Monday, August 07, 2006

Why Skepticism? (Part 2)

Alright, I'm finally getting around to this. When reading what I have below, please keep in mind that I haven't done much work with formal proofs (I'm a scientist, not a mathematician), and this is aimed at a less math-savvy audience, so there may be a few things in it a pure mathematician wouldn't like. This part of the Why Skepticism? mini-series deals with philosophical skepticism; that is, the doubt of any form of knowledge.

I'm going to make an incredibly bold claim here. I intend to mathematically prove the Agnosticism is the only viable philosophy (if one accepts some basic tenets of Mathematics and Logic). You're of course invited to argue with it (and by argue, I mean make formulated arguments, not simple denial), but please make the effort first to understand it, and question any part you don't understand.

I was Wiki-surfing at work the other day, and I came across this page (it's long, so I'll be summarizing the key parts): Gödel's incompleteness theorems

The first theorem it contains is: "For any consistent formal theory that proves basic arithmetical truths, it is possible to construct an arithmetical statement that is true but not provable in the theory. That is, any consistent theory of a certain expressive strength is incomplete."

The statement refered to here generally takes the form "This statement cannot be proved within this paradigm."

Here's the proof sketch of this given in Wikipedia:

The main problem in fleshing out the above mentioned proof idea is the following: in order to construct a statement p that is equivalent to "p cannot be proved", p would have to somehow contain a reference to p, which could easily give rise to an infinite regress. Gödel's ingenious trick, which was later used by Alan Turing to show that the Entscheidungsproblem is unsolvable, will be described below.

To begin with, every formula or statement that can be formulated in our system gets a unique number, called its Gödel number. This is done in such a way that it is easy to mechanically convert back and forth between formulas and Gödel numbers. Because our system is strong enough to reason about numbers, it is now also possible to reason about formulas.

A formula F(x) that contains exactly one free variable x is called a statement form. As soon as x is replaced by a specific number, the statement form turns into a bona fide statement, and it is then either provable in the system, or not. Statement forms themselves are not statements and therefore cannot be proved or disproved. But every statement form F(x) has a Gödel number which we will denote by G(F). The choice of the free variable used in the form F(x) is not relevant to the assignment of the Gödel number G(F).

By carefully analyzing the axioms and rules of the system, one can then write down a statement form P(x) which embodies the idea that x is the Gödel number of a statement which can be proved in our system. Formally: P(x) can be proved if x is the Gödel number of a provable statement, and its negation ~P(x) can be proved if it isn't. (While this is good enough for this proof sketch, it is technically not completely accurate. See Gödel's paper for the problem and Rosser's paper for the resolution. The key word is "omega-consistency".)

Now comes the trick: a statement form F(x) is called self-unprovable if the form F, applied to its own Gödel number, is not provable. This concept can be defined formally, and we can construct a statement form SU(z) whose interpretation is that z is the Gödel number of a self-unprovable statement form. Formally, SU(z) is defined as: z = G(F) for some particular form F(x), and y is the Gödel number of the statement F(G(F)), and ~P(y). Now the desired statement p that was mentioned above can be defined as:

p = SU(G(SU)).
Intuitively, when asking whether p is true, we ask: "Is the property of being self-unprovable itself self-unprovable?" This is very reminiscent of the Barber paradox about the barber who shaves precisely those people who don't shave themselves: does he shave himself?

We will now assume that our axiomatic system is consistent.

If p were provable, then SU(G(SU)) would be true, and by definition of SU, z = G(SU) would be the Gödel number of a self-unprovable statement form. Hence SU would be self-unprovable, which by definition of self-unprovable means that SU(G(SU)) is not provable, but this was our p: p is not provable. This contradiction shows that p cannot be provable.

If the negation of p= SU(G(SU)) were provable, then by definition of SU this would mean that z = G(SU) is not the Gödel number of a self-unprovable form, which implies that SU is not self-unprovable. By definition of self-unprovable, we conclude that SU(G(SU)) is provable, hence p is provable. Again a contradiction. This one shows that the negation of p cannot be provable either.

So the statement p can neither be proved nor disproved within our system.

Now, that's a little interesting, but the second Incompleteness Theorem is the important one here: For any formal theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.

Let p stand for the undecidable sentence constructed above, and let's assume that the consistency of the system can be proven from within the system itself. We have seen above that if the system is consistent, then p is not provable. The proof of this implication can be formalized in the system itself, and therefore the statement "p is not provable", or "not P(p)" can be proven in the system.

But this last statement is equivalent to p itself (and this equivalence can be proven in the system), so p can be proven in the system. This contradiction shows that the system must be inconsistent.

What this means: If a paradigm claims it's consistent, it's inconsistent. A paradigm can be proven to be consistent only from the perspective of a more powerful paradigm. (Seen from the rephrasing of the second theorem to "If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.")

A little note at this point: Inconsistent basically means that it includes contradictions, such as the paradigm being able to prove both p and not-p. Note that we can use "inconsistent" as a rough synonym to "flawed" here.

Now, here's where we expand it to the human mind. Everything you know, including every law by which your conscious and subconsious minds work and every perception in your life can be treated as such a paradigm (assuming that your mind is logical enough to accept basic arithmetical truths and the definition of natural numbers, though even without that the arguments probably still apply). This leads us to the following two possibilities of how the mind sees itself:

1. Your mind cannot prove its own consistency. In this case, your mind accepts the possibility that it might be flawed, meaning there could be a mistake somewhere. Therefore, there must be doubt.

2. Your mind can prove its own consistency. In this case, as we've just proven, your mind is either inconsistent or incomplete, splitting into two more possibilities:

2a. Your mind is inconsistent. There's something you're wrong about, but you aren't admitting or noticing it. You should have doubt, but you don't.

2b. Your mind is incomplete. This means that a more complete paradigm can then judge whether or not your mind is consistent. Since we cannot go beyond our mind, we have no way of knowing whether it would judge our mind to be consistent or not, therefore it might be inconsistent, so we might be wrong about something and should therefore have doubt.

"Alright," you might be saying. "So the mind as a whole can't be proved to be consistent. But what if I isolate a portion of my mind, and use the rest to check for that small portion's consistency?"

That's perfectly fair, and you can then prove the consistency of that portion from the point of view of your mind as a whole. But there's a problem here: You can't prove the rest of your mind is consisted, because of the proof above. And if you can't prove it's consistent, it might be inconsistent. And if it might be inconsistent, you might have made a mistake in checking the consistency of that smaller part. And with this possibility comes the possibility that that smaller part actually isn't consistent. So no part of your mind can be free from doubt.

But there's a potential problem. If this proof is sound, it casts doubt on even itself. So what's the point of having it then? Well, its existence and failure to have been disproven might be said to shift the burden of proof to anyone who wishes to claim that absolute knowledge is possible. If they cannot disprove this, they leave open the possibility that their absolute knowledge may not be absolute, and with that comes the possibility that it may be wrong.

Okay, so what the hell is the point of that anyway? What does this deal with that my "dark closet" in Why Skepticism? (Part 1)? This deals primarily with claims of non-evidence-based faith, those people who claim to just know from something inside of them the truth of God's WordTM. This proof shows that it's impossible for them to know this for sure, however much they may wish otherwise.

"I was working on a flat tax proposal, and I accidentally proved there's no God." - Homer Simpson

Interverbal said...

Hi,

I just wanted to say that this was by far my favorite entry in the circle a few weeks back.

Thanks for posting it.

Infophile said...

And thanks for saying that; it's always nice to get positive feedback.

Bob said...

"But there's a potential problem. If this proof is sound, it casts doubt on even itself. So what's the point of having it then?"

This is precisely what irks me when I come across examples of self-referential incoherence. Pointing out that a theory, or proof, is self-referentially incoherent doesn't exactly render its conclusion false, though highly suspect, as with fallacious arguments. It's difficult to determine what the strength of such a charge is, and when it is warranted. I've looked into it and found plenty of examples. I just need to get around to posting about it.