My quest to make some sense out of randomness, logic and Islam led me to the Theory of Incompletion. It is a beautiful theory on its own. It says how any theory or any law that we might have now can be nullified with new discoveries and invention and goes on to state that logic in itself does not exist: which did not lead to randomness (randomness itself had been known for a long time) but made it completely unavoidable in modern day Physics.

I hope, if you are still reading this, know that you can not predict exactly what’s going to happen because nature itself is non-deterministic: you can only predict probabilities. Thats where randomness becomes just so important. Poor Einstein, he was never able to prove that there are some hidden variables which could perhaps bring back the good old days of deterministic Newtonian Physics. Einstein was a physist, he was never scared of randomness but he still firmly believed that there must be something that could eliminate randomness, a true challenge to the Theory of Incompletion.

My own research that I have often been whining about is about randomness in time series and I somehow stumbled upon something else which can possibly be used to quantify the Theory of Incompletion: The number of wisdom.

So what is the number of wisdom? Chaitin discovered a number (called Ω, ‘Omega’) with the amazing property that it is *“perfectly well-defined mathematically, but you can never know its digits, you can never know what the digits in the decimal expansion of this real number are. Every one of these digits has got to be from 0 to 9, but you can’t know what it is, because the digits are accidental, they’re random. The digits of this number are so delicately balanced between one possibility and another, that we will never know what they are!”* (OK, I myself dont know wat I just said!!! 😛 )

Lets track back a lil and see what it actually is: In simple words, Chaitin is only trying to say that if we can find the exact digits of the ‘number of wisdom’, all sorts of randomness can be eliminated and also prove that mathematics has no limits (we already know that Theory of Incompletion says that mathematics is incomplete and has boundaries: wat is true within one boundary will be false in some other boundary)

I am intrigued, fascinated, to say the least! 😀

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Interesting. Amazing. Fascinating. This is one of my favorite topics of research too… have you read Chaos (http://www.amazon.com/Chaos-Making-Science-James-Gleick/dp/0140092501?)

Actually, The Chaitin number (i.e the probability of halting) is different for each program, which means it is a different number for each of these programs.

But given infinite programs and infinite executions they will converge to a certain number which would be the Caitin constant, which will be a real number between 0 and 1. But there are other numbers like this like pi and the Golden ratio, (Granted we do know some of their digits).

I dont see how Chaitin constant fit into the theory of incompletion and Physics overall, its purely mathematical and Computer science.

wowww…everything flew right over my head…which branch of knowledge is this?

@ Ali

Chaos and Butterfly effect are both very intriguing for me at least but I still havent read much about em. i surely do plan to waise.

@ Safi

the number of wisdom that I am referring to in the post is the Chaitin’s constant. According to Chaitin himself, “

My work attempts to go further in the direction pioneered by Turing and Post.“.They deduce incompleteness from uncomputability, I deduce incompleteness by using arguments involving information, complexity and randomness. In a nutshell, I argue that the world of pure math is infinitely complex, but any math theory only has finite complexity, hence incompleteness.And the best example of infinite complexity in pure math that I have been able to come up with are the bits in the base-two numerical value of the halting probability Ω, which show that some math facts are true for no reason, they are true by accident.@ Pinky

its pure mathematics and theoretical computer science.

P.S. for further reading, please refer to The Limits of Reason

I am asking how are you relating Chaitin’s constant with Physics and Theory of Incompletion? I mean this relates strictly to Theoretical Computer Science and Information Theory. If your reasoning is analogous, please explain the analogy.

Oh.. sorry. Read your comment before it got updated.

long long sentences makes it harder to comprehend english. try breaking it down for easier reading ..

(trying to be constructive …..)

reading twice, the article does make a little sense, but i fail to understand one thing.

what exactly is the objective here? what answer are we looking for?

are we looking to scientifically prove God’s plans for the universe?

are we looking to find, what the future holds for us?

and how exactly did you equate chatin’s theory with incompleteness thoery?

maybe math is not the fundamental science, maybe math is just a tool to fundamental science.

@ Safi

I think I have already answered your question 🙂

@ Yousuf

ummmm….u have been here before I think. If not, then welcome to the blog.

now coming to your suggestion, I can only try to write in shorter sentences, but cant promise that.

And now to the questions.

1. If the name of post is giving the perception that I am trying to scientifically prove God’s plan then I think I need to clarify that this is something that Einstein said. He was someone to come up with the Theory of Relativity which demolished the theory of Absoluteness but deep inside he believed that there was one Absolute: God and also that God had planned everything out already and that is wat he meant by saying that ‘God isnt playing dice with the universe’

2. This post basically comes as a consequence of my own research. You might want to read the post on Incompletion to have an idea.

3. For the relationship between Omega and Theory of Incompletion, see my reply to Safi’s comment above.

4. Yes, I totally believe that Mathematics is only something that can

helpus understand the world around us, it is incomplete itself and thus cannot help deduce everything. In other words, trying to prove everything logically is not the way out. Our knowledge is very limited. Obviously, there must be some sort of logic behind everything that happens but as yet, we cannot comprehend all the reasons.Probably God’s biggest fear is being predictable.

wow this is so much information after dealing with a heavy duty client that i had to slap my self hard to grasp the subject.

well done.

no one gets me that way 😛

Take some rest, girl

You had hard day.

@ Vaqas

God has no fears. He is the hidden treasure and He wants us to find out about him and that is the reason why he intrigues us and encourages us to as questions and then try to find out the answers. If he had given all the answers, life would have been worthless.

@ americanising desi

oh then i am sure this wasnt exactly the best post that u should have been reading after dealing with a heavy duty client 😉

@ Asma

🙂

I have been putting off reading this post because I knew that it would have something I would be stirred to respond to. The time when it was published, I had some deadlines ‘looming large’ which convinced me that it would be fatally extravagant to consume my extremely limited abilities on anything other than designated. So now that I am somehow over with it, here are a few observations of mine for your perusal:

– Could you clarify how do you find it apt to phrase the theorem as “any law can be _nullified_ with new discoveries”? The theorem rather says something like:

“In any formal (note this word!) consistent mathematical system, there are truths which cannot be proved within the framework of that system but they nevertheless are true”

Correct me if I am wrong, Goedel didn’t engage directly with the business of falsifiability of a ‘formal, consistent’ system. Though the ‘possibility’ of such a system or its truths getting nullified is an indirect corollary but don’t you think that the way you put it predicts an absolute occurrence of such an event?

– “…goes on to state that logic in itself does not exist..”

Forgive me if that is dumb, but I completely missed you there. What is that supposed to mean?

– About the connection between presence of randomness in physics and the incompleteness, I think there is no essential relationship between the two. Though I am not sure but it might be correct to say that randomness is a manifestation of the incompleteness in physics, the discovery of hidden variable theory would still not necessarily contradict Godel. If you think deeper, even Newtonian mechanics was incomplete. It had certain ‘truths’ like the absoluteness of time or de-localization of gravitational effects which, ‘even if true’, had no means to be proved by within the framework of deterministic physics.

When Einstein was working for a theory of everything, I really cannot believe he was thinking to necessarily contradict the implications of mathematical incompleteness. It had always been there even in things which conformed to his philosophy of nature. Take for example special relativity, something about which it could be said that he carved it completely out of his philosophical model of nature. He imposed speed of light as the limit on maximum velocity of an observer only because it will otherwise violate causality. If someone doesn’t get it, it usually means that an observer A in one frame would be able to see another observer B in a different frame doing something even before B actually does it! You see, he preserved causality by maintaining it as a ‘truth’ which is ‘unprovable’ even if it ‘intuitive’ or wanted by anyone’s ‘commonsense’. So I very much doubt if he ever wanted to do away with incompleteness. All he wanted was to have an explanation of nature which despite being incomplete was inline in her working with his philosophy. And that philosophy, by the way, was borrowed in large chunks from Leibniz. But lets leave it at that.

On a completely different note, is there an inherent randomness in the time series thing you do or we just assume it to help simplify the system? To explain what I mean, the outcome of the famous coin flipping experiment is treated as probabilistic even though in the eyes of classical mechanics it could be perfectly predicted. But doing that would necessitate the knowledge of initial state like the velocity we flip the coin with, value of ‘g’, air friction and blah blah. So to make things simple we just assume that the experiment is probabilistic, carried in unbiased way and then both outcomes are found to have equal probability. So I want to know if your time series thing is probabilistic in this sense or there is an essential randomness like in modern physics?

And lastly, thank you so much for sharing information of Chaitil’s work!

god, that is a loooong comment. i’d read it in detail and then get back to it 🙂

@brickwall: Agreed on all accounts…. But what I would like to add is that every system is essentially based on unprovable, indivisible, assumptions which are called axioms, laws and/or programs/modules (as Chaitin puts it very aptly). And these are in essence truths found not via proof but common sense and experimentation. The modular approach as you proposed in the case of the coin is very much needed in order to solve problems (in the time span they need to be solved in.)

Now I think what she mean by new discoveries nullifying old concepts is not in actuality related to these truths but really our penetration into the nature’s inner cogs itself. The simple example being the history of discoveries related to the atom. First Dalton’s indivisible atom, then Niel Bohr’s Deterministic model (principal, orbital, magnetic and spin numbers… etc etc) then Heisenburg and Shrodinger’s wave thingy and finally the new quarks (Standard Model). It goes to say that even now when you come to think of it, Dalton’s theory still holds in the modular sense and so do others. Only our depth of knowledge has given us new insights which leads us to change those axioms or laws we so dearly loved, with new ones underneath the layer above them.

As analogous to physics, this would some what give a glimpse of what incompleteness really meant…. That human knowledge will always have unanswered questions…. provided the old ones get answered first. Such as now, we have nt still been able to account for a few particles in the standard model…. I think they are gravitons, tachyons(which may seems like a fallacy) and gluons…. Again I and Godel would agree that when these answers or rather particle are discovered, another set of questions would pop out too… (which in one way or another prove and unprove the existence of a higher power, but that is besides the point.)

Our friend here may have gone overzealous here by saying logic may not exist… But yes, somethings are better answered by emotions and imagination than logic againg for reasons of modularity, which is actually what Gregory Chaitin stands for… That sometimes especially in mathematics, it is better to solve problems with imagination than logic which as Imre Lakatos and Godel claim (and proved) may not be so incorruptible as it seems.

I hope I have said (and typed everything right as I m in no mood of proof reading such a long comment.) But hey man I kinda like the way you think.

@Leena: Read the comment! will ya? Dont be Absar!

Uhhh why the heck is my comment under moderation?

I dont know yar, but its there…happy? 😀

as for reading the comment, i sure will

obviously…but over the weekend!Safee:

Thanks for the appreciation man, I am humbled.

I too agree with everything you said. But my point was that Godel’s incompleteness theorem possibly had nothing to do with physics and its randomness. Relying on axioms or experimental results might be the weakness, if we may say that, of most of physical and mathematical models conceived. But there is at least one prominent exception: ‘Formalism’. It is a philosophy of mathematics in which objects themselves and rules have no meaning at all. An axiom is merely a point from which we proceed according to some rules ‘we set’. Hence, mathematical objects and rules are regarded no more than the objects and rules of games. A very good example of such a system is a computer program. However complex operations or simulations it may do, it does it without attributing any meaning or significance to them. To a program, it is all either addition or the set of data which this operation is performed on according to certain rules.

David Hilbert, the famous german mathematician, was may be the most famous proponent of formalism and had put a complete restructuring of mathematics along these lines as one of the objectives of his ‘program for mathematics’. So what makes people so keen on such an understanding of mathematical logic? Well, people may have different reasons due to their own preferences but for some the viability of such a philosophy of mathematics had a very deep significance. If mathematics, that purest of intellectual activity, could be defined ‘completely’ and ‘consistently’ with formalistic start, it would prove once and for all that rational systems of thought are, at least in principle, viable or may necessary alternatives to their counterparts based on intuition, axioms, revelation etc. One very famous philosophical school which was eagerly looking forward to the success of Hilbert’s program was Logical Positivism. I don’t know its whole ancestry but suffice it to say here that it had found some very strong proponents in a group of thinkers which came to be known as ‘Vienna circle’. And Godel, despite having completely orthogonal views, was a very silent member of that group during his graduate studies.

Khair, forgive me for the irrelevant ‘ramblings’ may be, but that is to show that Godel’s incompleteness theorem ‘only’ took on the problem of viability of a formal mathematical system and proved that if it is consistent then it cannot be complete. To some, it was amazing, but to a good many others, it was one heck of a bummer!

Tell me if none of what I said above makes any sense : )

No man it makes perfect sense, but I have to slightly disagree with your presumption that theorem of incompleteness does not affect physics. In fact most of the General theory of relativity agrees with the theorem, probably because Godel was one of the most greatest yet unsung contributors to it. (Godel was Einsteins good friend at Princeton Institute for Advanced Study) More so Godel proved almost single handedly executed time in front of the eyes of the physics elite, which of course they did not like very much,. And thus his name has been shunned to obscurity.

However, as Gregory Chaitin explains in his article in Scientific American, And rule of law or axiom of computer program would not be able to predict the probability of the failure of a given set of rules (which may apply to any field of science actually). Which would inturn mean the this ‘problem’ (as in math problem) is incompressible (to an equation). Thus it is the underlying thought at the core of both physics and mathematics which are two feuding brethren fighting over the same thing.

You should read Godel’s work and Imre Lakatos man! Beautiful! They are both like the renegades of logic! I just started reading some stuf and its really cool!

Khair, Vienna circle? I thought I was the only numb-skull reading shit like thzat! I got my hands on a book call A world without Time By Palle Yourgrau and its written like a biography.

The problem is that I think it was only Einstein and Godel who had figured out what this world (in physics and Mathematics) really is. After that people just wanted to make themselves relevant it their shadow. The formalist (David Hilbert, whose philosophy was rooted out and obliterated by Godel…) remained that cuz it was their religion. The Positivists found their own niche and any other whatever-ist remained as if nothing ever happen, yet the world and our understanding of it was complete revamped by these two fellows. And I think no one, even now really understands the things these two talked about and showed us. Maybe we were nt ready for it.

Khair coming back to point and in short, I think Theorem of incompleteness really does effect Physics and is not restricted to mathematics only, even though mathematicians seem to ignore it ever existed.

I thought about reorganizing my thoughts but guess that will be too repetitive for the handful who are following this exchange while those who were smart enough to have given up on me already are not going to be bothered anyway. So here we go where we left it:

I will have to ask you for explanation or links as to why you think General relativity is in agreement with incompleteness theorem. I am aware of Godel’s birthday gift to Einstein in the form of a solution to the equations of general relativity which described a universe without any arrow of time. But I think it would be a mistake to consider it synonymous to the agreement of General relativity with incompleteness. Both belong to completely different domains and have little in common to relate to. There is nothing which General relativity can posit which could be said to be the business of Incompleteness theorem. If anything, General relativity is deduced out of the Principle of Equivalence – an empirical fact – that serves as an axiom thereby rendering it a non-formalistic theory. But this is in no way unique to General relativity. The whole lot of post-gallileo physics is constructed like that, be it Newtonian or its successor.

After that, I think it would be slightly inaccurate to say that Godel’s solution of General relativity was unduly banished into listlessness. General relativity gives us just (sets of) equations much like Maxwell’s theory of electromagnetism. In both cases, these equations themselves don’t contain any unique solution. We make different assumptions like co-ordinate symmetry, boundary conditions, homogeneity of medium parameters etc. and then deduce solutions which are valid only when those specific assumptions hold. Now what has happened here is that Godel made certain assumptions and then showed that under those special circumstances, equations permit existence of a universe without any arrow of time. But have we really been able to empirically verify that our universe fits into the assumptions Godel made? If not, and I dare guess so, we should not be surprised if the idea was not taken up earnestly by the scientific community. It is similar to Victor Veselago proposing negative index materials back then in 1967 but not given a serious consideration because at that time it was considered a mere fancy of imagination to have such materials. But now that it became technologically feasible everyone is upbeat about it. I think this example explains quite well why Godel’s work is still looking to get more than just a nod in amazement.

But just in case I am missing something and to be better able to see where you are coming from, could you conjure any alternative formulation of general relativity or any theory of gravitation that could be said in conflict with incompleteness theorem?

Coming now to the implications of Chaitin’s results, my understanding again goes in tangent to yours. I am not so comfortable with the seamless extension of his results to physical sciences. In fact, he himself doesn’t attempt to do it despite saying that he has been into physics community too. Rather, all he does is to encourage mathematical community to come closer in approach and expectations to the ones existing in physics!

While I cannot talk about his thoughts, my own hunch is that his result was based on Turing’s proof for computer programs which are examples of a formal system, something completely orthogonal to the model of physics. Until we come up with an analogue of Turing’s proof for experiments carried out in physics, I won’t be very upbeat about the implications of Chaitin’s results on the same. Even the unpredictability of quantum physics doesn’t match with it since it can tell us very definitely the probabilistic outcome of an experiment.

I have actually read some tad bits on Godel and his works here and there. One such work is the Palle’s book you suggested. However, for a much better appreciation and understanding, I would recommend reading Rebecca Goldstein whose book has been reviewed by Chaitin too on his website. Simply a masterpiece! But it does appear that Imre Lakatos is unfamiliar to me. No matter, shall read up on him in due time, insha Allah : )

To close this don’t know how long comment, I would like you to suggest some concrete example of how physics changed after the proof of incompleteness given by Godel. Lest someone repeats it, lack of hidden variables is not a relevant example. It was established not because of incompleteness theorem but due to Bell’s inequality… Khair, I hope that we will now be able to take it forward instead of falling in a circle : )

I would agree that Chaitin has nt been able to apply his incompressible probability to (Quantum) physics which would mean that incompleteness theorem may only be an intellectual (mathematical) issue. But intuitively and generalize in does apply to physics even though physicists may not agree to it. How do physicist calculate probabilities for particles? They do so by experimentation, and by Heisenberg’s uncertainty (which if presented correctly, offers pretty much the same conclusion as a incompleteness theorem would, by the way, Godel and Heisenberg sort of disagreed on Heisenberg’s take on this) Which simply mean the a particle can not be predicted to a certainty in either position or momentum to a multiple (frequency) of Planck’s constant. Thus making Plank’s constant the smallest quantity of measurement, which in Physics (which is based on experimentation AND equations) any equation (rules or axioms on maths) would not be able to predict both position and velocity of a particle at the same time, which would only leave experimentation at a physicist’s disposal.

Would this not be an incompleteness in Physics? Granted it does nt actually show that an equation or system would be completely wrong, or false. Just that any formal system of equation has a limit to its truth in the physical domain. (Neil Bohr’s system still hold true and is in use in chemistry to define orbitals for electrons)

These are just my thoughts and feel free to criticize, as I think criticism taken correctly allows us to grow. But all this mumbo jumbo only holds of quantum physics where general relativity still does nt apply to any extent. But again as Chaitin says, it would be futile to propose a theory of everything or Grand Unification Theory as it would still have exceptions and those exceptions would pretty much again prove the incompleteness. Even the current Standard Model has two many exceptions (undiscovered particles). I think am starting to move in a circle again….. but my point being that the universe is not compressible to be put into a single equation or a system of equations.

So to take things forward I would say that I ll have to present to you how GR and TI conform with each other. For which you ll have to wait for a while as I m busy with exams…. (That would make for a Long Long post) But I will definitely post it.

You seriously want death-by-discussion don’t you? : )

Your thought process and ingenuity in making connections between things apparently discrete are remarkable. But unfortunately, it might appear to you I am only being one rigid Brickwall to bang your head against in angst! To tell you honestly, you made me seriously reconsider my opinions don’t know how many times although in the end it always failed to reflect in my final conclusions. Resilience? : )

Coming to your present post, equations in modern physics, in principle, CAN predict us definite probabilities. We don’t have to resort to experiment table at all. Schrodinger equation, assuming we are smart enough to account for complex system values, can tell us all we need to know about probabilities. In fact, I for one, think it to be an extremely tricky business to go to the lab in order to determine the probability of ‘essentially’ random events. I have my own somewhat related thoughts on it which is why I had asked about the time series thing in the first post. But to keep issues bloating out of control, lets make it wait for now.

Then again, I think you are slightly less accurate in thinking we calculate probabilities using Heisenberg’s equation. It is actually not the business of Heisenberg’s equation to determine probabilities. All it imposes is the ‘limit’ on the ‘product’ of uncertainties between two complementary variables like position and momentum or energy and time. Now probably (hehe) you know it already but I am having an ambiguous feeling reading your words so here is a just-in-case clarification that it is perfectly possible to reduce to zero the uncertainty interval of measurement of one variable, say position, at the cost of making the interval of the other, here momentum, infinite. In probabilistic terms, we would say that we have a probability ONE (certain mean value with zero standard deviation) for measuring the correct position and a uniform probability distribution (certain mean value with infinite standard deviation) for measuring momentum implying that momentum can take on any value. And the preceding should also make it clear that in special circumstances at least we are able to calculate or measure one of the two complementary variables with absolute certainty. I would like to know your thoughts on how does any of this parallel with Chaitin’s findings?

Then you question whether this is not an incompleteness in physics? And my answer is pretty much the same as before, this is an incompleteness for sure but we don’t need to know Godel’s theorem to realize it. It was known and understood before Godel presented his proof. IT, only emphasized the need to rely on axioms and dismisses the possibility of having a complete, consistent, non-axiomatic system of rational thought. But physics was always free of these ills. As I said in the first of my posts, whether we go to the predictable era of classical mechanics or the present haziness of modern physics, the principles Godel wanted to bring ‘mathematical’ community to have always been endorsed by physical community. Just to share something with you that once again you may already know, here are is a link which neatly briefs up all the axioms of quantum mechanics:

http://www.ecse.rpi.edu/~schubert/Course-ECSE-6968%20Quantum%20mechanics/Ch02%20Postulates%20of%20QM.pdf

In my opinion, incompleteness theorem will have an implication for physics only if it tells us something about it which is yet untold by any of the theories of physics we have derived independently of Godel’s proof. That is why I asked you to conjure an alternative formulation of GR (of course qualitatively) which if accepted would negate the need of having an axiom-based approach.

Lastly, it was Einstein’s idea to have a unified theory of physics for everything. A lot of people, though in minority perhaps, don’t subscribe to it for their own reasons. But then I don’t understand why do you regard Einstein to have truly understood physics while disagreeing with his ultimate objective? On a personal note, I am myself not very keen on Einstein’s manifesto for physics and don’t know much on Godel’s ideas anyway.

Take your time and come back once your exams are over with. I also need some breather to pay more attention to my lab : )

Cool. I know and the hiesenburg’s principle, in that if you predict position (i.e that you ‘measure it… which when you do, you expose the said particle to a photon which would if measure the position accurately will inadvertently change the momentum i.e changing the velocity i.e changing the position for the future again in this case probably a higher energy state. dont worry I kinda had to pay attention in chemistry class. I had to give a test every week to my dad.) Thoerefore you can either measure one, position or velocity (i.e momentum) and even that for a particular present time! Seems like you are a physics guy so I cant stand a chance to propose my lame ideas for scrutiny!

Khair! But the thing is the underlying probabilities still exist and even schrodinger’s equation employ too many constant (? I think?) to be REALLY be completely from of these improbabilities. The real problem is, even though some of these probabilities CAN be calculated with equations but the nature’s improbability itself is incompressible! Like say I have a transistor (90mm CMOS PNP) and say I want to predict the flow of electrons and holes through it I can do that to a good certainty with 1) some classical equations and 2) high accuracy eqautions which employ quantum mechanics to calculate (I believe you must have seen such tool especially for 40, 65 and 90 mm fab technologies cuz at that small a level the quantum effects really kick in). But even if I be so accurate and put all the constants to there max precision I wont still be able to completely eliminate the natures intrinsic improbabilities cuz my own measurements would be creating them, wont they? So I cant really monitor and observe the system with out changing and inputing more probabilities into it, would I?

As far as Einstein’s unification was concerned, he proposed A unification theory, not THE unification theory. Theoretically you would be able to make THE unification theory which could describe the ‘verse… (I think the standard model already does…. somewhat.) but there would be exceptions to it and nobody really like that (especiall programmers and designers like me) What einstein want to do was see the effects of gravity at the quantum level (again I think…. I m never sure)… But the curent mathematical and physical communities made it into a holy grail!

Sure Lets not talk about it for a while! (Please post a smaller comment to this!) My head is already spinning like an electron!

Sure, this will be a short one and I won’t contest anything I differ with you on : )

I am essentially an engineer but my education contained some heavy dozes of physics and mathematics. Sometimes my dislike for being completely dependent on others interpretations made me solve some physics texts too during pastime. But I do aspire and try to someday be able to switch into some physics infested business. Lets see how soon it can be effected, if ever.

Have a nice exam : )

Oh come on! You really did nt have to do that! HEhehe Just kidding!

Would like to know any insight of yours…. or corrections to mine though.

oh my god, such long comments!! Brickwall and Safi…i dont think i need to interfere 😀

anyway, i just finished reading the discussion, its been very informative and as Safi has pointed out, I am just trying to say that with each new discovery, each new theory (regardless of whether it is established as a truth or not), we get mesmerized by the boundless knowledge of Allah (SWT). no matter how much we know, humans can never know enough. And i dont really mean that logic doesnt exist at all…its just a term which we have not been able to understand completely. u know there’s a very popular phrase, “prove it logically”…a question which usually pops up from somewhere in a lot of religious discussions. Try asking these people what they mean by logic in the first place.

Btw, I am reallllyyyy sorry for not having responded earlier as I have been busy but I guess you guys arent really missing me 😉

Safi,

Sorry for the late response. I was stuck solving a problem in my project which today turned out to be the silliest thing I could imagine. Khair, before I proceed, I want to know what are your plans about calling it a day?! I was willing to lead but see you lured me back in : )

Even if all what you say about the influence of measuring devices is true, it is still a shortcoming of our computational power which makes the underlying probability of the result irreducible for ‘practical purposes’. But here we are talking about probabilistic determinism ‘in principle’. If you take into account the effect measuring devices introduce into the system besides doing the act of measurement itself then you should be able to perfect your calculations. Unlike Bell’s inequality which precludes the existence of hidden variables, there is no theory or principle which could bar you from being able to do that.

Regarding Einstein’s motives, I am not entirely sure nor have been much keen as I noted earlier but marrying general relativity with quantum mechanics was certainly a thing he wanted to achieve dearly. But about Chaitin’s remark on the theory of everything, well the sense in which he says it is once again I think already accepted in physics community. To a mathematician, a theory of everything would mean something that doesn’t hinge on the ‘crutches’ of axioms or empirical facts. But when physicists use this word, they understand that it includes relying on empirical evidence which has no logical explanation otherwise. By the way, do you realize the oxymoron in saying A (or The) theory of everything that contains exceptions? : ) And no, standard model is not an everything theory by any standards.

Leena:

I kinda knew what you meant by whatever you said. Just wanted to see your thought process better, something which essentially brings me to blog world. Thanks for giving me the space though on your page.

Well, even though every bone in my body is telling me to not reply, I have to, the blogsphere is pretty much my whole social interaction for the day! (Yeah I dont have a life outside the computer).

Although the principle applies and requires measuring devices. But you have to understand that, in physics, we cant ‘predict’ anything without measurement. It would be pointless. I mean what will you be predicting? Is there a point to predict that a certain variable would take up certain value at a certain time? No. When in physics dept, do as the physicists do! Talk about particles! And when we talk about particles, we NEED measuring devices. otherwise it will turn into pure mathematic which in turn IS dictated by incompleteness. (You do realize, we are going round and round in a circle.) So even though it seems cool to isolate the hiesenberg’s principle from measuring devices, but you see there is no world without measuring devices or in short instrumentation. And believe me there can be an instrument which can give you a better resolution than the planck’s constant. In other word it IS NOT our inability to measure, it is actually nature’s inability to support such measurement.

As for TOE, lets keep that out of the loop for a while. This is already getting too cluttered. Lets finish one topic at a time.

@Leena: Would you be participating anytime soon?

And may I dare to ask what the heck is your project? And what do you do on the whole? You are nt a theoretical physics guy working on Particle Accelerators, are you?

OK lets try to break out of this circular loop. Tell me pointwise what you think about:

– The difference between the impossibility of calculating probabilities in principle and the same thing but only in practice due to lack of computational power. Mind you that in the case of former, you can have infinite computational power at your disposal and yet it is impossible.

– I claim that Godel’s work only talks about the need for having axiomatic or empiricist approach. He maintains that we can never ‘prove’ everything based on sheer logic. That is as far as it goes. And in physics, whether it is Newton/Einstein/Shrodinger and who not, everyone agrees to it. A reading of wiki entry on Godel’s work or anywhere would confirm and if not, please share.

– Chaitin, however, does prove some lack of computation. But again that is a lack of computation ‘in principle’ and not due to a lack in computational power. But in physics, all the examples you are giving me fall in the latter category, something which it is very cruel to identify nature with. That is why I say lets take some precautions before seamlessly applying Chaitin’s result on physics. The good mathematician didn’t see any connection and that should be enough of a caveat for those of us who are not even deep into Turing machines or know how far its model corresponds to nature’s own statistical nature. English is a funny language eh?

– I don’t see the point you have when you say without measurement it would be all mathematics dictated by IT. The irreducible probabilities result of Chaitin was derived for Turing’s machine. Otherwise, my calculator gives me quite definite result when I ask it to compute the probability of heads or tails in the simplest of experiments. But if you want it extended to physics, just explain how the universe corresponds to Turing’s machine. Otherwise, lets forget it for a while and get some peace back to our lives. How’s that sound? : )

– About nature’s inability to support a measurement which doesn’t affect it, this is once again true in itself but doesn’t put on us a theoretical limit from computing something. It is a practical limit which is not dictated to us by Godel, Chaitin or any of the theories in physics. For something to have effect on our understanding of the state of affairs in universe, it has to come by way of theoretical limits. A case in hand being Bell’s inequality or even Heisenberg’s principle.

That should be sufficient for now.

I am a student of optics/applied physics and my present project is actually Msc thesis on digital holographic displays.

You know, it was so tempting to tell you that I am into theoretical physics. That could have shied you away from furthering this no-ending back and forth :p

But on a serious note, don’t equate finishing a topic with reaching a consensus. Though the proverbial joke has shamelessly tied it up to donkeys, it is actually not unusual for humans to realize the significance of something awhile after studying it. That is why I think that sharing of ideas should generally be an end in itself. Reaching instant or before-we-say-byes consensus can only be an ideal.

Yeah agreed.

Probabilities are computed… for each experiment. As Chaitin says, it is neither impossible to compute them nor do you need excessive computational power to calculate them. But the ARE incompressible i.e. You cannot merge them into an algorithm. As he goves the example of the favorite constant pi. You can have a recursive algorithm to calculate pi which can keep dividing the circle into segments. with each iteration you ll get a better approximation of the constant, i.e the task is compressible. Probabilities in more cases, and especially the one Chaitin introduces that whether a system will actually halt i.e run into an exception is not a reducible task, yet however it is constant for the universe. I hope I explained this correctly.

What impact does that have in physics? That if you have a system of laws, and apply them to compute a particular case, You cannot “predict” that the case would run into an exception i.e literally. That was what I mean by an exception. That the computational part of physics would never know whether a ToE would ever run into an exception and there WILL BE exceptions to it.

I have a few books that I can share if you like. I dont want to write their names and I have nt read all of them yet, (Guess I m as illiterate as they come.) I m reading some other book. I might do a post on it if I have the power to understand it.

What I was trying to say in the last comment was the Heisenberg’s principle IS a theoretical limit and NOT a practical consideration. So it although may not affect the inner workings of anture but it does affect our understanding of it and poses a limit on the certainty of our understanding. Tell me if I m wrong in this.

Ok I wrote this comment in a hurry so I hope I covered all the basis. But I ‘d redo this when I come back. Hologhraphic displays! Cool! I m kinda working on planar interactive displays! The key here being interactive. Not as cool as LASER based hologram!

Looks like you have said it so here goes my take on this:

I think you are mistaking by simplifying the process of making a system with some set rules and then thinking that results obtained in that system are applicable to other systems with different sets of rules too. In the case of Turing machine, we cannot make a statement about halting probability. But I give you another system which is formed with set of rules as they were in Newtonian mechanics and then ask to ‘predict’ the possibility of a process coming to halt. Do you really think it would still be that hazy?

But here you can validly argue that the correct analogy ‘could’ be between program in Turing’s machine and our rules/theories in our universe. And the conclusion would be that we cannot predict that our theories would not run into exception. However, could you explain to me then what does it mean to randomly pick a program in our case? What if I deterministically pick a theory and ask whether it will break or not? Can I then assign a probability? In fact, what does it even mean to say that the probability of exception for our physical theory is a well defined number but we cannot compute it? Why can’t we say that it is either true or not hence the probability of exception is 0.5?

All of the above is NOT to say that Chaitin’s results don’t apply on physics. Rather, it is just to show the pitfalls present in a seamless application of results across the lines without full knowledge of the correspondence of both Turing’s machine as well as physical models, if any.

Heisenberg’s principle is definitely a theoretical limit and talks about the essence of nature itself. However, it doesn’t help your case in establishing the irreducibility of probabilities being a theoretical fact. As I explained before, Heisenberg’s principle does NOT deal with the business of computing probabilities. It is just the standard deviation of the statistical process and that too can be zero making at least one of the two completely deterministic. And then, even if you don’t know uncertainty ‘interval’, you can calculate definite probability for an outcome don’t you?

Lastly, what do you mean by sharing books without writing their names? Are you going to post them? That would indeed be so generous of you : )

Well, you gota love your eyes man to realize that you are better off without laser based holograms :p

I ll try answering them one by one (so we can move forward?) and please do realize that these are dogmatic statements at best. You may want to read up and make your own opinion. As far as the books are concerned, if you have a dropbox acc, I can easily share them.

Para 1:

A turing machine is (I think) a fancy name computer people give to any other system. Every system takes an input and produces an output according to a transfer function. The difference in a Turing Machine is that the output is a program which defines a transfer function. (Linear, non linear, piecewise defined, whatever). So any system with a specific set of rules can pretty much be simulated by a Turing machine, provided that you program it correctly. Thus Newtonian system can (and is) be simulated by a Turing Machine (in entirety). Now if you program it correctly (i.e remove all human errors) The instances where the turing machine will give an exception will be the the points where the Newtonian System would fail. Do you agree to this? (Please realize we are talking about an ideal turing Machine, not a physical computer, thus please forget Machine epsilon, precision and other mumbo jumbo).

Para 2:

Ok computing a probability (beforehand i.e predict) would mean that its is compressible to an algorithm. An algorithm means that a program with n bits can calculate it. Suppose this program selects other n bit programs randomly and calculate the probability omega (Chaitin’s constant) of the programs to halt or not to nth precision. Now suppose that this program halts, that would mean That our whole house of cards cam tumbling down, which means that the problem is incompressible and no such program exists. I hope I put it correctly. This means although there is a probability AND it is constant through out the board, but the reason why we cant calculate it is that there is not algorithm for it other than to measure it for each program. The probability of each program (measured by running it) will only give us a part of the precision of this constant. To calculate the constant well need to run all possible programs (i.e infinite) to get the complete number! And I bet you do realize the futility of the endeavor.

Para 3:

Well that pretty much a stateent and I m in not mental or intellectual capacity (i.e that of a chimpenzee, as someone once called) to refute it.

Para 4:

Well I think Heisenberg’s principle exactly does that… why? Firstly lets see if my concepts are correct. suppose an electron moving say in a wire. To predict its future position, I ll need to know its current position and velocity. To measure something we have the concept of predictable and acceptable error, i.e valid reading. I launch a photon which strikes the electron. now according to Heisenberg, I would only be able to measure the current position OR velocity (i.e momentum), right? Which in turn means the even though I am certain where the electron is OR how fast it is moving AT A CERTAIN POINT IN TIME. Now since I know only one quantity i.e position OR velocity and I have almost no idea about the other (keyword being almost), I can only predict with a certain probability where the electron will be at a future point in time. Is this correct? If it is, then yes There is NO WAY IN HELL I can ever make that probability ever to go to zero! I can only be 99.99999999…% certain (Actually that is more Schroedinger and less Heisenberg!) Even for that to happen I can only measure the current (one) quantity, keyword being measure. I wont be able to do anything with the system until I have measure quantities. And I can only predict only for a case, not the case or all case! Am I right? or wrong? Or just simple a dumbass? Hehe.

Aaah ok, I officially have burned all the brain cells for my qouta for the day! I m off too sleep…. (If only I could sleep upside down and regenerate!)

*ERROR FIX*

The difference in a Turing Machine is that the output is a program which defines a transfer function.

*TO*

The difference in a Turing Machine is that the output is the output of a program which defines a transfer function.

*END FIX*

Ok I cant fix all the little error so enjoy comprehending my bullshit on your own! Hehe

Well, you will have to wait now for however tad thing I have to say. My hands are full with stuff already

Dude! Where did you go? Just when I was starting to have fun!!! Oh and I have one paper to go!

Logic, especially Aristotelian, has been accepted in all Islamic science, Islamic science so to say which presents to man many ways of reaching at truth objectively but not from just order of reality, which in modern science is reduce to only

Scientific Method. According to Dr. Osama Bakar, logic as developed by Muslims jurists, philosophers, and scientists, was cultivated ‘within the framework of a religious consciousness of the Transcendent. In their view, logic, when used used correctly and by an intellect that is not corrupted by the lower passions, may lead one to the Transcendent itself’.It may be incomplete but not if not treated as

sui generis(i.e. unique) and when not divorced from the Sacred knowledge.You dudeness!

I am very sorry for being away for so long and I hope that during this period you would have lost your fun and interest in this discussion : ) I was and still stuck with the nuisance of exams and thesis and may be even now I should stay out of blogs and crap. But khair, I have a moment of relief for now so here we go:

– I won’t say that I disagree to your description of Turing’s machine but I do have reservations where you suggest its function and the process of it generating exceptions and what does it imply. I wish I got time to read up in detail on Turing machine’s issue but that seems to be an impossibility for now so I will just say that I will consider your thoughts if you could either reference a professional mathematician (the title of a book would do) suggesting it or I find one myself. Till then, it is safe not to speculate.

But if I don’t something, as I said, then why then do I have reservations? Well, I will quote Feynman here. It was in one of the volumes of his super-famous Lectures on Physics (probably vol 2) where he remarked something to the effect: “Classical electrodynamics was a mathematically consistent theory. The only problem with it is that it doesn’t conform to the experiment” And I myself have some fair enough experience with classical electrodynamics to have a feel of what he said. Now what bothers me about your understanding of Turing’s machine is that if it is a mathematical system emulating or simulating sets of rules then how can it generate exceptions to its own rules? We didn’t conclude that classical electrodynamics must have been at fault by mathematical reasoning. It came at large, if not essentially, by way of experiments. Otherwise we could have keep playing with our rules ad infinitum. If Turing’s machine, despite all its capabilities, is still the game of playing by rules then it is unfathomable to me how it would indicate holes in classical mechanics or electrodynamics? However, as I said before, I will consider this possibility even if I don’t grasp it only if I can be sure that I am taking views of someone who has worked with Turing’s machine first hand.

– I think what I have said above obviates the need for now to continue thinking along what you said under the caption Para 2. It can be done once we know for sure what is a Turing machine and the scope of its operations.

– Well, lets recall that all this back and forth around Heisenberg’s principle is to prove or disprove the following proposition:

“In a quantum mechanical system, it is impossible to compute the ‘probability’ of the outcome of a measurement to perfect precision”

To recall it further, it was actually your proposition made to suggest an analogue between quantum physics and Turing machine’s lack of ability to compute even the probability of program termination. Now let me explain again why the arguments you build up in favor in your last post don’t add up.

You hinge everything around a ‘specific’ case of electron’s position measurement. Before I address whether it proves something or not, let me make it plain that at least ‘in theory’, we have static systems in quantum mechanics whose probability do NOT vary in time. It is in fact the very first case presented for the application of Shrodinger’s equation in pedagogical texts I have come across. What can these systems represent I don’t know. But their existence is considered to be theoretically viable in an ideal, unperturbed world or perfectly isolated systems.

Now about the particular example you gave, granted that it ‘could’ be that I cannot make the probability go zero but that is besides the point here. We are talking about the ability or lack thereof to compute definitely the probability itself in quantum mechanical systems and not to achieve a probability one or zero of a measurement. In Turing’s machine, it is impossible to compute probability precisely, let alone making a definite guess about the outcome. But my point is, and which is why I find the two things disparate, that in Quantum mechanics theoretically we CAN compute probabilities up to whatever decimal point they exist. To quote you so that you don’t get the impression that I overlooked your argument:

“If it is, then yes There is NO WAY IN HELL I can ever make that probability ever to go to zero! I can only be 99.99999999…% certain (Actually that is more Schroedinger and less Heisenberg!)”

First, it is not the question of making it go one or zero, as I have pointed out above too. Second, you statement about 99.9999…% itself doesn’t follow necessarily. Something which is neither one or zero could be 0.5 and not necessarily 0.55555555… And even if it is the latter, suppose, still there is no way I think you can prove that this 0.55555… is irreducible too. If you like doing mathematics yourself, I can give you a wave function which gives probability 1/Pi in a certain interval. It would be irrational but certainly possible to codify due to the presence of algorithms.

Just in case it didn’t become evident before, I am stressing on theoretical cases in response to your practical one because we are trying to see here the presence of a ‘theoretical’ limit on the computation of probabilities. The one like uncertainty principle or bell’s inequality. In order to find that out, practical specific examples won’t help. You would have to take a general system, like Chaitin did, and then see if it yields an overbearing principle not. And if you take my hunch, had there been such a principle, it would be common knowledge already and we wouldn’t need to debate it today : )

And ignore the typos. I have to do lots of things before I sleep and cannot afford to spend a second more on proofreading this!

Umer,

I would like to add something to what you said. While it is not possible for me to comment about the wholesale acceptance of aristotelean logic in Islamic sciences, I can surely say that there have been persistent, albeit through minority may be, opposition to it. Two such famous works are Tahafut Al-Falasifa (Incoherence of the philosophers) and Al-Radd Alal Mantiq’een (Refutation of the logicians). The first one is written by no other than Al-Ghazzali. And the second is done by the great Hanbali scholar Ibn Taimiyyah, unfortunately a slightly less known figure in Pakistan for various reasons. If the appraisals I have seen are correct, the use of greek logic in Islamic sciences could never revive (at least) to its old form after the shattering refutations of the said scholars.

I have an exam today so I ll post a detailed comment later tonight, but in the mean time you would want to read a few things on turing machines and Godel/Chaitin’s work on wikipedia (not to undermine your knowledge, but to be on the same page)

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

Please read the last part especially, actually that was what that got me hooked into all this crap.

http://en.wikipedia.org/wiki/Turing_test

This is interesting if you are into AI and stuff.

I would also put turing machine entry in wikipedia but that is unnecessary. All you need to know is that a turing machine is an (analog or digital) machine which can store a program, read it instruction by instruction, and execute each instruction as it reads it. bear in mind that the execution may or may not generate an output bit pattern.

And as for your last comment, currently in Islam, we are not allowed to question the core concepts even as a purely philosophical discussion, and without being able to do that (which is only due to the vested interests of various factions of ‘scholars’) logic is not supported, but in wider, more tolerant Islamic society, this would be true.

Personally, I perfer the old egyptian way of doing things i.e measurement before theorizing, than the greek way, which was mostly about switting around and talking about things. But I do believe the latter is more fun!

Oh come on!!!! Where did my comment go?

@kw/ls: You HAD to change the theme! did nt you?

Ok here goes nothing….

Ok an exception in a turing machine is pretty much an exception in the physical world, that the output of the machine cannot be ‘quantified’ (?). Like if I had a controller and gave it an unbounded waveform, it would go unstable, so in a turning machine it would also putput something which may not be in the output set. I may be wrong, or not describing it correctly. You may want to read up some literature and make your own concepts. (cuz mine are jst wierd due to lack of intelligence).

On turing machine, I think it is best if you read the turing church paper and just the wikipedia entry. Also I got the books on drop box, so if you make an account I can definitely share them. (In return you WILL share some material with me, for my generousity of course) Heheh.

Ok your point on the underlying flaw of turing machines and that it would indicate flaws in other systems such as electrodynamics or mechanics, is well put. Please bear in mind that we are talking about turing machines and not the computer. In a computer everything is quantifiable due to its digital nature. Where as the turing machine is just the mathematics model and does not imply digital or analog physical system.

Ideally, when you would program the turing machine with the rules of a system it should emulate the system ‘completely’ and consistently , and in effect becoming the said system. That is the beauty of a turing machine. Now the Turing machine would have all the traits of the system and also its flaws, and thus would be emulating its flaws. The problem is that, under certain conditions (maybe or maybe not) the program WILL halt, and those conditions would be relevant to the system as well, simple example is a divide by zero. Now in actuality, no system should give a divide by zero, due to its inherent unspecified value. but systems do take it into account. a turing machine would FAIL to compute. So yes Maxwell equations would have unbounded or unspecified or should I say unpredictable reults in certain cases. (Now are these case true for the real world or not, that is a different story.)

Ok your para two!….. I think you have NOT understood my point! There are compressible probabilities and incompressible probabilities…. The compressible probabilities are those that can be calculated prior to the event happening via an equation! incompressibvle probabilities cannot be calculated in a general case. That is you WOULD NOT be able to calculate the probability of a particular program failing in a turing machine via an equation. THAT IS WHAT CHAITIN’S OMEGA IS!!!

We Now coming to the quantum mnechanics…. you can calculate the *underline* probabilities of values (as opposed to exact values) to your hearts desire! But you would not be able to to predict where the quantum mechanics would FAIL and why while staying in the system. Now suppose to added a new law to your quantum mechanics and I believe it would be consistent with your previous system, plus removes the exceptions or FAILS of the previous system. The New system would have completely new exceptions…. AND THIS IS WHAT GODEL’S PROOF IS!!!! That each system would have its own Godel’s sentence, And while staying within the system you would not be able to prove it, i.e it will generate exceptions!!!

Ok I may be blabbering here, and I AM inconsistent most of the time! It is your job to keep me consistent my friend!

I think and hope I made sense!

@kw: DUDE I HATE this SMALL BOX to write comment in!

Your first post:

I know Turing machine up to what you just said. What I meant by not knowing it is the actual maths and working which a professional mathematician or computer scientists do. I firmly believe it to be necessary before we extend its implications and applicability to domains outside mathematics itself. Having done some tidbits in physics myself, I find it utterly disgusting when someone (you are not intended here) tries to use physical results in extra-physical terrain without having an iota of the methodology of its investigation, nature of proof and the meaning of assumptions implicit therein. A case in hand is the use of special relativity for proving moral relativism in philosophy. This is complete Bull! Give another example from a different angle, unless you have done mathematical theorems yourself even at the most basic level, you cannot come to truly appreciate the Platonist stand that mathematical objects have an objective reality of their own and blah blah.

So I expect some prominent mathematician to come along and explain how an ideal Turing machine can help me determine the faults in our theories. Otherwise, to give an example, I cannot even comprehend how we can formulate quantum mechanical rules for Turing machine. It appears an oxymoron to even say that I can define rules for ‘inherent’ randomness of a system. And that is the reason why we think that we need Quantum computers to simulate quantum mechanical process. Anyway, consider all this rant to be coming given my lack of knowledge of the precise scope of Turing’s machine. Hence, don’t consider it something you should either agree or disagree to. You can simply overlook and bury this debate for once PLEASE! But I would like you to appreciate that I am not positing any of this upon the quantifiability or a lack thereof of Turing machine.

Then you mention about generating exceptions like unbounded results. And you do say it well actually. Let us leave it there for now until I am satisfied about my aforementioned apprehensions on Turing’s machine and its applicability to Physical theories.

I did understand what you mean by irreducible probabilities. All the claptrap in my last post was to show that time dependance of state function (say for electrons) do NOT necessitate a loss of ability to precisely compute probabilities. I have done enough of quantum mechanics to know it. And that is a very mild statement, I tell you. I actually have to resist saying that theoretically I should be able to bring into equations a system completely which if solvable in an exact manner would yield definite probabilities. And the uncertainty princple cannot do a dime about it! (please keep in mind my recap of how we got into this discussion). So yes, Chaitin’s constant is an irreducible number in mathematics but one will really have to go miles to show its existence in the world of quantum mechanics.

Then the failure of quantum mechanics. Before anything else, let me state that this is not tantamount to the presence of irreducible probability of electron’s position measurement. Now coming to the actual thing, I explained it before and would do it now again that Godel’s proof doesn’t deal with the falsifiability of (formal) mathematical systems. It talks about the presence of unprovable things, truths or falsehoods and it is NOT necessarily both of them. To show you what is mentioned in the wiki entry you referred before:

“Or one might say, no formal system which aims to define the natural numbers can actually do so, as there will be true number-theoretical statements which that system cannot prove”

Secondly, and again repeating it, Godel’s theorem is involved with FORMAL mathematical systems (notice the word in wiki entry too). Ones which are defined completely on logic and nothing else. Physics is already based on empirical truths or axioms which at times are not even attempted to be logically proved and hence doesn’t find any revelation in Godel’s results to the effect you are showing it to.

But there is something else from a completely different angle which to contain this discussion I was deliberately putting off till the hitherto-out-of-sight last post of mine in this thread! Reason being that Godel’s theorem was connected in this blogpost to the absence of hidden variables. But as you see, I am already having a hard time handling it with you! Anyway, it pertains to what the wiki says in the end quoting Hawkings and another physicist on the implication of Incompleteness for physics. In my opinion, there is just a possibility and not a necessity that we wouldn’t find a complete theory of physics, though for all practical purposes we would or might still not be sure which is the case at any given time. Unfortunately, I cannot afford to broach a completely new discussion on it here. But if you can tolerate to see where I come from on this subject, do a google search on “Unreasonable effectiveness of mathematics in natural sciences” and read up all the crap churned out starting from Wigner’s original paper. I believe it essential to first settle the nature of mathematics at least when it is applied in natural sciences as well as what do we think the physical universe to be and all the crap. Forget anything else, I find it totally baffling to think that under the premise of science’s closed universe, we human beings can possess or gain access to logic which can predict phenomena taking place at an astronomical distance away from us. But please, now don’t start arguing over this I am not presenting it for discussion but to creep you out in the hope that you would blacklist me for someone unworthy of having any reasonable discussion with and leave!

Aww… the above was a response to both your first and second posts. Sorry for inaccurate opening there.

I just remembered some things I forgot to say. I don’t have an account with drop box nor ebooks to share so apparently I cannot benefit your generosity

And about logic in Islamic sciences, you are taking it into completely wrong direction. You are probably talking about the superiority of the beliefs of one sect over other. That is what I infer when you suggest that various sects have their own interests in keeping it as such. However, what I and probably Umer were referring to is completely different. Since I am already exhausted due to what preceded in this thread, I would very shortly say that it relates to use of logic and the extent of it in understanding or explaining Allah’s attributes as they are mentioned in Quran and ahadeeth and a lot of blah stuff. It was essentially done after the greek logic crept into Islamic society. Now in case if you still find it hard to see any difference from what you said then just trust me if you can(!) that it has nothing to do with understanding or questioning the beliefs of a sect in the sense we understand it today at least in Pakistani context. Most of these things are present in hardcore texts of aqeeda which are either not translated into urdu yet or have been very little publicised, something I am actually very happy about!

If you have read turing machine entry in wiki you would have already pondered on all the mathematical description that you have wanted. For me to explain in short, a turing machine has in input a program from its program space and the input that the program itself takes. it iteratively parses through the program while working on the input to generate an output pattern.

Suffice it to say, there is nothing much to a turing machine, and the math involved is fairly simple, relating to basic set theory.

Ok I get it, where you are coming from, you are persisting that the empirical data gathered by the physical world to contruct basic laws and the axioms in mathematics are different, and lead to different system, and in the case of latter a formal system. I m with you till here, but I believe that the axioms formed in mathematics are also based on observation rather than just logic, otherwise they wont be axioms but theorems to the more general axioms underlying beneath them. So at the core, both mathematics and physics are in essence based on observations of the environment. It may sound lame, but the basic example is the 2+2=4. You actually explain it to a child by putting two and two objects in front of him or her, and counting them. otherwise there would be no logical way simple enough to prove it. Hell burtrand russel made it even more harder in that principia mathematica of his.

Quantum computers are just WAY too cool!!! But are turning machines none the less.Only capable of more than the digital computer we have these days. So even though the computational power may be increased but the mathematics will not change. Thus the irreducible probabilities that would occur then would still remain that way, even more so, as these probabilities are more concerned with turing machines themselves.

So the problem really is that, now Physics has delved so much into theoretics that it may have turned into a formal system… Sure we have been able to prove some of those, but the proofs came later than the formal systems themselves, which as you say are divergent from the essence of physics.

Ok but everything is besides the point. One other thing besides the point is the ability to calculate and reduce the probabilities of quantum mechanics, of course as quantum mechanics is a formal system, based on a language and grammer would be able to calculate the supporting probabilities. But the probability that the actual system has failed can still not be calculated. Now as you said, why would this system fail? It would not be because of uncertainty or bells inequality or hidden variables…. it would be because of incompleteness of the theory system itself. Now you are going to say, why would this sytem be incomplete in the first place? well thatwas the obious obviated by Godel, That a system would be consistent but never complete. The Godel sentence in the system CANNOT be removed, and to remove it would mean addition to the system thus developing a new system with its new Godel sentence. Now again, why would this be applicable to the empirical system? Well the need is obvious, That we interpret our observations on not just our perception to what we see, but how we see it. all measurements and observations are open to interpretation, though the rules of mathematics dictate us to come to a logical consistent conclusion according to Occam’s Razor. But The observations always predict future observations, the the probability if these observations can not be made certain by calculating the probability but only by performing the observation. (Calculating the probability of the turn, but rather playing the term and rolling the dice.) This is not the nature of just physics or mathematics but science and life itself.

“Which is probabily why Allah says that we are here for an examination.”

I hope I was coherent and consistent.

As for Umer and his coment, I was just being an ass that I m. I m a cynic at best and I tend to demarkate my territory when an optimist come along. I m sorry for that. Yes I agree with you and I m going to agree with you without typing or retyping anything more.

@kw/ls: You should be proud? This has snowballed into something ….. which I have no idea what to call!

As I now think about it, I feel that in the absence of body language, my previous comment could have appeared rude. Was that so? In case, I am sorry. I am extraordinarily poor at passing dry humor and almost all of the time end up offending people. One day I hope I would give up the struggle but till then people would have to bear with me a little!

So do you prefer me to rejoin to your post? Please understand that I am throwing you an opportunity to break this vicious back and forth cycle with the satisfaction of having put the last word : )

Hahahaha…. Dude! Go ahead. I m free for most part so go ahead, this is getting interesting anyway!

And If you were rude, then I was probably vulgar!

Look, for once understand that I know and appreciate it to the extent you have told here about Turing machines. What I am looking for is some expert coming along and shedding light on how it is possible to program a Turing machine to simualte a quantum mechanical system. Why is that so unique? Because in the presence of ‘inherent’ randomness and not the pseudo-randomness which we are made so accustomed to during our engineering education I just don’t know how can we even come up with rules to simulate individual process exactly. That is at least partially why we felt the need of having a quantum computer. The two would be interesting no doubt but it appears to me that they just might not be able to come along together on each and everything.

And I find you off the mark where you say that BOTH mathematics and physics are based out of observations. In simple cases like 2+2=4 it would be true but in a general sense there is nothing debarring a mathematician from making some hypothetical starting point, set down some crazy rules, and then see how it turns out if the rules are followed meticulously. You probably didn’t read or take into account the issue of interpretation of mathematics, which is visited upon in the debate of unreasonable effectiveness of mathematics too that I refered you earlier.

Then again, I find it hard to agree that by delving into theoretics physics has somehow become a formalist system. There is just no way that physics could ever become a formal system. A formal system is not something which involves rigorous mathematical derivations. This is something done in Logicist, Platonist and laguagistic worlds too. In fact, mathematicians belonging to any school arrive at the same equations and go through the same rigour of derivation. What distinguished one from the other is the ‘interpretation’ they give to this process of mathematical thought. And in that regard, physics doesn’t qualify to be placed in any category in toto. That is why there exists a debate and division of opinion in the scientific community. Though being engineers I know it seems very trivial to us as well as surprising to think what keeps them from just accepting mathematics as a tool and move on.

From what I said above, I hope it would be clear to you why I need not to respond what you said further about quantum mechanics and its falling apart. It is all based on the assumption that QM is a formalist system which as I said before couldn’t be more inaccurate. There is no issue of consistency and completeness in QM in the sense it is for a mathematician approaching mathematical systems from a formalist standpoint. While they are always troubled and disputing about the meaning/interpretation or worth they are to give to the essence of their work, a physicist is always sure and clear. And in fact, this is the reason why Chaitin says that Godel’s results deal more with Logic than Mathematics itself. Nevertheless, as I acknowledged in the last post, there *could* be some implications for Physics too but from a totally different perspective than the one you see it to be. But you have left me with little energy to talk about anything of my own thoughts : )

And I just cannot imagine how it could be getting interesting for someone. If anything, it is getting more and more insane with each comment that we post! No wonder, nobody but the two of us are reading it now. Nonetheless, my offer remains!

Ok I might have gone a bit overboard as I agree to a lot of things that you said, and to tell the truth I am just disagreeing with you on certain issues just to make this even more ‘insane’. I agree why quantum mechanics and physics on the whole can not be a formal system, (but if you do delve into it, you would see that just because of the physicists’ love for Newton and their inherent affinity to equations rather than statistics have turned it pretty much into a somewhat ‘formal’ system, where some of the ‘rules’ are just kept alive for the sake of inadequecy of not letting go, again you must realize, I m taking this dramatically), And dude! As long as you ‘ll keep posting, I keep on replying! I kinda like this insane topic and I cant really talk about it in front on normal people, otherwise they ‘ll probably send me to the loony bin.

So coming back to it, most are arguments are only for the sake of an argument as I do realize the Godel’s incompleteness would not directly apply to physics. As you said that you have certain thoughts of yours, I ‘d be more than happy to hear them, cuz then you ‘ll be obliged to hear my wacky ideas too!

But I do believe that Physicists should realize by now that Newton was not a god and should start taking heed to those who may have refuted his methods if not his thoughts, Liebnez for example, on whom all of Godel, Turing, Lakatos and Chaitin’s studies as based on. Heck if it had nt been for Liebenz, we would nt have had a computer. (I may be biased here as I was just reading some stuff on the subject and I tend to take some of the writer’s ideas to the head for a little while until making my own up, what can I say I m stupid and unoriginal.)

Ok I better write this code soon or my sup is going to kill me, so I ll write a bit more later on, till then do punch in those ideas that you keep telling me about, or not telling me about!

One thing that I do like to add to this already dense discussion…

Mostly during the 20th century and now the 21st century, almost all mathematical axioms, and postulates of physics were put in retrospect to either previous laws or definition, Rather than the other way around, which was the way for the greater part of the human history. Which is actually a good approach but only sometimes fail to give viable and practical results.

To give the example (not of course of the impracticality) is the Minkowski space which actually came after the definition of special relativity, only to give Lorentz transformation and Special relativity the mathematical framework they needed.

I think I dont need to give you the examples of impracticalities, which are quite numerous in both relativity and quantum mechanics. The only problem is that these associations again predict certain observations at the axiomatic level that may or may not be true (as I said before), thus these theories would be more of a mathematical exercise rather than that of physics.

And one final thought before I turn this over to you, Nowadays, most physics is first done at the blackboard with mathematics and then translated over to the physical world later on. Again this would make mathematic and formal systems again the cores of our physical world rather than the other way around.

In short, these days, scientists, especially physicists want formal system before they go out to do experiments and prove the formal systems. his is probably because the costs of experiments is very very high compared to what it was a hundred years ago, which again puts mathematics and all its not so juicy formalism in the foreplay.

Which would again kick Godel’s incompleteness back in.

Here is a definition of formal system relevant to our discussion from wikipedia, my most convenient to quote source at the moment:

http://en.wikipedia.org/wiki/Formalism_(mathematics)

“According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other contensive subject matter — in fact, they aren’t “about” anything at all. They are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation (or semantics).”

Please notice an official disenchantment from attributing anything to the meaning of rules or axioms themselves. While you think 2+2=4 is true because it is ‘obvious’, a person who subscribes to formalism would agree to it only in the framework of rules of addition operation we set beforehand. Otherwise, to him, it wouldn’t bother if 2+2=5 given that you come up with another set of consistent rules of work. Hence, to them, mathematics is no different in essence than, say, the game of chess. You have some rules, you have some objects with predefined properties and without bothering to think whether it makes any real sense you want to compute the outcome in different circumstances.

Can Physics ever take this form where it would be reduced to playing with meaningless rules and essentially inexistent objects? Not really, at least so far. To repeat what I tried to say before (I don’t have time to go back and read), it is not the mathematical level of rigor we delve in that makes something formalistic. But the underlying interpretations we give to the exercise which makes it so. Physics starts off with axioms that are true by virtue of observation or any other thing for that matter (yes even philosophy). But a formalist starts off with assumption that it is quite absurd to ask the question of being true or not the axioms and rules he uses. But this system of mathematics raises other problems like completeness and etc. which is where Godel’s theorem kicks in. But there is just no way that due to extensive application of mathematics Physics would become a formal system. If it were true, then let alone physics we would not have any other elaborate interpretation of mathematics like Logicism, Platonism, Language etc.

Now what I actually came here in for was to know others thoughts on what we mean when we say that something inherently random but yet converging to a mean value. I would personally like to think that in the face of essential randomness, it is meaningless to make permanent expectations about things like mean, variance, probabilities etc. To give better perspective, in classical mechanics we used to resort to statistics because the precise details of the systems were too sketchy to work out. So our results converging to some value would just mean that the rules working underneath are such that they favour particular outcome with certain variance and blah blah. But in quantum mechanics, we get to *gulp* that there are no rules anymore because of the absence of hidden variables. Everything, in its own particular instance, is random to the hilt. Yet, system converges to a particular value. Why? How is it fathomable?

With that said, I will leave the issue of the implication of IT on physics at least for now. I am not up with energy and time and additionally, you can find a lot if you google with the keywords I suggested you before.

Since you appear not to be doing it on your own, here is a kind of a push:

http://www.mathshelper.co.uk/Unreasonable%20Effectiveness%20of%20Mathematics.pdf

And please, I am quite a normal person like anyone!

Ok I m going to put a post on my blog containing some of the stuff, naturally I ll password protect it so I ll send you the password by email.

Ok about Formailism, A general formal system can have the most wacky of statement as axioms then build upon them using theorems, but a subset of these formal systems are utilized by Physics to put its observations and fundamental predictions into a language that can be used to transmit the system across the community but also to give the system a set of bounds and properties. Again I would give Minkowski’s space and Later Einstein’s field equations as examples. The formal system DID came about after the observations. But lets get past this, I want to make my point and move on to your question. The point is Even though the formal systems are products of pure mathematics but they are employed at a very fundamental level in Physics as well. While these systems have an associated Godel’s sentences. Now who is to say that the Godels’ sentence will or will not affect the physical aspects of the calculations done with that formal system (BTW, formalism associates only the semantics and linguistic, yet MOST formal systems pilfer from the domain of the ‘contensive subject matter’, again Take any formal system employed in physics, and since I m going to use this example in the near future, thermodynamics is the case and point.

Furthermore, I talk about the systems devised in retrospective conforming to glitches found in the previous standing system. Again an example is thermodynamics, whhich abstracted the transitive law to the zeroth law and minus first law of thermodynamics. Please bear in mind that either of these law are somewhat the inherent Godel’s sentences in the systems. (Actually if you take one then the other becomes the Godel’s sentence, I maybe wrong here), neither can be exactly proven through observations and were added later to the classical system just to ‘formalize’ the system. Again, I m human and I maybe wrong. Would be honored if you could correct me.

What I think that in quantum mechanics, the probabilities are no longer actual probabilities, but really abstract variables just are inserted only to easy the calculations. The case and point would be the quantum computer and the idea of qubits, which are essentially variables, associated with a probabilistic nature, but variables nonetheless. Since probabilities are associated to either particle or wave nature of photons/electrons, and we now know that neither are exclusive to each other at the microscopic level but infact inclusive and equal to each other, thus would imply the the associated probabilities be again variables rather than ‘probabilities’, as each property associated to the ‘particles’ (Or shall we call the something else to avoid confusion?) are quantized, which would make the probability distribution again a variable which can hold a value between 0 and 1. I hope I m making sense. (Or not)

As Heisenberg’s Uncertainty principle and his matrix mechanics propose, One CANNOT under any circumstances, predict the position and momentum of a particle. (Which I cannot understand one could do as you propose in your previous comments) but rather calulation a probability distribution, or cloud for these variable, (which in turn, turn the probability into the main variable rather than the position of momentum, as one CANNOT measure either at a given time, which in actuality form the basis of Heisenberg’s work, a detailed briefing is given on wiki). As position and momentum are unobservable, and only their effect is observable (i.e the energy emitted or absorbed.) Thus one can only be certain or the probabilities and there end states. (Again I maybe wrong in my interpretations.)

So this would make and abstract quantity as your variable of interest as I satated before rather than the (again) ‘unobservable’ quantities. This would imply that your system’s probabilities were never ‘too sketchy to work out’, but rather your system was based one probabilities as the physical (or rather classical) quantities underlying beneath these probabilities are simply no longer the inputs.

Again this is my opinion and understanding.

Ok…. TYPOs! again, I hope you understand what I just vomited out!

I will have to hold your progress from the very beginning of your post. How did you conclude that physics employ ‘formal’ mathematical systems? : ) Why not we use Platonic interpretation of mathematical systems which Godel and in present times Roger Penrose susbcribe to? And of course these are the two famous and not exhaustive examples.

And OK, for the sake of argument I will accept your idea that physics is built on ‘formalist’ interpretation of mathematics. Now Godel will have relevance but only insofar when we need to give a ‘logical prove’ the ‘consistency’ of these systems. But as far as physics goes, do we even need to do it on the same pitch as mathematics? Why not say that instead of doing it logically, we will do it empirically? Yes, for a mathematician who is mesmerized in the closed world of his logical systems that will amount to nothing. But as far as the ‘business’ of physics is concerned, should it not be sufficient? I will like to know your thoughts on this.

I honestly fail to understand your train of thought in the example you gave of thermodynamics. But about your conclusion I should like to caution you that in my humble opinion, and I am very firm on it, you can NOT formalize physics until you deprive it of its business to talk about real things in physical universe. So even though I don’t really get your example, if you think that thermodynamics today or at some stage is officially turned into a pure figment of imagination then you would be right in thinking that it has been formalized. Otherwise, that is an impossibility. Just because I cannot observe the atom itself doesn’t mean that atomic model is synonymous to formal systems in mathematics. There are more other ways to empirically justify the physical existence of something without actually seeing it. To give a crude example, I don’t see you yourself but that doesn’t mean that my understanding about your existence far away is any way formalistic or non-empirical.

You might be making sense in your thoughts on the variableness of probabilities but embarrassingly I once again fail to get to the bottom line of it : ) It would help I think if you explain your case with the help of simplest system possible which would contain no or minimum details and complexities other than the quantum mechanical nature itself. To suggest you one, measurement of the polarization of light or photon can be used here. Suppose you have a 45 degree polarized light and at the detector we have vertical and horizontal channels and we want to see which channel will the photon be detected in. If we calculate the probability, we would say that there is equal probability for detection in both channels though the detection of individual photon is ‘inherently’ random event. Now if the event is indeed inherently random, then what does it mean to be able to calculate the probability? At this point, you might like to go through again the prelude I gave to this question in my previous comment ‘probably’ : )

About uncertainty principle, you can refer to any authentic source and find out that it says that one cannot predict with perfection SIMULTANEOUSLY under any circumstances BOTH position and momentum or any other set of canonical variables. If we perfectly want to predict one, we make the other absolutely uncertain. Having done quantum mechanics myself to some extent, I am very sure of what I am saying. But I would nevertheless be happy to learn your source of understanding.

And by the way, I don’t get why you want to make password protected posts on this? Or in fact, why does it even bother you to the extent that you would make posts on this? And last but not the least, why do we love Godel? : )

Ok unstacking the stack here so last question first.

I m fascinated by Godel and the works of mathematicians, and to tell the truth I dont know why. That maybe the correct answer, infact maybe I m fascinated because I dont really understand it. As for the password protected thingy, I was talking about the sharing of the books that I promised. Any since we are pilfering someone else’s blog, I would like to unburden the person of our banter here.

Ok uncertainty principle SPECIFICALLY states that you cannot measure the two related variables simultaneously. It also states that measuring one after the other even in seperate yet same experiments are not equal. If I need to explain this, I can do so by the example of scalar multiplication where commutation holds. i.e ab = ba. But in the case of matrix multiplication (which is exactly what is used in Heisenberg’s uncertainty principle) AB != BA generally. Again the only thing you can be certain about would be again the probabilities, i.e test every point of the existence of the particle and use uncertainty principle along with the exclusion principle to develop a probability distribution and the state of the particle. I took this right out of wiki, I know that it is not the most authentic source, but it is written for layman like me.

Dirac (and formerly Schroedinger) approach the problem by assigning wave functions and taking the wave nature of the particle into account. But the essentially do the same thing. Dirac however generalized the two approaches as a special cases of his equation.

Ok, I might have been thinking out loud about the probabilities thing, and my thought rarely make any sense to me either. I think what I wanted to say was that, we know for a fact that measuring something that we cannot measure or trying to see what we cannot see if a futile endeavor. But there is a rather diverted route that we can take which will (eventually) lead us to the same endpoint, which is the study of effects rather than the real thing. Now the only effects that we can observe for certain are as opposed to intuition the probabilities of the states of these particles rather than there actual characteristics. So it would make these probabilities the inputs (or outputs) to your systems rather than the characteristics. So if they are the inputs to your system, which seem to be abstract quantities would this not turn your system into a mathematical model rather than a concrete empirical system? Because, as the history of quantum mechanics will show, that there are not only numerous ways to interpret this data but also sometimes seemingly illogical ways to do so. (The main reason of conflict between Heisenberg and Schroedinger was that, Heisenberg was comfortable with the concept of quantum jumps, in which case a particle could occupy one position in space and the other in two consecutive instances of time. which would have either quantized time, or make the particle magical to appear in two places at once). The emphasis here is on the seemingly rather than illogical, merely because we are not actually measuring the states but rather hypothesizing about them, later seeing if the predictions are true. In fact most of our hypotheses are based on our own prejudice to accept or reject certain notions, as is in the case of Schroedinger, who called these quantum jumps absurd.

So getting back to point. our inputs are usually the iterative points in the domain that we are working on (in the quantum mechanics) we test each of these points for their probability and generate a distributing, rather than a trajectory as would intuitively be our goal.

Ok, as my understanding of the quantum mechanics goes, you cannot ‘observe’ a single photon and generate a probability as to which axis it would strike. Infact this was one of the major issues in quantum mechanics in the double slit experiment with a single electron demonstrates the inseparable particle and wave nature of an electron (or photon in this case) in fact even with a single photon you would get the same result you get with multiple photons i.e 45 degree polarization. You can see this in the double slit experiment. Again this problem is do to our inability to observe the quantum world directly and that we have to deal with the probabilities. In fact the mere act of measurement would collapse the wavefunction to that very experiment. (not my words, taken from wiki entry on Copenhagen interpretation, which again is one interpretation)

I think you would now see my point here, and I would not have to type any further.

My thoughts on the third para. Physics, depends on formal systems, but is itself not a formal system, the conclusions that we come to after the experiment are open to interpretation and logic, and where most of us observe the Occam’s razor (again a mathematical tool). In actuality, your idea of physics would turn it into a precise instrument for measurement only, which is not so, physics also makes prediction as every science does (although those predictions and based of observed facts coupled with logic, again mathematics) and this predictive attribute is what makes it useful and interesting. If Physics was restricted only to observations, we would not be talking about quantum mechanics at all, as you would not be able to observe the quantum level at all without collapsing it. So with the example of thermodynamics I was not saying that it has turned into a figment of imagination. Infact thermodynamics is one of the most concrete of studies which deals with not equations but actual characteristics of matter involved. No gas is arbitrarily treated as an ideal gas and have the Gas laws applied. Mechanical engineers use tables for each gas recorded to predict its effects in a certain system. It cannot go more empirical than this where the only mathematical tool is the Euclidean geometry is the form of interpolation and extrapolation. But yes, it has been turned into a formal system with its specific set of rules and semantics, lets call it a real formal system if it pleases you to differentiate between the pure mathematical formal system that can exist right out of imagination. And the introduction of the zeroth and minus one laws is the proof of this transformation, that although the derivation of the system started out from an empirical stand point, but a formal system was required to give it weight and turn it into a tool to be used in the future. Otherwise we would be deriving the whole system for each experiment. So yes, these systems are not synonymous to the mathematical formal systems, but are a subset of that greater set of formal systems nonetheless, where the issues regarding these formal systems would again inherently apply, but not to those axioms generated empirically, but those introduced to make the system consistent, again the case and point would be the zeroth law of thermodynamics. I hope I m making sense and correct. Wallahu Alam.

Qouting you…

And OK, for the sake of argument I will accept your idea that physics is built on ‘formalist’ interpretation of mathematics. Now Godel will have relevance but only insofar when we need to give a ‘logical prove’ the ‘consistency’ of these systems. But as far as physics goes, do we even need to do it on the same pitch as mathematics? Why not say that instead of doing it logically, we will do it empirically? Yes, for a mathematician who is mesmerized in the closed world of his logical systems that will amount to nothing. But as far as the ‘business’ of physics is concerned, should it not be sufficient? I will like to know your thoughts on this.

AND I AGREE!!! This IS the point that I have been trying to convey all along and in fact this is what Chaitin proposes for Mathematics! That Mathematicians should take a more empirical role than sometimes absurdly logical as you so correctly say! I hope we had come to an agreement on this earlier, and we would nt have had to type all this!!! This is the exact essence of Leena’s post if you read the two articles she posted!!! (Thank ALlah!!! woooh!)

As to your first line! AGAIN I AGREE and THIS is what Chaitin said in that interview! Mathematicians need to come out of their obsession with David Hilbert and His Formalism… That is actually Imre Lakatos Preaches and you should read him…. I ll surely put up the books soon. Please post a comment on my blog somewhere so that I actually do it rather than just talk about it… I tend to forget things.

I will go in the queue like fashion so FIFO : )

What you have read about uncertainty principle doesn’t contradict what I have been saying all along. AB != BA in general in matrix algebra and so do in quantum mechanics. In particular cases, you can find quantum systems where AB = BA when both A=B. You can read up more on it here:

http://en.wikipedia.org/wiki/Quantum_state

To quote the relevant section with some capitalizations of mine to stress the important part:

“For any fixed observable A, it is generally possible to prepare a pure state σA such that A has a fixed value in this state: If we repeat the experiment several times, each time measuring A, we will always obtain the same measurement result, WITHOUT ANY RANDOM BEHAVIOUR. Such pure states σA are called eigenstates of A.”

And please remember that above was brought into the discussion ONLY to show that it is possible to have definite, perfectly calculable probabilities in quantum mechanics in stark contrast to Turing machine’s halting program problem.

Moving on now and on to what is actually too abstract a passage for me to pinpoint what your goal had been. Quoting the range of it:

“Ok, I might have been thinking out loud about……………………we test each of these points for their probability and generate a distributing, rather than a trajectory as would intuitively be our goal.”

First and foremost, if it is an understanding based either on the work of a physicist or the results of your own doing hardcore quantum mechanics then it could be fine. Otherwise, if it is based on what you understood from qualitative texts combined with a lot of musing then I humbly suggest a very high probability of error in my sight. As I said before, I really don’t get what you are trying to get at but I will surely like to say that it reminds me of works by people who tried to justify the presence of probabilities in quantum mechanics through the framework of classical mechanics. It was something like oh we cannot measure the real thing directly as it is without bringing disturbance which at such a level tend to produce randomness etc. This was of course in the early days of quantum mechanics when this debate was possible. The present view doesn’t talk about measurement, along which you are trying to thing. Right now, we understand that the randomness is ‘inherent’ regardless of the fact that quantities are measured directly or indirectly through their effect. It is one of the bitter pills of quantum mechanics but it really has to be swallowed if you want to avoid contradictions in building the theory in its advance form.

Then:

“Ok, as my understanding of the quantum mechanics goes, you cannot ‘observe’ a single photon …. taken from wiki entry on Copenhagen interpretation, which again is one interpretation)”

Yes, you are right that we cannot observe a photon irrespective of the fact whether it is single or in torrent. But that is why I said we have detectors! And secondly, when we talk about probabilities, we mean to give probabilities to photon’s state AFTER the measurement and not before that. So collapsing of state as in copanhagen interpretation has no bearing on my question or the scenario therein. To tell you the outcome we get, as it is a well studied case, photon is detected randomly in both horizontal and vertical channel. If the experiement is performed long enough, now how long it should be is a different thing, we see that the result indeed approach 1:1 for both channels just as the theory predicts us. But really, you can forget about measurements, quantum mechanics and everything and just tell me what does it mean, in your understanding, to say I know the probabilities for outcomes even though the physical process is inherently random. Remember what I said about it making sense if the physical process is considered in the light of classical mechanics since it actually never entailed any inherent randomness.

Coming now to your example of thermodynamics. First, I disagree that physics depends on formal systems. What can be said with certainty here is that physics depends on mathematics. And that is it. Now whether mathematics is a formal system or not is seriously under dispute as their are other vying interpretations of mathematics available. As I told you before, mathematicians like Godel and Penrose believe that mathematics is a Platonic system and not a formal one. Secondly, I don’t say that physics doesn’t predict. It does, but its predictions have to live up to the touchstone of observation for it to be valid science. Newtonian mechanics could not which is why it fell off the pedestal. What I said about thermodynamics becoming figment of imagination was in response to your insisting on the adjective ‘formal’ for it which if not redefined from its original definition of mathematics would indeed turn anything into figment of imagination. Khair, now you have done it as “real formal”. But that is unnecessary labour, don’t you think? As long as formalism itself is struggling to get a certificate of authenticity in its own area that is mathematics, it will be a disservice to physics to name it or the interpretations of its theory after formalism as in “real formal”.

Now the last thing, I cannot be more surprised at suddenly finding you in agreement! It is not the good luck but because it indicates to me that I have been grossly misinterpreted all along : ) Khair, not only that I never contested that mathematicians should or should not take empirical approach, I think we never had a discussion on what Chaitin is saying ‘per se’. We always wanted a connection into physics, to understand its chaos and to that effect this is what I already said quite long ago:

“Coming now to the implications of Chaitin’s results, my understanding again goes in tangent to yours. I am not so comfortable with the seamless extension of his results to physical sciences. In fact, he himself doesn’t attempt to do it despite saying that he has been into physics community too. Rather, all he does is to encourage mathematical community to come closer in approach and expectations to the ones existing in physics!”

See, I did say it in passing without challenging that mathematics community should give up on doing everything by logic according to Chaitin. What I was actually trying to say in my last post was that EVEN if we accept the formalist interpretation of mathematics, Godel’s theorem might still not kick in when mathematics is applied in physics at least in the same way that it does in pure mathematics.

And if that is the essence of Leena’s post then I never objected to it : ) You can revisit the very first comment of mine on this post in which, among other things, it talks about her making connections between the randomness in physics with Godel’s theorem. And I still stand by my opinion that she is wrong on all accounts there. But no where did I try to say anything about Chaitin’s results ‘per se’. If anything, it was only their implication on physics that I ever talked about.

Khair, I apologize if I disappointed you by not letting you long with the pleasure of being in agreement. I was actually confused if I should let go of it or not but then decided that it is certainly fine for me in general if I cannot convince someone of my opinions but it is not a good habit to have people not understand my opinions at all. So still do you want me to drop comment? I suggest it would have been really exhaustive so lets rethink if we really need to continue this madness : )

You ll have to wait a wile for a response.

ahem ahem…trust me safi, i really tried to read all the comments, i think ill muster up courage and come back some time later 😉

@ Brickwall,

Ibn Taimiyyah was recommended for every Muslim to be read by M. Iqbal to understand Islam, and to become better Muslims than to became detached-from-emotions-and-cold-blooded-scholars and mere philosophers. That’s the way Iqbal was, he would refer not to himself but to the dead scholars for these matters, for only in dead ones can we put our faith that, at the least, outwardly, they died as good Muslims.

Apart from his works on logic and the sort of more scholarly and technical works, please tell me about his books he wrote for people with simple faith who wanted to understand and propagate the inner and outer significance and meaning of Islam. I would be grateful to for that kindness.

@bw:

either I m as dumb as I think I m or your article on Quantum states proves my point. for posterity, let me qoute so you may not need to move back and forth.

“When performing a certain measurement on a quantum state, the result generally described by a probability distribution, and the form that this distribution takes is completely determined by the quantum state and the observable describing the measurement. However, unlike in classical mechanics, the result of a measurement on even a pure quantum state is only determined probabilistically.”

“Now suppose that we start the particle with a random initial position and momentum. (For argument’s sake, we may suppose that the particle is pushed away at t = 0 by some apparatus which is controlled by a random number generator.) The state σ of the system is now not described by two numbers p and q, but rather by two probability distributions. The observables P(t) and Q(t) will produce random results now; they become random variables, and their values in a single measurement cannot be predicted. However, if we repeat the experiment sufficiently often, always preparing the same state σ, we can predict the expectation value of the observables (their statistical mean) in the state σ. ”

“In quantum theory, even pure states show statistical behaviour. Regardless of how carefully we prepare the state ρ of the system, measurement results are not repeatable in general, and we must understand the expectation value of an observable A as a statistical mean. It is this mean that is predicted by physical theories.”

“However, it is impossible to prepare a simultaneous eigenstate for all observables. For example, we cannot prepare a state such that both the position measurement Q(t) and the momentum measurement P(t) (at the same time t) produce “sharp” results; at least one of them will exhibit random behaviour.[2] This is the content of the Heisenberg uncertainty relation.

Moreover, in contrast to classical mechanics, it is unavoidable that performing a measurement on the system generally changes its state. More precisely: After measuring an observable A, the system will be in an eigenstate of A; thus the state has changed, unless the system was already in that eigenstate. This expresses a kind of logical consistency: If we measure A twice in the same run of the experiment, the measurements being directly consecutive in time, then they will produce the same results. This has some strange consequences however:”

So if I m understanding this correctly, you still cant have pure states i.e now exact states for a particles at the quantum level. And the method you propose for fixing states essentially fixes nothing, in fact iterates the same experiment multiple times to get a probability distribution (as output as opposed to the pure states) which is what I said earlier about your system having inputs and outputs as probabilistic/abstract variables rather than real quantities.

This again turns into a more philosophical debate on what is actually real and what is abstract. What quantities are real in the sense of our everyday life and what are abstract, and where physics is just a manifestation of reality… which would again turn into a more existential debate, as to what actually describes our universe (or simply ‘verse) mathematics or physics?

Lets not get into that.

Just for kicks… read this

http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.0147v3.pdf

As for formal systems and physics, sure, physics is not exactly a formal system, lets agree of this and forget the whole Chaitin and Godel thing.

I wanted to write a more detailed comment and discuss this in depth, but I am taking this course and it is starting to take its toll on me.

Safi:

So if I m understanding this correctly, you still cant have pure states i.e now exact states for a particles at the quantum level. And the method you propose for fixing states essentially fixes nothing, in fact iterates the same experiment multiple times to get a probability distribution (as output as opposed to the pure states) which is what I said earlier about your system having inputs and outputs as probabilistic/abstract variables rather than real quantities.

First of all, you are not dumb. At least not as far as the discussion is concerned. It is actually due to the density of information wrapped up in physical equations. I always held it, honestly, and now I am getting firm on it that modern physics cannot be understood from qualitative texts. You have to have a real flavour of it otherwise you will keep looking for simplistic situations either in the form of “rationalizing” the presence of randomness or believing that it MUST be ALL random.

The passages you have quoted do not negate my stand. It only says that even after performing iterations, pure quantum states still remain describable only through probability distributions. Now before concluding that I just regurgitated your idea, let me explain something here.

A quantum mechanical system is seen through some “observables” and EACH of these observables in general have their OWN probability distribution. So it is ONE quantum mechanical STATE of the particle but MULTIPLE probability distributions corresponding to each observable. As long as ANY of these variables are probabilistically defined, the system is said to be probabilistic. And it turns out, owing to uncertainty principle, that there will never be the case when we have no state left to be defined probabilistically.

What I was saying before was, and which is present in the passage I quote in the previous post too, that I can have an arrangement in which ONE observable, say A, will become deterministic. Keep measuring it, and only it(!), again and again and each time you will get the same value which even in the formulation of quantum mechanics is perfectly predicted (certain mean, zero variance). However, the cost of this is that the other canonical variable, say B, constrained with observable A through uncertainty relation becomes perfectly unpredictable (infinite standard deviation). In technical terms, we can say that we have reached an eigen state of the pure state! : )

Therefore, though it is true to say that randomness is part and parcel of quantum mechanics on the whole but generally speaking, that doesn’t apply to every SINGLE thing (or observable) contained therein. If you do not agree to my views, then how would reconcile your understanding with the passage I quoted before from this very article? Specifically, the following:

“If we repeat the experiment several times, each time measuring A, we will always obtain the same measurement result, without any random behaviour. Such pure states σA are called eigenstates of A”

See, it mentions an absolute collapse of uncertainty in the measurement of A in quite uncertain terms.

As for the paper you referred, tell me if it attempts to connect Godel’s theorem with the absence of hidden variables? In case, I will like to find time to read it. Otherwise, let it bury here. As I said, I am not denying altogether any possibility of the applicability of Godel’s theorem on physics. And I consider it even if we take mathematics to be a non-formal system, like many mathematicians of yore at least. This is something I have said before as you would recall. My objection, however, was on the specific thought process through which you were trying to correlate the two. But lets not get into that again. Our posts are long enough already : )

Umer,

I wish I had read Ibn Taimiyyah’s works myself : ) Most of my knowledge of him is derived either from his opinions quoted here and there or from his followers among the latter-day scholars. Kinda like almost all of India/Pakistan practices and studies hanafi fiqh but without reading any book of Abu Haneefa himself, rahimahullah.

But of course the parallels are not so straight between the two. While Abu Haneefa RA didn’t author much (I have heard only one Al-Fiqh Al-Akbar on aqeeda), Ibn Taymiyyah was a prolific writer and teacher and a lot of the things he or his students wrote from his lectures have reached us down. But unfortunately, most of them are hitherto in Arabic. And this is precisely the reason that still keeps me practising with my Arabic in the hope that one day I will become fluent enough to read his and his student Ibn Al-Qayyim’s works in Arabic : ) Insha Allah, soon I will be over with my present preoccupations and then will have time to actually do a lot of things.

Coming to what you can do now, there is at least one book in English (and I hope in Urdu too) published by Darussalam titled “The right way”. Another compilation which appears beneficial for simple folks likes us is “Ibn Taymiayah expounds on Islam”. Unfortunately, as I said before, I never got time to grab any of the two for myself so far.

If you like, I can try telling you about some scholars who are held to have followed in the footsteps of Ibn Taimiah. Theirs will be a good starting point, insha Allah. Just drop me an email if you are interested.

It is very intersting. The whole theory seems to be incomplete. Nobody is so sure about the origin of the universe – how it started, where from it evolved, and so on. But definitely one thing seems to be certain that there is a basic rule that is followed everywhere. Things are fixed. Nature somehow knows how to proceed and it always proceeds that way! Thanks for the nice article.