Physics: Decimals can't be infinite because the space between must end.

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.

*head explodes*

Huh. I was just reading a chapter in "Everything and More", a short history of the concept of infinity and how it has been grappled with mathematically since the Greeks (by David Foster Wallace, the notoriously discursive fiction writer and essayist, who apparently is a bit of a math nerd and finally gets to put his love of footnotes to good use).

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).

well, .9999... == 1

Edit: don't mistake this to mean that it will reach this point. See other comments.

I'm good with regular math, but anything that takes me beyond algebra makes my head hurt.

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.

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.

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.

No. All fields are aesthetics on fundamental thoughts that human's are capable of having, but this does not mean that one field is the same as the others. They build upon each other, but they are not the same. Chemistry is NOT Physics, rather it uses Physics.

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...

Good clarification hardeharhar.

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.

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.

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.

[QUOTE]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?


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.

[QUOTE]Originally posted by CosmoNut
.....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...?!!!

Um...

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.

A few personal observances that objects do actually touch:

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.

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.

Again Mathematical abstractions have no physical meaning in and of themselves.

Again. The universe is just one vastly large set of mathematically describable interactions. How the subsets of mathematical interactions are are parsed into disciplines doesn't change that.

Abstractions are just a set of artificial assumptions about the actual interactions which may or may not be correct. As long as your assumptions are correct for the scale you are examining, your described outcomes will be correct as well. All physical sciences are engaged in discovering ever more correct abstractions (assumptions). The fact we don't know them all doesn't mean something at a less granular level doesn't actually happen - like objects touching.

Everything else is obfuscating bullshit hiding reality behind a layer of insider lingo. Go ahead an believe otherwise if you want but that won't change the fact that things actually touch.

Not all math is applicable to "reality."

Period.

You can continue to deny this, but if you ever have had a conversation with a real mathematician doing research in modern topics, you will quickly realize that math has advanced far beyond simple descriptions of reality.

Math doesn't depend upon confirmation in the real world, and that is a testament to its history and proofs.

Continuing to deny this prevents you from understanding the depth of math.

Edit: You don't discover assumptions. Science is actually founded on the principle of minimizing the number of assumptions needed to understand a system. In fact, the exact opposite of what you argue...

I should also say that it is a semantic point with regard to objects touching.

Is your cut off at the 70% orbital overlap level or the 50%?

This is kind of where this is going, but I've always kind of assumed that "objects" don't really "touch" at all, and that to assume that they did was to presume the "thingness" of the object itself.

[QUOTE]Originally posted by hardeeharhar
I should also say that it is a semantic point with regard to objects touching...Is your cut off at the 70% orbital overlap level or the 50%?


Precisely why I brought in the point of not having to go into quarks and string theory and all that, and brought up orbital probabilities*. What is touching? What is empty space?

I say (I assume, define personally) that things touch because there is evidence of electron repulsion. I touch my mouse because I feel a force between the mouse surface and my hand surface brought about by repulsion of the two molecular electrons. I define/ assume touching as the feeling of repulsive forces.

%Of orbital overlap of quantum electron probalities of molecules is beyond our ability to measure at this stage, right?

*I forgot electron orbitals that we know about come about from starting with the Hydrogen atom. At this stage we still can't work out electron probability 3d-space for molecules? Wow... lots of work still to be done.

[QUOTE]Originally posted by midwinter
This is kind of where this is going, but I've always kind of assumed that "objects" don't really "touch" at all, and that to assume that they did was to presume the "thingness" of the object itself.


Currently in our science the "thingness" of the object is assumed/ proven by the process of observation - ie, there is someway to prove that it is actually there. Eg. the pattern observable when atoms are smashed together, you can then "prove" that subatomic particles such as bosons and mesons and those kind of particles exist.

It "started" at a Newtonian level, we know the moon is a "thing" that exists because we can observe it, and also we can observe the effect of forces exterted on/ exerted by the "thing". In this case gravitational forces and light reflection is proof of the "thingness" of the moon.

About objects touching, that goes back to semantics as hardeeharhar says of how you define touching -- electron orbital probabilities, string theory, and again, how to define empty space....

[QUOTE]Originally posted by hardeeharhar
Not all math is applicable to "reality."

Period.

You can continue to deny this, but if you ever have had a conversation with a real mathematician doing research in modern topics, you will quickly realize that math has advanced far beyond simple descriptions of reality.

Math doesn't depend upon confirmation in the real world, and that is a testament to its history and proofs.

Continuing to deny this prevents you from understanding the depth of math.


This is great that you have mentioned this. Math for Math's sake does exist as a field of study in and of itself.

Using Maths for physics problems is Applied Math, but there is Math just for Math -- something I feel is so abstract, but I suppose I'm glad people do it because eventually with all those theorems and stuff it can be Applied to useful problems in the "real world".

Yes -- a "real mathematician" doing advanced Masters and phDs and post-doctoral work in Maths researches all sorts of stuff, especially the super weird crap of trying prove theorems that have not been proven yet. Theorems and proofs and all that jazz. Mindf*ck stuff if you're not in that field.

Just look at this from our Apple cheerleaders Wolfram Research:
http://mathworld.wolfram.com/FermatsLittleTheorem.html

Eg. Fermat's Theorem in this case. On the page there is no application to the "real world" although Applied Mathematicians have probably used it in various areas. There are links to other stuff there that's "pure mathematics"
("SEE ALSO: Binomial Theorem, Carmichael Number, Chinese Hypothesis, Composite Number, Compositeness Test, Euler's Theorem, Fermat's Little Theorem Converse, Fermat Pseudoprime, Modulo Multiplication Group, Pratt Certificate, Primality Test, Prime Number, Pseudoprime, Relatively Prime, Totient Function, Wieferich Prime, Wilson's Theorem, Witness".)

But, they can send a man to the moon and solve hundreds-year-old theorems but why can't they make my feet smell nice

[QUOTE]Originally posted by addabox
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.


I agree to the extent of what I understand. CosmoNut is trying to ask what he is "missing". He has not defined a "proffessional" Paradox like the kinds the "scientific community" is dealing with.

CosmoNut, IMHO, this is what you are "missing"

1. You propose a situation whereby two objects are moving towards each other, each time moving half the distance between them. Yes, at some point in time, the space between them will become sooooo small that essentially Newtonian physics is no longer relevant, we get into the Quantum physics realm and all the "weirdness" associated with it - which we have tried to describe above. This small distance is not as small as you think, we can think in terms of one Angstrom. Ten billion angstroms equal 1 meter. So AFAIK there is no specific definition where Quantum physics take over but certainly once you hit 1 Angstrom and less Quantum mechanics and the weirdness of atomic- and subatomic-particle interactions start to apply. At this stage the "Newtonian-style view" of an electron orbiting the nucleus like the earth around the sun is total rubbish.

2. You made a huge jump in tying together the maths side of things and the physics side of things. You started with saying, let's assume 0.000000000000....000001 approximates to essentially zero. Then you JUMP to the conclusion of these numbers relating to a scale of space between two objects. Then you jump again to the conclusion that "I approximate 0.0000000.....0000001 to zero therefore the real-world space between two objects at that level must also be zero". So like it has been said before, currently in our understanding there is a Physics "real world" situation and we use Applied Maths to tackle the issue. Going the other way round does not make sense in this case because you are taking an abstracted Mathematical idea and then "dumping" it onto the real world. The title of this thread itself is problematic because the way you defined it -- taking an abstracted mathematical situation and then "duct-taping" it onto a physical situation.

3. "Decimals can't be infinite" ... Who says so? If we just look at the Mathematical implications of that, again this is the third issue of what you are "missing". Why is 1+1 = 2? Only because by convention. Mathematicians do all sorts of weird stuff in the "pure maths" area. For example, depending on what they are trying to do and the conventions of the field they are working with, they can say Decimals are infinite or decimals are not, they can say all sorts of things, depending on the problem they are working on. Just take Pi - we think of it as 22/7 and it just goes on forever. But actually 22/7 is GREATER than Pi.
http://en.wikipedia.org/wiki/A_simpl...2/7_exceeds_pi
I would assume then from this that those programs that use computers to derive all the values of pie, do not just take 22 and divide it by 7. They use other formulas to work out the millions of digits or whatever. Stupid irrational numbers, so irrational


Disclaimer: Again, as is my understanding at this point in time and I can't believe my brain is still handling at least 10% of this mindf*ck stuff.

http://mathworld.wolfram.com/ZenosParadoxes.html
"The dichotomy paradox leads to the following mathematical joke. A mathematician, a physicist and an engineer were asked to answer the following question. A group of boys are lined up on one wall of a dance hall, and an equal number of girls are lined up on the opposite wall. Both groups are then instructed to advance toward each other by one quarter the distance separating them every ten seconds (i.e., if they are distance d apart at time 0, they are d/2 at t==10, d/4 at t==20, d/8 at t==30, and so on.) When do they meet at the center of the dance hall? The mathematician said they would never actually meet because the series is infinite. The physicist said they would meet when time equals infinity. The engineer said that within one minute they would be close enough for all practical purposes."

I've taken the engineer-style approach in my statement no.1 above. At the end of the day it comes down to our individual/ fields-of-study approaches, assumptions, perceptions and observations of distance and time.

Extra Credit: Two objects are 1 metre apart. They move towards each other, by one quarter the distance separating them every one second (i.e., if they are distance d apart at time 0, they are d/2 at t==1, d/4 at t==2, d/8 at t==3, and so on.) Given d = 1 metre At what time t will the distance between the two objects be 1 Angstrom (d/10billion)?

Which of you 1337 pe0pl3s can answer this (my head would burn up in an instant the second I tried to solve this). I suspect this is an easy question though for Maths peoples. I also suspect the answer is *not* 10 billion seconds.

Duh!!!

Will you quit trying to conjure up bogus points to argue against? Then state them as if I am daft because I must believe them? They are completely your own!!!

I say all reality is describable by math, even though we don't know how to do all those descriptions yet. Which has no direct conflict with your Not all math is applicable to "reality.". That's like saying, right after someone points out all squares are rectangles, -- Not all rectangles are squares!!! OMG this is so wrong!!! -- You might as well claim the price of kumquats in Jakarta disproves my point!

I couldn't give a rats ass about pure mathematics for the sake of pure mathematics (or the price of kumquats in Jakarta), this thread happens to be about applied math and a seeming paradox that really isn't a paradox at all.


Semantics. Assumptions in an equation, known "factors" in an equation, both are exactly the same thing because in physical reality we don't ever actually "KNOW" everything about everything. The terminology all just depends on which technical philosopher you most recently happen to agree with.

The entire scientific method hinges on testing the hypothesis to enough technocrats satisfaction that we then collectively "bless" the latest "discovery" of something that has been there all along. We treat the blessed idea as a "fact" for lack of anything better and employ it as an "assumption" in just about every use except when we describe it directly, then we label it as a "theory" in the finest print possible and get on with making the next "discovery" that we can use as an assumption in the NEXT "discovery"! See that last sentence? Whatever we "discover" today is just one of our assumptions in what we are trying to "discover" tomorrow.

Anyone who argues with that hasn't been reading or publishing enough lately!

Between the 33rd and 34th seconds the transition will occur. When in between is entirely dependent on the acceleration characteristics applied to the objects between the sample points. Figuring the remaining distances at the whole seconds is just binary vs base 10 math: 2^(-n) <= 10^(-10) solve for n. 10 seconds and a spreadsheet do that quite effectively.

NERDS!!!

My point exactly, put more succinctly.


This is why I said lots of people have a problem with string theory. The issue is that string theory is primarily a mathematical abstraction and is so complex that just about any observation, whether valid or not, could potentially be explained. (Sort of a deus ex machina, but we're not getting into that here).

Most of the phenomena that string theory propose we're completely unable to measure at this point, which has led many to criticize string theory as untenable. It's mathematical abstraction taken beyond the point of description, into a realm of creation, almost. Every now and then, the mathematical description has to be tempered with reality.

Slashdot had an article on this some time ago.

[QUOTE]Originally posted by Hiro
Between the 33rd and 34th seconds the transition will occur. When in between is entirely dependent on the acceleration characteristics applied to the objects between the sample points. Figuring the remaining distances at the whole seconds is just binary vs base 10 math: 2^(-n) <= 10^(-10) solve for n. 10 seconds and a spreadsheet do that quite effectively.


Lets assume acceleration to be constant. Since it is d/2 at 1sec, d/4 at 2sec, d/8 at 3sec, decceleration can be calculated. Extra Credit Part 2: What is the deceleration in metres per second squared?

Cool. Extra credit for Part 1 to you. Yay! But partial extra credit only. Show your working on how you solve for n ..!

I went for a swim earlier and it dawned on me, yeah, just a matter of 2^x = 10billion. Then just incrementally guess x and calculate the result.

Using a spreadsheet and "guessing" x and seeing how close it is to 10billion is kinda cheating though. Did you just do this?

How would you solve for x without using a spreadsheet/ calculator? Show your working on paper (since symbols don't show up properly on these pages), scan it, and post it.

[QUOTE]Originally posted by midwinter
NERDS!!!


*sigh* after all the sex, drugs and trance music this is the 1% of my brain and knowledge I use to have that is left over to be barely able to process this. Chemistry, electron orbitals, polynomial maths, Newtonian physics, integration and differentiation. I've totally forgotten how to calculate acceleration/deceleration... Phew \

I'm just dumbfounded that a question about Zeno's Paradox went all sub-atomic and shit.

[QUOTE]Originally posted by addabox
I'm just dumbfounded that a question about Zeno's Paradox went all sub-atomic and shit.


"all sub-atomic and shit" Well, we just came to a pragmatic approach. Two objects start moving at each other, 1/4 of the distance each second. Initial distance between two objects is 1 metre at 0sec. At 1sec, distance 0.5 metres, 2sec, distance is 0.25sec, etc.

Will they ever touch? Well, around 33seconds, the distance between the two objects is about 1 Angstrom, one-ten-billionth of a metre. Quantum mechanics start to come into play then. From 33seconds onwards that's where it gets "all sub-atomic and shit" because the definition of "touching" then comes into play.