Logic and layers… the logic
of layers / logic and language.
(Logic
and Language I)
Always two; never only one.
(Reason alone… language alone…
perhaps only one, language, but in language one is never alone, and with many
languages… many reasons).
Logic within the bounds of
language alone.
If
natural language allows us to think anything, subject to restrictions of
grammar, word order, well-formedness (all honoured in the breach by the
creative use of language, in poetry and rhetoric), does it not nevertheless
need further restriction (logic, semantics, pragmatism): the rigour of an
artificial and specially created (formal) language to eliminate all trace of
fancy from our thought?
Is
there one such that can replace the vague amorphousness of natural language？
No
one (apologies to Parmenides). Nothing (sufficiently) unitary. Singular.
Self-consistent.
No ‘one ‘. Only two. Directly
from zero to two. But what kind of counting is this? The logic of layers.
*
Philosophy
is made of layers or fields, we might even say viewpoints, each with their own
apposite set of concepts, initially read as dominant in their particular field,
but whose field of operations are often expanded into other fields, claiming
priority, as it were, over their neighbours, over other levels. Imperialising.
So
at one extreme, the concepts of metaphysics claim priority over logic et al as too narrow, as excluding too
much. And at the other extreme, logic claims the last word, relegating all else
to the dustbin of the incoherent, the un-sayable, mere untidiness of thought.
In between lie the intermediate zones peopled by such pretenders as ontology,
epistemology, phenomenology (which three are usually regarded as ‘metaphysics’
by practitioners and opponents alike), and then we have the sociology of
knowledge, history of ideas, philosophical history and genealogies, and
assorted specialisms such as the philosophy of mathematics, logic, the
philosophy of religion, political philosophy, and of course, ethics, together
with its various applications (justice, rights, ecology, fairness, etc).
If logic reduces gibberish,
then metaphysics increases scope. If the latter asks the big questions, then
the former questions the big questions. Descriptive modes show evolution and
contextual factors, while a variety of other approaches proffer various
starting points as privileged (from phenomenology to psychologism, from
rhetoric to ethics, from ecology to economics (and from Gödel to God)). Pick
your own poison.
So
each level in effect prioritises itself and de-prioritises the others… finding
the source of fault in the others to lie in their starting positions. Viewpoint,
position, level, each proposing an implicit hierarchy. Each vaunting its own
advantages.
Metaphysics accuses logic
of only being able to produce tautology (so only being able to pronounce on
highly restricted areas - where not limited to producing quantitative forms of
knowledge; but not any other kind … not least its own justification, which must
lie elsewhere…).
Logic would deny
metaphysics its scope; maintaining that this range is self-created and
self-justifying, producing imaginary entities without limit (language or
reason, left to itself, left to ourselves, will produce anything we want it to
– if only we are clever enough…). Sophism.
The
other regions of philosophy, all have their own rationale for excluding or
demoting the others in favour of themselves. Each announcing itself as the
privileged staring point… yet what else can they do, for to do otherwise would
immediately plunge them into self-contradiction, via the paradox of the Liar
(by deferring the privilege of being their own starting point, they declare
their own insufficiency). As in giving the Other the last word, the paradox at
the heart of the Romantics’ valourisation of Nature (as opposed to Culture or
Language), as of our dependency on an a-historical ‘outside’ for our ‘foundations’
(be they heaven, eternity, or universals). So each level, each region, evinces
its own particular strengths… and its own weakness. A case of blindness and
insight perhaps?
Logic,
to be able to be rigorous, to be able to elide the errors that arise in the use
of natural argumentation or natural rhetoric, uses axioms (the same is true of
any artificial system of language from mathematics to computer languages).
These axiom must protect the system from error production by making its
workings transparent and providing a foundation which is unassailable.
Yet
the axioms that logic uses to protect itself from the production of rational
unintelligibility may themselves be used against itself. The two taboos that
guard the gates of reason are the prohibition on any reference to a meta-set
and to any form of self-reference (the varieties of axiomatisation available
are pretty much all reducible to these two). In effect we may not make
sentences which involve reference to a final or limit case, ‘meta-sets’, nor
may we employ any reference by an element in a set to itself (the paradox of
self-reference or self-grounding) - both of which risk leaving us with both A
and not-A, or infinite repetition or regress (‘bad infinity’) in both
‘directions’ (‘out’ and ‘in’, ‘outer’ limit case and ‘inner’ self-reference).
Yet it must be admitted that this restriction does bar a spectacular amount of
nonsense from our argumentation. Unfortunately it also defines as non-problems
many, even most, of the things we find worth arguing about…
Furthermore,
it appears that (an axiomatised) logic damns its others (other ‘logics’ as well
as other arguments) for breaking the rule of ultimate exterior or intimate
self-reference whilst itself committing this error in excluding them (founding
exclusion, definition against, and only permitting ‘truth’ to emerge through
reference to itself alone ((self-reference as transgression)). Paradoxically,
it is precisely by excluding all that is outside of itself that it finds itself
left alone in an incestuous self-engendering, so subject to the paradox of
infinite regress.
One
problem with logic is that everything outside of itself can function as its
meta-set (it relies, finally, upon ‘the rest’ of language, which came first);
so exclusiveness again functions as if a reference to an element in a set; so
again the self-reference taboo appears to have been broken. Also logic’s
insistence on itself as the only valid starting point (reason alone) is prior
to its own logical operations, so also an act from outside… a deus ex machina. But one which must not
be referred to… for if it is, infinite regress again results… as the exterior
element is ‘taken in’ so leaving ‘something else’ to further play the role of
‘outside’, of an infinite extension of context. This kind of thinking is best
left to natural language, whose job it anyway is…. And perhaps the use of
natural language to philosophise (think, construct or criticise) is the best
description of what it is to do ‘metaphysics’?
However,
in the case of metaphysics, there is no exclusion, rather the total inclusivity
of metaphysics is its problem, which already means that the taboo of
self-referentiality has been breached… everything is always already inside. But
if everything is always already inside then, in envisioning it as such, we are
talking at the extreme outer limit, at the margin, even at a god-like point in
our ‘totality’ of perception, so ‘outside’ the putative totality… (but once
‘said’ it is, in turn, ‘brought in’, etc). Again reference to a meta-set is
inevitable (as is infinite repetition ‘outwards’ in dealing with this, as we
keep taking up a position ‘outside’ the previous one…). So are the two alike;
our two taboos, the two key or fundamental axioms, are they two facets of the
same problem, two faces of the same knot, twins begat of the same womb? A case
of words needing a fixing point outside, or an autochthonous birth, a rooting
which (the slide-ability of definition inside…) language denies… so always
revealing a ‘fault’ somewhere later along the line…? Or is this rather a
question of limit as the problem of the unlimited potential of language,
leaving any artificial or second order language always (already) in
contradiction with the first order language which supports it; but which it, in
an act of matricide (or patricide), sought to supplant….
So no ‘grounding’, no
‘rooting’, or ‘beginning’ point; rather only endless tautology. A saying of
that which we already know? (Plato would have approved of this…) So on yet
closer examination even these limitations, this sacrifice made for clarity, the
gift of (in both senses) tautology to the world, this too appears on closer
examination to fall into contradiction when we examine further its relation to
its origins, its context, and (crucially) to its inner consistency.
In
fact it would appear that any attempt to seek justification (for rational
domination over things and language) from the ‘outside’ results in a
contradiction (the reference to an exterior, anterior, to a meta-set); just as
any attempt at total rigour only leads to ever greater contradiction, as
self-referential claims become self-undermining. The logics of ‘external’ or,
indeed, ‘eternal’ justification are dealt with by deconstruction and
post-foundationalism; just as the limits of internal coherency mapped by logic
as source of the ’last word’ meets all the contradictions as listed from Kant
to Gödel, as well as the problem of excluding context and the embedding of discourse
owed to life (in the later Wittgenstein’s critique, later Habermas, the problem
of infinite con- or co-text in ‘deconstruction’, etc.).
So
the axioms we use to make logic work (as a second order language, along with
other formal languages such as grammar and math, number theory and programming
languages) also apply to our choice of realm, or region of philosophy or
thinking. Reference to an outside and (an equally infinite) self-reference,
both operate equally well as a critique of logic itself (as opposed to being
its saviour) as of the field of thought in general. Reminding us of the limits
of ‘beginnings’ in a world that has (always) already begun…
A field whose paths are
tortuous, self-undermining, prone to leaps and loss, all natural, even necessary;
source of our creativity. Not to be
feared if we only remember to check (in both senses of the word) their
potentially infinite meanderings in terms of their reference or in an empirical
moment (which is how theoretical physics works). And for some human
requirements cohesion alone will do.
In
part it is the logic of the starting point that is the problem; we always have
to begin with the language we have -and then try to customize it- or pretend we
have escaped its ‘problems’ when it persists as a substratum to which the
formal language can be reduced, but which this formal language can not, itself,
subsume, leaving the natural (received) language as having the ‘final word’ –
in practice a ‘final word’ endlessly deferred (see Russell and Whitehead with
respect to the attempted mathematisation of language: in fact only the reverse
procedure turned out to be possible; the natural language is the ultimate
meta-set - so not reducible to a more limited range of expressions… we may
tidy-up ‘bits’ but they are only ‘bits’ and not the whole, the latter is always
a moving target). And so not only perpetually deferred, but perpetually
subverting… as the natural language is always ‘more powerful’ than all attempts
to customise it… (as a restricted set artificial languages must defer to
natural language for their explication and starting point, so subverting their
claim to logical priority (they are ultimately dependant on that which they
seek to overcome)).
So
we are left observing from within – in transgression of the taboo on
self-reference. Logic, by contrast would try to view language from without, an
existential taboo (playing god, condition of enunciation) as well as a logical
taboo (reference to meta-set).
Any
reference to a meta-set also points to a banned logical topology ‘outside’ (a
taboo on the name of god, on gazing upon the face of god) where the infinite
reach of the exterior zone, its permanent ‘otherness’, or un-reachability,
echoes the infinite repetition of the logical operation ‘upwards’ where one
zone once reached implies another without. Whilst mirror-wise we find that
self-reference, as self-definition, is suggested by any reference to an
incestuous inside… Twin references to the horizons of sense, twin surpassings
of sense, as the mapping (of the limits) of sense.
Family
resemblances: Let us look again at our two limitations and their parallel
operations in other areas of discourse. The first, reference to a meta-set,
resembles the act of grounding outside, the ‘logic of the outside’, anchoring
outside, the rhetoric of eternity, transcendence, deus ex machina… (by the act of looking out, looking up, for a
timeless place for our eternally true first cause, dooming ourselves to the
eternal repetition of this operation). The view from without (language).
The second, self-reference,
or reference to an element in a set as part of that element’s definition,
resembles all attempts at self-grounding, the issues of embeddedness and
relativism, of immanence, of autochthonous-ity and self-given teleology. Always
assuming what we should have previously proved… (by the act of looking in,
looking down, to found, to seek ever deeper for the foundation, dooming
ourselves to the eternal repetition of this operation). The view from within
(language).
The first, a psychological habit, putting the
foundation outside; the second, our permanent condition within language…
…of ones self within language, but then to
position one self without, outside, language, (leaving one self within)
impossible - but necessary; one self in and one out, two… (source of every
paradox, within language and in logic).
As
a staring point both axioms position themselves on either side of a
relationship to language: one notes the contradiction at the heart of
positioning oneself ‘outside’ language, yet how to operate without some such
point of ‘reference’ (humans appear to need some such point, no matter how
fictional, illusory or provisional, not least to ‘ground’ or ‘guarantee’ ethics
and values); the other notes the impossibility of any absolute starting point
within language itself (the restriction the application of the axioms would
involve, only restrict the sayable to an even narrower set of sentences within
language).
An ever receding ‘outside’
as each becomes an ‘inside’: an ever narrower set of sentences…
Which
may not be such a bad thing. For the resort to tautology after axiomatics opens
up the path to second order or artificial languages. Actually very useful (akin
to ‘bracketing-out’ in phenomenology and the elision of the complexities of
process in formalist or structural analysis): especially if further tempered by
reference and empiricism. But is not the same true for ‘pure’ reason with its
potentially infinite spread - also requiring an (external) empirical moment.
Reason as careful reason (not too metaphysical) or a looser logic (not too
excluding) in alliance with a sense of (illogical) reference or an empirical
moment. But the necessity of Reference and/or Empiricism already denies Logic
and Metaphysics their dominance (‘Reason alone’) and acts as a corrective to
both modes of thought in their moments of excess. (‘Cohesion’ is the other path
of ‘tempering’ available to us; a path that returns us to natural language and
(all) its resources – yet this move itself may also be seen as requiring
tempering, by reference or empiricism as appropriate).
So
it is that logic also appears to be, at once, caught in incestuous
self-reference, and, at the same time (like metaphysics), is found to be
sitting firmly on the ‘foundations’ provided by reference to the infinite realm
of the meta-set. Perhaps this is what natural language (our language) naturally
does. Perhaps, indeed, this is what our minds do – the lack of limitation being
our cue to creativity, to imagine the moving of mountains, to thinking afresh…
(as well as producing mountains of gibberish). So perhaps a non-normative, but
a ‘descriptive logic’, as used in natural language (with all possible
contradictions), would seem to be the most useful thing. Just as we use both
A=A AND A=not A (especially when it, ‘A’, becomes B, or when we cross a level,
‘up’ or ‘down’), according to context… and still manage to understand what we
mean.
As evinced in
‘contradictions’ as banal as the Zeno paradox, that of the tortoise and
Achilles, where the finitude of a distance is contrasted to the potential
infinity of its subdivision; however what should be contrasted is the finite
time it takes to cross the given distance and the potentially infinite
subdivision this distance may be subject to. The latter (level) in no way
affects the former, so no contradiction…
Axiomatised tautology is
already a problem (tautologous system plus limits, Wittgenstein’s last word in
the Tractatus, the ‘last word’ of the ‘early Wittgenstein’, demonstrates the
irrelevance of pure reason in its most perfect recent incarnation, the propositional
calculus). But after Gödel demonstrated the insufficiency of all systems, does
this not mean that even axiomatised tautologies are open-ended… (so in-practice
imitating their parent, natural language). And does this strange and unwelcome
fact not mean that they are even less reliable than thought, despite their
surrender to tautology and the insurance policy of axiomatisation? Or does it
mean that they always imitate natural languages in the (distant) last resort…
and so may be more useful for being so…?
So
the choice once was: An axiomatised tautology versus … that which the axioms
(attempt to) forbid: a starting point rooted ‘outside’ of itself (a reference
‘up’ to its meta-set – which may, in turn, become infinite); or the equally
infinite digging ‘down’ (another ‘bad infinity’) of self-reference as a
starting point; a rooting ‘inside’ of itself. Yet according to Gödel, all
axiomatised languages ‘unwind’ at a certain point; contain a ‘hole’, a
‘lacuna’. So returning the formal languages to the state of natural languages
(even if at one ((useful)) step removed). An uncanny doubling back not unlike
the return to the arrow of time of thermodynamics, into a physics previously
thought to be (theoretically) reversible (as in the potential shrinking of the
universe after its expanding); an irreversibility courtesy of chaos theory (as
part of the pattern evinced by the ‘strange attractor’). Or by the arrow of
time implied in our differing perception of past and future within the
experience of the eternal present? So it is that both unaxiomatised ‘natural’
logic (‘metaphysical’ or taboo-breaking) and post-axiomatised logic (those
logics which have ‘unraveled’), both forms of reason, all return us to natural
language, and so are finally open-ended…evolving as (or with) natural language
itself. The conclusion is that reason in natural languages can have no end; is
open-ended; is infinite (given sufficiently wide-ranging and interesting
material). Where we decide to ‘stop’ the argument is occasioned by where we are
and who we are (as is the case in limiting the numbers after the decimal
point). And that is were the argument is… (in perpetuity…).
(Indeed the
‘axiomatisation’ of formal languages is precisely the limitation that prevents
their use in the examining of themselves, their origin and context of use
(their outside as meta-set) and their self-justification (their inside as
self-reference as ’bad infinity’). Whence the tautology… the ‘closed’, or
formal aspect… which when recognised as such (as a tool made for a task within
a wider field, as provisional restriction of range is a tool of many sciences,
not least linguistics) is fine. But the abrogation of the whole field,
including the denial of (the existence off (the question of)) its origins and
context is perhaps the true transgression of an axiom… Its virtue, that of
appropriateness and limitation to the task it was designed for - or can do
without stretching credulity or denying the obvious).
So
back to ‘reason alone’ as ‘language alone’, but (this is another issue)
requiring empirical/referential correctives (so another kind of reference to an
outside…). Lesson: whatever mode of argument is used; reference to fact,
testability of hypotheses, test of language in context, pragmatics of use; all
or some (one) of these modes will be required at some point.
(Yet
by language alone… clearly implies circularity within language, a hermeneutics,
like deconstruction, or, more broadly, historically, the circulation of
intellectual fashions, of ‘positions’ that reappear periodically under a
different name; the argument is where we stop the definitions and argument;
each ‘stop’ however (necessary) risking an exclusion that leaves something
important unsaid…. So the process of debate (which we call philosophy) circles
on (or perhaps that should be ‘spirals on’ ((‘upwards’))…).
Gödel
has shown that all attempts at consistency lead to contradiction; just like
inner self-reference, self-grounding is but another name for the internal
definition of axioms. Likewise the meta-view is also implied in any overview
with the aim of consistency… the consistency of the … whole – but from whence
do we see this? Which position is outside (of ourselves) just as which place is
prior (to ourselves). Or better, which point of view is outside of all of our
points of view (but which we can nevertheless know) and which thought is prior
to all our thought (but which we can nevertheless think)). Damned if we do and
damned if we don’t (the promise of tautology frays if we demand consistency…). In
plain English, the usefulness of such logical operations is limited to areas
apposite to their (largely quantitative) mode of mapping. And both types of
axiomatisation (and their impossible avoidance) require a place which is not
‘this’ place (not this level) require another level such that we can think it
and not be able to think it. And so A becomes not-A when two such levels are
collapsed into one (proposition). So again, one limit turns into the other. One
becomes not-one. Always (already) two.
Zero
to two; the ‘eternal present’ and its Other (others…). Court on which the game
is played.
So the state of play is one
where we recognize out dependency on that which we can not believe (its
insufficiency). If axioms once gave us formal languages which restricted
meaning and field of operations (above all supporting the quantitative), so
differentiating itself from ‘metaphysics’, then now such languages are shown in
the last analysis also to be liable to the same contradictions and aporia,
self-confusions and under-minings, as natural language (naturally used reason,
logic or rhetoric).Which if they were a part of natural language, which they
are, is precisely as we would expect it to be…
(None
of which is a problem if we regard formal languages as man-made, as
‘formalisms’ eternally indebted to their parent, natural language: if we regard
all our logic as the result of an ‘hardwired’ foundation, or ’intuitionism’
(perhaps closest to Kant) then we are bound on an endless search for the best
verbal formulas for this internal (and unreachable, indeed infinitely
regressing, Grail). But if we require that the efficacy of logic, of numbers,
is found to rest on an objective exterior, their ‘Truth’ (aside from
immediately suffering from the paradox of the reference to a meta-set) then we
find ourselves quickly in the realm of faith alone, a latter day Platonism…
blind belief (a return to blind assertion by any other name)).
So finally back to two; two
layers. ‘Ours’ and the ‘other one’, the one we are now ‘inside’ of , and the
one posited ‘outside’; indeed the ‘now’ position and its other, to which we may
move, or may have moved through, an other which may be ‘before’ or ‘after’ – in
another uncanny parallel with our basic, indeed inescapable, levels of temporal
experience; the ‘eternal present’ and its windows opening onto the different
levels of the past and future. Together with the generalisation of the ‘eternal
present’ into the ‘Absolute Outside’ of ‘eternity’, offering us the basis of
the binary sub-divisions that dominate (and often disfigure, that is,
configure) our fundamental ways of thinking…
Two
layers: If ‘O’ (’zero’) gives a monism (as viewed from within) as viewed from
within there is no ‘one’ else (not ‘
No Parmenadian One, only
none (No-one) none; or two.
AND two… (where 0 and 2
equal two… as in two forms of temporality, past and future, or two levels,
‘above and below’ or’ in and out’ or ‘in-front and behind’… )
Despite
the various models and forms, in mathematics and in logic, of sets, axioms,
propositional calculus et al with all
their careful working out and construction and reconstruction of formal
languages, all problems finally resolve into, return to, can not avoid the
problem of: the reliance on the other, the requirement that there be two, a
minimum of two layers. With the concomitant problem; reference to the top limit
and to inner self reference, to tautologise or not to tautologise (and so fall
into contradiction… but it now seems that even tautologised systems must do
this, must be incomplete). So perhaps these twin problems are both one, an
expression, two expressions, of their difference; (adding a level up and adding
a level down, meta reference and self reference, are they both aspects of one
problem, the adding of a level, the final reliance on another level) the adding
of a level by focusing upon one level alone (‘inner’, or ‘outer’); the problem
occasioned by the movement between two levels, across a field of difference
that is nevertheless required (in turn) to make the logic or other formal
system work.. And un-work (not work). At the same time…
So
self-reference does not remain on the ‘same’ level, but immediately pushes one
down to another level (‘down’ as opposed to ‘up’, as ‘up’ is more apposite to
references ‘out’) yet we do not think of going in ‘up’ to another level for
foundation or ground (intuitively ‘foundation’ or ‘ground’ should be ‘down’…
yet we use ‘up’ because of its sense of outside or eternal verity - a sense
inherited from myth, religion and our sense of the descent of the vertical,
part due to light, part to gravity). Going ‘up’ or ‘down’ a level suggests
symmetry: yet is going ‘down’ a level for grounding, as opposed to going ‘up’
one for such, really symmetrical (or like the arrow of time with respect to the
past and the future, asymmetrical, regardless of our imaginary wonderings to
and fro…)? Perhaps the key point is that if one level is said to support
another level it must be prior (that it equals the past, like the assumption of
the truth of a sentence at first sight, awaiting contradiction) implying
something must be a next stage means it must follow (that it equals the future,
like the assumption of falsehood in a sentence which has been found wanting,
the second or following step). Yet is it that in other readings, other
examples, both ‘up’ and ‘down’ directions may include layers that may be read
as prior? The present as ever-present event horizon suggests all (else) either
as past, as logical implication points back to causes, or as future, in the
move, imaginary, but also implied, to an event that has not occurred - to
results (with non-reversibility, perhaps taken as an axiom, as to the position
of the present as moving only ‘forward’, away from what was to what has only
been imagined, or supposed, often incorrectly…). Either/or: present only in
what is non-present. (Semi-present). Framed by the present. Either/or: could it
be that this pair is the product of this (temporal) relation or intuition?
Asymmetrical: the
usefulness of this image is: as clearly informing us that there is no place to
hide (the foundation). Both this and an(any)other level all are fallible (are
not final). All places (levels) are liable to infinity or contradiction.
Symmetrical: mirror images;
so reducible to one difference (with two references (or a difference further
divided) on either side, up and own, in and out). Finally one kind of infinity
or contradiction?
Or (again) asymmetrical,
two kinds on either side of a difference, two because different in kind, not
(finally) isomorphic. Their difference finally offering two kinds of infinity
or contradiction?
Two
layers, one as containing the other, upper ‘meta’ limit as contradiction
(infinite progression upwards or ‘out’), and inner as self-preferential
contradiction (infinite progression inwards or ‘down’). If in parallel, then no
relation. So one must contain, ‘nest’, the other… whence the origin of these
axioms or contradictions….
But
strangely irreducible to one, the same difference, the ‘same difference’, a
contradiction in terms, the importance of binary relationships to human
thought…? A bit like the intuitive sense of non-collapsibility of past and
future (and vice-versa) of temporal experience. Both are semi-present in our
experience or time, but one ‘has been’ and one ‘never arrives’ with a line ‘connecting
them’ drawn through the ‘eternal present’, a line that is uni-directional… So
both are finally not reversible or reducible to one. Like inner and outer
limits? Or does the popular confusion of subjunctive and indicative, of what we
wish for and what actually happened, indicate a zone of over-lapping?
Back
to natural language… as the quest for the holy grail of an artificial language
(formal language) designed to offer fault-free calculation or calculus have
foundered on the rock of tautology (expected of a formal language which is, by
definition, closed) but also on the (Gödelian) joker in the pack of a ‘hole’ or
contradiction (axioms and tautologies not withstanding) which emerges to
sabotage the Holy Grail, with the impurity of an unavoidable inconsistency
(turned positive by some, seeing in this ‘leaking’ of closed systems an
approach to the open-endedness and so creativity of natural languages). Leaving
the latter also as imitators of the ‘faults’ of the former. So when all the
forays into number theory, logic and other formal languages (grammars, computer
languages and Artificial Intelligence) have convinced us only of the secondary
nature of such languages, their fictionality, contingency and provisionality
(all tempered in terms of their use, which finally oversees the decision to use
or reject) in sum their dependency upon the natural language which is required
to understand them, and which they can not encompass (it possesses infinitely
more power that any of them – together with the concomitant ability to create
an infinite amount of nonsense too…)… So when all is said and done, we are
returned, both for comprehension and for the general communication as to their
worth, to the realm we have evolved over thousands of years and can all use with
marvelous ability, our mother tongue (and for those fortunate enough, for the
-truly- bilingual, a shadow language (which also implies being bi-cultural)).
So it is that Gödel finally trumps Wittgenstein by showing that artificial
languages are in the end like the natural languages (naturally enough as it is
the latter that spawned them) with respect to the diseases that their
formalized offspring were to be the cure. (Wittgenstein had, along with others,
shown the limitations of formal languages as their strength, tautology; yet in
so doing leaving all the really interesting questions of the human condition to
a realm ‘beyond logic’…so echoing the findings of twentieth century pragmatic
linguistics with respect to meaning). Yet in this Wittgenstein may be read as
having the last word, as, in his latter philosophy, he finally leaves
significance, or meaning, in the hands of our language and its contexts, its
use - again echoing the findings of pragmatic linguistics with respect to
meaning…
If
all the fuss only results in tautology and self-contradiction, well then (Kant
and Hegel had already dealt with this) thinking goes on - in need of a little
training perhaps, and certainly in need of testing, but not replaceable by a
narrower – putatively, error-free form…
Otherwise put: from the
‘inner secret’ to the ‘outer limit’; a range of human experience only subject
to rarification (the path from mysticism to logic).
Therefore
the solution to the ‘problem’ or maybe even ‘scandal’ of the impossible choice
between using tautologised second-order languages or rejecting these self-same
languages is the recognition that the incompleteness that, it now appears, is
endemic to formalised thought, is, in fact, endemic to all forms of thought, to
all kinds of language (formal and natural) and furthermore that, in terms of
creativity, they may well be essential
to all thought, but not to Truth (necessary but not sufficient). So therefore
the most useful strategy would be to differentiate between the ‘good’ and
‘bad’, useful and non-useful, manifestations of the two main contradictions or aporia that result from such
incompleteness, to differentiate between the ‘in and out’, or ‘up and down’,
‘bad’ infinities and other forms of open-endedness that effectively breach the
gold standard of A=A, such that we can use them effectively (as we do in the
use of ‘fuzzy logic’ in the programming of… domestic
appliances – a not inconsiderable taming of the putatively iconoclastic
‘breaking of the bonds’, or twin transgressions, of logic…). Useful, because
unavoidable.
Logic and layers within the
bigger picture. The determinations of species–being + the determinations of
culture = our experience, our perception. The foregoing + the open-endedness of
natural languages and the creativity this permits (a permission guaranteed by
the isomorphism between natural and artificial languages, by their open-ended
nature) = the possibility of new thinking, of a ever-fresh conceptualization of
a ever-changing world.
Last
word on the non-isomorphism of the two sides (‘up’ and ‘down’, ‘out’ and ‘in’).
Is any attempt to reduce them to a single difference doomed? Two viewpoints, or
two languages, or two levels are required; but the difference between the two
is never quite one (we are never quite locked in to the ‘prison-house’ of one);
we are left showing two ‘directions’… which never quite permit collapse into
one. Irreducibles, like (infinite) repetition, show us that we have ‘hit’
something, what that might be is quite another matter… (indeed a matter for matter
itself, our constitution as mind or ‘spirit’ on the foundations of a prior
material level…).
(So like the ‘arrow of
time’ a part of our metal hardware, our way of dealing with, of forming,
‘difference’?)
Within
(the bounds of) language alone.
Copyright Peter
Nesteruk, 2012.