Gregory Chaitin Lecture Mälardalen University 2005 Pt 4

[gadionson1]

nice video..nice

[czubinm]

I do have to note to you that we people are inconsistent. We can for example believe in two things that contradict each other. However these things do obviously conflict with physical laws. We do not define what reality should be. Ex: Someone might not believe in gravity, but gravity won't care.

And still! you can model this is mathematics without contradictions. As an example: A quantum turing machine(i.e. a computer) can simulate the physical laws and every human being with all their flaws.

[czubinm]

Secondly, math is very rigor but we still allow to make conjectures (unproven mathematical statements) until proven consistent or false. However ideas should have be refined until they are rigor. But Chaitin is making an argument for his digital philosophy. But this is much outside of mathematics and not widely accepted. We still work on continuos mathematics, arithmetic hierarchy, turing degrees etc.

[czubinm]

To be very clear about this: if your formal theory has a paradox then it's inconsistent. You cannot make it go away by adding it as an axiom. Worse, anything you have shown to be true in your theory is automatically incorrect. For example if I allow a paradox in elementary arithmetic then I can prove 1 + 1 = 3 and 1 + 1 = 4 and 1 + 1 != 3 etc. The theory has become completely useless.

If there's a paradox in your theory then in it's inconsistent and should be modified to be consistent.

[agentredlum]

I would like to state 3 examples relative to our discussion. Plato imposed his restrictions because he wanted to find flaw in the Pythagoreans, Liebniz imposed his restrictions because he wanted to uncover flaws in his peers philosophy and Chaitin imposes his restrictions because he wants to uncover flaws in Arithmetic! "hey! I'm just an innocent victim here, I was attacked by a coked up whore and a crazy dentist!" (12 monkeys)

[agentredlum]

Should we disregard Euler's brilliant results because he wasn't as rigorous as K.F. Gauss? If only Gauss was more like Euler, our mathematics would be more advanced by 300 years. These are some of the reasons why, in my opinion, restrictions do more harm than good. Essentially restrictions are for pessimists looking to find flaws. When truth and understanding are concerned I prefer to remain an optimist.

[agentredlum]

Just kidding...we need some rigor but not too much. Look at someone like L. Euler. Every time I read some 'experts' explanation of some of his beautiful and brilliant arguments 'they' are quick to point out the 'shaky' part of the derivation. Euler posed no restrictions on himself, in a way he did say to hell with rigor, and look at all he accomplished (product formula for primes and 1/s^2=pi^2/6) many results too numerous to list here. e^(i8)=Cos(8)+isin(8

[agentredlum]

Here is where you and I differ. I say this with the utmost respect for you because I consider your goal a most noble one and your arguments convincing. You claim that imposing restrictions is a good idea (Chaitin and 99.9% of mathematicians and scientists would agree with you.) I claim that imposing restrictions is NOT a good idea because I think that they lead to paradoxes and wasted effort. TO HELL WITH RIGOR!

[agentredlum]

I think Russell got his paradox from the barber of seville paradox. If you insist that the barber be a male citizen of seville then it is impossible for him to shave ALL the men of seville who do not shave themselves. The problem arises when he considers himself subject to the rule. If he shaves himself then he must not shave himself, if he does not shave himself then he must shave himself. Notice he does very well with everyone except himself. No one bothers to point THAT out.

[agentredlum]

Here's what I'm afraid of, 'throwing the baby out with the bathwater' One should not throw away a theory because of a few paradoxes. One should do their best to make the theory as consistent as possible and then make a list of all paradoxes arising from the theory. If someone finds a new paradox, add it to the list. If someone comes up with a theory that explains exactly the same but with less paradoxes then that is a better theory.

[agentredlum]

The set of all elements that are not members of any set. Suppose # is not a member of any set, do you put it in the set? If you do it has now become a member of a set so it does not belong there. If you don't then it is a member of no set so you must put it in there. You lose either way but that doesn't show that mathematics is inconsistent, it shows your set is self contradictory.

[agentredlum]

What kind of universe would this be if it did not allow for paradoxes? Mathematicians miss the point, they prove that a mathematical system that can give rise to arithmetic is not paradox free so they are saddened  because they feel that their quest for 'perfection' is impossible. They don't understand what perfection is...perfection must allow for paradox and contradiction

[agentredlum]

I do not disagree with you but I think that physical theories must necessarily be more complex than what they describe. E=MC^2 can mean many things unless you include an explanation about what it is you are talking about. The harder the theory the longer the explanation. The wonderful idea in Russell's paradox is that a perfect system MUST allow for paradoxes within itself. (Everyone misses this point)

[czubinm]

But yes, just allowing for any random kind of axioms and studying these systems is a valid way of moving forward. Stephen Wolfram did explore this in NKS and found these systems to have undecidability much faster and easier. It might be we select our math to be the most expressive untill we encounter a huge problem and this causes a paradigm shift(or revolution) in our thinking.

[czubinm]

Second I do think limiting ourselfs to a particular set of restrictions(in math) can help us move forward untill we understand enough to break down the walls. There's a trap if we simply try to allow for everything in a formal system. Namely our formal systems could easily be inconsistent and all our hard work to be invalid(see Russell's paradox).

[czubinm]

As an example, general relativity can be written on my hand however understanding it requires me to grok it, play with it, read lots of books etc.

[czubinm]

I think we have to make the distinction between human understanding and what should constitute a theory. Surely you don't want a physical theory to more difficult than the things it wants to describe?

However if I want to explain a theory to someone then I have to relate it to them in human concepts or new concepts(you have to grok it). Which usually contains far more information than the theory we're studying.

[agentredlum]

my point is that we should be suspicious about theories that impose restrictions. You and I might be living on other planets by now if plato had not strangled mathematical creativity. It took mathematicians over 2000 years to prove the circle cannot be squared using plato's restrictions. In the meantime so much effort was wasted on the calculation of an impossible task. Horrible!

[agentredlum]

It really bothers me when mathematicians say that a lot of fruitful mathematics has resulted from trying to square the circle by straightedge and compass alone. If you look at history Plato KILLED greek mathematics by imposing his restrictions! Most mathematicians were too busy trying to square the circle (by a dull and stupid process) to do anything else. Creativity must NOT be restricted.

[agentredlum]

He has discovered these irreduceable objects because he has imposed upon himself the limitations of Liebniz. He has done to himself what Plato did to the Pythagoreans when he suggested that they should be limited to only straightedge and compass. It's like being told to mop the floor with a toothbrush.

[agentredlum]

I guess the biggest problem I have with his theory is his statement 'to understand is to compress' Chaitin is in search of efficiency, a noble quest to be sure, but isn't he (and many others) who is engaged in wishful thinking when he claims god (nature) has simple laws?

[agentredlum]

TOE (sorry) If 'understanding' is what you want then simplicity is not the way to go. 'Beauty' is subjective and unnecessary, truth is most important to understanding. I try to divorce myself from wishful thinking concerning science, philosophy, or mathematics. Example: I say 'rainbow' and chaitin gives a nice explanation in 2 paragraphs then you discover that some physicist has written a whole book on rainbows, which one gives a better understanding?

[czubinm]

If you still want to make your theory more complex you'll have to add extra information. If this extra information can be tested and verified then it predicts more events and remains simpler. If it cannot be verified than it's more metaphysical. Moreover it's up to you and your philosophy if you want to believe in something that can never affect us. (causally disconnected)

[czubinm]

However our theories are used to explain _all_ particular events versus only a single event. So they are more complex than a single event but they explain readily every event in more simpler terms and predict new events.

[czubinm]

Let's suppose your theory is more complex than the actual event. Without loss of generality, let's assume your theory contains every bit of information that constitutes that event. Now every vital bit of information has been included (as you wanted) yet it's still no _more_ complex, it is equal. To make it more complex you'll have to add extra information.

[agentredlum]

"an explanation has to be simpler than what it explains" Why?Cause Leibniz said so?In the 17th century Leibniz had no idea of what we know now about complexity.Physicists have fallen into this same trap looking for a simple TOA though nature thwarts that effort at every turn they still cling to the idea of simplicity with voracious tenacity.They are going to get their wish but their simple TOA will be simply wrong

[czubinm]

@agentredlum I hope you mean the TOE(instead of TOA).

I'm going to make this post in several comments(length).

First of, why would you want to make it more complex? Makes no sense to me except for some wishful reason to have more beauty or something. I'll give an argument for simplicity though it may conflict with Leibniz since I haven't read his works.

[agentredlum]

By the way, I am not interested in making B more compact...I want to make B more COMPLEX. Not by inserting unnecessary components, misinformation or vccuous data, but by giving more vital information because I know that if I omit vital information my explanation is more compact but it is worse not better. What a crock! You cannot prove anything by adding it as an axiom! If you try it then you will find many who are eager to point out your circular reasoning.

[agentredlum]

Listen, you tube allows only a compact set of characters so it is very hard to put my point accross. You misunderstood my point 5 months ago, it was meant as a reductio ad absurdum, that is to say that his notions (compact is good, God has simple laws) if followed destroy the very notion of understanding which he thinks his 'lifes work' is putting on a more solid foundation. He is destroying the very thing he wants to protect!

[agentredlum]

No way Chaitin! You are going the wrong way. In order for an explanation to be effective it has to be more complex than the thing you are trying to explain. "To understand is to compress" nothing can be further from the truth. Some explanations are better than others but more important than size is quality. What makes you so sure that god starts with simple laws? In fact if you look at the evolution of life on this planet it mostly tends toward GREATER complexity

[agentredlum]

This is my opinion. Leibniz was trying to create structure in natural philosophy (all sciences) so in order to be spared from bogus claims from his peers (misinformation) he instituted restrictions to explanations (theories) but he himself could not explain his philosophy adequately to others because he made his explanation too compact basically gutting his own theory I seriously doubt that any intelligent thinker would sacrifice understanding for compactness

[agentredlum]

o-kay, how can you be sure that B is the explanation of A when you can't explain (understand) B? I think the idea he talks about (the explanation MUST be more compact than the thing you are trying to explain) is philosophically wrong. Compact explanations are usually not much good because they leave out info which is needed for the explanation. Chaitin (and many others) are taking Occam's razor to the extreme and if it agrees with Leibniz then in my opinion he was wrong too.

[agentredlum]

Take a system of information (call it A) and find the most compact way to explain it and let this compact way be smaller than the original system. Call this compact way B. You use B to understand A. According to his claim you have now 'understood' the original system but I would like to point out that since B is the most compact way to understand A you can't understand B since you can't break it down therefore you are using something you can't understand to explain something you don't know.

[czubinm]

@agentredlum In other words you're trying to explain the explanation. Since you cannot do that you'll just have to to be contend with B as fact where you cannot find some law or "why" to further explain B. B however still remains the explanation of A independent whether or not you can refine B.

Or in simpler terms don't try to fuck up Leibniz definition of a physical law or mathematical proof.

[raguks]

Interesting but sometimes vague.