Believing the truth

by Olivier Coudert

July 19, 1996



One day, between a glass of champagne and a straight scotch, Gilles and I had a discussion about truth and knowledge. I can't remember how it started; too much alcohol and cigarettes, but likely it came from some sarcastic comments from Gilles (I'm wondering why I added 'sarcastic' before 'comments from Gilles'; I guess I like redundancies:)) about some crazy/stupid/asshole/brainless people who believe they'll change the world by forcing other people's minds to their own truth, or who believe they're right, and (redundantly) that they have the truth. Ahum.... If you recognize yourself in this description, please get out of this page quickly if you don't want to have your beliefs ruined, and check out http://moron.believer.com/~kill/them/all to see whether you're on the list. If you're not sure, stay, be curious, and you're free to discuss the arguments as much as you want....

Well, as I said, after 'moulte' (look it up in a French-English dictionary; I'm not in the mood today) glasses, we started chatting: What is truth? What I am going to explain (well, say I'm gonna try) is that truth is tighly linked to the language, and that the language is self-limited. Consequently, absolute truth does not exist after a certain level of "abstraction" and "complexity" (this level is indeed fairly simple). Today I only use intuitive notions to outline this conclusion. Next time I'll play with more powerful tools to go a little bit further.

Okay, let's try to show that the concept of truth is MUCH more complicated than most of the people may think. Of course, there are several ways to do it, more or less "effective", but I'm going to do it in an easy way for this time.

As far as I know, expressing anything needs a language. You can have a very poor language that expresses very simple notions, like the one you use with a microwave, which is nonetheless sufficient to correlate time and heat in an oven. The key point that I want to address can be stated as follows:
  • As soon as a "language" is "powerful" enough, there are notions expressable in this language that can neither be "proved" to be "true" or "false". Going further, it exists, sentence X, such that you can BELIEVE that X is true, and I can BELIEVE that X is false, but NONE of us can PROVE he is right, and BOTH of us are still CONSISTENT (i.e., non-contradictory, i.e., no basic flaw in the deduction).
Some of the words above have been quoted, because they need a proper definition. I'll only give intuitive ideas.

"Language" stands for any language that you may have heard of, e.g., French, English, Latin, Sanscrit, mathematics, or the language you use to program your VCR. The purpose of a language, which is nothing but a very formal set of sequences of symbols (characters), is to capture the "outside world". In other words, one maps the "words" of the language on "objects" and "concepts" (a chair, a bird, being hungry, love, peace, etc...). This mapping is called an interpretation, and we will say that the outside world is a "model" of the language.

Having a language and an interpretation that makes what you are interested in talking about (the outside world) a model of the language, you can start playing with the language and "deduce" facts from assumptions. This is called logic. Here starts the story of the mathematicians' dream of having a language from which one could derive all the theorems (i.e., the "true" facts). Pay attention here: We will call a "theorem" a sentence that can be derived (deduced) in your language. We will call a "truth" something that exists in your model, here the outside world. A first question:
  • Is a theorem a true thing?
Well, if it is not, then one says that your language is inconsistent, that means that it doesn't fit what you would like to modelize, since you can deduce in your language the truth of false statements. It is obviously worthless to have such a language... So we assume that we only have consistent languages, i.e., any theorem (i.e., a sentence that can be deduced as true in your language) is also true in the model (e.g., the outside world).

The key question is:
  • If my language is consistent, is a true thing also a theorem?
In other words:
  • Can something true always be deduced?
The answer is NO for "powerful" enough languages. Here, powerful enough means for example that you are able to manipulate the following concepts: integers, addition, multiplication, predicate. That's enough!!! Beyond this point (and I guess that for anybody who masters these concepts,) "truth" is not always a question of deduction and thoughts. It is also a question of BELIEF.

Two consequences:
  1. It is IMPOSSIBLE to know all the truth. Not because you are short in time, but because of your own language. There are out there some truths that we will never know (at least if we use recursive languages). But I just do not see how a non-recursive language can fit into a human brain --as said Russell, a brilliant mathematician with a good sense of humour, "if God exists, he's not recursive"). One says that our language is INCOMPLETE, i.e., it cannot modelize the complete truth of the model; e.g., our world. In short, one can assert that:

    • We know that we don't know.
    • What we know that we don't know, we can believe or not.

    Beyond this point, confusion between truth and belief is easy to do....

  2. Going a step further, one can exhibit (I won't do it here, I need more formal tools than the English language) sentences X that are not only non-deductible, but that you can decide ARBITRARILY are true of false, without adding inconsistency. Thus, as I said at the beginning, you can decide that X is true and I can decide that X is false. But you won't have any way to prove your point. You just has a BELIEF --as I have--, and it must not be confused with a TRUTH.
Thanks for reading...

...but do not take my statement as it! Discuss/refute/fight it, and I'll answer.



       Olivier Coudert speaks French and English but, above all, UNIX. A mathematical artist, he concocts algorithms like Gerswin composed music, with so much passion that Silicon Valley lured him away from France to do as he so pleases. Most of the time he writes award-winning mathematical papers for esoteric conferences. And, when he is back down to earth he occasionally contributes to Swans.


Please, DO NOT steal, scavenge or repost this work without the expressed written authorization of Swans, which will seek permission from the author. This material is copyrighted © 1996, Olivier Coudert. All rights reserved.


Published July 19, 1996
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