Zero !!!

MathsIsFun · 2008-07-21 19:08:58

Yep, take this to a new topic ... it could get complicated!

Ricky · 2008-07-22 04:47:16

Topic made:

http://www.mathisfunforum.com/viewtopic … 174#p95174

bossk171 · 2008-07-23 16:26:39

Thanks for the new topic, back to the old one...

Could we just define 1/0? The same way we define the sqrt(-1)? My guess is no, because we haven't, but why? I suspect it has something to do with 0 not being a real number (I mean real in the literal sense not the mathematical).

It just seems to me that many times in the past there were plenty of problems that were "un-defined" and we just added more definitions until we could do the problems. It's my understanding that Irrational, negative and Imaginary numbers we all rejected at one time, and yet no more. Why not do the same thing with 1/0? Create "ostentatious" numbers (my word).

Ricky · 2008-07-23 16:59:26

Could we just define 1/0? The same way we define the sqrt(-1)? My guess is no, because we haven't, but why? I suspect it has something to do with 0 not being a real number (I mean real in the literal sense not the mathematical).

In order to do so, you'd have to break some rules that you really don't want to break. Like we talked about in the other thread, there is absolutely no way such a "number" could respect the distributive property.

It just seems to me that many times in the past there were plenty of problems that were "un-defined" and we just added more definitions until we could do the problems. It's my understanding that Irrational, negative and Imaginary numbers we all rejected at one time, and yet no more. Why not do the same thing with 1/0? Create "ostentatious" numbers (my word).

In the current formulation of mathematics, they aren't just "created". We can actually construct them. For example, I will give a short description of the construction of the rational numbers. This will assume that the integers have been constructed. Likewise, we could construct the integers from the natural numbers, and the natural numbers from axiomatic set theory. But again, here we are assuming the integers are constructed with all their familiar properties.

Define a set, [a, b] (a and b integers, b nonzero) and an equivalence relation ~ as follows:

[a,b] ~ [c,d] if and only if a*d = b*c

Remember that multiplication can be defined for all integers, but division can not (since 3/4 is not an integer). Thus, this definition is in fact properly defined.

Now we shall state that a rational number, a/b, is shorthand for [a,b]. So for example, we can say by the above equivalence relation that 3/4 ~ 6/8 since 3*8 = 4*6. Notice that this holds for any pairs of rational numbers that are (intuitively) equal. We also identify a single integer, a, with the equivalence class [a, 1], or rather a/1. We can now go on to define what it means to multiply, divide, add, and subtract two numbers of the form [a,b].

Similarly, the real numbers can be constructed out of the rational numbers with use of either Dedekind cuts or Cantor sequences. The complex numbers can be constructed out of the real numbers using splitting fields. Both of these are rather advanced in areas of analysis and algebra, respectively.

Now of course, it wasn't always like this. We came up with our axiomatic system so that it would give the properties that everyone up to the late 19th / early 20th century had implicitly assumed.

That isn't to say that you can't just come up with a new number. The important thing is for that to be useful. The construction of i (imaginary) for example came out when people starting looking at polynomial equations that had no solutions. You won't find the fraction 1/0 come up naturally, so there isn't too much a need for it. At least yet. If you find a use, if you can find it important to solve a problem (even if it has already been solved in a different way), then perhaps it is new mathematics worth studying. Again, I highly doubt such could happen from 1/0, but you never know.

bossk171 · 2008-07-23 18:08:50

I get it, I think. Some of the formalities are a little over my head, but I get the gist of what you are saying.

On a note of irony, Ricky, my musing are really a direct result of your (awesome) sig. When I first read it, I chuckled a little, but now I take it very (too?) seriously. Now, every time I encounter a mathematical problem, I think to myself, "where is that not a problem?"

George,Y · 2008-10-28 16:54:13

1/0 if existed, it would be infinity.

But infinity is paradoxy. So 1/0 couldn't exist.

However, derivatives mean 0/0. (google Berkley Newton)

George,Y · 2008-10-28 17:03:04

In fact, a rational can mean two things, one is fraction, the other is simply ratio.

In the latter sense, 1/0 could happen theoretically, but in reality, it never did.

Just think of gravity formula G=kMm/r^2

If r takes zero, would the gravity be infinity? sure it would be.

But simply, simply, the gravity law breaks when r is small. Nature doesn't allow paradox.

I am just stating the first part of my article recently submitted to philosophy of science, I hope them like it.

George,Y · 2008-10-28 17:07:58

"It just seems to me that many times in the past there were plenty of problems that were "un-defined" and we just added more definitions until we could do the problems. It's my understanding that Irrational, negative and Imaginary numbers we all rejected at one time, and yet no more. Why not do the same thing with 1/0? Create "ostentatious" numbers (my word)."

You really think the world in a simplified view like since Mr. Bush is president, everybody likes him.
Indeed, some mathematician accept surreals which includes infinity and infinitesimal, for real.
But the intuitionist only accept rationals.
Nevertheless, both of them are minorities, as you may guess.

Ricky · 2008-10-29 07:20:17

1/0 if existed, it would be infinity.

Can you prove this?

However, derivatives mean 0/0. (google Berkley Newton)

I took a quick glance, but can't find what it is you're referring to. Nor even what the above means.

In the latter sense, 1/0 could happen theoretically, but in reality, it never did.

Pray tell, what do you mean by the word "theoretically"?

Indeed, some mathematician accept surreals which includes infinity and infinitesimal, for real.

Do you deny their usefulness in mathematics? Do you deny that they have been used to solve problems, most notably in game theory?

But the intuitionist only accept rationals.

Why?

But infinity is paradoxy.

This is entirely incorrect. Infinity is used to solve many problems. Integrals for one are defined by infinite sums. Functional analysis which has yielded results that help out science and our study of natural uses infinite dimensional vector spaces. Differential equations which is based upon infinity (mostly because it's based on limits, integrals, and derivatives) is the foundation of modern classical and quantum physics. All of this stuff wouldn't work if infinity was a paradox.

Just because you think it is a paradox does not mean that it is. Indeed, if all these things work because of the properties of infinity, then it would be a rather huge cosmic coincidence that things that work are founded on incorrect principles. I don't have enough faith to believe that to be the case.

TheDude · 2008-10-29 07:27:23

George,Y wrote:

Just think of gravity formula G=kMm/r^2
If r takes zero, would the gravity be infinity? sure it would be.
But simply, simply, the gravity law breaks when r is small. Nature doesn't allow paradox.

Not exactly, the law of gravity doesn't break. It's just that your formula, which follows Newtonian physics, is incorrect. 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).

George,Y · 2008-10-29 10:49:40

"Not exactly, the law of gravity doesn't break. It's just that your formula, which follows Newtonian physics, is incorrect. 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 makes the sense it breaks at micro level to me. Though I don't mean in macro level it is 100 percent accurate.

Yes in fact it is not correct even in macro level, but only in micro level do people find its incorrectness. And even it looks correct in macro level, it is doomed to show its incorrectness at micro level. It's like the last straw.

Pray tell, what do you mean by the word "theoretically"?
I mean in gravity formula when d=0 F=?
And another question, how much energy is needed to generate absolute zero degree?

Can you prove this? 1/0=infinity
Yes I can
through 1/a monotonically increases when a decreases, given a is non negative.
Thus 1/0 must be above any number because 0<a Again don't tell me you don't define 1/x when x is not 0, this problem happens in the context above.

"Do you deny their usefulness in mathematics? Do you deny that they have been used to solve problems, most notably in game theory?"
Very funny Ricky, I wish I don't mean to deny this, but could you possibly give out how real infinity is played in game theory? And probably it is the potential infinity at play. And again, game theory is only a theory, it doesn't mean reality. A use of infinity in it doesn't mean it exist in reality. Marginal price is prevalent in Economic theory, but it doesn't mean commodities can be cut to any small.

Why?
Because indefinite division is unabtainable through on-going finite cuttings.
Kant believes in general Zenon's paradox.
Since no one can give out when they finish counting infinite decimals, intuitionists don't believe this structure exists in reality. They have the reason to doubt if nature has ways to fiinish counting that too.
Again Ricky, find a philosophy of mathematics and do some reading, don't be so ignorant to think everybody agrees on your standard analysis textbook.

"then it would be a rather huge cosmic coincidence that things that work are founded on incorrect principles. I don't have enough faith to believe that to be the case."

Well I have, and I have given the example before and now. Microly anything is discrete, however macroly a simple formula describes the law more and more acurately.
Just to think you do not have pi when you have only ten water molecules at play, but the cylinder volume formula calculates well when you have tons of water molecules to form an approximate water cylinder in a glass.

"0/0"
Sorry it is Berkeley's paradox. This time you won't miss it.

Ricky · 2008-10-29 11:43:04

Can you prove this? 1/0=infinity
Yes I can
through 1/a monotonically increases when a decreases, given a is non negative.

You not only just assumed that 1/0 exists, you also assumed that 1/0 has the very property you wish to prove. Proof by assumption isn't allowed. Please prove that 1/a monotonically decreases (indeed, this is what you meant) on the nonnegative numbers. You may assume that 1/a monotonically decreases on (0, 1) if you wish, I of course accept that.

Very funny Ricky, I wish I don't mean to deny this, but could you possibly give out how real infinity is played in game theory? And probably it is the potential infinity at play.

Surreal numbers and combinatorial game theory

And again, game theory is only a theory, it doesn't mean reality. A use of infinity in it doesn't mean it exist in reality. Marginal price is prevalent in Economic theory, but it doesn't mean commodities can be cut to any small.

You miss the point, George. If a theory in mathematics helps humanity in some way (whether it be economical or just to understand the universe we live in), then throwing it out would hurt humanity. And you can yell till you're blue in the face about philosophy and the real numbers and infinity, but at the end of the day, if the theory helps humanity, then we're gonna keep it. At least until we replace it with something else.

Infinity helps humanity.

Since no one can give out when they finish counting infinite decimals, intuitionists don't believe this structure exists in reality.

In the words, of Hardy, "Imaginary universes are so much more beautiful than this stupidly- constructed 'real' one...".

Well I have, and I have given the example before and now. Microly anything is discrete, however macroly a simple formula describes the law more and more acurately.
Just to think you do not have pi when you have only ten water molecules at play, but the cylinder volume formula calculates well when you have tons of water molecules to form an approximate water cylinder in a glass.

The problem George is that you will need a super computer to compute even the simplest of integrals if we assume values to be discrete and extremely small. By assuming things are continuous (infinite), we get a much nicer theory that allows us to do things. If you can't use your discrete system to preform science (and you can't), then your entire system is utterly useless, no matter how much it may reflect the "real" world, whatever that is.

"0/0"
Sorry it is Berkeley's paradox. This time you won't miss it.

I can find several things that people call the "Berkeley paradox", but none seem to do with 0/0. Perhaps it goes under another name as well? Why can't you find a link to it and just post it here?

George,Y · 2008-10-29 12:09:56

we get a much nicer theory that allows us to do things. If you can't use your discrete system to preform science (and you can't)
Well I need not to do that, but at the very moment I use calculus, I only acknowledge it as a good approximation tool instead of fact.

Please prove that 1/a monotonically decreases (indeed, this is what you meant) on the nonnegative numbers

For gravity case, I can prove this.

What is gravity G at d=0? (Indeed here I assume /0 exists and I assume G=kMm/d*d holds for d>=0)
Then that gravity is not equal to any number, since if it is equal to G1, G1 goes with d1, and Gravity is the same as G1, but smaller than 4G1 obtainable. So from 0 to d/2, not monotunosly decreasing. G<4G, increasing

And secondly I can also prove G(0) is not smaller than any number

And conclusion is it is larger than any number, infinity.

Ricky · 2008-10-29 12:30:08

I only acknowledge it as a good approximation tool instead of fact.

Then you accept the existence of infinity in mathematics because it allows you to accomplish things? In the end, this is all I would ever ask for. I personally believe that mathematics is a description of a pure universe rather than our real one, and that our real one is an approximation to that other universe. But to me, existence in the pure universe means that it does indeed exist.

For gravity case, I can prove this.
What is gravity G at d=0? (Indeed here I assume /0 exists and I assume G=kMm/d*d holds for d>=0)
Then that gravity is not equal to any number, since if it is equal to G1, G1 goes with d1, and Gravity is the same as G1, but smaller than 4G1 obtainable. So from 0 to d/2, not monotunosly decreasing. G<4G, increasing
And secondly I can also prove G(0) is not smaller than any number
And conclusion is it is larger than any number, infinity.

I edited this post after George posted below because I had realized I read his argument incorrectly.

Indeed here I assume /0 exists and I assume G=kMm/d*d holds for d>=0

Then what is a number over 0? You can't just assume that it exists, and you can't just assume that the formula for gravity holds at a number that just doesn't make any sense.

George,Y · 2008-10-29 12:36:57

it will be like
0 d/2 d
G 4G G

thus from 0 to d/2, G is increased to 4G
but from ∈ to d/2, ∈>0, we know it is decreasing
Uh hah, how does this happen? Where is the darn turning point?

Moreover, I can prove G(0) is not smaller than any number. And I think you don't need me to do so, do you?

Thus it is fair to conclude G(0)>any positive number, which is infinity.
Well if you only think infinity is a symbol ∞,
you don't really what it means, Ricky. And of course you assume I cannot prove out a symbol ∞.
It has a meaning beyond its name.

George,Y · 2008-10-29 12:46:23

Then you accept the existence of infinity in mathematics because it allows you to accomplish things?

No, I accept N grows larger to an extent, but not further larger, or epsilon grows smaller to an extent, but not further smaller. I accept just throwing off the error term since its impact is minimal.
I accept spending money doesn't mean I accept spending more and more endlessly. You distort the point.

"by observation, it wasn't handed down by god."
Ha, what is handed down by god? Your infinity?

You're also assuming gravity functions continuously at 0, which of course does not have to be the case (considering that "gravity" doesn't make sense at d=0).
You are again trying to block my premise. My proof is against gravity formula is global to 0.
And very funny, if you agree it does not exist at zero, can you give a distance where the formula fails? Do you want to claim it is valid everywhere but just not 0? You are playing a dodge ball. And good news for me is nature denies the above assumption.

Even at very close distance, the formula is plausible because there is doubt whether such huge force can be at balance while the energy of the universe is limited.

Ricky · 2008-10-29 12:46:50

Where is the darn turning point?

Turning point? It doesn't even have to be continuous at 0. But again, we're no longer talking mathematics George. Gravity has very little influence over real analysis.

Well if you only think infinity is a symbol ∞,
you don't really what it means, Ricky. And of course you assume I cannot prove out a symbol ∞.
It has a meaning beyond its name.

Indeed, it does have meaning. But just because it has a meaning does not force us to say that 1/0 is infinity. We can talk about 1/0 having a wide variety of meanings. The only reason why you've been able to state it is infinity is that you think it makes sense. You want things to be continuous, and I don't blame you. But there is a difference between want and it should be, to mathematically it "has" to be. You need to realize this difference. It is the difference between the pure mathematical universe to this real one. This difference you have already acknowledged.

Ricky · 2008-10-29 12:50:02

No, I accept N grows larger to an extent, but not further larger, or epsilon grows smaller to an extent, but not further smaller. I accept just throwing off the error term since its impact is minimal.

This is pretty much exactly what an analyst means when he says "as n goes to infinity".

And very funny, if you agree it does not exist at zero, can you give a distance where the formula fails? Do you want to claim it is valid everywhere but just not 0? You are playing a dodge ball. And good news for me is nature denies the above assumption.

First off, gravity doesn't work on the quantum scaled because of our physical description of it. It is not a mathematical failure that occurs, but a physical one.

And yes, 1/x is continuous on the interval (-infinity, 0)U(0, infinity).

George,Y · 2008-10-29 12:57:15

This is pretty much exactly what an analyst means when he says "as n goes to infinity".
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.

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!

I guess you can start a religion, Mathematicism, Ricky. Don't know if the ancient Greeks agree with you if their souls live to today.

George,Y · 2008-10-29 13:08:14

This is pretty much exactly what an analyst means when he says "as n goes to infinity".
I am just very curious Ricky here do you mean n really gets to infinity?
How about it never does so the series {0.9, 0.99, 0.999, ...} doesn't gets to 1 anyway?
when the difference is {0.1, 0.01, 0.001, ..., 1/10^n, ...}

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.

√3 is thus only a symbol for a number which has yet to be found, but is not its definition. This definition is, however, satisfactorily given by method as, say
(1.7, 1.73, 1.732, )
G. Cantor 1889

George,Y · 2008-10-29 14:16:21

I personally believe that mathematics is a description of a pure universe rather than our real one, and that our real one is an approximation to that other universe. But to me, existence in the pure universe means that it does indeed exist.

I got what you mean, Ricky. Sigh....

All_Is_Number · 2008-10-29 17:02:05

Ricky wrote:

It is not a mathematical failure that occurs, but a physical one.

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.

Please, tell us, from the perspective of the photon, what is the distance between The sun and the Earth? From the perspective of the photon, how long does it take to travel between the sun and the Earth?

TheDude · 2008-10-30 00:16:55

All_Is_Number wrote:

Ricky wrote:
It is not a mathematical failure that occurs, but a physical one.
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.
Please, tell us, from the perspective of the photon, what is the distance between The sun and the Earth? From the perspective of the photon, how long does it take to travel between the sun and the Earth?

It's simply a matter of semantics. When Ricky said "physical failure" he meant, I think, that our mathematical interpretation of the physical world breaks down. Of course it's the mathematics that break, since they are a man-made construction, while physics is just observation of a pre-existing universe.

What I mean to say is that math is just a tool. It is the physicist's job to use it correctly. When a physicist comes up with a mathematical model of the physical world and that model breaks down, as it does in the case of Newton's gravitational formula at distance = 0, it would be fair to call that a physical breakdown. It may not necessarily be true, since the physical world can't literally break down, but it also doesn't mean that math is incapable of describing such a situation. It's just that the mathematical model in question is inaccurate.

All_Is_Number · 2008-10-30 03:55:45

TheDude wrote:

All_Is_Number wrote:
Ricky wrote:
It is not a mathematical failure that occurs, but a physical one.
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.
Please, tell us, from the perspective of the photon, what is the distance between The sun and the Earth? From the perspective of the photon, how long does it take to travel between the sun and the Earth?
It's simply a matter of semantics. When Ricky said "physical failure" he meant, I think, that our mathematical interpretation of the physical world breaks down. Of course it's the mathematics that break, since they are a man-made construction, while physics is just observation of a pre-existing universe.

I agree that it's the mathematics that break down, but I don't think I misinterpreted what Ricky said, based on his assertions in another thread.

What I mean to say is that math is just a tool. It is the physicist's job to use it correctly. When a physicist comes up with a mathematical model of the physical world and that model breaks down, as it does in the case of Newton's gravitational formula at distance = 0, it would be fair to call that a physical breakdown. It may not necessarily be true, since the physical world can't literally break down, but it also doesn't mean that math is incapable of describing such a situation. It's just that the mathematical model in question is inaccurate.

No, it's not fair to say that it's a physical breakdown. The model only applies when d > 0. On the scale Newton worked with, distance cannot equal zero. It isn't inaccurate because of that. Maths may be capable of describing such a situation, but that does not imply that mathematicians know how to do it.

TheDude · 2008-10-30 04:05:31

All_Is_Number wrote:

No, it's not fair to say that it's a physical breakdown. The model only applies when d > 0. On the scale Newton worked with, distance cannot equal zero. It isn't inaccurate because of that. Maths may be capable of describing such a situation, but that does not imply that mathematicians know how to do it.

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

I would also say that it isn't the job of the mathematicians to come up with accurate models, that's the physicists' job. That's why I would call it a physical breakdown, even though it's obviously the math that's wrong. 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.

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. An example would be Newton having to invent Calculus in order to continue his work in physics. That was a mathematical breakdown because the math did not exist at the time to describe what he was observing.

But I feel like we're just arguing semantics at this point, which I find to be an extremely boring subject to debate.

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