Science Student 9 Posted February 14, 2014 I am really confused over the general idea of the summation of infinitesimals of some quantity. For example, can anyone show me mathematically how an infinite number of dm can either equal 5 kg's or 10 kg's? I can understand basic calculus. 0 Share this post Link to post Share on other sites

Sensei 941 Posted February 14, 2014 For example, can anyone show me mathematically how an infinite number of dm can either equal 5 kg's or 10 kg's? Who said mass is infinitesimals? If you have 10 kg of Carbon-12 isotope it means it's 10,000 g / 12 g/mol = 833.3 mol. 833.3 * N_{A} = 833.3 * 6.022141*10^23 = 5.01845 * 10^26 atoms of Carbon in 10 kg. You can read more about it http://en.wikipedia.org/wiki/Avogadro_constant http://en.wikipedia.org/wiki/Amount_of_substance http://en.wikipedia.org/wiki/Mole_%28unit%29 1 Share this post Link to post Share on other sites

studiot 1834 Posted February 14, 2014 (edited) Your question brings up an interesting bit of history because Archimedes was the first to successfully study this question and he wrote down his mathematical treatment in the middle of the third century BC. Unfortunately this document (called The Methodf) was lost in antiquity and only rediscovered in 1910, when and old parchment was cleaned. Anyway suppose we consider this thought experiment: Take a ruler and pencil and draw a thin line 25mm long. Draw another line right along side the first line so that you cannot see any gaps between them. Continue drawing lines for about three hours. You will then have a rectangular area on your paper. No imagine sharpening you pencil and repeating the experiment. It will now take you many more lines to draw the same rectangle, say six hours work. Sharpen again and repeat. Perhaps you can see where this is going. The thinner the line the more you need to create the area until. Until the line is so thin the number is ver very large indeed. This is what is meant by tending to infinity. This was exactly the process by which Archimedes derived his famous mensuration formulae, and the process by which we add up a very large number of very small contributions to create a whole. It is well known that we can add up an infinite number points to obtain a finite total. Mathematically that is what taking limits is about. However you ask how can an infinite number of points add up to different values? infinite number of dm can either equal 5 kg's or 10 kg's? Well the simplest way to see this is to look at and compare a couple of infinite series. [math]\sum {_1^\infty \frac{1}{{{1^2}}} + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \frac{1}{{{4^2}}} + \frac{1}{{{5^2}}}.............} [/math][math] = 1.645[/math][math]\sum {_1^\infty \frac{1}{{{1^3}}} + \frac{1}{{{2^3}}} + \frac{1}{{{3^3}}} + \frac{1}{{{4^3}}} + \frac{1}{{{5^3}}}.............} [/math][math] = 1.202[/math] You can see by direct term by term comparison that these two series have the same (infinite) number of terms, but their sums to infinity are different. Edited February 14, 2014 by studiot 3 Share this post Link to post Share on other sites

Science Student 9 Posted February 15, 2014 (edited) Your question brings up an interesting bit of history because Archimedes was the first to successfully study this question and he wrote down his mathematical treatment in the middle of the third century BC. Unfortunately this document (called The Methodf) was lost in antiquity and only rediscovered in 1910, when and old parchment was cleaned. Anyway suppose we consider this thought experiment: Take a ruler and pencil and draw a thin line 25mm long. Draw another line right along side the first line so that you cannot see any gaps between them. Continue drawing lines for about three hours. You will then have a rectangular area on your paper. No imagine sharpening you pencil and repeating the experiment. It will now take you many more lines to draw the same rectangle, say six hours work. Sharpen again and repeat. Perhaps you can see where this is going. The thinner the line the more you need to create the area until. Until the line is so thin the number is ver very large indeed. This is what is meant by tending to infinity. This was exactly the process by which Archimedes derived his famous mensuration formulae, and the process by which we add up a very large number of very small contributions to create a whole. It is well known that we can add up an infinite number points to obtain a finite total. Mathematically that is what taking limits is about. However you ask how can an infinite number of points add up to different values? Well the simplest way to see this is to look at and compare a couple of infinite series. [math]\sum {_1^\infty \frac{1}{{{1^2}}} + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \frac{1}{{{4^2}}} + \frac{1}{{{5^2}}}.............} [/math][math] = 1.645[/math] [math]\sum {_1^\infty \frac{1}{{{1^3}}} + \frac{1}{{{2^3}}} + \frac{1}{{{3^3}}} + \frac{1}{{{4^3}}} + \frac{1}{{{5^3}}}.............} [/math][math] = 1.202[/math] You can see by direct term by term comparison that these two series have the same (infinite) number of terms, but their sums to infinity are different. Wow, that's so amazing!!!! Edited February 15, 2014 by Science Student 0 Share this post Link to post Share on other sites

Implicate Order 16 Posted February 15, 2014 (edited) Excellent explanation Studiot. Often the mindset refuses to budge when dealing with the term inifnity and it takes the form in the mind of being an actual quantitative 'thing'. Any time you need to hammer home the principle that infinity is un-defined as opposed to a quantitative thing, they should refer to your explanation here. Archimedes was a clever fellow not to mention how he resolved Zeno's paradox......wow!! I often wonder how close he was to understanding the quantum. Edited February 15, 2014 by Implicate Order 0 Share this post Link to post Share on other sites

Someguy1 3 Posted February 17, 2014 (edited) I am really confused over the general idea of the summation of infinitesimals of some quantity. For example, can anyone show me mathematically how an infinite number of dm can either equal 5 kg's or 10 kg's? I can understand basic calculus. It's not possible for infinitely many physical things to make up a kilogram. If you take a kilogram of any material substance and subdivide it finely enough you'll end up with a large but finite number of subatomic particles. There are only finitely many quarks in the universe, far less than 10^100 in fact. It's true that there are infinitely many points in a mathematical line segment, but that has nothing to do with the physical world. Edited February 17, 2014 by Someguy1 0 Share this post Link to post Share on other sites

swansont 7124 Posted February 17, 2014 It's not possible for infinitely many physical things to make up a kilogram. That's not what the question was, though. Calculus does not assume physical objects are involved. 0 Share this post Link to post Share on other sites

Someguy1 3 Posted February 17, 2014 That's not what the question was, though. Calculus does not assume physical objects are involved. I suppose the OP would have to comment on that. But how many mathematical points does it take to make a kilogram? The question is absurd, is it not? Once the OP said kilogram, physics was implied, not math. 0 Share this post Link to post Share on other sites

studiot 1834 Posted February 17, 2014 (edited) I suppose the OP would have to comment on that. But how many mathematical points does it take to make a kilogram? The question is absurd, is it not? Once the OP said kilogram, physics was implied, not math. No, physics was not implied, maths was explicity called for since the OP not only posted in the maths section as opposed to the physics one, but further chose an area of pure maths over applied maths. In any case your argument that an infinite number of physical points cannot make up 10 kg is suspect. A vast area of physics, including most of classical physics is predicated upon the premise that you can indeed either infinitely divide a finite piece of matter or alternatively assemble a finite piece from an infinite number of parts. This underlies classical statics and dynamics, continuum mechanics, and even the angular momentum of quantum particles is derived from continuum mathematical analysis. Can you tell me any reason why, if each of the numbers in my series posted above was a mass coefficient, I could not assemble 1.6 kg from the first series and 1.2 kg from the second and any other value by suitable scaling? Edited February 17, 2014 by studiot 0 Share this post Link to post Share on other sites

Someguy1 3 Posted February 17, 2014 (edited) No, physics was not implied, maths was explicity called for since the OP not only posted in the maths section as opposed to the physics one, but further chose an area of pure maths over applied maths. In any case your argument that an infinite number of physical points cannot make up 10 kg is suspect. A vast area of physics, including most of classical physics is predicated upon the premise that you can indeed either infinitely divide a finite piece of matter or alternatively assemble a finite piece from an infinite number of parts. This underlies classical statics and dynamics, continuum mechanics, and even the angular momentum of quantum particles is derived from continuum mathematical analysis. Can you tell me any reason why, if each of the numbers in my series posted above was a mass coefficient, I could not assemble 1.6 kg from the first series and 1.2 kg from the second and any other value by suitable scaling? How many real numbers does it take to make a kilogram? The fact that the OP is confused about the distinction between math and physics is not a reason to amplify his confusion. Rather, it was an opportunity to clarify his misunderstanding. I know of no theory of physics that incorporates infinitely many tiny objects as constituents of matter. As I noted, there are far fewer than 10^100 quarks in the universe. That's a relatively small finite number. That's my take on this. There is a huge amount of confusion online regarding the distinction between math and physics. And telling someone that you can add up infinitely many tiny things to make a kilogram of some physical substance is truly and deeply wrong. But let me ask you this. Do you actually believe that a kilogram of gold is made up of 1/2 k plus 1/4 k + ... ? I assume you understand that this is false. But perhaps I'm misunderstanding you, and you actually believe this is true. If it's false, then you've confused and misdirected the OP. If you believe it's true, state that clearly so that I understand better where you're coming from. Edited February 17, 2014 by Someguy1 0 Share this post Link to post Share on other sites

studiot 1834 Posted February 17, 2014 (edited) Are you saying that the following are not true and should not therefore be taught to final high school / tech college / first year undergrads as they have been for more than the last hundred years? [math]\bar x = \frac{{\iiint {\rho xdxdydz}}}{{\iiint {\rho dxdydz}}}[/math] [math]{I_{xx}} = \int {\rho ({y^2} + {z^2})dv} [/math] Edited February 17, 2014 by studiot 0 Share this post Link to post Share on other sites

Someguy1 3 Posted February 17, 2014 Are you saying that the following are not true and should not therefore be taught to final high school / tech college / first year undergrads? [math]\bar x = \frac{{\iiint {\rho xdxdydz}}}{{\iiint {\rho dxdydz}}}[/math] [math]{I_{xx}} = \int {\rho ({y^2} + {z^2})dv} [/math] That's a complete non-sequitur. I asked you two questions: 1) How many real numbers are in a kilogram? 2) Do you literally believe that a kilogram of gold can be decomposed into infinitely many parts, each part of mass 1/2^n kilograms for n = 1, 2, 3, ...? 0 Share this post Link to post Share on other sites

studiot 1834 Posted February 17, 2014 (edited) That's a complete non-sequitur. I asked you two questions: Actually it is on topic since it addresses the OP. You have already been told by a moderator that your approach is off topic. Although you have not answered my question, I will answer yours (1) The question is flawed since the units are different on either side of the equation. Couch your question correctly and I will try again. (2) For the purpose of many physical models (already described) yes that is true. It is not a belief system, but a matter of definition. Edited February 17, 2014 by studiot 0 Share this post Link to post Share on other sites

Someguy1 3 Posted February 17, 2014 (edited) Actually it is on topic since it addresses the OP. You have already been told by a moderator that your approach is off topic. Although you have not answered my question, I will answer yours (1) The question is flawed since the units are different on either side of the equation. Couch your question correctly and I will try again. (2) For the purpose of many physical models (already described) yes that is true. It is not a belief system, but a matter of definition. I'm afraid that if I was sanctioned by a moderator, I missed it, I have no private messages. Did I miss something? Or are you confusing me with someone else? I certainly have no desire to transgress the etiquette of the forum. The OP asked how many infinitesimals are in a kilogram. This (to me) is an opportunity to explain to the OP the distinction between math and physics; and not to obfuscate it. So ... how many infinitesimals are in a kilogram? I really don't understand your point of view at all. Nor your refusal to respond to whether you truly believe that a kilogram of physical stuff can be decomposed into infinitely many pieces of size 1/2^n. (ps) Earlier you were adamant that the OP was asking about math. But he said kilogram. And I wonder if this is the source of our different viewpoints. A kilogram is a concept from physics. There is no such thing as a kilogram in mathematics. I am thinking that you don't agree with that. Because earlier you claimed that a statement involving kilograms was a statement of math. But it can never be. The word kilogram is part of physics and definitely not part of math. Edited February 17, 2014 by Someguy1 0 Share this post Link to post Share on other sites

studiot 1834 Posted February 18, 2014 (edited) Posted 14 February 2014 - 05:26 PM I am really confused over the general idea of the summation of infinitesimals of some quantity. Actually that was the question. The kilograms were an example, as shown by the opening words "for example". The infinitesimal in the example was dm, which dimensionally correctly refers to mass. Since you have revised your question (1) from how many numbers make 1 kg? to How many infinitesimals (I will take that as dm) make 1 kg? and I said I would answer a properly posed question here is my answer. (1) An uncountable number. Perhaps I should point out here that is because the domain is R^{3.} For the series I posted on the other hand the answer is a countable number. I already did answer your second question, quite specifically. You still have not answered any of mine. Edited February 18, 2014 by studiot 0 Share this post Link to post Share on other sites

swansont 7124 Posted February 18, 2014 You have already been told by a moderator that your approach is off topic. Told by a physicist and former physics teacher. I'm discussing the topic, not moderating. (Mods can't do both) —— The OP mentions dm, i.e. the infinitesimal one would have in doing an integral, as an example (the overall topic is one of summing infinitesimals. It's not that cryptic to me that physics is not the focus of the question). The answer to the conundrum is that, as I said, calculus does not obey the restrictions of physics. One is free to divide a mass into an infinite number of differential mass elements. The mention of summing infinite series addresses the issue raised: some series converge, and the utility of integral calculus is that we are dealing with ones that do. 0 Share this post Link to post Share on other sites

N S 0 Posted February 25, 2014 (edited) Simply taken, I think what Science Student asked can be answered and the ongoing discussion can be resolved in this way: Maths is not equivalent to the real world. It is imaginary. It helps our brains think of physical solutions easily. Thus, we can assume a particle of, say, 1 kg as being made of infinite particles of mass dm; while if we consider a physical object of 1 kg, we can assert that it is made up of finite atoms, molecules, which in turn are made up of finite entities. But if we apply calculus on it and its mechanical properties, we would almost be accurate. Since maths does not portray the real world exactly, the result will be idealised and approximate. Edited February 25, 2014 by N S 0 Share this post Link to post Share on other sites

swansont 7124 Posted February 25, 2014 But if we apply calculus on it and its mechanical properties, we would almost be accurate. But since maths does not portray the real world exactly, the result will be idealised and approximate. Yes. We wouldn't get the right answer if we used some average density parameters but the volume was half that of a hydrogen atom, for example. The user has to ensure that the math is reasonably applied to the problem. 0 Share this post Link to post Share on other sites

Science Student 9 Posted March 6, 2014 It's not possible for infinitely many physical things to make up a kilogram. If you take a kilogram of any material substance and subdivide it finely enough you'll end up with a large but finite number of subatomic particles. There are only finitely many quarks in the universe, far less than 10^100 in fact. It's true that there are infinitely many points in a mathematical line segment, but that has nothing to do with the physical world. But it might be true for energy or space-time. If we look at energy or space-time as continuous like the real number line, then maybe infinitesimal energies or infinitesimal distances apply. A physicist once told me that a Planck's constant is not necessarily the smallest possible quantity of energy, just the smallest detectable quantity of energy. 0 Share this post Link to post Share on other sites

Someguy1 3 Posted March 7, 2014 (edited) But it might be true for energy or space-time. If we look at energy or space-time as continuous like the real number line, then maybe infinitesimal energies or infinitesimal distances apply. A physicist once told me that a Planck's constant is not necessarily the smallest possible quantity of energy, just the smallest detectable quantity of energy. If that's true, then there must be uncountably many points in a one centimeter line segment. If so, exactly how many points are there? The Continuum Hypothesis would then be a proposition with a definite truth value in the physical universe; and we would expect physics postdocs to apply for grant money to do experiments to find evidence either in support or in opposition to the truth of CH. I've heard of no such grant applications. There's no evidence that there are infinite sets of points in the physical universe. On the contrary, there are far less than a googol (10^100) quarks in the universe. Of course it's entirely possible that we'll all feel differently after another few hundred years of scientific research. But there is no evidence in contemporary physics that the world is made up of an infinite number of dimensionless points. Edited March 7, 2014 by Someguy1 0 Share this post Link to post Share on other sites

swansont 7124 Posted March 7, 2014 If that's true, then there must be uncountably many points in a one centimeter line segment. If so, exactly how many points are there? The second question does not follow from the first being true, which it is. Number of points ≠ number of objects, be they atoms or quarks. 0 Share this post Link to post Share on other sites

Someguy1 3 Posted March 8, 2014 (edited) The second question does not follow from the first being true, which it is. I was not able to parse that. Are you saying that there are uncountably many points but I'm not allowed to ask how many there are? Someone suggested that the universe is a continuum. But we know quite a lot about continua from math. We know, for example, that they must contain uncountably many points, if the real numbers are taken as a model for the continuum. Is there some model for a continuum other than the real numbers? One that contains countably many points? I'm assuming that we are examining the notion that physical space consists of points, which are to be taken in the same sense as mathematical points. Identifiable with real numbers, for example. Dimensionless. So we have an uncountable set of points in a given region of space. I am asking, what is the cardinality of this set of points? If space is modeled accurately by the real numbers, then there are 2^Aleph-0 points in, say, any finite-dimensional region of space, bounded or not. I am asking if that's Aleph-1 or some other Aleph. If the game is to assume that actual, physical space is accurately modeled by the real numbers, then this becomes a question of physics, subject to experiment. I am pointing this out, in order to cast doubt on the idea that space is accurately modeled by the real numbers. It's not currently a mainstream idea. The mathematical reals are a continuum; but the physical universe is generally taken to be quantized. Of course this is not the last word on the matter, but it's what the preponderance of experts believes right now. But if space truly consists of a real-number-like structure, then the puzzles of set theory become matters of physics: subject to experimentation; and having a definite truth value in our universe. Of course I could make the same argument about the Axiom of Choice. And if AC turned out to be true about the physical universe, then the Banach-Tarski paradox would be a true fact about physical things! Edited March 8, 2014 by Someguy1 0 Share this post Link to post Share on other sites

studiot 1834 Posted March 8, 2014 (edited) the Banach-Tarski paradox would be a true fact about physical things! etc I am not sure whether this post puts you for or against the idea that you can add up infinitely many mass points to make 5kg or any other value. It seems to me that the B-T decomposition allows one to carry out uncomfortably many such constructions, rather than bar you from them. Incidentally, whilst is does not detract from the mathematical flow about the B-T decomposition, attempting to apply it unqualified to the physical universe does. The B-T decomposition assumes that all 'points' form an equivalence class. That is any one point can stand in for any other. In particular it ignores the well ordered principle of the reals. This is OK in appropriate context, but definitely not OK in physical space. The real numbers 1.0 and 3.0 may be equivalent in one sense, but they are not the same. This does not mean that one is in superior or preferable either, so relativity is preserved. What the B-T decomposition is saying is that you can map all the points between 1 and 2 to all the points between 3 and 4 or 5 and 6 or even 0 and infinity on a one to one basis. So you can, in that sense, create a 'copy' of all the points in the interval [1, 2 ] within the interval [3,4] by such a mapping. But it does not say you can construct two (multiple) copies of all the points within the interval [1,2] within the interval [1,2]. You require another interval to perform this. Edited March 8, 2014 by studiot 0 Share this post Link to post Share on other sites

swansont 7124 Posted March 8, 2014 I was not able to parse that. Are you saying that there are uncountably many points but I'm not allowed to ask how many there are? Yes. If the number is uncountable, there is no answer to the question "how many are there?" Someone suggested that the universe is a continuum. But we know quite a lot about continua from math. We know, for example, that they must contain uncountably many points, if the real numbers are taken as a model for the continuum. Is there some model for a continuum other than the real numbers? One that contains countably many points? As has been explained several times, the question of a model for the universe is moot. Calculus is not modeling the universe. It's a mathematical tool. 0 Share this post Link to post Share on other sites