To all concerned colleagues,
I appreciate the fact that discussions should be conversations about
issues,
but this particular issue and in particular the critique cited in my
posting
warrant extended exposition in order to show the reasoning upholding the
critique.
I am thus quoting from specific articles, the first phenomenological,
the second
analytic-logical--though they are obviously complementary as befits
discussions
in phenomenology and the life sciences.
EXCERPT FROM:
SELF-REFERENCE AND GÖDEL'S THEOREM: A HUSSERLIAN ANALYSIS
Husserl Studies 19 (2003), pages 131-151.
Albert A. Johnstone
The aim of this article is to show that a Husserlian approach to the
Liar paradoxes and to their closely related kin discloses the illusory
nature of these difficulties. Phenomenological meaning analysis finds
the ultimate source of mischief to be circular definition, implicit or
explicit. Definitional circularity lies at the root both of the
self-reference integral to the statements that generate Liar paradoxes,
and of the particular instances of predicate criteria featured in the
Grelling paradox as well as in the self-evaluating Gödel sentence
crucial to Gödel's theorem. Since the statements thereby generated turn
out on closer scrutiny to be vacuous and semantically nonsensical, their
rejection from reasonable discourse is both warranted and imperative.
Naturally enough, their exclusion dissolves the various problems created
by their presence. . . .
VII: THE GOEDEL SENTENCE
Following a procedure invented by Gödel, one may assign numbers in some
orderly way as names or class-numbers to each of the various classes of
numbers (the prime numbers, the odd numbers, and so on). Some of these
class-numbers will qualify for membership in the class they name; others
will not. For instance, if the number 41 should happen to be the
class-number that names the class of numbers that are divisible by 7,
then since 41 does not have the property of being divisible by 7, the
class-number 41 would not be a member of the class it names.
Now, consider the class-number of the class of class-numbers that are
members of the class they name. Does it have the defining property of
the class it names? The question is unanswerable. Since the defining
property of the class is that of being a class-number that is a member
of the class it names, the necessary and sufficient condition for the
class-number in question to be a member of the class it names turns out
to be that it be a member of the class it names. In short, the number is
a member if and only if it is a member. The criterion is
circular--defined in terms of what was to be defined--and consequently
not a criterion at all since it provides no way of determining whether
or not the number is a member.
The situation is obviously similar for the class-number of the
complementary class of class-numbers--those that do not have the
defining property of the class they name--since the criteria in the two
cases are logically interdependent. The criterion of membership is
likewise defined in circular fashion, and hence is vacuous. In addition,
the criterion postulates an absurd analytic equivalence, that of the
defining property with its negative. The question of whether the
class-number is a member of the class it names is unanswerable, with the
result that any proposed answer is neither true nor false. In addition,
of course, any answer would generate paradox: the number has the
requisite defining property if and only if it does not have it.
As might be expected, the situation is not significantly different for
the class-number of classes of which the definition involves semantic
predicates. Consider, for instance, the class of class-numbers of which
it is provable that they are members of the class they name. The
question of whether the class-number of the class is a member of the
class it numbers is undecidable. The possession by the class-number of
the property requisite for membership is conditional upon the question
of whether it provably possesses the property, with the result that the
question can have no answer. Otherwise stated, the number has the
defining property of the class it names if and only if it provably has
that property. In these circumstances, the explanation of what it means
for the class-number to have the property has to be circular in that it
must define having the property in terms of having the property. The
vacuity that results is hidden somewhat by the presence of the
requirement of provability, but while provability might count as a
necessary condition, in the present case it cannot be a sufficient one.
In fact, its presence creates a semantically absurd situation: the
analytic equivalence of having the property and provably having it. The
statement of the possession of the property by the class-number in
question is consequently both vacuous and semantically absurd, hence an
undecidable pseudo-statement.
The analytic equivalence of the number's having the property and
provably having it has a further and quite interesting consequence. In
principle, since the equivalence is analytic, it explains what it means
to say that the class-number in question has the requisite property,
that is, it explains what is being said by the statement that attributes
the property to the number. What the statement is saying, according to
the equivalence, is that it is provable that the number has the
property, which is to say, it is saying of itself that it is provable.
Thus, the statement is self-evaluating. It is not, strictly speaking,
self-referential since it contains no designator, and so cannot refer to
itself. However, it mirrors the self-referential statements of the sort
discussed earlier in that it predicates a semantic property of itself
(or at least purports to do so).
In these circumstances, it is not overly surprising to find that a
sentence having a vacuously defined semantic predicate of provability is
ambiguous or leads a double life. It may be used to express either of
two statements, a pseudo-statement that purports to evaluate itself as
provable, or, a genuine statement that evaluates the pseudo-statement,
which genuine statement is, of course, false since a pseudo-statement is
in principle not provable. The two statements, genuine and pseudo, are
not the same statement. The two have distinct truth-values, but the
basic point is that they differ in intended meaning. In the
pseudo-statement, the statement itself (that a particular number has a
particular property) is a part of the meaning of the pseudo-statement,
while in the genuine (but false) statement, it is not.
An analogous situation obtains in the case of other classes involving
semantic predicates. If the term 'heterological' that figures in the
Grelling Paradox were defined as applying to those words of which it is
false that they are heterological, then the resulting Grelling statement
(the statement that 'heterological' is heterological) could be plausibly
interpreted to be self-evaluating. It would be analytically equivalent
to the statement that it is false that 'heterological' is
heterological--an equivalence that may be read as saying that the
Grelling statement says of itself that it is false. This second
statement would, of course, find itself expressed by a sentence that
leads a double life.
Of particular interest for the purpose of understanding the error that
invalidates Gödel's theorem is the case of the class-number that names
the class of class-numbers that are not provably members of the class
they name. Once again, the question as to whether the class-number that
names this class is a member of the class it names is unanswerable. The
statement that the class-number possesses the required defining
characteristic is a criterially deficient predication, and hence a
pseudo-statement. In addition, the statement is analytically equivalent
to the statement that the class-number's possession of the defining
characteristic is not provable, and so may be viewed as saying of itself
that it is not provable. It is thus self-evaluating, and when stated in
this form, it is expressed by a sentence that leads a double life. As a
result, any formal system that admits and purports to accommodate a
criterially deficient predication of the sort will also require the
elaborate supplementary machinery found necessary to accommodate
self-referential statements: a three-valued logic, a procedure for
determining which instantiations of predicates (or substitutions into
propositional functions) produce pseudo-statements, and some notational
device for distinguishing pseudo-statements from the genuine statements
that are their sentential doubles. As we shall now see, in view of the
similarity in structure of the above statement to the Gödel sentence,
analogous remarks apply to the latter.
VII. THE GÖDEL SENTENCE
In his well-known theorem Kurt Gödel purports to show that any formal
system of classical logic equivalent to that of Principia Mathematica to
which arithmetic constants and the axioms of arithmetic (Peano's) have
been added, will contain sentences that are undecidable--that is,
sentences such that neither they nor their negations are provable within
the system. To this end he introduces a provability predicate defined
syntactically as membership in the set of sentences that are immediate
consequences of the axiom-sentences. Since the provability predicate
applies to sentences rather than statements, to avoid confusion it is
best termed 'a derivability predicate'. As in the arguments of the
previous section, Gödel has a number assigned as a name to each class of
numbers according to its rank in an ordering of the various classes of
numbers. Roughly characterized, the undecidable sentence figuring in the
theorem (the Gödel sentence) states that a particular class-number
satisfies a particular one-place propositional function that defines a
class of numbers. A little more precisely, it states that a particular
class-number has the defining characteristic of the class it numbers,
which class is the class of class-numbers such that the sentences
stating that the class-numbers possess the defining characteristics of
the classes they name are not derivable. In his informal introduction
to his theorem, Gödel points out that the sentence may be read as
stating via its Gödel number that a particular sentence, itself, is not
derivable.
The crucial line of reasoning in the theorem strongly resembles the one
found in the Liar. It runs roughly as follows: if the sentence were
derivable, it would have to be true, hence say something true, and
hence, as it says, not be derivable--which contradicts the assumption of
its derivability; if the negation of the sentence were derivable, then
since the sentence states its underivability, it would have to be not
underivable, hence derivable--with the result that both the sentence and
its negation would be derivable, a contradiction. As with the Liar, each
of two possible alternatives generates a contradiction, although in the
present case the consequence is not paradox but undecidability--
undecidability in the form of a sentence of which neither its truth nor
its falsity is derivable in the system. Gödel reasons that since the
undecidable sentence apparently states something true, its own
underivability, the system contains underivable true sentences, and
hence is incomplete.
The Gödel sentence is concerned with derivability rather than
provability, or sentences rather than statements. As a result one may
plausibly question whether it is vulnerable to the criticisms directed
above against criterially circular predications and self-evaluations.
While the Gödel sentence clearly differs from the latter, it is possible
nevertheless to raise the question of its legitimacy. Gödel himself
simply assumes that the sentence is legitimate--which, of course, it is
in the narrow sense that it conforms to the formation rules of the
system in which it figures. However, it does not follow that it is
legitimate in the broader sense that the interpreted sentence makes
sense. As we saw earlier with self-referential statements and
criterially circular predications, sentences that are apparently
well-formed may in fact express nonsense. The Gödel sentence may well
express just such a pseudo-statement, and have nevertheless been
admitted into the formal system through an inadequacy of the formation
rules. Gödel dismisses the possibility of faulty circularity on the
grounds that the sentence states only that a certain well-defined
formula is unprovable, which formula turns out after the fact to be the
one that expresses the proposition itself. Yet, an answer of the sort
will not do. Where circularity results from a substitution, being
adventitious and well-formed according to the rules do nothing to remove
the circularity. A statement with a circularly defined predicate is
semantically vacuous, and hence not a genuine statement. Thus, the
question of the meaning of the sentence, the statement it expresses,
calls for serious examination.
A first rather curious fact that more careful scrutiny brings to light
is that the most obvious reasons for thinking the sentence meaningful
are actually inconclusive. For instance, it might be found tempting to
argue as follows: that any particular string of symbols is either
derivable from the axiom-strings or not, and hence since the Gödel
sentence asserts that a particular string is not derivable, whether true
or not, it must at least be meaningful. However, the reasoning begs the
point at issue. If the Gödel sentence is not meaningful, then its
assertion that it is not derivable is not meaningful. It is a
pseudo-statement that may appear to state something but cannot in fact
state anything.
For the same reason, it would be question-begging to reason that since
the Gödel sentence states something true, its own underivability, it
must be a genuine statement. If the sentence makes a pseudo-statement,
it does not state anything, and so cannot state anything true. Reasoning
of the sort simply assumes (as does Gödel) that the sentence is
meaningful, and so fails to show that it is.
In contrast, there are two compelling reasons for deeming the Gödel
sentence not to be meaningful. The first of these reasons is that any
attempt to explicate the meaning of the string of symbols of which the
Gödel sentence is composed finds that meaning to be a complex whole of
which the meaning of that same string of symbols is a constituent. Any
explanation of its meaning turns out to presuppose what it is supposed
to explain. The situation differs from those discussed earlier in that
the explanation is given in terms of a string of symbols, a sentence,
rather than the purported meaning of the symbols. The presence of a
sentence creates the illusion that there is no vacuity; a statement may
be vacuous but a sentence is something perceptibly concrete.
Nevertheless, the situation remains essentially the same as those
considered earlier. The question being asked is whether the sentence is
meaningful, and that question cannot be answered by appeal to the
concreteness of the sentence. Such a line of reasoning would rule any
string of symbols whatever to be meaningful. Ultimately the situation
comes down to the following: the Gödel sentence is meaningful if and
only if the Gödel sentence is meaningful. Despite the shift from
statement to sentence, the meaning has been given a circular definition,
which, as we have seen, can only generate semantic vacuity and a
pseudo-statement.
The second reason for denying meaningfulness springs from a more
general consideration. The formalization of arithmetic together with its
metalanguage is presumably a formalization of the arithmetic and
metalanguage that occur in natural languages, in particular, in English.
Its translation back into English must be possible, and make good sense.
In English, one does not speak of sentences being true or of sentences
being derivable, but of statements being true, and of statements being
provable. The only cogent translation of the Gödel sentence back into
English is a statement that asserts its own unprovability from the
axioms of arithmetic and the laws of logic. Precisely such a
self-evaluation of unprovability was examined earlier and found to be a
criterially deficient predication, a pseudo-statement that is neither
true nor false. On its intended interpretation, the Gödel sentence does
not express a meaningful statement.
The basic point is that for a formal system to qualify as a
formalization of some discipline, it must admit of translation back into
the language of the discipline it purports to formalize. The point is
one that it is easy for logicians to overlook. The logic practiced in
formal systems is a form of what Husserl terms 'consequence-logic' or
'logic of non-contradiction', that is, the concern is with what follows
from certain statements in accordance with given rules, and not with the
truth of the statements (Hua XVII, pp. 15-6, 58-9). In addition, as
Husserl notes with regard to mathematics, it is customary for the formal
system to be treated somewhat like a game in which strings of symbols,
depending on their form, are derivable or not derivable from other
strings according to rules. The signs in the system have, like chess
pieces, "a games meaning" that replaces the arithmetic or statemental
meaning for which the signs are actually doing duty (Hua XVII, p. 104).
Nevertheless, if the game is to allow any conclusions to be drawn about
the discipline being formalized, its strings of symbols and its rules
must be interpretable, which means translatable back into the original
language. In the case of Gödel's formalization of arithmetic, a
particular sentence, the Gödel sentence, translates into a
pseudo-statement. Such a sentence can hardly provide a sound basis on
which to build a persuasive proof of the incompleteness of formalized
arithmetic.
Matters are not improved if the Gödel sentence is replaced with a
simpler one, one of the sort suggested by Kripke that uses a proper name
to refer to itself and to say that a particular sentence, itself, is not
derivable. Any such sentence has nothing to do with either arithmetic
or the metalanguage of arithmetic, and so its presence in a system of
formalized arithmetic is quite unwarranted. More importantly, the
definition of the name it contains is circular. It defines the name in
terms of a sentence that contains the name, which name is not as yet a
name since the point of the definition is to make it one. It would be no
less nonsensical to declare 'Gorg' to be a name for the word
'Gorg'--although in fact there is no such word since, prior to the
definition, 'Gorg' is a mere string of letters. Furthermore, the
sentence in question should in principle be translatable back into
English if it is to be considered a proper formalization of what it
purports to formalize. On translation, the sentence becomes a
nonsensical self-evaluation of unprovability. The Kripke sentence is
thus no improvement on the Gödel sentence.
EXCERT FROM:
THE LIAR SYNDROME
SATS Nordic Journal of Philosophy, vol. 3, no. 1
VI. GÖDEL AND SENTENTIAL SELF-REFERENCE
Kurt Gödel's well-known theorem, widely termed 'Gödel's Theorem',
demonstrates that any formal system of classical two-valued logic
augmented with the axioms of arithmetic and a portion of its own
metalanguage will contain sentences that are undecidable in the
system--sentences for which neither they nor their negations are
provable within the system. The metalinguistic evaluations are made
possible through a provability predicate defined syntactically as
membership in the set of sentences that are immediate consequences of
the axiom-sentences. Since the provability predicate applies to
sentences rather than statements, to avoid confusion it is better termed
'a derivability predicate'. The undecidable sentence figuring in the
theorem, the Gödel sentence, says that a particular sentence, itself, is
not derivable. Thus, the undecidable sentence responsible for the
incompleteness apparently states something true, its own underivability.
The paradoxical line of reasoning central to the theorem also strongly
resembles the one found in the Liar. It runs roughly as follows: if the
sentence were derivable, it would have to be true, hence say something
true, and hence, as it says, not be derivable--which contradicts the
assumption of its derivability; if the negation of the sentence were
derivable, since the sentence states its underivability, it would have
to be not underivable, hence derivable--with the result that both the
sentence and its negation would be derivable. As with the Liar, each of
two possible alternatives generates a contradiction, although in the
present case the consequence is not paradox but incompleteness.
The Gödel sentence figuring in Gödel's proof states that a particular
number satisfies a particular one-place propositional function that
defines a class of numbers. In Gödel's formal system a number is
assigned as a name to each class of numbers according to its rank in an
ordering of the various classes of numbers. Roughly characterized, the
Gödel sentence states that a particular class-number (the class-number
of the class of class-numbers for which the sentences stating they
possess the defining characteristics of the classes they number are not
derivable) has the defining characteristic of the class it numbers (that
of the non-derivability of the sentence stating its possession of the
defining characteristic of the class it numbers).
Clearly, since the Gödel sentence, on its intended interpretation,
states the underivability of a certain string of symbols, rather than
the unprovability of what is said, it is not vulnerable to the reasoning
presented earlier against statemental self-reference. Sentential
self-reference is widely and plausibly esteemed to be a harmless
operation. In this spirit, Saul Kripke has contended that by
interpreting elementary syntax in number theory, "Gödel put the issue of
the legitimacy of self-referential sentences beyond doubt; he showed
that they are as incontestably legitimate as arithmetic itself." Kripke
is obviously right when 'a legitimate sentence' is taken to mean a
formula of the formal system that is a well-formed formula according to
the formation rules of the system. However, the important issue is
whether such sentences are legitimate in the sense that they make good
sense on their intended interpretation, rather than express dubious
statements that inadequate formation rules have failed to exclude. Gödel
makes no attempt to show that the interpreted Gödel sentence makes sense
(nor does Kripke); he seems simply to assume that it makes sense given
that it is well-formed according to the rules of the system. The
assumption hardly commands automatic endorsement, since, as we saw
earlier with statemental self-reference and criterially circular
predication, sentences considered to be well-formed may in fact express
nonsense. The issuing of a certificate of legitimacy should be
contingent upon the results of closer scrutiny of the meaning of the
Gödel sentence.
To clarify matters, let 'E' and 'e' represent some normal class of
numbers (such as the class of even numbers) and its class-number, and
let 'D' represent an underivability predicate. Let 'N' and 'n'
represent respectively the class and class-number of all classes such
that the sentence stating that the number has the defining property for
membership in the class it numbers, is not derivable. The necessary and
sufficient conditions for each of the two class-numbers, e and n, to be
members of the class of class-numbers, N, may then be stated
respectively as follows:
(10) Ne df D('Ee')
(11) Nn df D('Nn')
The statement of membership conditions in (10) is clearly not circular.
The same is not obviously the case for the statement of membership
conditions in (11). Indeed, on further inspection, the alleged
legitimacy of the Gödel sentence, the left-hand side of (11), becomes
quite suspect.
For instance, it might be found tempting to argue as follows in favor
of the claim that Nn, the left-hand side of (11), should make perfectly
good sense. What it states is equivalent to what is stated by the
right-hand side, the underivability of a particular string of symbols,
'Nn'. Since a string of symbols is either derivable from the
axiom-strings or not, a statement asserting it is not derivable must be
meaningful, and hence be a genuine statement. Given the equivalence of
the right-hand and left-hand statements, the Gödel sentence must also
express a genuine statement. However, such a line of reasoning begs to
point at issue. The question is whether the sentence 'Nn' makes sense.
If it does not, then the left-hand statement of (11) does not, and so
neither does the statement equivalent to it, the right-hand side of
(11). The latter must then be a pseudo-statement, one that appears to
assert the underivability of a particular string of symbols, but one
that in fact cannot assert anything. Thus, in assuming that the
right-hand side of (11) asserts something, the argument presupposes what
it purports to establish.
For the same reason, it would be fallacious to claim (as Gödel does)
that the Gödel sentence states something true, its own underivability,
and then to argue that since it states something true, the left-hand
statement of the equivalence must also be true, and hence a genuine
statement. If the sentence makes a pseudo-statement, it states nothing,
and so cannot state anything true. Such an argument simply assumes (as
Gödel does) that the sentence makes a genuine statement, and so fails to
show that it does.
In point of fact, there are two excellent reasons for thinking the
sentence cannot make a genuine statement. First, the predication on the
left-hand side of (11) is meaningful only if the statement on the
right-hand side is meaningful. The latter is meaningful only if the
string of symbols 'Nn' is a string of symbols that expresses a
meaningful statement. If the string 'Nn' expressed nonsense, then since
it is also the Gödel sentence, the latter would not make a meaningful
statement. Thus, the meaningfulness of the predication, Nn, is
conditional upon the meaningfulness of the statement expressed by 'Nn',
which is to say, itself. As a result, the predication is criterially
circular. The situation echoes that of the Grelling paradox: the
attribution of a particular predicate to a particular individual fails
to make sense. In the case of the Gödel sentence the circularity is less
apparent because the relevant statement is defined in terms of its
sentence rather than in terms of itself. However, the shift from
statement to sentence fails to avoid circularity since the question
still arises as to whether the particular string of symbols is
legitimate in the sense of expressing a genuine statement.
The second reason for thinking the sentence illegitimate is no less
decisive. If the formalization of arithmetic-plus-metalanguage is to be
considered a faithful rendition of arithmetic-plus-metalanguage in
English, its translation back into English must make good sense. The
exception could only be a situation where the formal system employs some
peculiar idiom in order to correct an incoherent English one. Such
appears not to be the case. It is true that English speaks of the
provability of statements rather than of the derivability of sentences,
but it manages to do so without collapsing into incoherence. Talk of
sentences being true, or false, or derivable, has its source in what is
convenient for logicians, and not in the incoherence of some English
idiom. In these circumstances, the only cogent translation of the Gödel
sentence back into English is a statement asserting its own
unprovability, as in (7). Such a statement is a pseudo-statement
afflicted with the Liar Syndrome, one the negative effects of which are
neutralizable in English with appropriate precautions.
Thus, the Gödel sentence is properly judged to be illegitimate. It
makes a pseudo-statement, and consequently should never have been
admitted into a formal system that is two-valued, and hence unequipped
to accommodate such sentences. Moreover, since a pseudo-statement says
nothing, the argument in Gödel's Incompleteness Theorem fails appealing
as it does at two crucial points to what the statement says.
The theorem cannot be rescued by an appeal to the services of the
simplified version of Gödel sentence suggested by Kripke, a sententially
self-referential sentence constructed through the use of proper names
for sentences. The definition of such a sentence may be represented as
follows, with 'n' representing a sentence name:
(12) n =ds D'n'
Clearly, the statement expressed by 'n' has nothing to do either with
arithmetic or with the metalanguage of arithmetic, so its presence in a
system of formalized arithmetic is quite unwarranted. In addition, a
definition as in (12) succumbs to charges analogous to those directed
above against (11). First of all, 'n' is a meaningful name of a sentence
in a two-valued system only if the right-hand side of (12) is a sentence
that expresses a meaningful statement, and the latter is the case only
if the 'n' on the right-hand side is the name of a sentence that
expresses a meaningful statement. Thus, the meaningfulness of the name
'n' has been made to depend in circular fashion upon the name 'n' being
meaningful. The situation is not unlike that of declaring the word
'Gerg' to be a name for the word 'Gerg', whereas prior to a definition
it is a mere string of letters, and not a word. Likewise, in (12) 'n'
may name a name only if 'n' is already a name and hence designates
something.
Secondly, a formal system that is a formalization of the arithmetic and
metalanguage given in a natural language should in principle be
translatable back into that language if it is to be considered a proper
formalization of what it purports to formalize. Since the only cogent
translation back into English of the concept of derivability is that of
provability, the interpreted Gödel sentence becomes a nonsensical
self-evaluation of unprovability as in (9) above.
Thus, the shift from statemental self-reference to sentential
self-reference is, from the point of view of present concerns, of less
than dubious utility. Statements that are self-referential and
predicates that are criterially circular in the sentential mode may be
represented as follows, where the predicate '' represents any
sentential semantic predicate:
(13) p =ds 'p'
(14) Nn df 'Nn'
(13) is, as it were, the sentential rendition of (2), while (14) is
that of (8). When transformed into their sentential correlates, the
pseudo-statements that instantiate (2) and (8) become sentential
evaluations that instantiate (13) and (14). Certainly, in discussing
formal systems it may be useful to speak of sentences rather than of the
statements they make, but otherwise the transformation yields no
significant gain. If syntax faithfully reflects semantics, as it should,
the formation rules of the system must screen for definitions and
instantiations that generate sentences expressing statements afflicted
with the Liar Syndrome. Contradiction is the price of failure to do so.
Any system that contains both semantic predicates of some sort (of
truth, provability, possibility, necessity) and names or designators of
statements, sentences, or classes, must, if it is to avoid unnecessary
problems, screen for failures of instantiation and substitution salva
significatio. It must be suitably equipped either with formation rules
that eliminate any resulting nonsensical and irrelevant statements, or
with a notation that prevents confusion of the pseudo-statements with
the genuine statements that evaluate them. The system that figures in
Gödel's Theorem fails to do any of this.
VII. IMPLICATIONS
The puzzles attendant upon self-reference have over the years generated
a wide variety of extravagant claims. Although in view of the above
findings the error of these claims is obvious enough, a brief spelling
out of the obvious is perhaps not amiss.
The widespread tenet that a formal language cannot contain its own
metalanguage without generating paradox is quite overstated. It is true
only of certain formal languages, those lacking the machinery necessary
either to eliminate certain pseudo-statements or to accommodate them in
a three-valued system equipped with disambiguators. The Liar provides no
grounds to speak, as has Hilary Putnam, of "giving up the idea that we
have a single unitary notion of truth applicable to any language
whatsoever ... ," and hence of giving up any notion of a God's Eye View
of the world, and embracing a general Antirealist or non-Objectivist
account of human knowledge. Indeed, it would be astounding to find such
claims warranted. English has been serving as its own metalanguage for
an impressive length of time without requiring the services of hermetic
levels of truth, and without collapsing into incoherence.
Gödel's Theorem is often understood to show that any system of
formalized arithmetic must be incomplete. In addition, it is not
infrequently touted to have other far-reaching implications. John
Stewart, for one, has argued that Gödel's Theorem undermines an
Objectivist epistemology and supports transduction, the view that
subject and object exist only in their relationship to each other.
Michael Dummett deems the theorem to show "that no formal system can
ever succeed in embodying all the principles of proof that we should
intuitively accept." Likewise, Roger Penrose takes it to show that in
mathematical thinking "the role of consciousness is non-algorithmic,"
and that "human understanding and insight cannot be reduced to any set
of computational rules."
As concluded above, Gödel's Theorem is made possible by a failure to
either exclude or accommodate sentences that express pseudo-statements
on their intended interpretation. Such a situation provides no obvious
support for the claim that mathematics has no firm foundation, and hence
none for Antifoundationalism or for Antirealism. Nor does it reveal some
deep feature of mathematical thinking, a feature that eludes capture in
a formal system. Such a feature may well exist, but evidence for it must
be sought elsewhere. Finally. it cannot reasonably be claimed to reveal
some remarkable capacity of the human mind: self-reference. The latter
simply generates nonsense. A capacity to lapse into nonsense, however
proficiently exercised, is hardly a very awe-inspiring human trait.
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