

duality

(e.g.,

adjunction)

even _{in the presence of}

disduality

(e.g., non‐equivalence).

From this

_point

of

_view,

the differencebetween

_duality

and

_disduality

may be considered relative and

continuous,

the transition between them

being gradual.

Disduality

is not

_anything

uncommon. If _you have more models than

theories,

or

if you have more theories than

models,

_you have

disduality.

If _you have more _spaces

than

_equations

can

_represent

as solution spaces, or if _you have more

_equations

than

spaces can

distinguish,

_you have

disduality.

If_you have more

reality

than

language

can

express, or_youhavemore

language

than

reality

can

differentiate,

_youhave

disduality.

The

entire

_enterprise

of science

is,

in a_{way, about}

elucidating duality

or

disduality

between

formulae and solutions

₍

\mathrm{i}.\mathrm{e}., substantival entities

satisfying

them),

just

as

philosophy

has

centred around the dualism between the

_epistemic

and the ontic. The

_{duality/disduality}

between formulae and solutions is

_{crystal‐clear}

in

_logic

and

_geometry,

as seen in the

formal

_{correspondence}

between

_logic

and

_algebraic

_geometry,

and it further holds up for

analysis

and

_physics

aswell. Given the

Schrödinger

equation,

for

example,

_youcanthink

of the Hilbert space of

solutions,

which inform us of the microstructureof the

_quantum

universe. What if the

_Schrödinger

_equation

is non‐linear? You have

just got

an infinite‐ dimensional

_simplectic

manifold as the solution space. Given the Einstein

equation,

_you can think of the manifold of

solutions,

which tells _you about the macro structure of the relativistic

universe.5

If there are not

_enough

solutions to realise

_equations,

or if

thereare not

_enough

_equations

to formalise

_solutions,

you have

disduality

(otherwise

you

have

_complete

_duality),

and

_knowing

about that is a

gain

in the

enterprise

of

science,

as Gödel

incompleteness

served as a fruitful theorem for later

developments

in different

fields.

_Disduality

is a

general

idea of the limit of the

epistemic

or the ontic. In Godel

incompleteness,

what is

_incomplete

is the

epistemic,

and

_yet

in

_principle,

it can be the

other_way around. And indeed itis thecase in

_quantum

theory

that what is

incomplete

isthe ontic as

_{non‐commutativity}

tellsus

(that

isto_say,

_reality

is

_incomplete

rather than

quantum theory

is

_incomplete).

_Duality

and

_disduality

are not

_fancy

rhetoric,

but

_they

do

_pin

down the fundamental

meaning

and limit of the scientific

_enterprise.

_Although

this sounds like abold

claim, nonetheless,

it is

arguably supported,

and to some extent

justified, by

numerouscases of science in which

duality

and

disduality play

central rôles.

We can evenshed new

light

onthe so‐called frame

problem

in

(philosophy of)

artificial

intelligence

from the

_disduality

_point

of view. It is concerned with the fundamental limitation of the

_{computational theory}

of mind.

My

abstract formulation of the frame

problem

is asfollows:

\bullet Dimensions of

reality

are

possibly infinite;

\bullet Need \mathrm{a}

(finitary)

frame to reduce

_possibly

infinite dimensions and to

_identify

the

finitary

scope of relevant

information;

\bullet Need \mathrm{a}

(finitary)

meta‐frameto chooseaframe because there are

possibly infinitely

many

frames;

\bullet This meta‐frame determination process continues ad

infinitum.

Here_everyframe is assumedtobe

_finitary,

as_{every formal}

_system

isassumedtobe

finitary

(i.e.,

recursively

axiomatisable)

inthe standard formulation of

_{incompleteness}

theorem. This

argument

applies

to any sort of

_finitary

_entity,

and so, if the human is a

finitary

entity

then it

_applies

tothe humanaswellasthe machine. What is essential in the frame

problem

is the finitude of

_beings.

From this

_point

of

_view,

theframe

_problem

isabout the

'_{What thestructureof}

a_{physical theory}ishas beenacentralissue in recentdevelopmentsof structural realisminthe

analytic philosophyofscience. There could be different solutions. Forone_thing,the structureofa_{physicaltheory}maybe