Wednesday, April 26, 2006

The Boundary of Logic

It can be shown logically that logic will only go so far. You can not arrive at ultimate truth by logic alone. Logic itself shows this to be true.

I have recently discovered Godel's Incompleteness Theory.

Proposition: This statement is false.

For those who are not mathematicians or philosophical geeks like myself, you will think about this for a few minutes, shrug your shoulders, and then go about your business. But for me, as a mathematician, an engineer, and a quasi-philosopher/theologian, I want to get to the bottom of this. I want to understand. What are the full mathematical implications of Godel's Incompleteness Theory? I know that in any mathematical system, there are some postulates that are true that can not be proven.

Let G say, "The above Proposition is true."

In a computer program, would G return 0 or 1 (or would the program go into an infinite loop and evenutally crash)? In a digital circuit, if G activated an LED when it was true, and turned it off when it is false, would the LED be on or off?

It seems like, at this point, you have to leave the world of mathematics and enter the world of philosophy or theology or something meta-mathematical. The problem for me, when I leave the world of mathematics, and try to reason everything through, I am actually bringing with me everything I have learned about mathematics. Thus, I never leave that world. Thus, I am trying to force the meta-mathematical world into the mathematical world.

Some might use this to say, "See! Postmodernism is true." But that would not be an intelligent and reasonable conclusion. Godel didn't believe that. He was a Platonist. He believed that mathematics is describing some kind of reality, and not empirical reality, but something more concrete than that.

http://www.edge.org/3rd_culture/goldstein05/goldstein05_index.html

8 Comments:

Blogger elvisfromeurope said...

Dan,

Hi, interesting problems. Could that metaphysical reality possibly be ........ gasp ........ GOD ?????

3:57 AM  
Blogger elvisfromeurope said...

In a less sarcastically tone,that papers says:

" Wittgenstein seemed to be saying the latter; that there are aspects of reality that exceed our ability to express them."

Of course, reality will always change and there are thousands of things that exceed our PRESENT DAY ability to express, know, understand etc.

In the year 1800 no one could express quantum physics or know computers or many other things. In the year 3000 there will be many things that exist but we today can't express.

5:04 AM  
Blogger elvisfromeurope said...

try out this PERL program:


$statement=0;

while($i<1000){

if ($statement==0) {$statement=1}
print $statement;
if ($statement==1) {$statement=0}
print $statement;
}


you actually get a clock cycle....

7:44 AM  
Blogger Dan said...

That PERL program is not describing the problem. The program is simply causing the statement to continuously change states. It is not really reflecting the logical problem.
In your program, the state of the statement depends on the point of time it is in in your while loop. But in the problem I posed, time is a non-issue.

Do you follow?

9:53 AM  
Blogger Dan said...

Metaphysics is certainly the study of what we know to be real and how we know it to be real.

Epistemology is the study of what we know to be true, and how we know.

Ethics is the study of what we know to be good and just, and how we know.

All of these fields of study are related to each other. What we believe about what is real, what is true, and what is good determines our worldview. The final authority on which an individual depends and in whom we trust to determine reality, truth, and goodness is that individual's god.

So, what is your final authority? The God of the Bible? Yourself? "Common Sense?" That which the state dictates? The majority of people in your country? The majority of people in the world? A pet rock?

Perhaps you are unsure about the final authority of truth, goodness, and reality, and thus you are a worried and anxious individual. Or perhaps you are unsure and apathetic. If that is the case, then without knowing it, you are probably distracted by your temporary comfort, and you are not realizing that you are depending on something (money, democracy, the "goodness" of man).

If you are not exalting the God of the Bible as the ultimate source of truth, reality, and goodness, then you are bowing down to another idol. If you don't trust in God, then you are trusting in something else - and living out an illusion.

The illusion will not last forever. God will reveal himself. I hope and pray that when He does, then all the readers will receive Him.

Today, if you hear His voice, do not harden your hearts. Today is the day of salvation.

10:15 AM  
Blogger MarcoConley said...

Dan,

So happy you stumbled upon to the Godel issue. The paradox you cite is known as Russell's paradox, and it play's a role in the incompleteness theorem.

But then incompleteness theorem is even more amazing than just logical paradoxes. The incredible thing, the unbelievable thing about the incompleteness theorem is this:

Any information system will have facts, which are true, which it can never ever prove as true. Other things are false, but it can never see them as false. And some things simply are neither true nor false, but both, or neither.

and this is true for ANY information system. for a perfect computer, for the smartest person's brain, or for the mind of god.

There is no perfect thinker, there is no absolute truth. So close that we, as human, need never worry about it, but still.. it's a fascinating thought.

I'm sure I'm not conveying the wonder of it all. I wish I could just upload thoughts to people's brains.

3:46 AM  
Blogger elvisfromeurope said...

Not for the mind of god. That is way past anything we can possibly think.

We can simply invent any kind of logic, any kind of reasoning anything and simply force it to make it work. So true, false and anything else is always relative within the system containing it.

Imaginary numbers make no sesne, how can you have a number that multiplied itself gives you a positive ? We just invent the formalism and operate on it. So we can invent another symbol and say it is the symbol for contradiction, where it is both true and false, and go on from there.

5:16 AM  
Blogger Dan said...

Joe,

I take issue with you (big surprise). I would suggest that we invent the symbols. But the concepts behind the symbols were discovered, not invented. Mathematics is representing and describing something real in the metaphysical world.

Marco,

There are absolutes. The mind of God is beyond tracing out.

7:13 AM  

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