Zero !!!

All_Is_Number · 2008-10-30 04:41:34

TheDude wrote:

Actually, the model never applies. It is never correct, at best it's a very accurate approximation.

Being correct and being applicable are two different things. If they were the same, it would be impossible for us to know anything quantitative about the world around us.

I would also say that it isn't the job of the mathematicians to come up with accurate models, that's the physicists' job.

It's the tool that doesn't work, not the physicists. It's in no way a physical breakdown. The breakdown is strictly mathematical.

My point is that math is (probably) capable of describing what happens at d = 0 (assuming such a situation is physically possible, which I'm not sure it is), it's just that the physicists haven't come up with an accurate model for that situation.

Newton's gravitational theory is not the only place in which the limitations of mathematical knowledge become apparent. Those limitations must be overcome in the science of mathematics before they can be overcome by the other disciplines of science.

If, on the other hand, math was not advanced enough to be capable of describing that kind of situation then I would call that a mathematical breakdown.

I eagerly await your explanation of the exact definition of division by zero. Until we have one, it is indeed a mathematical breakdown.

TheDude · 2008-10-30 05:20:20

All_Is_Number wrote:

It's the tool that doesn't work, not the physicists. It's in no way a physical breakdown. The breakdown is strictly mathematical.

I don't understand how. Let's say I model gravitational strength by

. Can you call that a mathematical breakdown when my model doesn't represent reality? Like I said this is all semantics so I suppose you could, but I never would.

Newton's gravitational theory is not the only place in which the limitations of mathematical knowledge become apparent. Those limitations must be overcome in the science of mathematics before they can be overcome by the other disciplines of science.

I never said it was, I was just referring to the example given many posts ago now. I fully agree that math can and does break down, my only point is that when you talk about Newton's gravitational formula I don't understand how you can say that math breaks down when the model is incorrect to begin with. If anything, the fact that you can't evaluate it when d = 0 is a signal that your model is wrong and needs to be fixed, not that math somehow needs to account for 1/0.

neonash7777 · 2008-10-30 08:35:09

A good number of fields of mathematics accept 1/0 = ∞. And it's actually meaningful for solutions.

It's called the Riemann sphere "R union {∞}" and has a few number of other names too.
Whereby -∞ and ∞ are the same number as ∞.

It makes sense if I have one apple and make groups of 0 apples, how many groups can I make. Well here's your group of zero apples and here's your group of 0... so ∞ many people can get a group of 0 apples!

Granted you have to be careful when using this idea of infinity, but so-what, you have to be careful when using 0 as well. If you accept the notion of a historical infinity (one where you keep track of where it's derived from) math works quite well with it.
For example, Normally
2 * ∞ = ∞, but instead keep the history of the infinity such that 2 * ∞ = 2∞

1/0 = ∞
2/0 = 2 ∞

By keeping the history of it, we can see later on what is possible to cancel out of it.

This also helps a lot because multiplication is finally a closed group like addition.
The concept of an identity and an inverse exist for all elements.

Also if you graph something like y = 1/x as x -> 0, y -> ∞. We might as well say when x = 0, y = ∞ where it's not a positive or negative, it's both. they are the same. It just makes sense.

All_Is_Number · 2008-10-30 11:47:50

TheDude wrote:

Let's say I model gravitational strength by
. Can you call that a mathematical breakdown when my model doesn't represent reality?

Typically, scientific models are not chosen arbitrarily as you have done.

Like I said this is all semantics so I suppose you could, but I never would.

Nor would I, since your model wouldn't work for any distance. I would call it an incorrect and completely useless model, albeit one that does, in fact, highlight an aspect of mathematics in for which we currently lack understanding.

I never said it was, I was just referring to the example given many posts ago now. I fully agree that math can and does break down, my only point is that when you talk about Newton's gravitational formula I don't understand how you can say that math breaks down when the model is incorrect to begin with. If anything, the fact that you can't evaluate it when d = 0 is a signal that your model is wrong and needs to be fixed, not that math somehow needs to account for 1/0.

It seems you don't understand Newtonian gravity. The model doesn't break down; it was never intended to be used at d=0. Such a scenario cannot exist on the macro-scale for which Newton created his model. Of course that doesn't change the fact that mathematics, at our current level of understanding, breaks down when we attempt to divide by zero.

TheDude · 2008-10-30 23:56:13

All_Is_Number wrote:

It seems you don't understand Newtonian gravity. The model doesn't break down; it was never intended to be used at d=0. Such a scenario cannot exist on the macro-scale for which Newton created his model. Of course that doesn't change the fact that mathematics, at our current level of understanding, breaks down when we attempt to divide by zero.

This is true, but it's also the reason I ask why you consider it to be a mathematical breakdown when the model isn't even intended to be used at such small distances. Yes, under certain situations that the model was never intended to be used it requires the impossible (division by 0), but I fail to see why you would blame it on the shortcomings of math rather than the shortcomings of the model.

George,Y · 2008-10-31 04:49:38

TheDude wrote:

All_Is_Number wrote:
It seems you don't understand Newtonian gravity. The model doesn't break down; it was never intended to be used at d=0. Such a scenario cannot exist on the macro-scale for which Newton created his model. Of course that doesn't change the fact that mathematics, at our current level of understanding, breaks down when we attempt to divide by zero.
This is true, but it's also the reason I ask why you consider it to be a mathematical breakdown when the model isn't even intended to be used at such small distances. Yes, under certain situations that the model was never intended to be used it requires the impossible (division by 0), but I fail to see why you would blame it on the shortcomings of math rather than the shortcomings of the model.

Well it is like this. When one time maths model fail to explain reality, we say it is coincidence. When again and again this kind of thing happens, are they just coincidences? There could be some fundamental reason that maths isn't able to universally describe university, and the reason is its logic flaw.

TheDude · 2008-10-31 04:55:48

Yes, that's true, but it's not relevant to the specific example of Newtonian gravity because the model has been shown to be flawed. You can argue that we don't have sufficiently advanced knowledge of math to combine quantum mechanics and general relativity to handle situations like singularities (which, last I knew, is true), but in the very specific case of Newtonian gravity it is the model that is at fault, not math.

All_Is_Number · 2008-10-31 05:13:29

TheDude wrote:

Yes, that's true, but it's not relevant to the specific example of Newtonian gravity because the model has been shown to be flawed. You can argue that we don't have sufficiently advanced knowledge of math to combine quantum mechanics and general relativity to handle situations like singularities (which, last I knew, is true), but in the very specific case of Newtonian gravity it is the model that is at fault, not math.

The flaw in your logic is your assumption that because the model isn't perfect, the mathematics must be. The flaw in the theory is additional to and independent of the mathematical breakdown, and make no mistake, the mathematics do break down.

Quantum gravity handles classical physics' singularities pretty well, btw, but that's a topic for a different thread.

TheDude · 2008-10-31 05:21:05

I can agree with that, but I would argue in turn that the mathematical breakdown at the very specific distance of d = 0 is inconsequential because the model becomes wildly inaccurate before that point anyway. I'm more concerned about the fact that the answers that the model gives at very small values of d aren't even close to reality than I am by the fact that eventually it doesn't give an answer at all.

All_Is_Number · 2008-10-31 09:59:14

TheDude wrote:

I can agree with that, but I would argue in turn that the mathematical breakdown at the very specific distance of d = 0 is inconsequential because the model becomes wildly inaccurate before that point anyway. I'm more concerned about the fact that the answers that the model gives at very small values of d aren't even close to reality than I am by the fact that eventually it doesn't give an answer at all.

Again, on the scales Newton was modeling, the theory worked very well. It models point masses, not massive objects with non-zero volume. Once the radius of the objects become significant compared to [i]d[/d], calculus must be used to calculate the gravitational attraction. Don't think that because this is disregarded in the formula's simplest form that this has not been long known (it was likely understood by Newton himself).

Ricky · 2008-10-31 10:47:47

To George:

But not. I deny indefinate increase in most times or indefinite decrease in all times. I have a stop point while the above has not.

Forgive the pun, but that's a rather small difference. The basic concept is still there. The only real difference is that the mathematican proves arbitrary difference whereas you have some magical value that is "small enough". Praytell, what precisely is that value?

Hmm, physics can fail, but mathematics cannot. Mathematics is perfect, the only imperfection is the reality. Even when they do not meet, it is other that should change but mathematics. Good point!

If you want to call it a mathematical model failing, I'm perfectly fine with that as well. But the point is it isn't the equation that has the problem, but what the equation represents is where the problem lies.

Don't know if the ancient Greeks agree with you if their souls live to today.

I wouldn't care. If they could make a compeling argument against what I believe, then I wouldn't believe it. It doesn't matter what they think, it only matters why they think it.

I am just very curious Ricky here do you mean n really gets to infinity?

George, I've said a million times that when we say "approaches infinity", we mean the value gets arbitrarily close as N gets sufficiently large. N only need exist, it does not have any restriction on the value.

Don't invite your Real number friend, Ricky. You know perfectly a Real number is nothing but restating a series. It defines the problem, rather than solving it.

I've also said a million times that if you want to disagree with the basic structure of mathematics, that's perfectly fine. But when someone talks about mathematics, it implicitly means that they are talking about said structure. You should be very clear if you wish to talk about a different structure, as great confusion can arise.

All_is_Number:

You seem to be of the opinion that not being able to do something in mathematics is a "breakdown" of sorts. If that were the case, then I suppose in your perfect mathematical system, you could do evreything. Therefore, "doing something" looses it's meaning and this system becomes utterly useless.

However, we do indeed want our mathematical models to breakdown whenever the physical description does not make sense. In our case, when d = 0, what we are asking is "how much force does a particle exert on itself?" This is a nonsensical question, and therefore, it is a really really good thing that the mathematical model breaks down at this point. It shows that our interpretation is in complete aggreance with our model. And in the case where d > 0 but signficantly small, our mathematical model makes sense but does not spit out the right values (at the quantum level). Therefore, we have found an error in our model which must be corrected.

It's a mathematical failure, not a physical one. Your assertion requires us to believe that because we don't currently understand how to do something in mathematics, that it cannot possibly be done.

If you want to call it a failure of the mathematical model, that is perfectly fine. I call it a physical failure because it is (typically) the physicst who comes up with the model. Case in point, the model breaks, not the mathematics behind the model.

But as soon as you accept that you are working in a field (ring), you can prove that division by zero is impossible. If you remove the assumption that we are working in a field (ring), then you can possibly have a division by zero. But you also want it to represent reality, and I don't see how you could do that if what you consider to be a "number" didn't form a field.

No, it's not fair to say that it's a physical breakdown. The model only applies when d > 0.

Correct me if I'm wrong, but we were talking about sufficiently small but positive d. In other words, the quantum level.

Maths may be capable of describing such a situation, but that does not imply that mathematicians know how to do it.

Is that the job of the mathematician or the physicst?

To Dude:

First, I would suggest forgetting the semantic argument that the model doesn't apply because it is an approximation. When we say "apply", I think everyone has a clear understanding that all models are neccesarily approximations, and so it is rather trivial to call them such. But perhaps it is reality that is only an approximation to our physical models?

A mathematician believes reality approximates mathematics. A physicst believes mathematics approximates reality.

My point is that math is (probably) capable of describing what happens at d = 0 (assuming such a situation is physically possible, which I'm not sure it is), it's just that the physicists haven't come up with an accurate model for that situation.

I don't even think d=0 makes sense, let alone is possible. Is it possible to taste the color blue?

If anything, the fact that you can't evaluate it when d = 0 is a signal that your model is wrong and needs to be fixed, not that math somehow needs to account for 1/0.

If your mathematical model breaks down precisely where what it represents breaks down, then this is indeed a good thing. d = 0, I believe our physical interpretation of "force" breaks down, you can't ask how much force a particle puts on itself.

Neonash:

This also helps a lot because multiplication is finally a closed group like addition.
The concept of an identity and an inverse exist for all elements.

Perhaps, but you loose the property of a field. And you are now saying that the inverse of infinity is 0? Ok, then first assume that we are using what you call "normal" infinity:

(2 * infinity) * 0 = infinity * 0 = 1

so

2 * (infinity * 0) = 2 * 1 = 2

Likewise, now assume that we are using the "historical" infinity:

2 * (infinity / 0) = 2 * 1 = 2

infinity * (2/0) = infinity * 2infinity != 2

In either case, you just destroyed the associative property, and I don't want to do any computations in a system that isn't associative. That would be a nightmare!

All_Is_Number · 2008-10-31 12:12:07

Ricky wrote:

In our case, when d = 0, what we are asking is "how much force does a particle exert on itself?" This is a nonsensical question, and therefore, it is a really really good thing that the mathematical model breaks down at this point.

On the macro-scale, that is indeed a non-sensical situation. It's not so much that it's a good thing the mathematics break down as it is non-consequential, since the model assumes d>0. There are other times when we are interested in that point, but don't have the tools to investigate.

I call it a physical failure because it is (typically) the physicst who comes up with the model.

That's a bit like saying inflation is the fault of the chocolate manufacturer because the price of chocolate rose with everything else. Mathematics are great, but they're nowhere near the perfection you seem to have attributed to them.

Case in point, the model breaks, not the mathematics behind the model.

The maths break down, even if the model breaks down also.

But as soon as you accept that you are working in a field (ring), you can prove that division by zero is impossible. If you remove the assumption that we are working in a field (ring), then you can possibly have a division by zero. But you also want it to represent reality, and I don't see how you could do that if what you consider to be a "number" didn't form a field.

Right. If we make certain assumptions then we can thereafter draw logical conclusions based on those assumptions. Progress, however, tends to come when we recognize that we are mistaken with our assumptions.

Correct me if I'm wrong, but we were talking about sufficiently small but positive d. In other words, the quantum level.

Okay, consider yourself corrected. BTW, at the quantum level, two particles are not prohibited from occupying the same space.

Maths may be capable of describing such a situation, but that does not imply that mathematicians know how to do it.
Is that the job of the mathematician or the physicst?

You seem to be under the impression that the two groups are mutually exclusive. Scientists often work in multiple disciplines of science. Perhaps you've heard of Calculus? Newton developed that to solve physics problems.

Last edited by All_Is_Number (2008-10-31 12:14:38)

George,Y · 2008-10-31 23:22:56

Ricky wrote:

The only real difference is that the mathematican proves arbitrary difference whereas you have some magical value that is "small enough". Praytell, what precisely is that value?

Ricky wrote:

George, I've said a million times that when we say "approaches infinity", we mean the value gets arbitrarily close as N gets sufficiently large. N only need exist, it does not have any restriction on the value.

Ricky, I do not have a physical evidence that all that approximation has a stop point. I do have some for particular formulas which in fact deals with a multiple of smallest divisions - the formula to calculate mass and electricity, for example

And what is more, I have logical evidence that the concept "arbitarily small" embeds "infinitesimal" as an essential part and hence is paradoxy and hence cannot exist even before any physical evidence is found.

Let's define a pair of quantities, either one is larger than the other at the same time. I do not need to list all possible two quantities to convince you that such pair cannot exist through pure logic. And do you buy it? Do you just say that because I cannot find all pairs of quantities to present them in front of you so that you believe there is some room for such a paradoxy pair to really exist?

There is priori knowledge beyond empirical knowledge, and this is LOGIC. You are free to make assumptions and make definitions, but your assumption and definition cannot be self-contradictory.

And the proof of potential infinitesimal containing real infinitesimal was the main topic of my essay, the gravity break down was just the appetizer. I regret not putting the argument of priori knowledge in it to present it more convincing.

And forgive me for not sharing the proof just for now. I can give a hint, my proof only use rationals, or fractions, without even a touch on Reals, and it doesn't have to, since Reals are found on the assumption of all fractions' existence.

Ricky · 2008-11-02 05:24:04

All_is_Number:

Right. If we make certain assumptions then we can thereafter draw logical conclusions based on those assumptions. Progress, however, tends to come when we recognize that we are mistaken with our assumptions.

Are you saying you believe that what we think of as a real number should not form a field?

Correct me if I'm wrong, but we were talking about sufficiently small but positive d. In other words, the quantum level.
Okay, consider yourself corrected. BTW, at the quantum level, two particles are not prohibited from occupying the same space.

So we weren't talking about sufficiently small yet positive d? Can you please explain this then (posted by TheDude):

It's an accurate approximation when you apply it to systems the size of planet or galaxies, but at the atomic level it is no longer accurate. That's when you have to start using things like quantum mechanics, and if you try to take r all the way down to 0 you get into uncharted territory (GUTS, string theory, and the like).

I think talking about what *actually* happens when d=0 is something we should avoid since it is still uncharted theoretical physics (as far as I know), but we all agree the mathematical model breaks at this point.

You seem to be under the impression that the two groups are mutually exclusive. Scientists often work in multiple disciplines of science. Perhaps you've heard of Calculus? Newton developed that to solve physics problems.

No, I'm not. But that doesn't mean you can't separate the two to a large degree. Certainly there are mathematical physicists, and mathematicians who alongside physicists. But they can wear different hats, one a physicist hat and one a math hat. I have provided earlier in the thread a criteria of how to access which hat the person is wearing. What I would like to see is criticism of this criteria, as I have failed to find any myself. But my knowledge of physics is rather limited.

Ricky · 2008-11-02 05:29:27

George:

And what is more, I have logical evidence that the concept "arbitarily small" embeds "infinitesimal" as an essential part and hence is paradoxy and hence cannot exist even before any physical evidence is found.

Actually, if you were to consider the concept of infinitesimals in the real numbers, then you would reach the conclusion that all infinitesimal are equal to 0, which is just a softer way of saying that they are 0. Again, we consider two real numbers to be equal when they are arbitrarily close.

Let's define a pair of quantities, either one is larger than the other at the same time. I do not need to list all possible two quantities to convince you that such pair cannot exist through pure logic. And do you buy it? Do you just say that because I cannot find all pairs of quantities to present them in front of you so that you believe there is some room for such a paradoxy pair to really exist?

No, you prove the trichotomy law for the order of the real numbers, and then conclude that no such pairs exist.

And forgive me for not sharing the proof just for now. I can give a hint, my proof only use rationals, or fractions, without even a touch on Reals, and it doesn't have to, since Reals are found on the assumption of all fractions' existence.

Can you provide the reason you refuse to share it?

George,Y · 2008-11-02 06:35:49

The reason why is that the proof is in that essay I submitted.

Math Is Fun Forum

#51 2008-10-30 04:41:34

Re: Zero !!!

#52 2008-10-30 05:20:20

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