i don't know if this is a smart question, but
a 3dimensional object, say a card box, are made of surfaces. (duh)
each surface is made of planes.
planes are made of lines.
lines are made of points.
but points are 0 dimensional!
are we actually saying that we can make a box out of NOTHING?
duh.
Centeris
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I may not be right but technically what makes the cardboard cardboard is the bonds holding the molocules together. Same thing for atoms and quarks. Your first post was correct in most ways. A line is technically made of points but they are CONNECTED points... same all the way back up in such a way that a surface is a bunch of connected points. There is no way to create a box (whether cardboard, or in the air) out of nothing.
exactly what ozo said
don't forget that a segment (or complete line) is an INFINITE number of points, which are in fact (as per Lebesgue's "Théorie de la Mesure") of length zero.
Saying, as you did, that an infinite number of zero-length elements sticked together is "making a non-zero length out of nothing" is a bit presomptuous ;-)
>planes are made of lines.
>lines are made of points.
I disagree with both of these statements.
A line has length, no thickness, it it a single dimensional mathematical concept - you can't actually have a line, you can only describe it.
A plane is a two dimensional mathematical concept - similar to a line, you can't have a plane. You can describe it as maybe lying on the surface of your box, but if you take the box away, the plane is either where it originally was or it moves with the box. It all depends on your description of the plane - whether you used the box co-ordinates to describe the plane, or the co-ordinates of the room you are in.
You are confusing mathematical concepts with reality. RobinD is correct. There is no such real thing as a "line". It's a mathematical abstraction. Same with "plane" or "point".
For example, using time (t) as a variable in seconds:
M = 5t + 6
Now, M is variable that changes with time. If you plotted M vs. t, you would get a straight line.
If you can point to a line in time in real life, I will agree that lines are real.
The words in the original question are dimensional descriptors based on the space you are using. They aren't real things.
if everything is just an abstraction, and we can't "map" it in real life, under what grounds are we concluding so many things from mathematics when the assumptions can't even prove themselves? i'm not asking about the technical definitions or other connotations such as "the assumptions are the building blocks so we can't prove it," but points, lines, and planes are, let's face it: undefined entities! no equation can do them justice.how do we define something using terms that are undefined just the same? oh i need an aspirin.
Centeris
I have to agree with alexcvt and RobinD as well. You are confusing mathematical abstractions with reality.
As for your above post:
> if everything is just an abstraction, and we can't "map" it in real life
But we can. I can take a coin, measure its radius and apply a simple mathematical calculation based on abstractions to get its circumference. I can then measure its circumference. The results of the calculation and the measurement will agree up to the precision of the measurement I made.
> under what grounds are we concluding so many things from mathematics
Because the conclusions are based on valid mappings between mathematical abstractions and reality.
> when the assumptions can't even prove themselves
What assumptions?
And what do you mean by prove?
The validity of the mappings is given by the fact that results gotten from calculations involving abstractions agrees with observed reality.
> but points, lines, and planes are, let's face it: undefined entities!
No they're not.
Points, lines and planes all have a mathematical definition. Only because they're abstract concepts and not observable real world entities doesn't mean they're not defined.
One can take a real 3D object, approximate it with planes and do some calculations with these planes to get results which apply to the real object.
> no equation can do them justice
What exactly do you mean?
> how do we define something using terms that are undefined just the same?
Again, what do you mean?
Greetings, Centeris,
Mathematics isn't a way to *create* reality, it's just one way to *describe* it.
Like, we could describe reality with words, but words are made of patterns of approx. 26 letters in some alphabet, that are abstractions and meaningless in themselves. But that doesn't mean reality is meaningless.
Imagine!
Hi Centeris
I believe your assumption concerning lines is questionable. Lines cannot be made of 0-dimensional points no matter how many times the word "infinite" is used. An infinite number of 0-dimensional points is still 0.
I would describe a line as an interval connecting two points and a plane as an area bounded by n lines.
EB
The term "nothing" is subjective. It is limited to our depth of vision. As we cannot even begin to understand with our human minds the complexity of the universe and it's smallest particles, we can name what we can't measure to be nothing. These are the limitations of language and communication, as well as math and science. So since nothing is just a name for something, then I say yes, we can make something out of nothing.
i have to disagree that points,lines and planes are NOT undefined. they are, because they're the fundamental building blocks of more complex geometries and mathematics. if they can be broken down to simpler components (especially a point) that disqualifies them from being the most basic. equations of lines and planes can only do so much as to describe their behavior, not their "being."
i'm just going to challeng the point that you can't creat something out of nothing
for the univers to be around as we think it is now and have the laws of thermodynamics to be correct matter and energy are just basicaly two sides of a coin, there are pleany of examples of how matter can become energy (cumbustion for example) true it's not a 100% conversion but that dosn't mean it can't be done, anyways that means in thery you can turn energy into matter taking something that has 0 mass and giving it mass and there is your 0 to 1 conversion
Centris,
It seems like you are confusing something here and use the concept of points, lines and planes in the wrong context.
Let's take points for example.
Even in mathematics, you will find different fields where "point" is defined differently. In this discussion, I was referring to points in geometry, and to be more precise, real Euclidean n-space. In this context, a point is a well defined concept:
In real Euclidean n-space a point P is defined to be an n-tuple of real numbers <r(1), r(2), ..., r(n)> where the real numbers r(1) up to r(n) are said to be the coordinates of P.
Lines and planes are also well defined - I won't give the definitions here but if you don't believe me let me know and I will state them.
> they're the fundamental building blocks of more complex geometries and mathematics
I agree that they are building blocks, but not the most basic ones. If you are looking for the most basic building blocks of maths then you'd be better off looking at Cantor's set theory and Peano's axioms. Pretty much all of maths is built on these.
> equations of lines and planes can only do so much as to describe their behavior, not their "being."
Being implies existence, and here's where I think your confusion comes from:
What does it mean for a point to "exist"? When working in 3-dimensional Euclidean space I can write down
point P = (3.2, 4.2, -1.5)
Now, does P exist? I have given no meaning to P other than its coordinates in some space I am working in. P doesn't exist physically, P exists solely as a mathematical construct. An abstraction. It's not even an abstraction of something in the "real world" yet, because I have not given you enough information.
So P exists, but not in the way people intuitively use the word "existence".
I could now refine my example and say P is the location of the top left corner of my desk. If I told you the origin of the coordinate system and the fact that the numbers given are inches, then P would have a meaning in the "real world".
But what about the existence of P? P is still a mathematical concept, an abstraction. But this time it is an abstraction of something, namely the location of the top left corner of my desk.
But what if I refine the example even further and map P to the part of my desk which is at the top left corner? I am mapping P (a mathematical abstraction) to something in the real world, but what is that something? Is it the atom that sits at the very top left corner of the desk and still belongs to the desk? But hold on, atoms have a volume, so it's not the atom that P maps to. How about the electron that's in the outer orbit of the atom? What if P maps to that electron at the specific point in time that would make it the top left electron of the desk? But hold on...do electrons have volume? And so on...
You can see where this is going. The problem is that we don't know whether the universe is discrete (i.e. there's such a thing as the shortest length, nothing can be smaller than this shortest length), or continuous (there is no shortest length). This is a question which has not been answered, although Quantum physicists have some interesting things to say on this matter.
So the bottom line is that we don't know if a point exists in the "real world" and any attempts in defining a point can only be done within an abstract concept, where "existence" takes on a different meaning. Using "real world" existence on these definitions is futile. Nothing meaningful can come out of it because you are mixing together two different concepts.
Well...I hope what I said made sense, if anything is not clear please ask...
Konst
Konst has mathematically reaffirmed the statement I made earlier, and that Centeris made in the beginning.
You should give the points to yourself, Centeris. You have answered the question twice and everyone continues to respond to the question with the same answer.
Forget the box.
Here is another example.
Human beings.
You can say that we have points and lines and such.
You can say that we were constructed from a wee little sperm and a tiny little egg. But what are those made up of ? The same things as that box, only in a different composition. No one has to build a human. The matter was created out of "nothing", yet here we are, points in space, an abstraction of reality, with no defined spacial relationship to the real world, debating a question that will remain unanswered if we try to answer it by typing words into a computer.
We started out as "nothing", and we are still, "nothing".
Man, now I'm not sure that I'm even here pondering this question.
For all I know, I could be an entity fabricated by the subconcious of Centeris, designed to disprove the existence of mysel...
my opinion: it's all the creative imaginations that those mathematicians had in their mind, that they need to use points, planes and lines to explain to yet-not-so enlightened people back during the old ages. they tried to use something imaginative to describe the real world.
however, modern scientists have proven that these terms (points,planes..etc) are not real and cannot be used to define accurate characteristics of a real object. for example, you may measure the length of a straight line (the distance between the first point and the last point forming that line) to a finite decimal places. e.g. 52.30 cm.
however, the real fact is that the actual length has infinite decimal places, because you can divide space for an unlimited number of times. e.g. 52.30423121124... right?
so, when we use point as a 0-dimensional object, we are actually limiting the number of times that space can be divided. the point might take 0.2000000 as a value of its magnitude so that when we measure something, we calculate n*points_magnitude to get a value with finite decimal places.
please correct me if i'm wrong. :]
the idiot genius~
what about cosmic strings, from my understanding they are one dementinal objects, and an event horizon is a plane not a 3 dimensinal object. also in order for the bigbang theroy to work out we need to live in a twelve demensinal spacetime. now not every object has 12 dimensions so if an object can have 4 of 12 dimensions why couldn't an object have 1 of 3 dimensions or any other combonation
Centris,
> cool down everybody i'm only aksing if you can make a card box out of nothing.
No, you were not simply asking if you can make a card box out of nothing.
In case you forgot, here's your initial question:
> a 3dimensional object, say a card box, are made of surfaces. (duh)
> each surface is made of planes.
> planes are made of lines.
> lines are made of points.
> but points are 0 dimensional!
Your first assumption was that a card box is made of surfaces. You then went to show that a surface is just a collection of 0-d points.
Your question then was:
> are we actually saying that we can make a box out of NOTHING?
To which the answer is a definite NO, the reason being that your initial assumption (a card box is made of surfaces) is incorrect. A card box is NOT REALLY made of surfaces - at least not of surfaces in the mathematical sense. You are mixing up two different concepts. I have been focusing on this throughout my posts.
You then went on to challenge the fact that points, lines and planes are undefined entities. I have shown in my posts why this is not so.
> surely there's an easy way to disarm this argument.
Yes there is, and various people have disarmed your argument by saying that you are confusing mathematical concepts with reality!
> wormholes and nth dimensions need not be discussed
Wormholes were not discussed. You did not take Kaosand's comment serious now, did you?
n dimensions were mentioned, but only because you said that points are undefined and I had to give you the definition to prove you wrong.
It seems to me that you did not understand all those posts that were trying to show you why your initial assumption is incorrect. You are saying we are thinking way to far...well, maybe it is you who is not thinking far enough. Look at your initial question, think about it carefully. Look at my posts and think about them carefully. Don't just read them. Try to follow my argument. If you don't understand some of it, let me know and I will clarify it. If you think it is wrong, then state what you think is wrong with it.
Konst
Greetings,
Konst, be gentile. Centeris already stated he didn't understand. It's quite possible his ideas are fixed so strongly that a couple of insights from those who think have the answer won't cut it.
It may be so difficult for him to grasp, that it takes a while to settle in and start making sense. From personal experience I know some maths is pretty hard to get around.
Imagine!
OmegaJunior,
You're right, my previous post does sound a bit harsh. I apologize to Centris if he feels offended - I did not mean to offend anyone.
I just didn't like the way Centris seemed to ignore my posts. I feel that they were not just random insights but structured constructs, each step depending on the previous and leading to the next. This should make it easy for Centris to point out what exactly it is that he doesn't understand. He could then state this and I would be able to comment on it.
His simply saying "focus on the answer" and ignoring the answer I tried to give annoyed me a bit to be honest.
Still, you have a very good point. Sometimes something is so obvious to me that I just can't understand why someone else doesn't get it. Maybe I should try to come up with an explanation that tackles the problem from a different angle.
Konst
Alright, different angle.
Centris, in the beginning you were mapping a card box (real world object) to surfaces (mathematical abstractions).
You then went on to do some reasoning based on these abstractions, namely that surfaces are collections of 0-d points. Since 0-d points have no volume and no length you drew the conclusion that, when you come right down to it, surfaces are made from nothing.
You then mapped this result you got from working with abstractions back to the real world, arriving at the final conclusion, namely that a card box is made from nothing.
This seems a bit weird, so you asked what's wrong with your reasoning.
There are two things which are wrong with your reasoning. One is a major flaw, the other a minor one. Let me try to explain in some detail:
Number 1 (major flaw):
Can you see the structure of your reasoning? You are taking something in the real world, mapping it to some abstract concept, do some analysis with this abstract concept, get a result and map this result back to the real world. In general, there is absolutely nothing wrong with this - it's what scientists do all the time:
REAL WORLD -> ABSTRACT CONCEPT -> REAL WORLD
Why do they do it? Because it is much easier to work inside an abstract system and analyse the real world within this abstract system.
Take particle physics for example. Scientists have been trying to find out about the basic building blocks of matter for ages now. At some point they arrived at the abstract concept known to us as atoms. They invented a whole framework which tries to explain how everything is just a collection of atoms and how they interact and how physical phenomena can be explained using the conecpt of atoms.
Reasoning inside this framework lead to predictions about the behaviour of real world objects. Experiments were performed to verify if the predictions are correct. They found out that some predictions were incorrect, so they changed the framework and made the theory match experimental data.
More experiments were made, more predictions were proven wrong, more changes were made to the theory.
Over time, more elaborate models for electrons were introduced, quarks were added to the model, more and more elementary particles were discovered,... the story is still unfolding.
So it's like:
REAL WORLD -> ABSTRACT SYSTEM -> PREDICTION ~> REAL WORLD
It's the last mapping (~>) that's very crucial. If experiments verify that the mapping is indeed correct and experimental data agrees with your prediction then your prediction was correct.
If many experiments are performed and your predictions are always correct, then you can assume that your abstract system is a good approximation to the real world.
Of course, you can never prove that your system is a PERFECT approximation of the real world. All you can do is perform more experiments and hope that your system doesn't break down and produce wrong predictions.
If it does, then you need to change it.
So let's look at your last mapping. In your abstract system (geometry) your result was "surfaces are made of nothing". While this itself is wrong (I will talk about this later, it is the Number 2 thing which is wrong in your reasoning), let's assume for now that it is correct and surfaces are indeed made of nothing.
Your last mapping is:
"surfaces are made of nothing" ~> "card boxes are made of nothing"
The statement "surfaces are made of nothing" is a statement about the nature of surfaces.
The statement "card boxes are made of nothing" is a statement about the nature of card boxes.
So you are implicitly assuming that the nature of surfaces is equivalent to the nature of card boxes.
But it's not. A surface is not equivalent to a card box, so your mapping is incorrect.
This is the crucial point. Does it make sense? Do you understand what I am trying to say?
Number 2 (minor flaw):
The statement "surfaces are made of nothing" is incorrect.
Surfaces are collections of points. A point, while it has zero dimension, is not nothing. If it were nothing, it wouldn't be a point.
So the correct statement would be "surfaces are made of 0-d points".
Still, to many people this does not make much sense. How can a surface or even a line have a certain length if it is made of 0-d points which all have no length?
Surely 0+0+0+0+....+0 = 0, no matter how many 0s you add.
So what's wrong here?
Nothing at all.
First, the fact that a point has 0 dimension does not stop me from taking infinitely many such points satisfying some equation, throwing them in a set and declaring this set to be a line.
Second, the main confusion comes from the fact that it is VERY tricky to get your head round the set of real numbers, despite its apparently simple structure.
Most people, when thinking about points and lines seem to be under the impression that a line is a nicely ordered collection of points, one next to the other, from the beginning of the line to the end of the line.
While the points are indeed well ordered, the notion of "next to" does not apply at all.
Take a real number, say 1.648. There is no real number which is next to 1.648. For AYN real number r, I will always be able to find another real number which is exactly between 1.648 and r - namely (1.648 + r) / 2.
So if 1.648 is a point on a line, there is no point next to it.
In fact, for any two distinct real numbers r and s, there are infinitely many real numbers between r and s.
I can now define the notion of "distance". I can say that the distance between any two distinct real numbers r and s is (r - s). If r and s are distinct, I can be sure that their distance is not 0.
So, even though a line consists of single points which have no dimension, a line can have a length because I am working with points from the set of real numbers whose structure ensures that the notions like length and distance make sense.
Right now you are probably thinking something along the lines of "WHAT???". Well, it wouldn't surprise me anyway.
Still, please take some time and think about it. I know it's not easy at all to get your head around it.
If something is not clear, please let me know and I will try to explain it in more detail.
Konst
For the kind comment of OmegaJunior,
Thank you for your concern. I appreciate it very much but i want to let you know that i wasn't offended at all. in fact i am very thankful that my question was taken very seriously, and i thank Konst, you and everybody who posted their comments.
For the kind comment of Konst,
I am very sorry if my last comment seemed to look like i was taking your comments for granted. i want you to know that i never ignored anyone's posts, especially yours. judging from your comments i can say that you really have spent time to make me understand your answer.
I am sorry if i didn't write this earlier, but i am asked this question because i am currently working on a paper that, in a way, "attacks" the way mathematics is being taught in schools, i will not expound on this any further.
This 3D-0D confusion is one of the questions that one student asked of his teacher. i was very disappointed that the teacher answered the question with a very verbose and complicated explanation. You were very right when you said that "sometimes something is so obvious to me that I just can't understand why someone else doesn't get it." A relatively easy (and obvious) question must have a relatively easy and obvious answer.
True, that this question was formed from wrong premises. and that this confuses mathematical abstraction with reality. but would you really expect a mere 6th grader to understand all this?
Of course i knew the answer to this question all along, but, how do you bring that explanation down to the level of someone who knows what surfaces, planes, lines and points are, but doesn't understand cosmic strings, euclidean n-space, and M= 5t+6?
Please understand tha I am not trying to make an impression here. I just want a 6th grader answer to a 6th grader question. i think i was wrong not to say this at the beginning.
Nonetheless I am not offended in any way. This is all just a part of something i'm doing.
To Innocence Lost,
Can you further simplify that answer? the one i'm looking for is something like that.
Peace out,
Centeris
Hi Centris,
Thanks for clarifying things. Now I understand why all the answers you got were not really satisfying.
Finding an explanation that can be understood by a 6th grader seems pretty tough. It's certainly something for me to think about, as I am not very good at giving simple explanations ;)
In fact it has happened to me a number of times that a child asked me a question to which I knew the answer but I just wasn't able to formulate it in such a way that the child understands me and at the same time I am not saying something that's incorrect.
This is extra hard when it comes to maths as maths requires you to be in a certain frame of mind when you want to really understand and make sense of it. Unfortunately, maths is taught in a way that makes it absolutely boring for kids (unless you're really into it, in which case it's still boring because you have to wait for the rest of the class to catch up) - and when something bores you to death, why would you make the effort of changing your frame of mind if you could just as well say it's all nonsense and I'm never gonna need it anyway?
> A relatively easy (and obvious) question must have a relatively easy and obvious answer.
This is true to some extent.
What does "simple" mean? I could give an answer to some question in just one single sentence. Does that mean the answer is simple?
I'd say not necessarily - the answer might be really simple for me (because I know what I am trying to say) but to you it just doesn't make any sense and in fact it looks like it's not an answer at all.
So it's more like
A relatively easy (and obvious) question must have an answer that is relatively easy and obvious to someone with enough background knowledge and the right frame of mind to understand it.
And it's exactly that background knowledge and frame of mind that's important. You will never find an answer to your question that is simple per se, you will always have to make implicit assumptions about people's background knowledge.
What this means is that you probably won't find an answer that every 6th grader will understand. Rather, if a 6th grader asks this question you should spend some time talking to him/her to find out how he sees the world, which words he/she uses to describe things and so on. You can then start to think of an answer that would start out in the child's frame of mind.
Of course, this is no easy task...then again, being a good teacher isn't an easy task either.
Good luck with your paper!
Konst
In your expert opinion, Konst, for a sixth grader with such capacity to ask that kind of question, what answer do you think will he understand? i was hoping that an analogy would do, minus all the technicalities of math. i already already have something in mind,but i think that should i switched shoes with the kid, i don't think i'd buy it myself. This is why i need everybody's help!
Regards,
Centeris
P.S. by the way Konst, my name is centEris, not Centris! :P
Hmmm,
The question now becomes very interesting.
How about telling the child to look up into the night sky.
In this day and age, even a 12 year old is aware that there are and endless amount of stars in the sky.
You could tell the child to use their imagination on a dark night (best beyond the lights of the city), and look up into the starry sky. Even better, print out a picture of a very starry night. Connect the stars(dots) to make a card box, just like a constellation. (In fact, there is a cool constellation called Caelum, fittingly defined: The Sculptor's Chisel).
Then go on to explain that everywhere inside that box there are stars, and beyond the stars, more stars.
And inbetween the stars there are planets, and asteroids and moons and spacejunk, and stuff we've never heard of.
And beyond the stars we don't even know or understand yet. Beyond that, possibly something else, and so on and so on. And on a clear day, when we point up into the blue sky, we may say there is nothing there, yet when night falls, we can use the same stars to make our box. If we took the space shuttle into outer space, we would find the box NEVER goes away.
Our box exists!
So although we may think that there is nothing, there is always something.
The same goes for the box in your hand.
You could even take the paper printout of the stars and wrap it around the box (visuals are always better for kids). The box in your hand is made up of the same things as the box up in the sky, and there is no end to what may exist inside the walls of that box, yet we may say nothing.
Just use your imagination!
Now this is just off the top of my head, so it may need some refining, but I think you, and the child should get the basic notion of what we are trying to explain here.
Hmm. Good question.
I.L.
Hi Centeris,
> P.S. by the way Konst, my name is centEris, not
Centris! :P
Ooops...sorry about that!
> In your expert opinion, Konst, for a sixth grader with such capacity to ask that kind of question, what answer do you think will he understand?
In this case I wouldn't think of my opinion to be an expert one, but anyway, I can try my best.
I know that analogies can work very well for kids, but in this case I don't think they'd be a good starting point.
I would first tell the kid that his/her initial assumption is not quite correct.
A card box is not MADE of surfaces, surfaces can be used to DESCRIBE a card box.
What I'd say next would depend on the kid's reaction to my statement.
If he/she asks something along the lines of "What do you mean by describe", I'd go into detail about how maths can be used to describe things in the real world. Here an analogy would work great. I could take the box and say I wonder how much water I could pour into it. Now is there a way to find this out without actually having to pour water into it because I don't feel like getting wet? And there sure is, because we can use maths to describe the properties of the box (length, height, width) and based on this calculate its volume which will tell us how much water would fit into it. In my calculations I have implicitly worked with surfaces, yet the surfaces of the box are not wet.
If it were correct that a box is made of surfaces then surely it would also be correct to say that finding out how much water fits into the box is equivalent to pouring water into the box - but I have just shown that this need not be the case.
Again, if the kid reacts in a different way, I would of course respond differently.
Konst
InnocenceLost,
I think your analogy is good, but I also think it is not an analogy to the actual problem.
The real problem is not the fact that it's hard to grasp that something is made out of nothing (which is hard to grasp because it is wrong). The real problem is that the kid thinks that a card box is MADE out of surfaces, which it is not.
Konst
This question seems to be going nowhere.
So do we have to limit our field of vision and say that there is such a thing as nothing? Then we must say that "space" is nothing. We must also say that a point (which is just a description of a point IN space, and has no "real" mass) is also nothing. Then I guess I can say that we can't "Make" (as "make" in the "real world" to me means to create or build) a box out of nothing (namely, air). If we say that "something" has to be a physical material (such as cardboard or paper or ice or whatever), then no, we can't do it. Unless of course we have supernatural powers and can use our mind to manifest objects out of no where. I hear Merlin could do that (but he didn't really exist in the real world).
Maybe we should watch "The Matrix" again and see if we couldn't get a little advice from Morpheus.
whatever
I.L.
Greetings, Centeris.
There is quite an easy way to show that the box isn't made of surfaces.
The kid in question already stated that surfaces don't have thickness, didn't he? Well, take a slide measure and show that every side of the box actually has thickness.
Explanation: the box is not made of dots *without* measure and mass, it is made of molecules *with* measure and mass.
Hoping this helps,
OmegaJunior
Your first entry was a real mind bender, but truly its not that complicated. With the original example of a box of cards. This is obviously three dimenional (it has length hight and width). When you reduce it to planes, vectors, et al, you move it towards theory. You say that points are zero dimensional, fair enough. But when you connect two points (make a vector), you make it one dimensional, you give it length. Carrying that on, you make a plane with two vectors (2-D), and finally put planes together (making them 3-D). In looking at the whole for its parts, you manifested the fallacy of composition. If you look at a box as a series of unconnected points, then yes, it is nothing, but if you look at the connected points as a unified whole, you have your 3-D box.
Theres my two bits, KyazarDude
for the kind comment of Innocence Lost,
you know what i like your analogy--it sounds pretty convincing to a young boy but i think he meant that "is it possible to make something exist from something (point)that doesn't, when he asked if he can make something out of nothing. i think the concept of nothingness is not the object of emphasis.
for the kind comment of Konst,
what if the kid does react differently-- how would you put it?
for the kind comment of KyazarDude,
i think i'd need a simpler (for dummies) explanation. if you have one (and i'm sure you do) please post it as well.
for the kind comment of OmegaJunior,
he can say that the thickness of the edge of the box is caused by different planes piled up on each other. i think. but i think that's a great way.
OK first of all, a plane in theory is 2-D, it has no depth, so stacking them would not achieve anything. If you piled a million 2D planes on top of one another, you would still have no depth (a million times zero is still zero)
And as for simplifying my first answer, here goes:
The concepts of planes, vectors, and points are generally used in theory. Your box of cards is not a theory, it is a concrete example. Your card box is made of cardboard, which is matter, it is not truly made of points and vectors and planes. You could DESCRIBE it with vectors, planes and points THEORETICALLY, but truly it is made of matter (cardboard), and thus the argument that it is made of nothing is invalid. Hows that?
Now that I have looked at the whole conversation, I realize that my explanation may still be two complicated. Luckily as a high school student, I am much closer to the age of the kid in question... Tell me Centeris, this 6'th grader, do they understand anything about the concepts behind points, vectors and planes? Thats really important to the answer, if they know that I might be able to help with a simple answer...
Until next time, KyazarDude
KyazarDude,
> OK first of all, a plane in theory is 2-D, it has no depth, so stacking them would not achieve anything.
> If you piled a million 2D planes on top of one another, you would still have no depth (a million times zero is still zero)
That's not quite correct, as I have pointed out many times now.
Ask yourself this question: What is a 3 dimensional space? Is it NOT a collection of infinitely many 2 dimensional planes?
It is.
For a more in-depth explanation, read my comments.
Konst
Konst, I don't think I'm wrong, a stack of 2D planes is still only 2D, the theory is that space is constructed out of perpendicular planes, not stacked parallel ones. I stand by my claim that if you were to stack a million 2D planes on top of one another they would still have no height (10000000 * 0 = 0, no matter what)
KyazarDude
KyazarDude,
> the theory is that space is constructed out of perpendicular planes
What theory and what space are you talking about here?
I'm still talking about Euclidean n-space in general and Euclidean 3D-space in particular - which is not necessarily "constructed" out of perpendicular planes.
You can define Euclidean 3D-space using planes. Take any plane P (this will represent Euclidean 2D-space). Points in this plane can be specified using two coordinates, x and y.
Take one such plane and label it P(z = 0). Take another and label it P(z = 1), ...etc. In general, we can label a plane P(z = r), where r is a member of the set of real numbers.
Euclidean 3D-space can now be defined to be the set
E3D = {P(z = r) | r is an element of R}
A point in this set now needs to be specified using 3 coordinates: z (specifying the plane), x and y (location in the particulare plane specified by z).
So Euclidean 3D-space can indeed be looked at as a mere collection of planes.
QED.
Konst
Looking at your post again, I can see where your confusion comes from:
> a million 2D planes on top of one another
Note the "on top". Of course you are right in saying that if you stack planes on top of each other they will not result in "height". It's like saying that a number of
points laid out next to each other results in length.
Nonetheless, a line can be looked at as a collection of points. Similarly, a cube can be looked at as a collection of planes.
Read my post that starts with "Alright, different angle".
The second part of it looks at why "next to" is a notion that does not apply when talking about points on lines.
It's the same with planes and cubes. The notion of "on top" does not apply here.
Konst
here we go agaaaaiiin............
For the kind comment of KyazarDude,
For a high school student, you sure know a lot of things, but maybe it will really help if we focus on giving an answer that a lay man can understand, because whatever conclusion you'd come up to, or should you be able to resolve what planes, points and lines are with Konst, the one who asked this question will not be able to get it anyway. Previous definitions have already been agreed on (or at least i think we have) and I believe that we'd want to be satisified with those for the mean time. I hope you understand.
For the kind comment of Konst,
I think you got it! I just want to ask for a question you think he'd ask after telling him your answer. i only need one--in all finality.
Regards,
Centeris
Hey Konst, let me reiterate that I am a high school student, I must admit I know nothing about Euclidian n-space and all that jazz. The theory that I have been basing my posts on is the theory I have learned (and hopefully understood) in OAC Algebra & Geometry (affectionately known as AlGeo). Your explanation is probably correct, but I'm afraid its totally lost on me!
Later, KyazarDude
I have been thinking about it, and at this point I must agree with Konst and KyazarDude. It is an age old debate.
I wonder if anyone here has the ability to define it it layman's terms. Reality in itself is an infinitely complex
equation, whether it be mathematical or simply philosophical. Truth is that there are many things in this "world" that take time to understand. This child at the age of 12 may not have the faculty to understand the answer, and the answer for now may have to be something that he/she does not truely understand. To expect this child to understand the answer would be expecting far too much. Maybe there is your answer.
Educate, build your way to the answer. It takes time. I hate to use a cliché, but Rome was not built in a day.
You may have to tell the child to be patient, the question is often as important as the answer.
I.L.
For the kind comment of InnocenceLost,
I was not asking for a definition, for should that have been the only thing i needed, i would not have asked this question, with all due respect. What I am asking for is a way for a 12 year old (who has the capacity to ask questions that we have been dealing with seriously) to understand how it is like. A true definition can be made as simple as possible, but not simpler (Einstein said that) I'm asking for a simpler *way* to make the child understand it now, without bombarding him the complexities or the prerequisite knowledge of math. should he have any questions beyond that, perhaps that will be the only time to make him realize that he needed to learn more about math to see why the answer (if anyone can provide) is such. not only have we given him an "enlightening" answer, but it will also stimulate him to study more beyond the lessons assigned. is it not one of the goals in teaching mathematics?
For the kind comment of KyazarDude:
Don't worry Dude, You're only in high school. I'm 19, I'm a college student, and even I can't grasp that concept fully myself.
For Everybody else:
Now do you understand why i need an answer "in plain english?" A High School Student can't even understand such concepts (i.e. Euclidean n-space) let alone make a 12 year old understand that.
Regards,
Centeris
KyazarDude,
I'm sorry, I forgot that you said you were a high school student.
I'm also sorry to hear that my explanation is lost on you...the problem is probably the vocabulary. In maths there's a ton of words and symbols that have a specific meaning in a specific context. If you don't know or can't guess that meaning, you will be lost.
If you want to you can tell me all the things in my post which confuse you and I'd be happy to explain in more detail. Don't be afraid that you won't understand it. I am sure you will, as the actual idea behind it all is quite simple once you found out how to look at it.
Konst
Centeris,
> I think you got it! I just want to ask for a question you think he'd ask after telling him your answer. i only need one--in all finality.
I think that it'd be most likely for the kid to ask a question about the conclusion I came to:
"If it were correct that a box is made of surfaces then surely it would also be correct to say that finding out how much water fits into the box is equivalent to pouring water into the box."
I am sure the kid would by now have seen that you don't need to pour water into the box in order to find out how much water would fit into it, which is a step in the right direction.
But the kid might ask something like "Why would saying that a box is made of surfaces be similar to saying that..."
And I'd say something like "Because the bottom line is that you simply can't pour water into a collection of surfaces. The box is NOT the same as the surfaces we are using to describe it. Saying that a box is MADE of surfaces is as wrong as saying that you can only find out how much water fits into the box by pouring water into it."
The fact that the box is not the same thing as the collection of surfaces is something that the kid will have to accept at first. In order to truly understand it the kid would have to be able to make high level abstractions and at the same time realise that such abstractions are not to be confused with the actual idea of which they are an abstraction.
Some children will be able to do this earlier/easier than others. It is not something you can teach them. All you can do is teach them everything they need to know in order to be able to understand it eventually.
> I'm asking for a simpler *way* to make the child understand it now, without bombarding him the complexities or the prerequisite knowledge of math.
> should he have any questions beyond that, perhaps that will be the only time to make him realize that he needed to learn more about math to see why the answer (if anyone can provide) is such.
So you're saying that you're looking for an answer the child will understand yet it may not know why the answer is such?
Isn't that a contradiction in terms? Surely knowing why something is the answer is part of understanding the answer, i.e. if you truly understand an answer it is implicit that you know why it is the answer, for if you didn't you wouldn't have understood it.
Now the problem with maths is that you can't start somewhere in the middle and expect it to make sense. If you want to truly understand something you have to first understand the theory it is based on.
I mean, suppose you want to learn the piano. You have taken a couple of lessons and now you want to start playing Mozart. It's not gonna work, is it? You need to practice and learn a lot firs, only then will you be able to play Mozart.
Konst
So allow me to understand.
You are writing a paper that "attacks" the way that math is taught in schools. As you did not expound on that any further, I surmise you hold a defined opinion on the matter.
Do you have a subjective opinion on this subject, with the intent to prove that maths are not neccessary to explain the question? It seems as though you are looking for an answer that only supports your thesis. In research we must keep an open mind for an answer that we may not have expected. At this point, it seems to me that the answer you are looking for is not going to appear. I have held much scepticism myself as to whether the answers for life's mysteries lie in math. Still I believe that it is not the answer for everything, but for the question you have posed, I think we must bow to the idea that math is the key to it's understanding.
Maybe you should consider that the outcome of your thesis is not what you had anticipated. This does not devalue your paper. On the conrtary, it is all in the name of enlightenment.
On the other hand, the answer may be out there, just not in the confines of this discussion group. I would be interested to hear it, should it exist.
If I have made a mistake in my assumtions (and it wouldn't be the first time), I appologise.
I.L.
For the kind comments of Konst and InnocenceLost,
I am sure that when you were taught in elementatry school that the circumference of a circle is C=(pi)r^2 and when pi is approx= 3.141592653589.... i don't think your teacher had to brief you with Boolean logic or showed you the derivation of the numerical value of pi using the squeeze theorem, right? but back at that time, did you not accept that value without question? have you tried solving that using calculus? of course not, because your capacity at that time cannot handle that. but it still made sense, didn't it?
this is why we need an answer at the "surface level." i am not saying that we put set off without the math. that's a terrible thing.what i mean is being "correct" will not do so much if the listener would not understand it anyway. and and in-depth analysis will follow, if and when he is ready.
if we are to teach mathematics, in my humble opinion, i think we must start with analysing something qualitatively, or by-concept, so to speak. after establishing that understanding, it is only then should the quantitative aspect of math be discussed in full detail. from where i come from, they don't do that very often. they even spoonfeed solutions to their students.
i am not against math (i love math--why else should i make this as my topic?) i am against the way how the knowledge is being passed onto the younger generations. i don't people making math a lot harder to understand than it really is. sometimes the concept behind formulas are not given emphasis, so their solutions are constrained by that formula. this trend does not open students' minds to critical thinking. this also applies to every subject/course we take, and i think you'd agree with that.
i'm trying to prove my point that math can be taught in an angle that the student will understand. in this particular question, i am asking for a way for a 12 year old to understand that a box is not made of mathematical surfaces, in his own language.
i hope i'm making sense.
Centeris,
I appreciate your post, but there are two things I want to say:
1. You cannot compare the question "Why is pi 3.1415926...?" with the question "Why is a card box not made of surfaces?".
Why? Because the pi question is a question that can be both asked and answered within the formal system of maths. You can say pi is pi because it is defined to be the ratio of a circle's circumference and radius. Not much to understand there really, it's just a plain and simple definition.
Now the box question is of a different nature. It is a question that is "outside" of the formal system of maths because it is a question about the relationship of formal systems with the real world. As with the pi question, a simple answer does exist, namely that a real world objects are not MADE of mathematical abstractions. The abstractions are used to DESCRIBE the real world.
This answer is simple, so why can't it be accepted like the answer to the pi question? Because to accept it you have to first understand the difference between abstract concept and reality. Once you accept that there IS a difference the question and its answer will indeed look simple to you.
As for an actual answer I would give to the child, I have done the best I could to come up with something. For my part, the question is answered.
Konst
For Konst,
As a matter of fact, you already have.
As for the others,
Is there someone else who has a better "answer" than what Konst said? I really think his answer is already good enough for me and is a solution to my question. If nobody else does, then I'll close the question and give full points to Konst. Are there any objections?
If there aren't any then I need someone to concur.
This is Konsts's answer that I find acceptable:
**********
" I would first tell the kid that his/her initial assumption is not quite correct. A card box is not MADE of surfaces, surfaces can be used to DESCRIBE a card box.
"...If he/she asks something along the lines of "What do you mean by describe", I'd go into detail about how maths can be used to describe things in the real world. Here an analogy would work great.
>>>>>>>>>>>I could take the box and say I wonder how much water I could pour into it. Now is there a way to find this out without actually having to pour water into it because I don't feel like getting wet? And there sure is, because we can use maths to describe the properties of the box (length, height, width) and based on this calculate its volume which will tell us how much water would fit into it. In my calculations I have implicitly worked with surfaces, yet the surfaces of the box are not wet.<<<<<<<<<
"If it were correct that a box is made of surfaces then surely it would also be correct to say that finding out how much water fits into the box is equivalent to pouring water into the box - but I have just shown that this need not be the case.
Will anybody object to that?
Centeris
It has been concurred, and thus Konst gets full points for a job well done. Thank you Konst for all your time and effort spent to arrive at the answer I am looking for.
Konst if you would be so kind please e-mail me at therealnecromancer@hotmail
>Name
>Highest degree attained
>school
>country
because i cannot take credit for your answer! I want to include you on my list of "consulted experts" in my paper.
InnocenceLost et al will also be acknowledged.
I had a great time talking with everybody. Thanks for helping!
Regards,
Centeris
KyazarDude,
Seems like a case of mistaken identity.
Approx. half way down, text in the middle of the page.
http://www-en.sergiobonell
Hey everyone, this actually haas nothing to do with this discussion, but there seem to be many knowledgable people here... I'm doing a research paper on punk and its origins, and right now I'm doing a musical timeline that leads up to punk and I need to know when disco, metal, and glam became popular.... Like I said, nothing to do with math, but I would apreciate any help!
Thanks, KyazarDude
What is reality anyway? If we think, we categorize our sensory input, aligning it to a internal world of images. In the best case, it correlates well with the real word, but often it does not, just look at the confusion around you. Even your intuitive concept of a cube and its facets is a vast simplification of reality, let alone its mathematical description.
So what is science? It is nothing more than a very restricted and incomplete way of describing reality. Our theorems should be consistent, based on repeatable observation of measurable phenomena, or, in case of mathematics, any consistent framework based on a set of axioms. Within its restriction it is useful (?), it reduces the confusion and we can build televisions and mobile phones with it, and sometimes, it is even beautiful.
It is only a tool. Just as any of our thoughts, don’t take it to seriously.
Business Accounts
Answer for Membership
by: CenterisPosted on 2003-01-09 at 19:01:37ID: 7698938
i tried to answer this question by challenging the premise:
a solid card box is a 3 dimensional surface made of ATOMS!
atoms are made of protons, neutrons, electons, (and some other elementary particles such as quarks).
but,
oh my god.
supposing that these particles are solid spheres, this means that they are still made of spheres
and spheres are made from points
the rest is history!
oh maaaaannnnnnn.
just tell me i'm stupid. tell me where i was wrong! :P