Tuesday, December 04, 2007

"Mind & Spirit" -> Misuse, Abuse, and Interesting Godel

Einstein said that the reason for going to Princeton's "Institute for Advanced Studies" was so he could take walks with Kurt Godel.

It is reported that the rest of the center's staff always wondered why the gregarious Einstein seemed to love the time with the quiet Godel. Perhaps, it was simply because he had found a mind as brilliant as his own.

When you start to think about metaphysics and math, you might suspect that Einstein, with all of his work that disclosed that time could slow down, or that light is both a wave and a particle would often get quoted to show that all things are relative, and there is no objective truth. However, as Alan Sokel discloses in his book, Godel is really the most abused. As a matter of fact, Alan devotes a whole chapter to how the liberal art and philosophy majors abuses of Godel's work.

So, who was Godel, and why is he so abused?

Somehow, Godel often gets cited as proving that "math is bad" and humans can't know anything because all math has contradictions.

This is maddening thought, and simply misreading the implications of Godel's work. (Which is repeated in Church's lamda calculus or Turing Halting problem.) What Godel pointed out is that our math is always going to be incomplete!

It might be worth just a bit of history on math here as it will help us understand how we got to this point.

It has alway been thought that if we had a "really good" set of axioms, that we could construct an air tight structure that would basically allow us to derive absolutely every theorem that was out there. In the early 1900s, the great mathematician David Hilbert declared that this was an important task that must be done. His call to action was called Hilbert's program.

Whitehead and Russel decided that they could layout a system that could answer Hilbert's call. This airtight system was the Principia Mathematica. However, every time they tried to come up with a system that could describe everything, they ran into paradoxes. A more mathematical way of saying paradox is "inconsistent." So, they couldn't find a consistent axiom system that you could derive everything.

So, what is an inconstant (paradox) system?

I think most people know about Russel's paradox, and I have mentioned this before.

Basically, it just means that you can set up a logic structure "sort of like" the following, and get a paradox (or inconstancy in math talk):

a. The next line is false
b. The above line is true

Obviously, both a and b cannot be true.

So how do you get out of a paradox? Well in our case above, we can simply say "don't allow a second line to describe the first line."

We solved the paradox! How? We outlawed it. So, to make sure that we can't accidentally create paradoxes, we come up with an axiomatic system to prevent them. Today, we use Zermelo–Fraenkel as our axioms (abbreviated ZFC), and it does not allow the first line to refer to the second line nor vice versa. Therefore, we have no paradoxes.

However, what was the consequence of outlawing the second line?

Well, this is the rub. If we want a system to basically describe everything, it turns out that we needed the device that we outlawed!

So, we have two choices:

1. Make up axioms that will have paradoxes that can be derived, but describe everything.

2. Make up axioms that can't describe everything, but don't have paradoxes.

Now, you might say, "isn't there a third option of a complete system with no paradoxes?" Dr. Godel proved the third option does not exist!

To give you the short version, mathematicians (and more importantly engineers) simply use #2. Crisis adverted.

So, while there are paradoxes in math, we neatly solve any "real world" issue by simply living with systems that are incomplete.

ZFC is the most common basis for derived works, and (cross our fingers) nobody has been able to create a inconstant result by ZFC. So, people use ZFC, and everybody can still do their PhDs, using this axiom set.

The rub is that there are things which may be intuitively obvious and may not be solvable in ZFC. (Although ZFC has shown itself robust, as we have solved Fermat's Last Theorem, Bieberbach Conjecture, etc. However, the point is that there will be things that are true that cannot be be proven (without having axioms that could be used to derived paradoxes.))

The more interesting thing to say is that math cannot derive "everything."

If you do some reading on the history of math, this is major flip-flop. Before Godel, paradox was absolutely not accepted in mathematics. The prevailing thoughts was "we will find a complete system that is consistent, and we will not stop until we do." After Godel, the writing was on the wall that to do any "useful" work, math needed to accept that all systems were incomplete (or face the problem of dealing with paradoxes).

After Godel, it was accepted that there could always be limits to our ability to derive everything from a given set of axioms. Steven Hawkings mused on this in a lecture stating that as Godel proven we cannot derived everything, perhaps we were at the limits our of ability to unify everything.

Now, how do we interpret this? I see three ways we could use it:

1. It says nothing. Math is a man made system that has no larger purpose than to be a tool to solved limited scope math problems. Math can't derive everything? Good, we never wanted it to.

2. It cracks the foundations of mathematics. Man can't know anything for sure, because if we can derive everything, we have to allow for inconsistencies.

3. Math is a tool that is 100% reliable in its limited scope. However, there are things which may be true that you can't prove from a limited set of axioms. This may have implications that not everything in the universe can be derived, and there may be theorems that are perfectly true that we can never prove.

The last way is the best way of using Godel for apologetics. In my younger days, I used to refer to this as God's dirty footprints across our smug self satisfaction. It proves absolutely nothing, but it does open the door.

I do think that it is somewhat interesting that the main problem with paradoxes revolves around self reference, and Christianity/Judaism is the only religion where the monotheistic God refers to himself as

"I am that I am" (or better translated "I be that I be")

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