## Friday, April 25, 2008

### A Godellian Approach to Infalliblilty

There is no need to continue reading this post, if you do not agree entirely with the following and have no doubt as to the common sense and meaning of each of the words that are used in these AXIOMS:

(AX1) A STATEMENT is any grammatically complete English sentence.

(AX2) Every STATEMENT falls into one and exactly one of three SETS:

(T) The set of statements that we have decided and agree are TRUE.
(F) The set of statements that we have decided and agree are FALSE.
(U) The set of statements that we have NOT decided and agreed belong to either Set (T) or Set (F).

(AX3) The operation of this exercise is first, to propose statements for consideration; and second, to decide and agree which of the three sets it belongs to.

DISCUSSION:

(D1) At the outset the set (T) contains exactly three statements that we agreed and decided (if you got this far) are in fact TRUE, namely Axioms (AX1), (AX2) and (AX3) which are all valid Statements, by inspection, and are decidedly and agreeably TRUE by our common sense and agreement of the meaning of "axiom".

(D2) For any new statement that is proposed, the first logical step to make is to decide and to agree whether or not it belongs to the set (U), that is, whether there is any logical hope or possibility that we can decide and agree about whether it definitely belongs to EITHER set (T) or Set (F).

(D3) I shall now demonstrate that the Set (U) is definitely not empty, that is the Set (U) is not the NULL set. Consider any paradoxical or logically inconsistent statement like the Liars Paradox:

(LP) "This very statement is false."

Assume first that (LP) belongs to the set (T), that is, that LP is TRUE. If (LP) is true then it must be FALSE, that is, it belongs to Set (F). But if (LP) is false, then (LP) must be true and so also belongs to set (T). And so on. Thus in order not to violate AX2, we are forced to decide and agree that (LP) belongs to one and only one set, the set (U) because we cannot decide that it belongs to either Set (T) or Set (F). Quod erat demonstrandum (Q.E.D.) -- The set (U) is not empty.

I believe, though we have not decided or agreed to this, that every paradoxical or illogical statement of this sort (LP) belongs to set (U), that is, we cannot logically decide which of the two disjoint sets, (T) or (F) it belongs to.

(D4) Consider another statement which is not a paradox but is nonetheless very strange:

(NAP): "This very statement is true."

Does (NAP) belong to Set (T), (F), or (U)?

I can't say at the moment, so let's hear your opinion in the Comment Thread.

(D5) I wish to propose for discussion the following Definition:

(D5D1) An INFALLIBLE statement is one that we decide and agree CANNOT belong to either Set (F) or Set (U), even if we cannot decide and agree that it belongs to set (T).

(D7) Is the set of all statements that don't belong to (T) (F) or (U) empty?

(D8) Is NAP an infallible statement? Am I the Pope?

(D9) Are there any identifiable DISJOINT sets of statements within the Set (U)?

(D10)
I know, I know...this is the weirdest Philippine Commentary posting ever! But I want to test the hypothese that seemed so clear in my head before but looks really strange to me now that in a quintessentially Goedelesque way:

INFALLIBIITY IS DECIDEDLY AND AGREEABLY TRUE BUT ALSO HERETICAL!

The basic idea is that for Papal Infallibility to be "logical" it has to stand outside the Deposit of Faith that is Divine Revelation (Sacred Scripture and Sacred Tradition). It is no longer Religion but Mathematics.

The Nashman said...

Let me guess, you are coming out of retirement to teach at Uni again....

DJB Rizalist said...

Teach others? NO! I only worry about one student. Me.

Richard said...

DJB,

You didn't even need to go to all that trouble. Even the Bible (and history, see Pope Alexander VI) concedes that all men are fallible. I've never understood the Catholic Church's claim that there is one man, who, coincidentally, happens to be their leader, who is not.

Richard

Jego said...

The Pope pronouncing ex cathedra has the power to instruct or order Catholics what to believe in order to remain Catholic. In other words, that authority can determine what is in the deposit of faith. It can therefore never be heretical unless one pronounces it so from outside the Catholic faith. You can't do it from inside.

blackshama said...

Old Catholics, Orthodox Christians and Anglicans believe that Papal Infallibility is beyond the deposit of faith.

The Orthodox consider Pio Nono's declaration of Papal Infallibility as a heresy. The Orthodox appreciate the logic of it though.

Jego said...

Yap. For the Orthodox, it is not the Pope that is infallible, but the Church. That's one of their problems with the Catholics. The Orthodox's infallible pronouncements come from Councils. They only have had seven officially recognized infallible ones if Im not mistaken, the last one was in the 8th century.

Dominique said...

http://www.rae.org/godel.html

As a third implication of Gödel's theorem , faith is shown to be (ultimately) the only possible response to reality. Michael Guillen has spelled out this implication: "the only possible way of avowing an unprovable truth, mathematical or otherwise, is to accept it as an article of faith."(8) In other words, scientists are as subject to belief as non-scientists. And scientific faith can let a man down as hard as any other. Guillen writes: "In 1959 a disillusioned Russell lamented: ÔI wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than anywhere...But after some twenty years of arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.'"(9)

DJB Rizalist said...

Dom,
There is a gulf of a difference between "scientific faith" as you call it and "Catholic faith".

That gulf is that of humility.

scientists accept, nay crave the possiblity of their being proven wrong, because everytime someone fails to prove them wrong their "faith" grows stronger that they are "right" and that their statement or hypothesis is true. And even when they are proven wrong, they are lead to some better and stronger truth, thus their "FAITH" grows stronger either way.

But Catholic faith, which proclaims itself infallible is full of hubris and does not admit the possibility of being wrong in the issues and matters addressed.

Thus IF they are ever proven wrong, it would be a HUGE and DEVASTATING disaster for their "faith".

Betrand Russell lost faith only in his own version of mathematic reality. Godel showed the way to a math that is stronger, better, more illuminating than what Principia Mathematica set out to do.

I also believe that if the Church were to adopt this scientific attitude, then it too would be stronger and less susceptible to utter discredit and desert.

I believe the ethical and moral teachings of the church do not need a foundation incredible and amazing claims. Maybe in the 1st century yes, but now we know that reality is far more amazing than claims of miracles like transubstantiation, resurrection and assumption.

Belief in these things are utterly unecessary, irrelevant and indeed deleterious to the practice of love and charity and goodness to our fellow man.

If I attack those "foundations" it is purely with the intention of substituting something stronger and less vulnerable to the attacks of those who hate the Church and wishes it be destroyed.

I believe her greatest enemies are within, not without!

DJB Rizalist said...

ben (blackshama)

BTW, thanks for the info on Orthodox Catholics. I am interested in the whole idea of infallibility being a form of heresy because it has a deep connection with Kurt Godel's theorems.

Recall that his first theorem proved that there are TRUE but UNPROVABLE theorems in every nontrivial formal system complex enough to include ordinary arithmetic.

If indeed we take the formal system in this case to be "the Deposit of Faith" then it is quite possible to show that the dogma of infallibility is infallibly true (provably true) but NOT within the axioms or deposit of faith, and thus HERETICAL.

It is a pure intellectual exercise that excites me because I never knew before very recently that there was this amazing connection.

Equalizer said...

Dean Bocobo,The Rizalist

Since Christ said the gates of hell would not prevail against his Church (Matt. 16:18b), this means that his Church can never pass out of existence. But if the Church ever apostasized by teaching heresy, then it would cease to exist; because it would cease to be Jesus’ Church.

PAX

The Equalizer

DJB Rizalist said...

Equalizer,
Even more amazing is that the POPE, in "further clarifying" the doctrine of Papal infallibility during Vatican II (after it had first been itself proclaimed infallibly by the BISHOPS in council during Vatican I) saw fit to distinguish both infallibiities from IMPECCABILITY

This was most necessary since it is historical FACT that even the RCC could not in 1968 deny, that there have been Popes (like the Borgias) who were clearly SINNERS of the most scandalous kind.

What this means is that the Church has already admitted that INFALLIBILITY of the Pope in his teaching office does not imply IMPECCABILITY of the Pope when it comes to his "personal acts".

A most interesting distinction because it means that even IF some solemn definition by the Pope is deemed infallible, the RCC has left open, precisely the possibility that it is HERETICAL or APOSTATIC!

Godel's theorems suggest to me that such PECCABILITY (sinfulness) in the case of infallibile proclamations can only be in the nature of heresy or apostasy because it would constitute a new AXIOM that MATHEMATICALLY cannot be in the Deposit of Faith to begin with!

So choose!
Logic or Peace??

Anonymous said...

"Colorless green dreams sleep furiously". Noam Chomsky

based on your axiom of what a statement is, will this qualify as a gramatically complete english sentence? noam says it does.

DJB Rizalist said...

anonymous,
According to Steven Pinker that is the only quotation of Noam Chomsky in Bartlett's compilation, if memory serves.

Taking English as a formal system it obviously qualifies as a complete English sentence, having subject and predicate respectively embellished with adjective and adverb.

Thanks for bringing it up. Am prepping a post on matters he addresses in Language as Instinct. A fascinating tome I 've also been reading lately.

Jego said...

Chomsky's theory has been dealt a blow by actual empirical evidence. That's science. It's not science if it's just stories made to sound scientific like the Multiple Universes ek-ek being peddled nowadays to account for the improbability of our universe to exist. And Im still not sure about that dark matter ek-ek, not to mention string theory. Right now theyre still stories with a lot of math in them IMO.

The abiding faith scientists have is in the orderliness of the universe. This is something they take as an unproveable given, their central infallible dogma; that the laws of nature exist and the entire universe must obey. The speed of light is c all over the universe, even those parts of it we can never have access to because of c.

cvj said...

Doesn't Papal Infallibility mean that the Pope is able to generate additional Axioms (AX4, AX5...etc)?

DJB Rizalist said...

cvj,
Yes! you can look at it like that. But that is why it is also heresy because the way the RCC and Vatican councils put it, such infallible definitions and proclamations are merely discoveries of truths already found in "the Deposit of Faith" namely Scripture and Tradition.

Anonymous said...

jego,

i've read the link. thing that surprises me is that the researcher thinks this tribe has no sense of numeracy. i wonder though if this hunting tribe has any sense of fairness when allocating their catch. the researcher was silent on this. which means that words do not a concept make.

DJB Rizalist said...

jego,
in his book, Language as Instinct, Pinker in fact cites many such examples of "primitive tribes" supposedly having no concepts of numbers and time, including several native American and African peoples, all in an effort to prove that they are so different from modern man, with his busy arithmetics and schedules, that they live in a paradisiacal state of bliss and oblivion to such concerns. But in every case, digging deeper showed those initial impressions to be wrong. Even in the case of the piraha, there is the suggestion that they at least know the difference between big and small, between few and many, and undoubtedly between young and old.

The central thesis of pinker and I guess chomsky (though I've not read him very much) is that language IS a universal instinct for describing common experiences, but that such descriptions can take on a surprisingly large and varied number of forms.

And of course one really only needs the most rudimentary concepts of NONE and ONE to build up a whole digital mathematics.

The reference you cite is a long way from "blowing up" this concept of language as instinct.

unless of course you are suggesting that those people are not human. :)

Anonymous said...

which brings me back to my point, djb: concepts--if you go by the universal grammar theory of chomsky--can be learned in any language. the most efficient form however will always be the mother tongue. if english is the universal language in science, as you persistently hammer (so be it), allow then filipinos to adopt lexicons in english, but explain the concepts in L1. that to me is effective bilingualism.

DJB Rizalist said...

anonymous,
I concede that concepts can be learned in any language. That's fine as a linguistic theorem. But we are really talking here about the practical problem of 22 million kids in the pubic schools and their need to learn algebra, trigonometry, physics, chemistry, biology, Shakespeare and Jose Rizal.

It's simply impractical and wrong headed to insist that just because concepts can be learned in Hiligaynon or Tausug that we ought to invest the money in training algebraists and authors of trig books in those tongues. On top of which we have to invent a whole new WRITTEN form to accomodate such "can dos".

A MEDIUM OF INSTRUCTION must be a written language, so this mother tongue hypothesis thingy is sticking our head in the sand about what really works and what our kids really need. it's all the cunning linguists mental mass disturbation.

public school ain't adult swim hour you know

Anonymous said...

that is because djb you insist in a monolingual instruction. and this where we differ. i buy bilingualism, hook, line and sinker. teach mathematics and science in the vernacular using english books. who says it can't be done--if practicality is all that concerns you.

you're practical argument borders on the economics alone and not pedagogical. educational research abounds suggesting learners do not react well to stimulus that does not appeal much to them, language included.

Jego said...

The reference you cite is a long way from "blowing up" this concept of language as instinct.

'Language as instinct' isnt Chomsky's thesis if Im not mistaken. 'Language acquisition is instinctive' is his thesis. This includes the acquisition of the mathematical part of language. The Piraha certainly didnt acquire and couldnt acquire a mathematical language. Language acquisition therefore isnt instinctive in the Piraha.

DJB Rizalist said...

Jego,
But the Piraha have a language. since the researchers only claimed they could not teach them arithmetic, at least as we practice it, and since arithmetic is not math, we cannot conclude they have not mathematical or even quantitative instincts. Indeed they report that they at least distinguished the "few" from the "many".

There is no fundamental exceptionalism that can imaginably be upheld for the Piraha, which I think is the main issue.

There isn't anything particularly unique in other words about ANY language at least among homo sapiens. It would seem the need and urge to communicate is a sense, an instinct, autonomic, like seeing or hearing without conscious effort.

Surely, you do not doubt that do you?

Jego said...

Sure. But we're talking about Chomsky's theory, not language.

Chomsky said the universal grammar is, well, universal. The Piraha shows it isnt universal.

"Everett, once a devotee of Chomskyan linguistics, insists not only that Pirahã is a “severe counterexample” to the theory of universal grammar but also that it is not an isolated case. “I think one of the reasons that we haven’t found other groups like this,” Everett said, “is because we’ve been told, basically, that it’s not possible.”"

from the New Yorker.

Who knows how we acquire language? The jury's still out. Pinker believes it evolved. Chomsky believed it just emerged. Both are just stories since we couldnt design a controlled experiment using real people because of ethical reasons. We have to make do with field research. Everett at least has actual observations to back him up. Chomsky just has a story. ;-)

DJB Rizalist said...

Jego,
I am perfectly willing to concede that Chomsky was wrong about the universality of some grammar that he defined. He's wrong on a lot of things anyway.

But my question really is do we believe something about language IS universal, so that all this "mother tongue" exceptionalism can be seen for what it is: anti-colonialism.

You see I think Filipino nationalists will never accept that English IS an integral part of their cultural heritage, in fact the biggest and most significant part. whatever injustices accompanied its introduction the reality of it in our lives basically forever just cannot be reversed any more.

In other words, we are just as well off with it as without it, and far worse of if we try to get rid of it now.

cvj said...

What's the chance of getting Benign0 to go to the Amazon and pester the Piraha about having no concept of time and number? In that respect, the Piraha are better off not understanding English ;-)

Anonymous said...

Jego: It can therefore never be heretical unless one pronounces it so from outside the Catholic faith. You can't do it from inside.

The Catholic Church is the body of Christ, but Christ is the head. The head is outside of the body. The Divine head pronounced that the Holy Spirit would lead the apostles to all truth. The infallibility pronouncement merely clarified what Christ already defined.