A Matter of Steam-Powered Math

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A Matter of Steam-Powered Math

Postby The Great Hippo » Thu May 22, 2008 10:46 pm UTC

I'm writing a story and a big question popped up. It concerns physics as much as math, but I thought this might be the best place to ask. I understand that this question is a bit too wide open for a very solid and scientific answer--that's all right. I'm more interested in what's possible.

When you divide by 0 in conventional devices, you either get a error or crash the program. That's because it's an illegal operation; computers can't handle it. You reboot the computer (assumedly!) and move on. But what happens if the computer isn't an electronic device, but a mechanical one?

That's to say--if we had, say, a device like Babbage's Analytical Engine hooked up and functioning, and proceeded to perform an illegal operation (take your pick), what sort of results would we get? Would it spout out an unintelligible answer? Would it enter a loop? Could it plausibly even break the machine? Are any or all of these a possibility, depending on the nature of the function? What happens when a mechanical representation of math encounters one of its nonsensical dead-ends?

I apologize in advance for getting my filthy non-math hands in your math forums. If this is a stupid question and I am a stupid person for asking it, let loose with your bitter vitriol.

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Re: A Matter of Steam-Powered Math

Postby GreedyAlgorithm » Thu May 22, 2008 11:05 pm UTC

IMO it is a stupid question but you are not a stupid person for asking it.

Computers are made of matter too. Steam doesn't make things magically explode. "Because it tried to divide by zero" is not an explanation for why a computer crashes except insofar as it might point someone to the real reason, like "if my 'divide' function has a zero as its second argument, something probably went wrong and I probably want the program to stop trying to continue, so in cases like that I have the program throw an error instead".

So you could say "well, to divide A by B, my machine repeatedly makes a pile of B marbles and moves them to a pot while adding a single marble to a new pot, until the first pot contains at least A marbles, and then my answer is the number of marbles in the second pot". In that case the machine would keep adding marbles to the second pot until (for example) the marbles started falling out of the full pot and maybe clogged up the machine.

Or you could say "I have the same machine as before except the lever arm that moves the pile of B marbles won't actually move until it has at least 1 marble." Then maybe the machine stalls.

Basically you can make up plenty of machines that will do different things, and then programmers will say "why didn't they just add such-and-such to prevent the problem??" :)
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Re: A Matter of Steam-Powered Math

Postby jaap » Thu May 22, 2008 11:10 pm UTC

The Great Hippo wrote:That's to say--if we had, say, a device like Babbage's Analytical Engine hooked up and functioning, and proceeded to perform an illegal operation (take your pick), what sort of results would we get? Would it spout out an unintelligible answer? Would it enter a loop? Could it plausibly even break the machine? Are any or all of these a possibility, depending on the nature of the function? What happens when a mechanical representation of math encounters one of its nonsensical dead-ends?


Mechanical division is usually done in by a long division, i.e. repeated subtraction of the divisor, usually shifted left. An attempted division by zero will therefore result in an endless loop, where it repeatedly subtracts zero or repeatedly shifts zero to the left. It might be as if it is 'stuck in neutral' as it were.

I don't know for sure how square roots or logs are done. If it were by a Taylor series approximation, then inputting a negative number would cause it to evaluate a non-converging alternating series. It would add a small number, subtract a larger number, add an even larger number, subtract a larger number still, etc. If it were programmed to do a particular finite number of terms, it would end up with a large positive or negative number that is obviously wrong.

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Re: A Matter of Steam-Powered Math

Postby UserGoogol » Thu May 22, 2008 11:29 pm UTC

The posters above said what I was going to say. There is no fundamental difference between the Analytic Engine and a modern electronic computer. They both work by cranking through algorithms, not by "physically doing" the division. If a machine ignored the rules of division, it would either go into an infinite loop or produce an inaccurate result.

But I did think of a machine that would work by physically doing the numbers and which would break, which I think is kind of neat.

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You have a metal rod on a hinge. That metal pole passes through a blue widget which can move up and down a second vertical pole which is fixed one inch away from the hinge, and that blue widget also has a second pole fixed to the blue widget which points horizontally. You have a second red widget which can move freely along the pole. To divide x by y you move the red widget to the point x inches to the right and y inches up from the hinge. The horizontal pole then moves to a position x/y inches above the hinge. If you tried to divide a number by zero, the hinges would be perpendicular, and either you would not be able to move the red widget into position or the machine would break. That said, this is also what would happen if x/y is merely very very large (since the machine would be of limited size, so the poles wouldn't line up in practice even though the lines in principle intersect) or if x or y is too large (since the pole wouldn't be able to reach over) so it's not too meaningful, but still.
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Re: A Matter of Steam-Powered Math

Postby antonfire » Fri May 23, 2008 4:13 am UTC

Essentially, whenever a machine (steam powered or not) reaches an illegal operation, it does whatever it was programmed to do. If the programmer is intelligent, it raises an error and either moves on or stops the program. If the programmer is an idiot, it gets stuck in a loop or returns a nonsense answer or something like that. If the programmer and designer are both idiots, I suppose something could go physically wrong.

It's possible that a prototype - one that the designer didn't intend to be used by just anyone, so ey could make some assumptions about what the operations would be - might break in some spectacular manner on an illegal operation, I suppose.
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Re: A Matter of Steam-Powered Math

Postby The Great Hippo » Fri May 23, 2008 4:44 am UTC

I was thinking of crank-operated calculating engines which are used to perform bank interest functions and store the resulting data. Someone particularly sneaky manages to figure out a way to submit a bank account with certain features that will cause an illegal operation to happen at a specific time (the machines are turned on for a bit every day to calculate interest), throwing the machines into an endless loop. The calculating engines cannot begin new functions until the previous function is reconciled; since the illegal function can never be reconciled, they have to be reset (meaning all data must first be recorded by hand, which takes an incredible amount of time).

I think I see where the error of my judgement came about, now. This has been very helpful!

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Re: A Matter of Steam-Powered Math

Postby Mettra » Fri May 23, 2008 11:52 am UTC

Just thought I'd make an additional comment to back up some things touched on above. The reason a program will crash if it divides by zero is basically because the programmer made a special case for it, or the programming language/module has a built-in case-specific fix. A programmer could program 0/0 to output 1 if he wanted. It's not that we give the computer the problem and let it decide on its own what the solution is. It has to have a program (designed by a person, obviously) that is designed to answer such questions. So the limit of computer knowledge is the limit of human knowledge in this case. Computers are actually quite low on intelligence, their strength comes in brute force + programming interface.

A physical case of dividing by zero might (oh internet gods forgive me) be a black hole. The fundamental object is a singularity which has no dimensions (it is a geometrical point) yet it contains mass so it must have infinite density. There is an event horizon around the singularity (like a sphere of inescapable force, hence the name black hole) but that is a detail - the singularity is the 'main event' if you will. I personally believe that such objects are unphysical, and that the singularities we have in black holes are not geometrical points but follow the Pauli exclusion principle and the Heisenberg uncertainty principle. But I have no theory or framework within which to demonstrate that this might be true. The only thing we have at the moment that describes black holes is general relativity so the idea is generally accepted for now. In fact, general relativity has quite a few disturbing aspects probably because it doesn't include any kind of quantum limitations - this is one of the main goals for theorists today, to combine quantum theories with general relativity.
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Re: A Matter of Steam-Powered Math

Postby SlyReaper » Fri May 23, 2008 2:03 pm UTC

Mettra wrote:Just thought I'd make an additional comment to back up some things touched on above. The reason a program will crash if it divides by zero is basically because the programmer made a special case for it, or the programming language/module has a built-in case-specific fix. A programmer could program 0/0 to output 1 if he wanted. It's not that we give the computer the problem and let it decide on its own what the solution is. It has to have a program (designed by a person, obviously) that is designed to answer such questions. So the limit of computer knowledge is the limit of human knowledge in this case. Computers are actually quite low on intelligence, their strength comes in brute force + programming interface.

A physical case of dividing by zero might (oh internet gods forgive me) be a black hole. The fundamental object is a singularity which has no dimensions (it is a geometrical point) yet it contains mass so it must have infinite density. There is an event horizon around the singularity (like a sphere of inescapable force, hence the name black hole) but that is a detail - the singularity is the 'main event' if you will. I personally believe that such objects are unphysical, and that the singularities we have in black holes are not geometrical points but follow the Pauli exclusion principle and the Heisenberg uncertainty principle. But I have no theory or framework within which to demonstrate that this might be true. The only thing we have at the moment that describes black holes is general relativity so the idea is generally accepted for now. In fact, general relativity has quite a few disturbing aspects probably because it doesn't include any kind of quantum limitations - this is one of the main goals for theorists today, to combine quantum theories with general relativity.


As far as I know, the singularity of a black hole does have finite, non-zero volume. It's just very small. So it's not dividing by zero, just dividing by epsilon, for every epsilon greater than 0...
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Re: A Matter of Steam-Powered Math

Postby antonfire » Fri May 23, 2008 3:23 pm UTC

The Great Hippo wrote:I was thinking of crank-operated calculating engines which are used to perform bank interest functions and store the resulting data. Someone particularly sneaky manages to figure out a way to submit a bank account with certain features that will cause an illegal operation to happen at a specific time (the machines are turned on for a bit every day to calculate interest), throwing the machines into an endless loop.
I think that that scenario would be reasonable, actually. It's not unimaginable that a programmer made a mistake somewhere that would leave the machines open to an exploit like this. It's not too different from some actual denial of service attacks. Especially if there's no such thing as the internet in your world, I imagine programmers wouldn't be as security-conscious as they are in ours.
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Re: A Matter of Steam-Powered Math

Postby BeetlesBane » Fri May 23, 2008 4:36 pm UTC

after quoting Mettra, SlyReaper wrote:As far as I know, the singularity of a black hole does have finite, non-zero volume. It's just very small. So it's not dividing by zero, just dividing by epsilon, for every epsilon greater than 0...

If the volume is non-zero it's not, strictly speaking, a singularity; it's a physical entity that approximates a singularity.

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Re: A Matter of Steam-Powered Math

Postby SlyReaper » Fri May 23, 2008 5:19 pm UTC

BeetlesBane wrote:
after quoting Mettra, SlyReaper wrote:As far as I know, the singularity of a black hole does have finite, non-zero volume. It's just very small. So it's not dividing by zero, just dividing by epsilon, for every epsilon greater than 0...

If the volume is non-zero it's not, strictly speaking, a singularity; it's a physical entity that approximates a singularity.


The word means different things in mathematics and black hole physics.
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Re: A Matter of Steam-Powered Math

Postby maninblack » Fri May 23, 2008 6:22 pm UTC

Mr. Hippo sir, I love your question and the idea in which it fits. As antonfire said if the programmers were not concerned with security any number of things could happen. I like the idea of your fictional calculators grinding away indefinitely, all the while adding interest to some well constructed account. But the machine could fail in any number of ways if there were no preprogrammed response.

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Re: A Matter of Steam-Powered Math

Postby gmalivuk » Tue May 27, 2008 7:34 pm UTC

SlyReaper wrote:
BeetlesBane wrote:
after quoting Mettra, SlyReaper wrote:As far as I know, the singularity of a black hole does have finite, non-zero volume. It's just very small. So it's not dividing by zero, just dividing by epsilon, for every epsilon greater than 0...

If the volume is non-zero it's not, strictly speaking, a singularity; it's a physical entity that approximates a singularity.

The word means different things in mathematics and black hole physics.

No, it really doesn't in this case. The *mathematics* of black hole physics, at least in the current state of the art, do in fact predict a point of infinite density and zero volume.
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Re: A Matter of Steam-Powered Math

Postby Robin S » Tue May 27, 2008 7:46 pm UTC

I think what SlyReaper was trying to say was that, yes, general relativity predicts a point mass singularity, but "the real world obeys quantum physics" and therefore any actual singularity would not be a point mass.
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Re: A Matter of Steam-Powered Math

Postby Token » Wed May 28, 2008 11:59 am UTC

Does quantum mechanics rule out point masses?
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Re: A Matter of Steam-Powered Math

Postby SpitValve » Wed May 28, 2008 12:10 pm UTC

Mechanical calculators exist. They were used before modern computers. And you could definitely divide by zero. It'd set the thing whirring crazily as it (presumably) attempted to continually subtract zero from a value. You had to pull the plug to stop it. My profs have anecdotes of walking down the corridor, and hearing the distinctive manic churning of a machine dividing by zero, following by swearing as the operator realises s/he has to start all over again...

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Re: A Matter of Steam-Powered Math

Postby BeetlesBane » Wed May 28, 2008 2:09 pm UTC

Token wrote:Does quantum mechanics rule out point masses?

It's been a long time since I took any quantum theory courses, but I suspect the Heisenberg Principle would cause srious problems with attempting to define a point mass.

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Re: A Matter of Steam-Powered Math

Postby Mettra » Wed May 28, 2008 4:26 pm UTC

Token wrote:Does quantum mechanics rule out point masses?


This is a very pointed (no pun intended) question. There are two different 'meanings' of point mass, one which is a convention in Newtonian mechanics and the other which is a 'real' physical object.

In Newtonian mechanics, physicists often use the idea of a point mass in order to more simply examine the consequences of some system taking place. For example, in collisions, one normally thinks about two objects colliding as point masses at first - this is just so we get a fundamental idea in our heads about what will happen. If object A collides with object B that is at rest, momentum will be conserved and we can normally do the math in our head. However, this isn't a true enough picture to actually use for most real problems. 3D objects (I guess even 2D objects) can rotate as well as translate, and they also have a geometric shape which normally ends up 'biasing' the directions and surface contacts of impact. Then there are other 3D properties such as material considerations - all of these additions complicate the real motion of the system. We use point particle analysis at first to get an idea of what generally OUGHT to happen.

Additionally, there are also very useful concepts such as the center of mass. The center of mass is a point (a real geometrical point) at which a physicist can assume all the gravitational force acts. It's as if the mass of the entire body is all clumped into that little point. If you toss the object forward and up, the body will rotate and have such a complex motion that it generally defies real analysis. However, the center of mass will just make a nice parabola. It's one of the most cheaty things that physicists can do. Turn some extremely complicated turning and twisting and flaying into a parabola. All the mass is not really at the center of mass, it's just a mathematical trick. I shudder at thinking how difficult it would be to figure out the gravitational effect on a body without using the center of mass.

But there is no such thing as a real point mass. It is utterly and unabashedly verboten by quantum theories, and these are our best theories with our best experimental agreement and stuff is pretty much 99.999% for sure everywhere. There are two main problems with point masses as a real object. These are two principles of quantum mechanics - the Heisenberg uncertainty principle and the Pauli exclusion principle.

The uncertainty principle says that you never know the position and momentum of a particle to a high degree of accuracy. The problem here is that we could measure the momentum of the point particle very very precisely but that's not allowed unless we don't know it's position very precisely (since it's a point, we know it's position fairly exactly) - that means, I think, that the particle would 'grow' in other words, it would effectively gain volume so that we're uncertain about its position.

The Pauli exclusion principle says that no two identical fermions can share the same quantum state. Now, there are two different types of particles, one is bosons the other is fermions. Fermions are particles like protons, electrons, and so on. Bosons are generally called force carriers. It's fermions that we generally think of as making up matter. So what's a quantum state... Well an example would be the 'orbitals' of electrons in atoms. Ever thought that it was silly for electrons (negatively charged) to orbit protons (positively charged)? Well it is. In fact, the only thing keeping the electrons from smashing down into the protons is the exclusion principle. Ever wondered why there's only a certain number of electrons allowed in each orbital? Pauli exclusion principle again. For example, the lowest orbital can only contain 2 electrons. These two are allowed since one of them has to have a different 'spin' than the other. If they both had the same spin, they would both be equivalent in the eyes of quantum mechanics and their wave functions might interfere destructively to ... 0. So they would just pop out of existence.

Also there's the additional headache of any point mass being automatically a black hole, since any amount of mass has a Schwarzschild radius > 0.
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Re: A Matter of Steam-Powered Math

Postby Token » Wed May 28, 2008 5:10 pm UTC

Mettra wrote:The uncertainty principle says that you never know the position and momentum of a particle to a high degree of accuracy. The problem here is that we could measure the momentum of the point particle very very precisely but that's not allowed unless we don't know it's position very precisely (since it's a point, we know it's position fairly exactly) - that means, I think, that the particle would 'grow' in other words, it would effectively gain volume so that we're uncertain about its position.

The uncertainty principle is a statement about probability distributions of operators on the particle's wavefunction, not a statement about the particle itself. It doesn't rule out knowing the position of a particle to any arbitrary degree of precision.

Perhaps a better question would be - is it meaningful to talk about the volume of a particle under QM?

(I should point out that I do have some knowledge of QM, and have not wondered about why electrons have orbitals since deriving it from first principles. Not enough knowledge to be useful, though.)
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Re: A Matter of Steam-Powered Math

Postby Mettra » Wed May 28, 2008 6:11 pm UTC

Token wrote:The uncertainty principle is a statement about probability distributions of operators on the particle's wavefunction, not a statement about the particle itself. It doesn't rule out knowing the position of a particle to any arbitrary degree of precision. [1]

Perhaps a better question would be - is it meaningful to talk about the volume of a particle under QM? [2]

(I should point out that I do have some knowledge of QM, and have not wondered about why electrons have orbitals since deriving it from first principles. Not enough knowledge to be useful, though.) [3]


[1] Not if you know its momentum to an arbitrary degree of precision. That's what I was saying. With a point mass, we could give it a very large mass and have an arbitrarily precise measurement of its momentum. By doing this we couldn't break the uncertainty principle, so we'd have to have some uncertainty about its position. Because of this fundamental uncertainty, it would appear to us as if it had some larger size, and if we measured it in sufficient dimensions, larger volume.

On the flipside, talking about measuring its position accurately gets confusing. It is a POINT mass, after all. When talking about the position of a point mass, you are inherently talking about infinitely precise position. If you weren't talking about infinitely precise position, then the position would have some error bars, and those error bars would translate into reality as 'the particle has some size/volume P'. This is my interpretation of a point mass in quantum mechanics at least. Maybe it doesn't even make sense to think about what they might 'look' like.

But to follow along that train of thought, if it's a point mass, it seems it would have to have infinite momentum. That isn't an real answer, so take it as you will.

I'd have to disagree with you and say that the uncertainty principle is in fact a statement about the particle itself. That's what gets people so upset about it in the first place.

[2] I'm not really quite sure what you're asking anymore, but it seems to me that volume or size in the quantum world is a given. Talking about lines or points or planes leads you to cylinders, spheres, and boxes.

[3] Then you'll have to forgive my spiel. I do it so much these days, it's like second nature. Since we're in the math forums, I thought it'd be more polite to assume less knowledge rather than more.
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Re: A Matter of Steam-Powered Math

Postby Token » Wed May 28, 2008 6:57 pm UTC

OK... try this. If I were to ask you to define what you meant by a particle's volume, what would you say?
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Re: A Matter of Steam-Powered Math

Postby Mettra » Wed May 28, 2008 7:14 pm UTC

Token wrote:OK... try this. If I were to ask you to define what you meant by a particle's volume, what would you say?


I'd make three length measurements along three orthogonal bases and multiply for a very coarse answer (some might call that 'size'). To get a better idea, I'd rotate my meter sticks in the appropriate directions by some angle 'a' and repeat as many times as I think is appropriate. I think you can imagine the effect the uncertainties have on the volume, and not just from innacurate measurement, but at a more fundamental level. That's sort of what I was getting at.

If you tried the other method, using the density equation and given some density, you'd have to find the mass, which also has the same kind of fundamental uncertainty.
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Re: A Matter of Steam-Powered Math

Postby Token » Wed May 28, 2008 7:36 pm UTC

That's more a possible method of measurement than a definition. I'm asking what exactly you MEAN by the volume of a particle.
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Re: A Matter of Steam-Powered Math

Postby rhino » Wed May 28, 2008 8:06 pm UTC

It's clear that there can't be a definition of volume in QM in the standard sense.

For a "normal" shape/object, the volume is (mathematically) defined to be the measure of the set of points that are part of the object.
(A measure in 3-dimensional space is a real-valued function on (some) subsets of R3 which obeys some important properties, like giving the correct volume to cuboids)
The precise definition of measure is fiddly, but there's clearly no hope of applying it here because "the set of points that are part of the particle" is not well-defined.

The best you could hope for is some integral based on its wavefunction, but I'm trying to imagine one and I can't see how even that could possibly make sense.

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Re: A Matter of Steam-Powered Math

Postby Token » Wed May 28, 2008 11:32 pm UTC

So if it's not possible to define what it means for a particle to have volume, it can't be said that it's not permitted for a particle not to have volume, surely?
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Re: A Matter of Steam-Powered Math

Postby Mettra » Thu May 29, 2008 3:30 am UTC

Token wrote:So if it's not possible to define what it means for a particle to have volume, it can't be said that it's not permitted for a particle not to have volume, surely?


I would say no to that. It's clear we're on shaky ground here, but instead of offering a positive argument, I can offer the negative one.

For any normal chunk of mass, we can say it is made up of fermions. Now you might argue with me about whether the uncertainty principle applies (I would say it does, and that it disallows zero volume), but the Pauli exclusion principle would most definitely apply to a hypothetical zero-volume chunk of mass. It just wouldn't happen in the first place.

Additionally, I would argue that the measurement method I used earlier is the best definition for volume that we have available to us. If you disagree and that rubs you wrong, I would say that unless you can come up with a better way, then the idea is going to devolve into a semantic uncertainty. I think the dimensional analysis of m^3 works fine here for a definition of volume.
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Re: A Matter of Steam-Powered Math

Postby rhino » Thu May 29, 2008 12:52 pm UTC

Mettra wrote:For any normal chunk of mass, we can say it is made up of fermions. Now you might argue with me about whether the uncertainty principle applies (I would say it does, and that it disallows zero volume), but the Pauli exclusion principle would most definitely apply to a hypothetical zero-volume chunk of mass. It just wouldn't happen in the first place.

Additionally, I would argue that the measurement method I used earlier is the best definition for volume that we have available to us. If you disagree and that rubs you wrong, I would say that unless you can come up with a better way, then the idea is going to devolve into a semantic uncertainty. I think the dimensional analysis of m^3 works fine here for a definition of volume.


When I said "normal object" I meant an object in some classical model of the world, in which volumes are well-defined. I didn't mean "a real-world object, assuming that QM is a perfect model of the real world".

In any case - what do you mean by "measuring a particle's length"? The only kind of answer that will satisfy me is some mathematical operation on its wavefunction. This is not just me being unreasonable - measuring position, for example, has a perfectly well-defined mathematical meaning in terms of operators. I hold that "put a meter rule next to it and see how long it looks" is obviously nonsensical in the context of QM, but even if you disagree I hope you'll see that it can't form the basis of a rigorous definition of volume.

Mods: maybe worth splitting this discussion off?

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Re: A Matter of Steam-Powered Math

Postby Mettra » Thu May 29, 2008 1:47 pm UTC

rhino wrote:In any case - what do you mean by "measuring a particle's length"? The only kind of answer that will satisfy me is some mathematical operation on its wavefunction. This is not just me being unreasonable - measuring position, for example, has a perfectly well-defined mathematical meaning in terms of operators. I hold that "put a meter rule next to it and see how long it looks" is obviously nonsensical in the context of QM, but even if you disagree I hope you'll see that it can't form the basis of a rigorous definition of volume.


I see more clearly what you mean now. The issue is fuzzy at best.

Just to be clear, I didn't literally mean 'put a meter stick next to it and measure it' I only said that as a thought-experiment-type analogy for which in the real world we would use a more clever technique.

I just had a thought 'apart from' a wavefunction operation. How about defining volume in terms of how many 'identical' particles could occupy the space? For example, we have a 'well-defined' size of an atom because of the electron orbitals. We could do that in reverse and use the exclusion principle to 'measure' the size maybe.
zenten wrote:Maybe I can find a colouring book to explain it to you or something.

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The Great Hippo
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Re: A Matter of Steam-Powered Math

Postby The Great Hippo » Thu May 29, 2008 5:18 pm UTC

SpitValve wrote:Mechanical calculators exist. They were used before modern computers. And you could definitely divide by zero. It'd set the thing whirring crazily as it (presumably) attempted to continually subtract zero from a value. You had to pull the plug to stop it. My profs have anecdotes of walking down the corridor, and hearing the distinctive manic churning of a machine dividing by zero, following by swearing as the operator realises s/he has to start all over again...


This also really helped me out; thank you. I'd love to learn more about mechanical calculators (going to look them up later today for more info).

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Re: A Matter of Steam-Powered Math

Postby BeetlesBane » Thu May 29, 2008 6:56 pm UTC

I'm wondering if what could be called a "Schwarzchild mass" (ie: a mass concentrated within a sphere having its Schwarzchild radius) might serve in place of a point mass. The radius would be insignificant in most classical scales, allowing the mass to be treated as concentrated at a point. When the radius became meaningful, if the spheres overlapped it would signal a problem.


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