DaveC426913 said:
The "job" is to know the position and trajectory of every particle at a given moment in time. If we live in a Newtonian universe, this is enough to know every event that precedes the present moment and every event that follows it as long as we know it to a high enough degree of accuracy. Thus, it would be knowing everything (is this sounding familiar?).
With respect, this shows your error in interpreting “what the job is” in this thread, and this is why I asked YOU to state what you THINK the job is, because your idea of what the job is clearly does not agree with mine. Read the first post in this thread once again. It quite clearly says :
Imparcticle said:
Is it possible to know everything?
………… one person can know all that there is to know about the universe.
The “job” is thus NOT to know things “to a high enough degree of accuracy” – it is to know everything – ie to an UNLIMITED degree of accuracy. And the job is not simply “to know everything in principle” but to know it in practice. (the question did not specify “is it possible to know everything in principle?” – it simply specified “is it possible to know everything?”).
Thus, we must ask : Are there any insurmountable impediments to our knowledge which would make it impossible for us to know everything? This means “practically insurmountable” impediments as well as “in principle insurmountable” impediments.
DaveC426913 said:
Now, we don't have to know about a particle's postion and trajectory to an *infiinite* degree of accuracy (which you are claiming, not me), and we don't have to know it *exactly* (your claim, not mine), and we don't have to reduce it to integers (your claim, not mine). We need to know about it only to a degree of accuracy that is (granted, very, very high) high enough to confidently know where that particle will be a billion or trillion years (or whatever you want) in the future.
The question is what is “high enough” to do the “job”? The “job” remember is “to know everything”, not simply to know most things approximately, or to an arbitrary degree of accuracy. If you simply wish to make approximate or arbitrary measurements and predictions, I can equally well argue that the HUP does NOT prevent me from doing the job – I can make approximate measurements and predictions even taking HUP into account. In fact, in the macroscopic world chaos is a much greater limitation than the HUP to accurate predictions. When you play billiards (pool), your inability to predict the ball’s trajectory more than a few collisions in advance is due to chaos, and NOT due to HUP. We rarely if ever encounter HUP limitations to our epistemology in the macroscopic world, but we almost always encounter chaos limitations to our epistemology.
But making approximate measurements, I repeat, is not “knowing everything”.
DaveC426913 said:
I hope we are now back on track. The “job” is to know everything, “all that there is to know”, and not simply to make an approximation to it.
Now let’s look at specifics. How accurately do you wish to measure the position of each particle in our snapshot of the universe? In other words, to how many decimal places do you wish to measure the position? You obviously (refer to your above statement) do not want to measure to an infinite number of decimal places. Let’s say that you wish to measure it to P decimal places, where P is an integer. You may say that P can be arbitrarily large – I will show below that P must necessarily be very small.
Now because you have “truncated” the real (true) position measurement by taking an arbitrary P number of decimal places in your measurement, this necessarily means that your future predictions will also be limited in their accuracy. Let us say that taking P decimal places in your original measurement of each particle allows you to predict those particle’s future positions at time t to an accuracy of Q decimal places, where Q is another integer. There will be a 1:1 mapping between values of P and Q – for any given integer P decimal places of original measurement there will be a limit to the accuracy of your future prediction at time t of Q decimal places. Yes P can be “arbitrarily large” if you have enough storage espace, but P is always an integer and Q is always an integer. There is an a priori limit to the precision of your predictions imposed by the value of P that you choose. This is an “in principle” limit, not simply a practical limit, imposed by your choice of P.
The same argument applies to measuring the velocity of the particle. The “real” velocity is a real number, and you must necessarily truncate this to P decimal places when you measure and record that number. The P decimal place number that you store is an approximation to the real velocity of the particle.
“So what” I can hear you say – “I can make P as large as I like – I can make it arbitrarily large”.
But can you? You now need to record these P decimal places of position for EVERY particle in the universe (the job is to know everything, remember?). If there are N particles in the universe this means you will need to record 6xN numbers (one each for the X, Y, Z coordinates, one each for the dX, dY, dZ components of velocity), and each of these numbers to to P decimal places of accuracy. Thus (assuming you are working in base 10) you will have 6xNxP base 10 digits to store and to work with. N is a VERY large number – possibly as large as 10^87.
Now you say “But I can compress the data by working with relative positions and velocities rather than absolute positions and velocities”
This saves you just 6xP digits out of your total of 6xNxP digits. So now you need to store and work with just 6x(N-1)xP digits.
Now the big question, which gets us back to how large P can possibly be : Where are you going to store all this information? You only have the universe itself to use as your storage device after all. Thus your “job” is to store 6x(N-1)xP decimal digits using just N particles. If P is equal to 1 or more decimal places, you simply will not enough enough storage space – not just in practice but in principle. Once again this is an “in principle” limit – not simply a practical limit. Let’s put some numbers into “see” the problem more clearly.
How accurately would you like to measure the positions and velocities? To 5 decimal places perhaps? This seems reasonable (but it would be grossly insufficient to then allow us to claim that “we know everything” – because in fact we would know and predict to only 5 decimal places or less, we would plainly not know everything).
OK, so let’s assume P is 5. N is probably around 10^87. Thus we end up having to store 6 x (10^87 -1) x 5 decimal digits, which is effectively equal to 3 x 10^88 digits. We must store these digits using 10^87 particles (I am not even going to bother going into the practical problems of storing just 1 digit using 1 particle). It simply cannot be done – it is impossible in principle. Using 10^87 particles and ignoring practical storage problems, we could not encode more than 10^87 digits (in fact the number would be much less than 10^87 digits) – but this means that our storage limitation – a limitation in principle not just in practice – forces us to record each position and velocity to LESS THAN one decimal place.
The only possible solution you might have at your disposal is to be able to store more data IF the data can be compressed – but this reqires an incredible amount of redundancy in the data, and we have NO reason to believe there is any redundancy in the relative position or velocity data of the particles involved. But no matter how much you compress the data, you will always necessarily store an integer number of decimal places for each particle - whereas the real particle's postion and velocity are denoted by real numbers each with an infinite number of decimal places. No matter how much you compress the data, infinity always remains infinity. You simply cannot, in principle, store the "real" position data, the best you can ever do is to store an "approximation to it". Thus there is ALWAYS an in principle impediment to "knowing everything".
I will cut some slack here and I will agree that the absence of data redundancy is a practical limit at the present time, rather than an in principle limit. But there is no reason to believe, given what we know about the universe, that this limit will ever be overcome. And even if it can be overcome, the real data is a real number and can NEVER be compressed to an integer number of decimal places. And this, dealing as it does with the "definitional differences" between real numbers and integers, is an In Principle limit and not merely a practical one.
We can continue to argue whether the lack of storage is an in principle or a practical limitation – but I will also remind you that the original question did not ask whether it is possible to know everything “in principle” – it simply asked whether it is possible.
Thus the correct answer to the question posed at the beginning of this thread, which was “Is it possible to know everything?”, is “No, for two reasons – lack of storage space, and the HUP”.
MF
