[FOM] Self-referentiality and Godel sentences

[Arnon Avron]

In order not to go into an infinite, fruitless exchange of messages
with Torkel Franzen or others, I will try to clarify in this message
once and for all my position concerning the "self-reference" of arithmetical 
sentences, and after this I shall do my best to force myself not to respond
to further postings on the subject (this does not apply to already
sent ones, like Friedman's posting directed to me).

Let us start with the following words of Torkel Franzen:

"The general pattern when we talk of a formalization A* in a formal
theory T (such as PA or ZF or ACA_0) of a statement A ("This sentence
is unprovable in T", "Every natural number has a unique prime
decomposition", Ramsey's theorem, the fundamental theorem of calculus,
etc) seems to be that A* is the translation into primitive notation of..."

 Now I wonder: can it be the case that Franzen does not see the deep
difference between his first example ("This sentence is unprovable in T") 
and all the other examples??? If so, then I have no choice but to
explain the obvious: All the other examples are clear 
mathematical propostions that every mathematician understands, without
a need for any formalization (to say nothing about a need for Godel 
numbering...). In fact most mathematicians (and not only mathematicians,
of course)  know these theorems, use them, find new proofs
for them - all without ever seeing 
any formalization or feeling any need for one. Indeed these theorems had 
been recognized as such before any formalization  took place. 

 Can the same be said about "This sentence is unprovable in T"? Has this
sentence for itself any reasonable meaning without formalization?? 
It should be  clear that in "formalizing" this meaningless sentence
we are *not*  TRANSLATING anymore a previously  meaningful mathematical 
fact into a certain formal system, but we are formulating a new 
mathematical fact (unknown before) within that system (what makes the 
formal sentence we construct meaningful, by the way, is the standard semantics
of the the formal system in which it is constructed - N, in the case of PA -
and we can clarify this  meaning to ourselves by a translation that goes in 
the opposite direction: from the formal to the informal).

  I am afraid that putting "This sentence is unprovable in T" in the same
list with the other examples might be due to the unfortunate fact that 
the same word in English ("formalization") is used in texts for both 
the translation of an informal proposition into a formal language (which 
is necessary for using the formal language) and for the arithmetization 
of syntax (or similar processes) which is something completely different 
(arithmetization of syntax is a case of developing some amount of 
formal mathematics in a given formal system
in ANALOGY to what is done in some other system, informal or even formal.
I shall return to this point when I react to Friedman's question(?) why
I dont believe Godel - as if I had ever doubted what Godel had said...).

To explain further the difference,  think of the 
criteria for judging whether a given formalization is adequate. In the 
case of "Every natural number has a unique prime decomposition" 
the basic necessary criterion (which is usually also sufficient
for all practical purposes) is that it should be possible to prove
that the meaning of the formal translation
(as determined by the intended, precise semantics of the formal
language used) is equivalent to the original, informal (or even formal) 
proposition. The proof may be formal (in our specific
example: it might be done in  a system in which we can directly
refer to finite sets or finite sequences) or informal - it does not matter
as long as it is convincing (devoted formalists: please ignore the
last remark!). It is even not very important what is the "equivalence"
concept used.  It might be a very weak one (according to which, e.g. 
any two true sentences, or any two theorems, are equivalent). The only 
important factor is that the equivalent can convincingly be established.
Of course, the easier is the informal proof, or the shorter/more natural
is the formal one - the better is the translation reflecting the
intensional content of the sentence - but mathematicians usually
have little reason to worry about this.

Does this criterion apply to "This sentence is unprovable in T"??
Can anybody seriously claim that a Godel sentence for T is 
equivalent, in any mathematically sensible meaning of the word
"equivalent", to the english sentence "This sentence is unprovable in T"???
The very fact that a Godel sentence for T is meaningful (and true,
if constructed in a natural way) while "This sentence is unprovable in T"
is meaningless excludes any possible form of "equivalence"! 

A remark before proceeding: I guess that some readers will be ready to claim
that "This sentence is unprovable in T" is indeed *true* because
any sentence not in the language of T is unprovable in T. Since
a Godel sentence for T is also true, we do have here according
to this interpretation at least 
the weak kind of equivalence I have talked about above. Well, I would ask 
such readers to consider "This sentence is provable in T" instead of
the the previous one (I trust that Franzen would not mind). The new
sentence will now be *false* (by the same argument) while by Loeb theorem,
its formal counterpart (that some people call 
"its foramalization"!) is *true*.

What makes it possible for a Godel sentence for T to be meaningful and true
althogh it was produced as a counterpart of the meaningless sentence
"This sentence is unprovable" (or even "This sentence is unprovable in T"-
the difference, which does exist, is not significant for my 
present purposes)? It is precisely the fact that the latter is essentially
self-referential, and has no other alternative meaning, while a Godel
sentence for T does not (and cannot) directly refer to itself - 
only to some (term for) a number which serves as its code according to 
a very concrete, peculiar choice of Godel numbering. Moreover:  a Godel 
sentence does have an independent meaning: the one determined by 
the standard semantics of the system T.  This independent meaning
is in fact  its *primary* meaning! 

Now at this point Friedman might protest:
"You don't believe Godel  when he says that he was
thinking about what self reference means in PA? And when he talks about the
liar paradox guiding him?" Well, I dont need Godel's testimony for this.
Even a stupid person like me can understand for himself that Godel was 
guided by the liar paradox, and that he was thinking of how to make
an *analogous* construction within PA (whether this is the same as
"thinking about what self reference means in PA" depends on the meaning
of "means in PA" here. I dont think that Godel would have quarrled 
with the way I have understood it, but had he done so I would 
not have accepted his view about what he has accomplished. I dont 
share Godel's Platonism either, so what? What does this have to do 
with *believing* Godel??).

Now analogy is an excellent methodology. It provides motivations, insights,
guiding lines, etc. The History of science in general and Mathematics
in particular is full with productive analogies (sometimes quite far-fetched
ones, sometimes even of a completely formal nature!). But analogy is not
identity. Moreover: practically every analogy breaks at some point.
This is also the case with the present analogy. The Liar sentence
is paradoxical (and meaningless, in my opinion). The analogous sentence
constructed by Godel is not paradoxical, and it
is meaningful and true. Describing both as "self-referntial" on a par
is misleading, and in my opinion completely misses the point
of what Godel has done. 

Let me use here some analogy myself. Like any analogy, it is far from 
perfect, but I hope that it would help clarifying the point. Suppose
my son is on a long trip somewhere at the other side of the Globe,
and one day I get an urgent message from him: "t1*s1+t2*s2=t3*s3",
where t1 is S followed by 1063 zeroes, s1 is S followed by 9 zeroes,
t2 is S followed by 31 zeroes, s2 is S  followed by 35 zeroes, t3
is S followed by 2663 zeroes, and s3 is S  followed by 3 zeroes.
If nothing had been agreed by us before, I might (with enough patience)
check and see that this is a theorem of PA. I might conclude that
my son does remember after all his travellings some fundamental 
school arithmetics, and even something from the course in Mathematical
Logic he took at the university. This would be comforting,
but  I might still wonder why has he found
it so urgent to show me these facts, and why has he chosen this particular
identity for this. Well, a day later I get the directions: evaluate t1*s1,
t2*s2 and  t3*s3; write the results in decimal notation, using
the letters O, M, Y, E, N, D, R, S as numerals for
the numbers 0,1,2,5,6,7,8,9 (respectively); replace the symbols "+"
and "=" with spaces. I  follow all these directions and end up
with the unsurprising message: "SEND MORE MONEY" (before Friedman
accuses me of pretending to be creative, I am quick to state that I have
used here a well-known puzzle). Well, does this discovery change the 
original meaning of the arithmetical sentence and the fact that 
according to its standard interpretation it is a true 
sentence of PA? Will a later swear of my son that he was guided by his
need of money when he wrote this sentence (or his actual success to get
the money using this sentence) change the primary meaning
of the sentence and reveals its "true meaning" (effective as its 
use as a code  might be)?? 

Well, I repeat that I am aware that the analogy can be attacked here
in dozens of ways. But the point it that nothing that requires a code 
of the sort just described can be taken as the primary meaning of a sentence,
and that all sentences of PA have their primary standard meaning, which is
independent of any goals, motivations, or guiding lines that a certain
person might have while  constructing them.

I come now to the last question: why do I insist on this point?
Why do I find  it so important? After all one of the things I usually
find as totally useless is to argue with people about the meaning 
of words. So the truth is that if Friedman, Franzen and others
would like to call "self-referential" a sentence of the form \phi(t),
where $t$ is a term whose value is the Godel-number of 
\phi(t) according  to some peculiar (but natural) coding, then I would
accept that it is "self-referential" according to their definition/standards
of self-reference. Had it been just a debate on the correct use of words
in our closed community I would not have devoted so many words
(and time) for this. I would have said: OK, call such sentences
"self-referential" if you want (but if you dont want to trivialize
the concept try to define "natural coding" in exact terms, because otherwise
every sentence will be "self-referential"), and I will use "truly
self-referential" or "strictly self-referential" or "essentially
self-referential" for what *I* understand as "self-referential". This
would be, I hope, the end of the debate. However: fortunately or unfortunately
(make your choice), Godel's theorems are well-known  also outside
our community (in the sense that a lot of people outside
our community  have heard of them, and many, many of them believe they
understand them). Everyone agrees that they have a lot of g.i.i.,
and so they are extensively used in arguments and debates about various
philosophical issues. Now many (perhaps most) people in the Philosophical
community have never really read a rigorous proof of Godel's theorem.
Most of them have only some superficial idea that Godel has used a variant of
the liar paradox - and they are taking it as such. As a result I have
seen more than once articles in respectable Journals of Philosophy
arguing, e.g.,  that the Godel sentence (not the  "Godel sentences",
needless to say) has no meaning at all (and so it is
not true) on the same ground that the liar sentence is meaningless...
(please look again at the message I have replied to in my first 
posting on the subject to see what dangers I have in mind!). Now
the main reason for all the misconceptions and wrong uses is that while
Friedman and  Franzen use "self-referential" in their 
extended meaning of this term, other people read and understand it in 
its straightforward meaning. This is why I think that
we, the mathematical logicans, should be very careful in what we say
and how we say it. The subject of Godel theorems is not a usual mathematical
subject in which mathematicians can use whatever terminology they like,
and nobody else would understand (or care about...)! This is precisely why
in a small popular book I have written in Hebrew about Godel's theorems
(and I will one day translate into English) I took as much pain
to explain the difference between Godel sentences and The liar as I
took in explaining the strong connection between them
(that Friedman has been so kind to remind me about).

  This has been my main motivation in this debate, but I would like to add
a second, deeper one. I believe that the duality between numbers and 
propositions is an essential ingreient of the world of Mathematics.
The similarity, even a sort of isomorphism, betwen the world
of arithmetics and the world of syntax is a fact, and an extremely important
one. Godel's proof  nd Hilbert's program are based on this strong
similarity. On the other hand the two worlds cannot be identified, and
their being two nonidentical things  is also an essential component of Godel's
proof. PA for itself is not enough for recognizing the truth of Con_PA.
One should go out of the world of PA into the world of propositions
(and back) to recognize this. It should indeed be obvious that
one just cannot do with only one of these two worlds. You need propositions
in order to talk about numbers. On the other hand 
propositions cannot be only about other propositions. Otherwise
we have only paradoxes and nothing with some true content. Moreover:
despite the similarity/isomorphism  there are
essential differences. Thus the generic concept of TRUTH applies only
to propositions. There is no such generic notion for numbers, and any
attempt to make the translation of this notion from propositions
to numbers can yield only partial "truth definitions" that are only
approximations of the generic truth concept (a concept that *transcenda*s the
isomorphism numbers/formal-propositions). So although it might
sometimes be technically convenient to "identify a sentence with its Godel
number" (as Feferman has done in some of his famous papers
and Lindstrom is doing in his book on incompleteness), it is conceptually
wrong.  I think that  it is important not
to confuse the two, and to be sensitive here to some
type distinctions: Propositions can talk about numbers. 
Numbers can code propositions (and other 
numbers). Numbers do not "talk" about  other numbers, and sentences
in the language of PA cannot speak about other sentences of PA
(in particular: they cannot be self-referential). Again: I am convinced
that it is impossible to have a full, satisfactory Godel Theory
without this duality!

Perhaps I finish with one more analogy, this time from Physics (but since
my knowledge of  Physics is not very deep, I will humbly accept any
attack on this analogy or what I say here). Everyone knows that energy
and mass are "the same thing" by Einstein Theory and Einsten's
famous formula. Yet this very formula can make sense and be useful
only if the concepts of mass and energy are not identified. I think
(again, I dont pretend to be sure) that modern physics cannot really
do with only one of them, and that there will always be things 
which can be said about mass but  not about energy, and vice versa.
Duality is a very strong weapon, and a fascinating phenomenon.