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About LinkBacksDo you think that, provided enough information, we could calculate truly anything? Come up with your own example, anything works.
September 21st, 2007, 02:42 PM
Do you think that, provided enough information, we could calculate truly anything? Come up with your own example, anything works.
Not sure I entirely understand your question but yes I think there is a lot that can be calculated given enough info. Just look at all the chem research being done on computers using MO theory. Although the power of current computers still limits a lot of that work forcing people to make approximations or just have the time to calculate it.
Yeah, I suppose I should have included that we'd have to have the proper technology.
Here's a crazy example... Do you think it would be possible to determine where every water molecule (supposing we could mark them to distinguish one from another) will eventually end up, when a drop of water hits a surface? We would know velocity of the drop, location of each water molecule within the drop, what water molecules were H-bonded to what other molecules, the EM force holding them together, etc. And please note, this is purely hypothetical.
I thought of this because of the fact that mathematics is so intrinsically linked with nature. I would think that if everything is governed by strict mathematical relationships, we really should, as I said, given enough information, and the ability to calculate it, and the knowledge of how to calculate it, be able to calculate anything (or at least all physical phenomena).
Yessssssss.
Oh HHHHHHH.
Yessssssssssssss.
But (sorry to cut you of there).
You IDIOT.
yOU very BIG idiot.
(NO, i ACTALLY LIKE YOU.....KISS KISS HUG HUG)
You are assuming you can calculate FASTER than DA GOD OF DA SPACE-TIME......DA awareness of DA LORENZ TRANSFORMATION REFERNCE.
hem. but really. to calculate everything, you'd need an infinite amount of calculations.
Think about it, there are only a countable number of programmes available (as you have a finite set of operations and thus a countable number of possible program lines. Since every program must be finite in length and as a countable union of countable sets is countable there are only a countable number of possible programmes) to calculate an uncountable number of real numbers. So there must exist some real number that no program count compute (i.e. give the n'th digit in its decimal expansion in finite time).
I recently read Hawking's "Brief history of time" and in it he discusses basically this question that given complete knowledge of an initial state could we predict (through calculation of some sort) everything that is to come. The problem with doing this is the uncertainty principle. It prevents us from knowing exactly an initial state. Also, as we see in quantum mechanics which uses the uncertainty principle we can often only give probabilities about an outcome.
i think the chaos theory applies. so that would be a no.
42
42 is the answer, but what is the question?
21 times 2Originally Posted by KALSTER
At the moment, no.
That is not just because of limitations in computing power, but in limitations of our physical understanding - As Supernothing rightly pointed out, quantum uncertainty (The uncertainty principle) dictates that measurements of systems are not deterministic, but rather probabilistic. (BTW, I have just started reading that hawking book, some really interesting stuff in there, recommended to anyone, phd in mathematics or not :P)
The measurement problem is an example of this uncertainty - The state(s) of a system will change when we attempt to measure it. By measure, I'm not referring to a physical probing of the system by apparatus, but rather a mere observation of the system by a consciousness. This leads to potential weird scenarios, the most popular being that if you put a cat in a box, Where no form of measurement of the cat is available, then the cat is both dead and alive at the same time! ...A superposition of quantum states.
Ok, I think a few people here are trying to be a bit too clever for this problem. Chaos itself is not a limit to computation - it merely bounds the "trust interval" of any numerical solution to a problem. However, you can get better and better solutions by including more and more significant digits - so chaos doesn't limit computation theoretically.
Secondly, quantum mechanics merely changes the questions you can ask and not the ability to compute them. I fail to see how the arguments provided so far actually limit the ability to compute, especially the talk about the uncertainty principle.
The limit to computation is fundamental to the idea of computation. There are only a countable number of programmes to compute a uncountable number of real numbers - so there must be some uncomputable real numbers lurking on the real line (in fact majority are). Chaitin's constant is an easy example of such a number.
i guess. the more information we can gather, the more precise we can make our calculations. but ultimately we could never have an absolutely precise calulation. we will always have the +/- error to deal with.
But surely, to calculate everything, to have all information, requires all questions to be asked... and answered?Secondly, quantum mechanics merely changes the questions you can ask and not the ability to compute them. I fail to see how the arguments provided so far actually limit the ability to compute, especially the talk about the uncertainty principle.
Think about it, there are only a countable number of programmes available (as you have a finite set of operations and thus a countable number of possible program lines. Since every program must be finite in length and as a countable union of countable sets is countable there are only a countable number of possible programmes)
I am not a mathematician, so countable sets don't form part of my day-to-day vocabulary. However, if one considers a program that is intended to do something quite simple, such as determining the area under a curve but, being a program, it does it numerically rather than by integral calculus, - isn't it the case that such a program could not arrive at an exact answer in a countable number of steps, whereas the use of integral calculus could give an exact answer quite simply in a short space of time?
What I am getting at is, whilst it could be true that numerical approaches would require an uncountable number of steps, is it possible that some other approach could arrive at an exact result in a countable number of steps?
As an example of a quantum system, consider an electron moving freely in space.
The electron will have a 'position wavefunction' which tells us the probability of finding the electron at any point in space.
It will also have a 'momentum wavefunction' which tells us the probability that the electron will have a certain momentum, or velocity at a given moment.
So long as the electron is moving freely in space it has an infinite uncertainty in its position - it could be anywhere. For an electron trapped inside a tiny hypothetical box, the uncertainty in it's position is very small, and we say it is localized.
If you know the position wavefunction, you can work out the momentum wavefunction via a Fourier transform and vice versa.
The nature of this relationship is that if the position wavefunction is localized, then the momentum wavefunction will be well spread out and vice versa. Thus the more certain we can be about the value of one of these wavefunctions (how accurately we can pinpoint the electrons position for example), the less certain we can be about the other.
The fundamental mathematics of this relationship therefore says, that we cannot know the precise position and velocity of the electron at the time, or any other properties of the particle. This behavior is inherent in relating two types of wavefunctions together. Since the wavefunctions tell us all information we can possibly have about the system (in probabilistic form) then this is the extent of the problem.
Programmatically, or otherwise, our universe 'does not permit' us to know with certainty the value of two or more properties of the electron at the same time.
You are raising a different question here - first you can get to the "exact answer" in a countable number of steps (which is the joy of first countable spaces, more jargon to through around lol). However, a countable number of steps is still too many to be useful - infinite running time is excluded on principle. Secondly, numerical integration is usually the best you can get for many functions - no amount of pen and pencil calculus will give you a closed form solution to the integral of e<sup>x^2</sup> for example.
As mentioned earlier, countability of the steps doesn't help. I think you are missing my point though - the number of total possible programmes is countable however the length of any one programme is finite.
The question should be what is a valid question - asking for both the position and momentum of an electron for example is not a valid question as it does not make sense physically (as the momentum and position operator do not commute). It's a kin to asking someone to draw a square circle.="bit4bit"]But surely, to calculate everything, to have all information, requires all questions to be asked... and answered?
:PThe question should be what is a valid question - asking for both the position and momentum of an electron for example is not a valid question as it does not make sense physically (as the momentum and position operator do not commute). It's a kin to asking someone to draw a square circle.
Well unfortunately I can't say that I fully understand all of the maths involved, but the principle above, which I've read from numerous sources in the past, seems physically to be quite clear. If the electron is moving through space, then it must have a momentum, and must have a position, at any point in time. Suppose our goal was to calculate both values for any given point in time? Quantum mechanics says we can't, and so this is an example of a system where any known calculation will not give us the desired result - both values.
Perhaps my understanding of this system is a little general, but nonetheless it will not allow me currently to believe anything other than stated above... so there!
To river_rat:
Thank you for your considered response. Shortly after writing my message above, I read it and thought, "what a load of nonsense I have just written". My reasons for thinking this seem somewhat different to those you outlined. I won't go into them for fear of writing a second load of nonsense!
However, I would like to pin down the nature of the problem. As I understand it, the question is, can we calculate any number with the limited set of operations (addition, multiplication and so on) at our disposal? For example, if these operations are represented by a, b, c ..... , can we generate any number N by operating on some starting number. N<sub>0</sub>, as in:
{a, b, b, c, a, ...}N<sub>0</sub> = N ?
I take it that the answer to this is no. If that is so, the question arises, what are these numbers that cannot be calculated in this way? Irrational numbers come to mind - are there others?
river_rat - I'll be honest and say i do not entirely understand your reasoning but it seems to me that you took at mathematical/computational approach while I was looking at it from more of a chemistry/physics basis. Either way we seem to have arrived at the same conclusion.
Thank you gentlemen for maintaining a thread in which you all discuss a subject in an objective, informative, critical yet supportive manner. What a change from the bickering and petty animosities on several other threads.
Maybe there is something in this mathematics lark after all.
Group hug. :wink:
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