Physics: Decimals can't be infinite because the space between must end.
I've pondered this off and on since high school:
I understand the principle behind the assertion that decimals never end and that you can (theoretically) infinitely halve the space between two objects and they'll never touch. A parabolic curve works on this principle, but really...
If two objects are travelling at each other, at some point the space between them must end so that there is NO space left and they are touching, right? By this notion, I'd think that a value of their distance apart MUST end. So it seems that at some point 0.00000000000000000...1 must become simply 0 when the objects finally touch.
--> <--
Surely someone way smarter than I am can explain this. What am I missing here?
I understand the principle behind the assertion that decimals never end and that you can (theoretically) infinitely halve the space between two objects and they'll never touch. A parabolic curve works on this principle, but really...
If two objects are travelling at each other, at some point the space between them must end so that there is NO space left and they are touching, right? By this notion, I'd think that a value of their distance apart MUST end. So it seems that at some point 0.00000000000000000...1 must become simply 0 when the objects finally touch.
--> <--
Surely someone way smarter than I am can explain this. What am I missing here?
Comments
Decimals are abstractions in math having nothing to do with reality per se.
Physics takes mathematical abstractions and tries to use its results to predict events -- but attempting to do the opposite -- find physical meaning in math is irrelevent.
You're basically describing "Zeno's paradox", AKA "the Dichotomy", of which there are several variants involving distances and times, but which stripped of the word problem narratives (which is a good idea to avoid some of the syntactical confusion arising out of moving between natural language and math) amounts to "successive halving a number approaches but never reaches some limit". Or , if you prefer, "the decimal goes on forever". The most common presentation is "how can I ever really cross the street if I must first go half way, and then half way again, and so on, with no end to the infinite succession of halves?" with "the other side of the street" representing the limit which is approached but never reached.
Another way to talk about the same thing is to consider the infinite density of the number line-- that for any two points a and b we can always define a third point c between a and b such that c =( a +b)/2.
The idea can be applied to any interval you like-- if I increase the resolution of of my measuring stick indefinitely, can I ever be said to have an actual "height", since there does not appear to be a final term in my series of ever more minute slices of space?
A good chunk of mathematical history is the story of coming to terms with theses "infinitesimals"-- the arbitrarily small amounts that separate the sum of a convergent series from its limit (which is to say a mathematically rigorous treatment of infinity), and I have neither the math nor the vB code typographical chops to represent the actual equations that deal with all this, so I heartily recommend Wallace's book (although for my money the math gets pretty dense at times).
However, it should be noted that a lot of the "paradoxical" nature of Zeno lies in the way natural language is ambiguous where mathematical language is not, and some shifty sliding between the two.
That is, Zeno is inviting us to regard an interval in the real world as being "made of" the infinite convergent series 1/2 + 1/4 + 1/8 + 1/16....., approaching but never reaching 1, whereas in fact in the real world the interval "1" (AKA "crossing the street") can be understood to simply exist (in the manner that things in the real world are allowed to do, as opposed to the abstract rigor of mathematical space), and that same convergent series approximately describes the a priori quantity "1". It's the inversion of the abstract descriptor with the thing itself that makes for the apparent weirdness. (For any mathematicians in the house I hope I'm not making to much of a hash of this, it does take a book to really nail down the movement from Zeno to transfinite math, and a lot of it isn't really "intuitively" graspable outside of the math itself).
Edit: don't mistake this to mean that it will reach this point. See other comments.
Originally posted by giant
well, .9999... == 1
Right, but the important thing to consider in CosmoNut's question is that in the case of "paradoxically" applying that to the real world you have to bear in mind that ".9999999...." is not a feature of the world, it is a feature of a mathematical description of the world.
So that "the reason" that .9999......"is equivalent" to 1 is a series of internally consistent mathematical statements regarding the abstract interval 0 to 1 on the number line, but if we start thinking of the distance to cross the street as being somehow "made" of (as opposed to "satisfactorily described by) .99999...... we immediately get into trouble.
CosmoNut is actually describing the inverse of Zeno's paradox, something like "since I already know that space can be successfully transversed, and mathematics purports to describe space, how is it that a finite world of closed intervals simultaneously supports infinite descriptors in what appears to be the same interval?"
Again, the problem arising if we are not very careful to distinguish between real world phenomena like "things colliding" and abstract descriptions of things colliding that move endlessly toward a finer and finer description of the point of collision.
Originally posted by CosmoNut
I've pondered this off and on since high school:
I understand the principle behind the assertion that decimals never end and that you can (theoretically) infinitely halve the space between two objects and they'll never touch. A parabolic curve works on this principle, but really...
If two objects are travelling at each other, at some point the space between them must end so that there is NO space left and they are touching, right? By this notion, I'd think that a value of their distance apart MUST end. So it seems that at some point 0.00000000000000000...1 must become simply 0 when the objects finally touch.
--> <--
Surely someone way smarter than I am can explain this. What am I missing here?
Here's the rub for the two objects traveling towards each other bit. They may be moving towards each other, but they are always aiming for a point short of the other object. The closer they get the slower they move towards the new version of the aimpoint which is updated to always be halfway there. The objects following the halfway there game never intend to touch, they always dynamically adjust their travel to conform to the halfway there game rules.
This way the seeming paradox is gone when we think of objects actually traveling towards each other that are not playing the halfway there game. If the objects were meant to actually touch each other they would not ever worry about the halfway there aimpoint. Then any smartass that tries to tell you the motion to get them to touch is mathematically impossible is full of crap because the smartass would have to introduce a bogus assumption that would eventually force a division by zero someplace in the real world math to force the paradoxical halfway there idea onto a constant motion object not playing the halfway there game. That bogus assumption and division by zero being the happy contradiction that that allows you to safely ignore the halfway there paradox when you are moving someplace in the real world.
Originally posted by hardeeharhar
Things never actually "touch" each other, but that really is besides the point.
Decimals are abstractions in math having nothing to do with reality per se.
Physics takes mathematical abstractions and tries to use its results to predict events -- but attempting to do the opposite -- find physical meaning in math is irrelevent.
Au contreaire! Physic IS math! Chemistry IS physics. Everything else follows off that. The problem is when someone make an assumption that is valid in one system but not another for some reason or another. Then all hell breaks loose, but the rules are all still purely mathematical and consistent. We can get to areas which we don't understand well enough yet and consistency starts to fall apart there, but that all comes back to assumption validity.
That is the study of Chemistry builds upon the study of Physics.
Remember the fields have nothing to do with reality all that much either, as in, the discplines are human inventions, and the scopes we limit them to are artificial, but that is another discussion for another time...
Concerning "smallest possible distances":
If you subscribe to string theory, there is a smallest possible distance, the size of the individual string. (link)
This implies that nature is in fact, digital, not analog.
String theory has been maligned a bit these days though. I'm not qualified to make any statement beyond "that's intriguing", in any case.
Oh, and Mac_Doll, you were right.
Originally posted by CosmoNut
So...has anyone actually given a real answer yet, or is it just a "paradox" and we leave it at that?
Oh, and Mac_Doll, you were right.
The answer is: objects in motion do not move through the decimal system. They move through real space. Cutting space into ever finer slices is a mathematical process intended to closely approximate the motion of objects in space, but it is not the motion itself, or the space itself.
So on the number line you can have one infinite series that approaches an upper limit of, say, 2, and another that approaches a lower limit of 2, and they never meet, despite getting ever closer forever.
But that is an abstraction designed to help us describe the world. Two baseballs hurtling at one another are described by such numbers, but not constrained. They are not obliged to perform Zeno's endless halving of the remaining space, because the space between them is not being generated by a formula-- it already exists, the point that the baseballs make contact is a real point in real space, and the mathematical description is after the fact.
Originally posted by addabox
how is it that a finite world of closed intervals simultaneously supports infinite descriptors in what appears to be the same interval?
I think that's it in a nutshell. It's true that if you do continuously halve the distance you travel, you will never reach where you are going but you just don't do that in the real world. If a street is ten steps wide and you take ten steps, you will reach the other side, assuming you aren't run over while you were too busy pondering the nature of the universe instead of watching for oncoming traffic.
Fractals follow a similar principle of bounding infinite detail in a finite geometric space.
So...has anyone actually given a real answer yet, or is it just a "paradox" and we leave it at that?
Addabox and Benzene have nailed it down pretty well. Your question is about what is the smallest possible space between two objects.
For example, one idea of what an "object" is:
There is a way to mathematically try and model that, but like we learnt in college(?) scientific models are just that - a way of trying to describe a situation, in this case, trying to describe mathematically a physics case.
So coming back to the question of the smallest possible space between two objects, we go down to what is the physical nature of those objects? Are you talking about atoms? Subatomic particles popping in and out of our dimension quantum-mechanics style? Then there is the String theory earlier mentioned which means "everything is connected there is actually no empty space"...
1. The above paragraph is my armchair-layman's interpretation of stuff.
2. I'm more inclined to disrespect the maths side of things because that's just a way to represent the "real world" whereas physics actually gets down to the nitty gritty of what IS the "real world". Maths for the sake of maths is too abstract for my liking.
.....So it seems that at some point 0.00000000000000000...1 must become simply 0 when the objects finally touch....Surely someone way smarter than I am can explain this. What am I missing here?
What is missing is what the mathematical description of the empty space is. If 0.000000000000000000000000000000000...001 = 0 is true on a physical level then some assumption has been made of that "empty space" - you have "put something in there (that something could be nothing*)" for 0.000000000...00001 = 0 if you are talking about objects occupying physical space.
*So we go back to essentially the physics question of "what is empty space", or "what would you put there" for 0.00000000000.0001 = 0 to be true.
Then there is the question of what is your definition of two objects "touching"
Come to think of it, even if we look at the atomic level of things and not go deeper into subatomic particle and string theory, CosmoNut, have you done "electron orbitals" at college? Unless this theory has been disproven recently, see, in high school we are taught that electrons "orbit around the nucleus made up of protons and neutrons".
This is actually not true. http://en.wikipedia.org/wiki/Electron_configuration
You see, in an atom, electrons occupy what is known as a "probability region" or "electron orbital space". That is, at any one point in time, we cannot predict where the electron actually is. It's not like predicting where the moon is as it goes around the earth. The "probability region" is only defined when you use some way to "observe" the electrons. For example in the image above the 3d spaces defined are where the electrons are most likely to be for those atoms (not going into shells and subshells too complex)
So two objects touching on the atomic level is an interesting question because if you take it as the electron orbit being the outer edge of an object, there is no such thing as an atom being a nice round ball so you can define it as touching another atom of being a nice round ball. In other words atoms are not nice round balls...!
1. Again, probably gonna get slammed by more in-the-know people.
2. But man, my first-year chemistry classes at college were
"fun" when I learnt about electron orbitals. It was like, WTF? Atoms are not these nice round balls...?!!!
We don't know what any multi-electron atoms' electron probability maps actually look like. There are higher order approximations that appear to imply that our use of the hydrogen orbitals are at least qualitatively correct for ATOMS, but when push comes to shove, QM calculations on MOLECULES are mostly high level approximations of systems of atoms.
Hydrogen filled with any number of electrons still looks like a ball -- as the electrons tend to avoid each other.
Regardless, when you push upon an object the electrons of your skin and the object start filling shared molecular orbitals pair wise until the energy of the next highest orbital is greater than the work being done to push the two objects together. Simply, the electron clouds of your finger and key repel each other.
1) That stupid Subaru totally ruined the back of my car.
2) A punch can, in fact, cause a broken nose.
3) You should always pay attention to the bat, while playing baseball with kids.
Hardeharhar hit the nail on the head. Since QM states electron clouds in terms of potentials, the idea of "distance apart" for macroscopic items is by definition, also quite fuzzy. The electron clouds will start to interact, and it is through this repulsion that we can exert force. There is a distinct, mathematically described potential that an electron belonging to an atom in my mouse will spend some of its time in my hand (even if they're technically separated by thousands of angstroms).
If you wanted to really get technical, you could measure distances by measuring between nuclei, but with brownian motion and the potential for quantum tunneling even that's rather sketchy. The point at which your objects become sufficiently small to truely "measure between them", you end up with a whole other level of weirdness.
Moral of the story: you can't be glib when talking about anything on a subatomic scale.
Originally posted by benzene
This thread is actually quite funny to read.
Hardeharhar hit the nail on the head. Since QM states electron clouds in terms of potentials, the idea of "distance apart" for macroscopic items is by definition, also quite fuzzy. The electron clouds will start to interact, and it is through this repulsion that we can exert force. There is a distinct, mathematically described potential that an electron belonging to an atom in my mouse will spend some of its time in my hand (even if they're technically separated by thousands of angstroms).
If you wanted to really get technical, you could measure distances by measuring between nuclei, but with brownian motion and the potential for quantum tunneling even that's rather sketchy. The point at which your objects become sufficiently small to truely "measure between them", you end up with a whole other level of weirdness.
Moral of the story: you can't be glib when talking about anything on a subatomic scale.
Let's not make the mistake of implying that the instrumental imprecision of defining the smallest possible unit of space time to be used in defining the boundary between objects has anything to do with transcendental numbers, except insofar transcendental numbers might be used to describe some aspect of such a unit.
The infinitude of irrationals and such is of a very different order of epistemology than the "fuzziness" of the world at very small scales, which in fact the point of "paradox" in the original post.