 |
 | Home | Contact Us | FAQ | Search & Site Map | Link to Us |
 | Sign In | Join | Other 45 Sites in Network |
Math Forum / Mathematics / General Topics / October 2008-- Wrong limits do not commute
An excercise, in mathematics, always consists of two parts: a question and an answer. Some teachers think that they are smart by asking wrong questions and nevertheless expect right answers to those questions. But, of course, the "right" answer to a wrong question is still wrong. Especially with limits, wrong questions are imposed upon students, in order to demonstrate, for example, that: limits do not always commute. Example 1, by Horand Gassmann, from "Why does everyone do it?": http://groups.google.nl/group/sci.math/browse_frm/thread/bfa9f4f2c780e6f f_n(x) = 1 if n <= x < n+1 = 0 otherwise Then int_{-infinity}^{+infinity} f_n(x)dx = 1, and hence lim{n->infinity}[int...] = 1, as well, while lim{n->infinity} f_n(x) = 0 for every x, and so int_{-infinity}^{+infinity} [lim...] = 0. Let's analyse this. We shall first unravel the meaning of these limits: int_{-infinity}^{+infinity} f_n(x)dx = lim int_(-L)^(+L) f_n(x)dx L->oo Therefore what's really going on is the following (n natural, L real): lim lim int_(-L)^(+L) f_n(x)dx ( = lim 1 = 1 ) n->oo L->oo n->oo lim lim int_(-L)^(+L) f_n(x)dx ( = lim 0 = 0 ) L->oo n->oo L->oo Where the answer between parentheses is the one expected by your maths teacher. Now Horand himself has published a picture that is associated with this little problem: -------- : : ---------------------^----------- ^ ------^---------------- f_n L n L n+1 L Apart from the limits, as long as we are in the finitary domain, then it's obvious that there exist the following possibilities for L and n: L < n ; L = n ; n < L < n+1 ; n+1 = L ; n+1 < L Now Horand says that it's possible to "fix" n and let L go to infinity or that it's possible to fix L and let n go to infinity. _I_ say that such a "fixed" real L or natural n is an undefined concept. A real is just a real and there's nothing "fixed" or "variable" with it. We have to live wit the fact that _both_ n and L approach limiting values and in _whatever_ fashion. So what is the proper way to express this idea? With other words: what would have been the _right question_ belonging to this excercise ? The following is my proposal: lim int_(-L)^(+L) f_n(x)dx min(n,L)->oo And now the _proper_ the answer is that this limit: does not exist. Or rather that it has a value between 0 and 1, not by coincidence the two extremes found by Horand's students (at least those who got an 'A' for their exam). In this case two limits do not commute and the associated right questioned limit does not exist. Hmm, would that be systematical? Example 2, by Horand Gassmann, from "Why does everyone do it?": http://groups.google.nl/group/sci.math/browse_frm/thread/bfa9f4f2c780e6f Let g_n(x) = 2.n^2.x if 0 <= x < 1/(2n) = 2n - 2.n^2.x if 1/(2n) <= x < 1/n = 0 everywhere else. These functions are triangular, and they all disappear outside of [0,1], so I can compute int_0^1{g_n(x) dx} = 1 for every n. And again, for every x, lim{n->infinity g_n(x) = 0. So again, the limit and the integration can't be interchanged. Sure, g_n(x) = 0 , sure. It's not the first time I've seen infinities suddenly disappear by mainstream mathematician's "ingenuity": http://hdebruijn.soo.dto.tudelft.nl/www/grondig/natural.htm#bv http://hdebruijn.soo.dto.tudelft.nl/jaar2006/ballen.jpg As Mr. Gassmann has admitted, this is a proper picture of the problem: http://hdebruijn.soo.dto.tudelft.nl/jaar2008/gassmann.jpg We see that the function g_n(x) becomes a sharp peak at x=0 for n->oo and that, geometrically, it certainly not disappears or becomes zero. The vemin is again in the false doctrine that it would be possible to "fix" x and let n go to infinity. Again, HdB says that such a "fixed" real x is an _undefined_ concept. A real is just a real and there's nothing "fixed" or "variable" with it. Such is even more obvious with the present function g_n(x) , where x and n are clearly intermixed, by definition. So we _can_ have: 0 <= x < 1/(2n) , 1/(2n) <= x < 1/n , 1/n <= x , and nothing prevents us from groing to limits _while_ this is the case. If we just do this, then what we get is what physicists know as a delta function: delta(x) = 0 for x <> 0 = any for x = 0 ; integral_(-oo)^(+oo) delta(x) dx = 1 It's easy to see that: lim g_n(x) = delta(x) n->oo Whatever definition might be the "right" one, a delta function roughly speaking is just a very large peak around x = 0 with area normed to 1. With the finitary domain in mind, there is no problem in understanding what delta functions actually are (even if they are a bit beyond x=0). Note that, in this case, it's _not_ even the wrong question. It's just that professors expect a wrong answer from their students. The second answer is wrong, namely, and the first one is right: lim [ int_0^1 g_n(x) dx ] = 1 = int_0^1 [ lim g_n(x) ] dx n->oo n->oo Example 3, from a (hypothetical, I've made it up) textbook. x^2 - y^2 Let F(x,y) = --------- . x^2 + y^2 To avoid trouble with the mainstreamers, we could define F at (x,y) = (0,0) as the mean value F(0,0) = 0 . As far as I'm concerned myself, I'd rather define F(0,0) as 0/0 = anything. And in view of the sequel that anything is restricted to values between -1 and +1. Proceeding: x^2 - y^2 x^2/y^2 - 1 lim ( lim --------- ) = lim ( lim ----------- ) = - 1 x->oo y->oo x^2 + y^2 x->oo y->oo x^2/y^2 + 1 x^2 - y^2 1 - y^2/x^2 lim ( lim --------- ) = lim ( lim ----------- ) = + 1 y->oo x->oo x^2 + y^2 y->oo x->oo 1 + y^2/x^2 So again, according to the textbook, the limits do not commute. x^2 - y^2 x^2 lim ( lim --------- ) = lim ( --- ) = + 1 x->0 y->0 x^2 + y^2 x->0 x^2 x^2 - y^2 - y^2 lim ( lim --------- ) = lim ( ----- ) = - 1 y->0 x->0 x^2 + y^2 y->0 + y^2 So again, according to the textbook, the limits do not commute. Unravel the limits. We will do so by introducing polar coordinates: x = r.cos(phi) ; y = r.sin(phi) x^2 - y^2 r^2 ( cos^2(phi) - sin^2(phi) ) Giving --------- = ------------------------------- = cos(2.phi) x^2 + y^2 r^2 ( cos^2(phi) + sin^2(phi) ) Now we can understand immediately why the professor's limits for x and y _indeed_ do not commute ! At (x,y) = 0 , this function is _singular_ and it assumes any value between -1 and +1. And the latter is the case everywhere else in the (x,y)-plane. So unless there is a clear path to infinity, any value between -1 and +1 would be good enough for an 'A'. See? The function is a circular wave, or rather the wave perpendicular to it and has period Pi. Here is an (ASCII) contour map of it. Notice the singularity at the origin. 0 -1 0 \ | / \ | / \ | / \ | / \ | / \|/ +1 ------------------- +1 /|\ / | \ / | \ / | \ / | \ / | \ 0 -1 0 Why not ask then _proper_ questions belonging to answers ? Here comes: x^2 - y^2 lim ( --------- ) = undefined (between -1 and +1) |(x,y)|->0 x^2 + y^2 x^2 - y^2 lim ( --------- ) = undefined (between -1 and +1) |(x,y)|->oo x^2 + y^2 With other words: let the absolute value of the vector (x,y) approach zero, or infinity, and see what happens. (Neighbourhoods and such ..) Excercise 4. x.y Let G(x,y) = --------- . I'm sure you can do this one by yourself. x^2 + y^2 Wrong limits do not commute. Almost dare to say now that non commuting limits are simply .. wrong. Any counter examples are quite welcome. Guess it's not so much superstition but merely a sense of reality that so many people let believe that: properly posed limits _do_ commute. Han de Bruijn >An excercise, in mathematics, always consists of two parts: a question >and an answer. Some teachers think that they are smart by asking wrong [quoted text clipped - 45 lines] >such a "fixed" real L or natural n is an undefined concept. A real is >just a real and there's nothing "fixed" or "variable" with it. Fine. So what? When people talk about L being "fixed" that doesn't mean that L is some special sort of real number, a fixed one as opposed to other reals that are maybe not so fixed, it's just meant to say that the _notation_ "L" refers to _one_ real number. It's a fact that for every real L the limit as n tends to infinity is one thing and for every natural number n the limit as L tends to infinity is something else. There you are, exactly the same fact expressed without the word "fixed". >We have >to live wit the fact that _both_ n and L approach limiting values and [quoted text clipped - 7 lines] >And now the _proper_ the answer is that this limit: does not exist. Or >rather that it has a value between 0 and 1, No, the limit does not exist. >not by coincidence the two >extremes found by Horand's students (at least those who got an 'A' for >their exam). In this case two limits do not commute and the associated >right questioned limit does not exist. Hmm, would that be systematical? Sounds like you've uncovered some deep conspiracy here. The idea that iterated limits are "wrong" and the limit as above is "right" is very strange. It _is_ a fact that if I(n, L) is a function of two variables and the two iterated limits are not the same then the limit as min(n,K) tends to infinity cannot exist. This is easy to prove, and very well known. >Example 2, by Horand Gassmann, from "Why does everyone do it?": >http://groups.google.nl/group/sci.math/browse_frm/thread/bfa9f4f2c780e6f [quoted text clipped - 24 lines] >real x is an _undefined_ concept. A real is just a real and there's >nothing "fixed" or "variable" with it. What doesn't exist is the point you're trying to make. Again, the fact is just that for every x, the limit as n tends to infinity is 0. When people say "for every fixed x" they're just trying to make it more clear exactly what is meant. >Such is even more obvious with >the present function g_n(x) , where x and n are clearly intermixed, [quoted text clipped - 13 lines] >With the finitary domain in mind, there is no problem in understanding >what delta functions actually are (even if they are a bit beyond x=0). There certainly is a notion of limit for which the limit of these g_n is a "delta function". So what? That's simply a different sort of limit. >Note that, in this case, it's _not_ even the wrong question. It's just >that professors expect a wrong answer from their students. The second >answer is wrong, namely, and the first one is right: > > lim [ int_0^1 g_n(x) dx ] = 1 = int_0^1 [ lim g_n(x) ] dx > n->oo n->oo Your notion of "right" and "wrong" is wrong. It's like those nasty professors are saying things about apples, and you're saying they're wrong. You agree that the things they say about apples are correct, but nonetheless they're wrong because they should be talking about oranges. It's a very strange sort of "right" and "wrong" - if I say something that's true it's wrong because _you_ say I should be talking about something else. >Example 3, from a (hypothetical, I've made it up) textbook. > [quoted text clipped - 38 lines] >y _indeed_ do not commute ! At (x,y) = 0 , this function is _singular_ >and it assumes any value between -1 and +1. You should really learn a little more about what those evil professors say. The limit of that function as (x,y) -> (0,0) does not exist (that's the correct way to express the same fact that you're expressing by saying the function is singular or that the limit is "anything". That limit does not exist). _Proving_ that that limit does not exist is a standard exercise that those nasty profesors ask their students to do. And the simplest proof that that limit does not exist is simply to note that the two iterated limits are not equal. Many of the mathematical points you raise in all this are correct, although many are expressed in very unfortunate language. But the things you say about what those professors say versus what they should say are hilarious - none of the correct mathematical points you raise are news to thos profesors, and they're not something that they've been trying to hide from their students. It's like you tell us that professors tell their students that cats are not dogs, and that it's wrong for profesors to tell their students this because dogs bark! Honest, it _is_ like that. Yes, dogs bark. But that's not news, and it's not something that professors have been misinforming their students about when they tell them that cats are not dogs. >And the latter is the case >everywhere else in the (x,y)-plane. So unless there is a clear path to [quoted text clipped - 45 lines] > >Han de Bruijn David C. Ullrich "Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.) David C. Ullrich wrote (answering Han de Bruinj): [many things, correct of course, about nasty professors and non-commutating limits] I understand why you would like to remind the innocent reader of the reality. But you lose your precious time feeding this troll, and I am sure no serious student is in real danger of confusion over those things. So, lets do math for a change. Same way as the g_n sequence does converge in some sense towards the "delta function" (which is not a function, but never mind), or that the sequence (1,-1,1,-1,...) converges towards 0 in some sense (Cesaro, say), is there a way, for instance, to get a "limit" to (x^2-y^2)/(x^2+y^2) (probably 0, of course) and, more generally, is there some general setting where all those pesky non-commutative limits at last commute? > David C. Ullrich wrote (answering Han de Bruinj): > [quoted text clipped - 5 lines] > sure no serious student is in real danger of confusion over those > things. I wish you were right .. Worse ! Serious mathematics is in real danger of confusion over those things. Well, not really over things as simple as these limits, but the generalizations of these things. (Take a look at e.g. point set topology) I became aware of these things at the time I studied Lie Groups at the Eindhoven University of Technology. <quote> My absolute favorite is a book I borrowed from the library many years ago and never succeeded in finding it back again. It's written by one of Lie's pupils: Georg Scheffers (1866–1945). It's written in German & titled "Differentialgleichungen von Sophus Lie" or some such like it. I still remember quite well that Georg Scheffers' book is abundant with lucid geometrical interpretations and very much readable. Not this book, but several others by Georg Scheffers about Lie groups can be found at: http://concise.britannica.com/ebc/article-9048172/Sophus-Lie http://0-www.search.eb.com.library.uor.edu/eb/article-9048172?tocId=9... Or try Google("Sophus Lie" "Georg Scheffers" "differential equations") </quote> The dark side of this story is: _topology_. Got stuck on it. Got *PLONK*ed by my superiors, at last. And this is my sweet revenge. > So, lets do math for a change. Same way as the g_n sequence > does converge in some sense towards the "delta function" (which is not a [quoted text clipped - 3 lines] > course) and, more generally, is there some general setting where all > those pesky non-commutative limits at last commute? You've clearly not understood that _wrong questions_ can be formulated within mathematics. The answer is: no. Han de Bruijn > > David C. Ullrich wrote (answering Han de Bruinj): > > [quoted text clipped - 67 lines] > can be formulated > within mathematics. The answer is: no. indeed. > Han de Bruijn >>An excercise, in mathematics, always consists of two parts: a question >>and an answer. Some teachers think that they are smart by asking wrong [quoted text clipped - 55 lines] > something else. There you are, exactly the same fact > expressed without the word "fixed". We would be on speaking terms here, iff the following would be the case: lim int_(-L)^(+L) f_n(x)dx = 0 with _one_ L (such that L < n = oo) n->oo lim int_(-L)^(+L) f_n(x)dx = 1 with _one_ n (such that n+1 < L = oo) L->oo The augmentations between parentheses are not necessary then. But this is _not_ what our teacher is doing. Because: lim lim int_(-L)^(+L) f_n(x)dx n->oo L->oo lim lim int_(-L)^(+L) f_n(x)dx L->oo n->oo So now the n that has been declared fixed suddenly becomes _variable_, i.e. more than one real number, and the L that has been declared fixed suddenly becomes variable, i.e. more than one real number. Which is not what we've agreed upon. Han de Bruijn > >>An excercise, in mathematics, always consists of two parts: a question > >>and an answer. Some teachers think that they are smart by asking wrong [quoted text clipped - 79 lines] > > Han de Bruijn I do not know how physicists think about it, but to mathematicians. In lim lim int_(-L)^(+L) f_n(x)dx n->oo L->oo n must be regarded as constant while determining the limit when L->oo. And after taking that limit while L -> oo, there is no L remaining in g(n) = lim_{L -> oo} int_(-L)^(+L) f_n(x)dx, if such a g(n) exists, to affect the value when considering the limit as n -> oo. >>>An excercise, in mathematics, always consists of two parts: a question >>>and an answer. Some teachers think that they are smart by asking wrong [quoted text clipped - 77 lines] >suddenly becomes variable, i.e. more than one real number. Which is not >what we've agreed upon. You're not making any sense. Read what I wrote again - there is nothing "agreed upen" when we say that the limit of g(n,L) for a "fixed L" as n tends to infinity is 0; that simply says that for every L the limit as n tends to infinity is 0. And it follows from that that the limit as L tends to infinity of the limit as n tends to infinity is 0. >Han de Bruijn David C. Ullrich "Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.) David C. Ullrich wrote, in response to HdB: > Your notion of "right" and "wrong" is wrong. It's like > those nasty professors are saying things about apples, [quoted text clipped - 4 lines] > something that's true it's wrong because _you_ say I > should be talking about something else. Yes. And this brings me to a general point. I've said the following in another thread, namely in response to Tonio in the article http://groups.google.nl/group/sci.math/msg/f51ee837a2fc0ee1 > Then you'd also have trouble with this. The power of mathematics is not > its generality, but its ability to deal with special cases. In short: mathematics is TOO GENERAL. Mathematicians are so affraid of missing something that the precautions they take are close to paranoia. Now I will be the last one to say that one should'nt carefully consider anything possible, but it should be a great relief if some kind of, say, mathematics BY DEFAULT could be available. And my _great dream_ is that such mathematics should reflect .. sense of reality. I could mention a lot of examples, but just one may be sufficient for our purpose: my old friend the 'scinc' function. http://groups.google.nl/group/sci.math/msg/c83b58832ecdff3f Any definition other than with sinc(0) = 1 will never have applications outside mathematics. That's why default mathematics should only contain _this_ definition and no others. How can I be so sure of this ? I'm so sure because the _continuous_ and the _discrete_, in _science_, are just two ways of looking at the _same_ thing. For example, an ideal gas is _continuous_ in Computational Fluid Dynamics, but it is _discrete_ in Statistical Mechanics. _Singularities_ arise in continuous functions where the discrete substrate can no longer be hidden. I became aware of this due to a discovery called "Fluid Tube Continuum", while I was working on apparatus for nuclear reactors, quite a while ago. With such a rough real world continuum, the place where the model breaks down, and the discrete becomes _visible_ can be pinpointed. It leads to the rather surprising result that such breakdown will happen if the "primary" mass flow within the apparatus exceeds a well defined, though approximately calculated value. See my website for more details: http://hdebruijn.soo.dto.tudelft.nl/QED/index.htm#ft I mean, before somebody says that default ("realistic") mathematics is a hopeless excercise, the _whole_ picture must be taken into account. In the somewhat desperate hope that I can convince someone, somehow .. Han de Bruijn > Any definition other than with sinc(0) = 1 will never have applications > outside mathematics. That's why default mathematics should only contain > _this_ definition and no others. You are quite free to define your own "default mathematics", so long as you do not insist that anyone else be bound by it. But the you must equally allow anyone else to define his or her own "default mathematics" without let or hindrance as long as they do not bind you to it. But there is a non-default standard for mathematics in general, and in that standard the definition of a double limit (two variables simultaneously) is different from the corresponding definitions for iterated limits (in which the limiting for one variable precedes the limiting of the other). > Wrong limits do not commute. Almost dare to say now that non commuting > limits are simply .. wrong. Any counter examples are quite welcome. Given A transistor dissipates power propotional to its linear dimension. a) for a fixed chip size what is the power dissipated by the chip as the transistor size gets very small? b) for a fixed transistor size what is the power disapated by the chip as the chip size gets very small? The limits in a) and b) are hardly wrong. However, c) what is the power dissipated by the chip as the transistor size and the chip size get very small? Has no answer. The related question, d) what is the dissipated power density as the transistor size and the chip size get very small? has an answer. At best you could say "In properly posed problems limits _do_ commute". So now all we need to do is define "properly posed problem. Oh I know, a properly posed problem is one in which limits commute. - William Hughes >>Wrong limits do not commute. Almost dare to say now that non commuting >>limits are simply .. wrong. Any counter examples are quite welcome. > > Given > > A transistor dissipates power propotional to its linear dimension. Let T = linear dimension of transistor, C = linear dimension of chip, P_T = power dissipated by transistor, P_T = power dissipated by chip, k = certain constant, N = number of transistors on chip. Then we have: P_T = k.T^3 , N = C/T^2 (approximately), P_C = N.P_T = k.C.T . Right? I hope so. My knowledge of electronics is a bit rusty. > a) for a fixed chip size what is the power dissipated by the chip > as the transistor size gets very small? lim P_C = lim k.C.T = 0 T->0 T->0 > b) for a fixed transistor size what is the power dissipated by the chip > as the chip size gets very small? lim P_C = lim k.C.T = 0 C->0 C->0 > The limits in a) and b) are hardly wrong. > However, > > c) what is the power dissipated by the chip as the transistor size > and the chip size get very small? lim P_C = lim k.C.T = 0 C->0,T->0 C->0,T->0 > Has no answer. Yes? (Suppose I don't have the right expression for small chip sizes ..) > The related question, > > d) what is the dissipated power density as the transistor size and the > chip size get very small? > > has an answer. lim P_C/C = lim k.T = 0 C->0,T->0 C->0,T->0 > At best you could say "In properly posed problems limits _do_ > commute". > So now all we need to do is define "properly posed problem. Oh I > know, a properly posed problem is one in which limits commute. ?? Han de Bruijn > >>Wrong limits do not commute. Almost dare to say now that non commuting > >>limits are simply .. wrong. Any counter examples are quite welcome. [quoted text clipped - 7 lines] > k = certain constant, N = number of transistors on chip. Then we have: > P_T = k.T^3 , nope P_T = (power per transistor) * (number of transistors) = (k * T) * (area of chip / area of transistor) = (k * T) * ( C^2 / T^2) = kC^2/T > N = C/T^2 (approximately), P_C = N.P_T = k.C.T . Right? > I hope so. My knowledge of electronics is a bit rusty. [quoted text clipped - 4 lines] > lim P_C = lim k.C.T = 0 > T->0 T->0 lim P_T = lim k.C^2/T = oo T->0 T->0 > > b) for a fixed transistor size what is the power dissipated by the chip > > as the chip size gets very small? > > lim P_C = lim k.C.T = 0 > C->0 C->0 lim P_C = lim k.C^2/T = 0 T->0 C->0 > > The limits in a) and b) are hardly wrong. > > However, > > > c) what is the power dissipated by the chip as the transistor size > > and the chip size get very small? lim P_T = lim kC^2/T C->0,T->0 C->0,T->0 Has no answer. So the limits are meaninful (for a fixed chip size as the transistor size gets small the power dissipated by the chip gets very large; for a fixed transistor size as the chip size gets small the power dissipated by the chip gets very small) but they do not commute. - William Hughes >>>>Wrong limits do not commute. Almost dare to say now that non commuting >>>>limits are simply .. wrong. Any counter examples are quite welcome. [quoted text clipped - 7 lines] >>k = certain constant, N = number of transistors on chip. Then we have: >>P_T = k.T^3 , Typo, must be: P_C = power dissipated by chip. I'm too hasty .. > nope Whew ! Sorry ! Ooops ! How could I misread the question so much: > P_T = (power per transistor) * (number of transistors) > [quoted text clipped - 3 lines] > > = kC^2/T Yeah, you're quite right, of course, if what you mean is: P_T = k.T and P_C = k.C^2/T . >>N = C/T^2 (approximately), P_C = N.P_T = k.C.T . Right? >>I hope so. My knowledge of electronics is a bit rusty. [quoted text clipped - 7 lines] > lim P_T = lim k.C^2/T = oo > T->0 T->0 lim P_C = lim k.C^2/T = oo T->0 T->0 So that limit _does not exist_. Right ? >>>b) for a fixed transistor size what is the power dissipated by the chip >>> as the chip size gets very small? [quoted text clipped - 4 lines] > lim P_C = lim k.C^2/T = 0 > T->0 C->0 lim P_C = lim k.C^2/T = 0 C->0 C->0 >>>The limits in a) and b) are hardly wrong. Apart from the fact that the first one (a) does _not_ exist. >>>However, >> [quoted text clipped - 5 lines] > > Has no answer. Correct: 0/0 is anything. But now we divide the power by the chip area, being C^2 , and get zero for the limit of the power _density_, right ? > So the limits are meaninful (for a fixed chip size as the transistor > size gets small the power dissipated by the chip gets very large; > for a fixed transistor size as the chip size gets small the power > dissipated by the chip gets very small) but they do not commute. Not at all. Your "meaningful" limit (a) does not exist, namely. Han de Bruijn On Sep 11, 10:48 am, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> wrote: > >>>>Wrong limits do not commute. Almost dare to say now that non commuting > >>>>limits are simply .. wrong. Any counter examples are quite welcome. [quoted text clipped - 42 lines] > > So that limit _does not exist_. Right ? Nope. The limit is oo (e.g. as the size of the transisor gets small the power dissipated by the chip gets large, and you can make this as large as you want). If you insist on finite limits take P_C = min(M, k.C^2/T) where M is some maximum power. - William Hughes [ whatever ] Thanks for the formula describing the power consumption P_C of a chip with linear dimension C containing transistors with linear dimension T. The formula is (apart from a constant): P_C = C^2/T . Here the quotient N = C^2/T^2 is the number of transistors on the chip. One thing should be noted: N is a natural. Meaning that William Hughes' formula is only valid for C = sqrt(N).T : straight lines through the origin in the (C,T) plane. And since the minimal value of N is 1, only the following are relevant: C = T , C = sqrt(2).T , C = sqrt(3).T , .. Consequently also T > 0 , T > 0 , C > T (last one greater or equal). C>0 C>T | / | / | / | / | / forbidden |/ --------------------- T>0 | | | The search for a MEANINGFUL limit could end in the following: consider the number of transistors on chip as fixed (therefore travel along one of the lines C = sqrt(N).T , for fixed N). What is the limit of P_C if the transistor size approaches zero then ? Answer: lim (N.T) = 0 . T->0 Han de Bruijn On Sep 12, 3:03 pm, umum...@gmail.com wrote: > [ whatever ] Ok, we will go back to accepting oo as a valid limit. > Thanks for the formula describing the power consumption P_C of a chip > with linear dimension C containing transistors with linear dimension [quoted text clipped - 7 lines] > Hughes' > formula is only valid for C = sqrt(N).T Nope. If you insist that N be a whole number the correct number of transistors is N = floor(C^2/T^2) Of course the minumum number of transistors on a chip is not 1 but 0. So the limits i: lim P_C = lim k* floor(C^2/T) = oo T->0 T->0 and ii: lim P_C = lim k* floor(C^2/T) = 0 C->0 C->0 are meaningful and do not commute. <snip discussion of another limit> Whether another limit is meaningful or not has nothing to do with whether i and ii are meaningful. - William Hughes > So the limits > [quoted text clipped - 6 lines] > > are meaningful and do not commute. I rest my case. Han de Bruijn On Sep 13, 2:36 pm, umum...@gmail.com wrote: > > So the limits > [quoted text clipped - 8 lines] > > I rest my case. Good. Note that you now agree that your statement "properly posed limits _do_ commute" is nonsense. - William Hughes > On Sep 13, 2:36 pm, umum...@gmail.com wrote: > [quoted text clipped - 16 lines] > > - William Hughes Might be wrong about the meaning of "I rest my case". Anyway, these are your words, not mine. Han de Bruijn On Sep 13, 3:16 pm, umum...@gmail.com wrote: > > On Sep 13, 2:36 pm, umum...@gmail.com wrote: > [quoted text clipped - 18 lines] > > Might be wrong about the meaning of "I rest my case". I assume it means that you do not indend to present any more arguments. As you have not presented any argument that the limits are not meaningful, I assume you accept the example. - William Hughes > On Sep 13, 3:16 pm, umum...@gmail.com wrote: > [quoted text clipped - 24 lines] > any more arguments. As you have not presented any argument > that the limits are not meaningful, I assume you accept the example. I accept the example as a confirmation of the fact that wrong limits do not commute. And the reverse is true as well. Han de Bruijn > I accept the example as a confirmation of the fact that wrong limits do > not commute. Since the limits in the example are not "wrong" this is nonsensical. - William Hughes >>I accept the example as a confirmation of the fact that wrong limits do >>not commute. [quoted text clipped - 3 lines] > > - William Hughes According to _common_ calculus, a limit that is infinite does not exist. So _your_ argument is nonsensical, always has been. Han de Bruijn > >>I accept the example as a confirmation of the fact that wrong limits do > >>not commute. [quoted text clipped - 8 lines] > > Han de Bruijn ********************************************************** Honestly and seriously, Han: stopping making such a an a.s of yourself...it already is embarrasing! (1) Lim n (n --> oo) = 00 exists perfectly well in "common calculus", just as lim (in) (n-->oo) = 0 perfectly well exists. (2) Lim sin(n) (n-->oo) DOES NOT exist. And both things are so because of MATHEMATICAL definitions. Period. Nobody cares whether it seems nice, neat, intuitive or whatever to you so or not, or whether one of your galore of brain halves tells you this or that. (1) and (2): two different things...and it'd be nice if you could FINALLY learn what's the difference between these two. Regards Tonio >>>>I accept the example as a confirmation of the fact that wrong limits do >>>>not commute. [quoted text clipped - 13 lines] > Honestly and seriously, Han: stopping making such a an a.s of > yourself...it already is embarrasing! I'm not ashamed in whatever respect. > (1) Lim n (n --> oo) = 00 exists perfectly well in "common calculus", False. n can approach infinity, but it's not a limit. > just as lim (in) (n-->oo) = 0 perfectly well exists. ?? ; don't even know what _that_ means. > (2) Lim sin(n) (n-->oo) DOES NOT exist. Agreed. > And both things are so because of MATHEMATICAL definitions. Period. > Nobody cares whether it seems nice, neat, intuitive or whatever to you > so or not, or whether one of your galore of brain halves tells you > this or that. It's just a figure of speech. What I really mean to say is: "in whatever way you're looking at it, you will always find the same result". There's not just ONE (formalist) way of doing mathematics. I vehemently protest against that view. > (1) and (2): two different things...and it'd be nice if you could > FINALLY learn what's the difference between these two. The only one who has to learn here is _you_. Han de Bruijn > >>>>I accept the example as a confirmation of the fact that wrong limits do > >>>>not commute. [quoted text clipped - 44 lines] > > Han de Bruijn- ************************************************* ***Sigh***...That last line in your post is, sadly, true, Han: the only one here that has to learn something is me. So let us begin: I give up. As someone once said: There's no blind man as the one who doesn't want to see. Regards Tonio > >>I accept the example as a confirmation of the fact that wrong limits do > >>not commute. [quoted text clipped - 6 lines] > According to _common_ calculus, a limit that is infinite does not exist. > So _your_ argument is nonsensical, always has been. Absolute piffle. Infinite limits are meaningful and anyway I pointed out a simple way to avoid the infinite limit limit and you said "whatever". - William Hughes > >>I accept the example as a confirmation of the fact that wrong limits do > >>not commute. [quoted text clipped - 8 lines] > > Han de Bruijn The right (iterated) limits lim__{x->oo} lim+{y->oo} x*y/(x^2 - y^2) and lim__{y->oo} lim+{x->oo} x*y/(x^2 - y^2) both exist and both equal zero, so it is HdB's argument that is nonsensical, and always has been. >>>>I accept the example as a confirmation of the fact that wrong limits do >>>>not commute. [quoted text clipped - 11 lines] > both exist and both equal zero, so it is HdB's argument that is > nonsensical, and always has been. When substituting polar coordinates (r,phi), the function becomes equal to x*y/(x^2 - y^2) = tan(2.phi)/2 . The WRONG iterated limits are thus eplained by the fact that tan(2.phi) is zero for phi = 0,Pi/2,Pi, .. But the function again assumes wild values for (x,y) approaching (oo,oo) from any other direction: oo 0 oo \ | / \ | / \ | / \ | / \ | / \|/ 0 ------------------- 0 /|\ / | \ / | \ / | \ / | \ / | \ oo 0 oo Han de Bruijn > > In article <eeec9$48ce4413$82a1e228$6...@news1.tudelft.nl>, > [quoted text clipped - 17 lines] > to x*y/(x^2 - y^2) = tan(2.phi)/2 . The WRONG iterated limits are thus > eplained by the fact that [...] What do you mean by "wrong limit"? Writing the word in capitals does not make anything to make it more comprehensible. What is a wrong limit? -- m >>>In article <eeec9$48ce4413$82a1e228$6...@news1.tudelft.nl>, >> [quoted text clipped - 23 lines] > > What is a wrong limit? Yeah, what is right and what is wrong .. In 'sci.math', these words are so often (mis)used that I've decided to (mis)use them as well. A _wrong_ limit is tentatively defined by the original posting in this thread and has been gradually refined with subsequent postings. Work in progress .. Han de Bruijn > >>>In article <eeec9$48ce4413$82a1e228$6...@news1.tudelft.nl>, > [quoted text clipped - 28 lines] > limit is tentatively defined by the original posting in this thread and > has been gradually refined with subsequent postings. Work in progress .. In the original post you say Wrong limits do not commute. Almost dare to say now that non commuting limits are simply .. wrong. Any counter examples are quite welcome. There is no other mention of wrong limits. So "wrong limits" to you are limits that do not commute. This definition has not been refined. You have been given meaningful limits that do not commute, quibbled, then ignored the limits when the quibbles were answered. - William Hughes > In the original post you say > [quoted text clipped - 6 lines] > that do not commute, quibbled, then ignored the limits when the > quibbles were answered. There have _not_ been given meaningful limits that do not commute. Han de Bruijn > There have _not_ been given meaningful limits that do not commute. The limits i: lim P_C = lim min(M, k* floor(C^2/T)) = M T->0 T->0 and ii: lim P_C = lim min(M,k* floor(C^2/T)) = 0 C->0 C->0 are meaningful and do not commute. - William Hughes > > In the original post you say > > [quoted text clipped - 15 lines] > There have _not_ been given meaningful limits that do > not commute. The way you write, it makes me suspect that whatever example I give of non-commuting limits you will deem it not "meaningful". Anyway, here is an example (which by the way, also illustrates the difference between pointwise convergence and uniform convergence). Consider the sequence of functions I -> I (I is the unit interval [0, 1]) (f_n) given by: f_n(x) = x^n Consider also the sequence of points (x_m) given by: x_m = 1 - 1/m Plugging x_m in f_n we get a double sequence given by: f_n(x_m) = (1 - 1/m)^n Computing the iterated limits gives lim_m (lim_n (1 - 1/m)^n) = lim_m 0 = 0 lim_n (lim_m (1 - 1/m)^n) = lim_n (1 - lim_m 1/m)^n = lim_n 1 = 1 Oh, look they do not commute. I am sure you will chime in now and tell us exactly why they are not "meaningful". Regards, G. Rodrigues > The way you write, it makes me suspect that whatever example I give of non-commuting limits you will deem it not "meaningful". Keep trying. > Anyway, here is an example (which by the way, also illustrates the difference between pointwise convergence and uniform convergence). Consider the sequence of functions I -> I (I is the unit interval [0, 1]) (f_n) given by: Always wonder how relevant those subtleties are, but anyway .. > f_n(x) = x^n > [quoted text clipped - 13 lines] > > Oh, look they do not commute. I am sure you will chime in now and tell us exactly why they are not "meaningful". Nice excercise for the weekend. Thanks. Han de Bruijn > Nice excercise for the weekend. Thanks. G. Rodrigues wrote: <QUOTE> Anyway, here is an example (which by the way, also illustrates the difference between pointwise convergence and uniform convergence). Consider the sequence of functions I -> I (I is the unit interval [0, 1]) (f_n) given by: f_n(x) = x^n Consider also the sequence of points (x_m) given by: x_m = 1 - 1/m Plugging x_m in f_n we get a double sequence given by: f_n(x_m) = (1 - 1/m)^n Computing the iterated limits gives lim_m (lim_n (1 - 1/m)^n) = lim_m 0 = 0 lim_n (lim_m (1 - 1/m)^n) = lim_n (1 - lim_m 1/m)^n = lim_n 1 = 1 Oh, look they do not commute. I am sure you will chime in now and tell us exactly why they are not "meaningful". </QUOTE> We all agree that the following (single) limits _are_ meaningful: lim_m (1 - 1/m)^n = (1 - lim_m 1/m)^n = 1 lim_n (1 - 1/m)^n = 0 But the iterated limits, indeed, are _not_ meaningful. The reason is, again, the false premise that variables can be "fixed" and after that be "variable" again. If you take m "fixed" then it _remains_ fixed. If you take n "fixed" then it _remains_ fixed. The _honest_ way to consider an iterated double limit though, is to let _both_ variables approach infinity in whatever way. A transform into polar coordinates does wonders again. Let m = r.cos(phi) and n = r.sin(phi) . Then, take the following limit: lim_r (1 - 1/(r.cos(phi)))^(r.sin(phi)) = lim_r [(1 - 1/(r.cos(phi)))^(r.cos(phi))]^(sin(phi)/cos(phi)) (*) Needless to say that the real thing is between the square brackets [] and we could also have written: lim_m [(1 - 1/m)^m] . Hey ! This is one of our favorites = e^(-1) Conclusion: lim_(m,n) (1 - 1/m)^n = e^(-tan(phi)) . Here: tan(phi) = n/m . Meaning that it assumes only positive rational values, in principle. The rationals are so dense in the reals, though, that it's not necessary to distinguish them from reals, in the problem at hand. So tan(phi) runs over the reals and phi does it as well. 0 1/e | / | / | / | / | / |/ phi --------------------- 1 | | For phi = 0 , the x-axis , we find e^(-tan(0)) = e^0 = 1 For phi = Pi/2 , the y-axis , we find e^(-tan(Pi/2)) = e^(-oo) = 0 We can see that these are equivalent to the single, and _meaningful_, limits above: lim_m (1 - 1/m)^n = 1 respectively lim_n (1 - 1/m)^n = 0 For the other angles (phi) we find values between these extremes (0,1) and thus we see that a meaningful limit lim_(m,n) (1 - 1/m)^n is NOT defined, unless we specify what the slope tan(phi) = n/m is of the ray that goes to infinity. In general, we must describe HOW infinity is approached. For phi = Pi/4 , the ray y = x , we find e^(-tan(Pi/4)) = e^(-1) Oh, I _know_ that the answer to an ill posed problem may seem correct. But as I've declared in the Original Posting, I'm not questioning the answers, but rather the _questions_. (And, oh yeah: the idea "fixed") (*) Note: the special case cos(phi) = 0 , phi = 90 degrees must be distinguished, but this is covered by one of the single limits above. Han de Bruijn > > Nice excercise for the weekend. Thanks. > [quoted text clipped - 33 lines] > again, the false premise that variables can be "fixed" and after that > be "variable" again. Whyever not? Isn't that exactly what crossed partial derivatives do? And occasionally reversing the order of taking partial derivatives makes a difference. Or is that another area in which HdB considers it to be WRONG? > If you take m "fixed" then it _remains_ fixed. But you can have it fixed for a whole set of different values of m, then put them together to get a function of m. Well maybe you can't but many people can. its not all that hard. > If you take n "fixed" then it _remains_ fixed. Not unless it has a particular numerical value which is both understood and understood not to change. There is no essential difference between a name for a constant and a name for a variable, except for the number of values that you allow it to represent. > The _honest_ way to > consider an iterated double limit though, is to let _both_ variables > approach infinity in whatever way. I find your way no more honest than one which is path dependent. I suppose in HdB-land line integrals are always independent of the path taken. > A transform into polar coordinates > does wonders again. Only by obscuring the view. The rest, being of little mathematical value, clipped. >>>Nice excercise for the weekend. Thanks. >> [quoted text clipped - 35 lines] > > Whyever not? Isn't that exactly what crossed partial derivatives do? Maybe, but _they_ don't first disappear into infinity. > And occasionally reversing the order of taking partial derivatives makes > a difference. Only occasionally. The whole of ThermoDynamics (example) works _without_ making that difference. > Or is that another area in which HdB considers it to be WRONG? > [quoted text clipped - 27 lines] > > The rest, being of little mathematical value, clipped. At your convenience. But yes, it's no rocket science, admittedly. Han de Bruijn On Sep 22, 4:44 am, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> > We all agree that the following (single) limits _are_ meaningful: > [quoted text clipped - 8 lines] > consider an iterated double limit though, is to let _both_ variables > approach infinity in whatever way. Are you always this clumsy? Look, the function f(n,m) = (1 - 1/m)^n maps N^2 to R (or Q if you prefer). Now you can perfectly well compute lim{m->infinity} f(n,m) for every n. This object is a function from N to R (or again Q if you prefer) and it is given by g(n) = 1 for every n. Functions from N to R are called sequences. Now you have an object for which you can compute lim{n->infinity}, and it happens to be 1. On the other hand, if you look at lim{n->infinity} f(n,m), you will find that this limit is another function from N to R(or Q), given by h(m) = 0 for every m, and again you can compute lim{m->infinity} h(m), only this time you get 0 as the limit. Hello, it seems to me, that Han de Bruijn is able, to perform these calculations but for some strange reason he thinks that the result indicates something bad. For him both limit processes should yield the same result, something like exp(-1). Since that isn't the case, he concludes that mathematics is wrong as a whole. I don't think, that anything could be done about that. Alois > Hello, > it seems to me, that Han de Bruijn is able, to perform these > calculations but for some strange reason he thinks that the result > indicates something bad. Heh, heh ! Some understanding is dawning .. > For him both limit processes should yield the > same result, something like exp(-1). Nope. I didn't discuss the answers. I've criticized the _questions_. > Since that isn't the case, he concludes that mathematics is wrong as a > whole. Nope. A question like lim_[max(m,n) -> oo] (1 - 1/m)^n is perfectly legitimate, as far as I am concerned, and you cannot deny that it _is_ a mathematical question. The answer is that this limit does not exist, though. A "good" thing about this limit is that m and n are not "fixed" or such nonsense and that m and n _both_ are free to approach infinity. You want to see a limit that's satisfactory in whatever respect ? Like this: lim_[ |(x,y)| -> oo] exp(-(x^2 + y^2)/2) ; the limit is zero. > I don't think, that anything could be done about that. One should come up with better examples, better than I've seen so far. Han de Bruijn > You want to see a limit that's satisfactory in whatever respect ? Like > this: lim_[ |(x,y)| -> oo] exp(-(x^2 + y^2)/2) ; the limit is zero. Not if x and y are allowed to be complex >>You want to see a limit that's satisfactory in whatever respect ? Like >>this: lim_[ |(x,y)| -> oo] exp(-(x^2 + y^2)/2) ; the limit is zero. > > Not if x and y are allowed to be complex Yeah, right. Default mathematics is with reals, oh well .. Han de Bruijn > On Sep 22, 4:44 am, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> > [quoted text clipped - 24 lines] > h(m) = 0 for every m, and again you can compute > lim{m->infinity} h(m), only this time you get 0 as the limit. Uhm, I thought that you had *PLONK*'d me. Can't resist ? Again, it's not so much the answer, but the _question_ that disturbs me. (Most of the time I can answer your questions "correctly" and routinely) What is right and what is wrong ? I could also have introduced a more euphemistic terminology and say the following. Limits that are asked for without awareness of Geometry might be called _blind_. Those are limits that are blindly asked for, without having a picture in mind. Then our working hypothesis is formulated as follows. Limits that do not commute are _blind_ limits. They are only formal, not geometrical, hence irrelevant for mathematics _as a whole_. Not to speak about any applications in sciences, such as physics. I've mentioned already the example in Thermodynamics, where not a single exception can be found on the rule that crossed partial derivatives always commute. Doesn't that mean anything to anybody? One could at least ask: WHY oh WHY ?! Han de Bruijn > > On Sep 22, 4:44 am, Han de Bruijn > <Han.deBru...@DTO.TUDelft.NL> <snip> > Again, it's not so much the answer, but the > _question_ that disturbs me. [quoted text clipped - 15 lines] > not geometrical, > hence irrelevant for mathematics _as a whole_. Why, for just once, can't you drop the crank act and act with a little humility. Your arrogance is plainly insufferable, particularly when it is borne out of sheer ignorance. Really, for someone who does not even understand double iterated limits, to pontificate on how something is supposed to be irrelevant to mathematics is mind-boggling. It is not double iterated limits that are irrelevant, it is your ignorance and posturing. Your nonsense about "blind limits" or the supposed lack of geometrical picture or their purely formal status or whatnot is not a reflection on mathematics as a whole but is simply reflection on your own limitations and prejudices. It just so happens that you mr. de Bruijn do *not* have the adequate intuitions about these elementary things. And no, mathematical intuitions are not just geometrical, there are also those of algebraic and analytic type (rough division). Regards, G. Rodrigues >>>On Sep 22, 4:44 am, Han de Bruijn >> [quoted text clipped - 21 lines] >>not geometrical, >>hence irrelevant for mathematics _as a whole_. You make scrambled eggs of someones else's posting quotes, huh? > Why, for just once, can't you drop the crank act and act with a little humility. Your arrogance is plainly insufferable, particularly when it is borne out of sheer ignorance. Really, for someone who does not even understand double iterated limits, to pontificate on how something is supposed to be irrelevant to mathematics is mind-boggling. It is not double iterated limits that are irrelevant, it is your ignorance and posturing. Your nonsense about "blind limits" or the supposed lack of geometrical picture or their purely formal status or whatnot is not a reflection on mathematics as a whole but is simply reflection on your own limitations and prejudices. It just so happens that you mr. de Bruijn do *not* have the adequate intuitions about these elementary things. And no, mathematical intuitions are not just geometrical, there are also those of algebraic and analytic type (rough division). And edit for your own is void of NewLines, Carriage Return - LineFeeds. But never mind .. I'll repeat my answer to Mariano for convenience. <QUOTE> Oh no ! This is a completely _objective_ definition. _Other_ people than me (i.e. Virgil, Rodrigues, Hughes), have said repeatedly that they find any geometrical intuition _irrelevant_. Only the formal algebraic result seems to count. Considering them as representatives of mainstream maths, this leads to the conclusion that common limits are developed without a picture in mind, without seeing, therefore BLIND-ly. Since geometry _is_ an integral part of mathematics, since ancient times, blind limits _are_ limited to the algebraic part of mathematics, and thus _are_ irrelevant, if mathematics is considered as a unification of algebra _and_ geometry, for mathematics _as a whole_. But it seems that you are pissed, because "blind" sounds pejorative. So does "false" in logic, and nobody has any problems with it.</QUOTE> Han de Bruijn > >>>On Sep 22, 4:44 am, Han de Bruijn > >> [quoted text clipped - 50 lines] > But never mind .. I'll repeat my answer to Mariano > for convenience. Please, bear with my limitations. Currently, I can only post via the mathforum interface, which is not exactly the best. > <QUOTE> > Oh no ! This is a completely _objective_ definition. [quoted text clipped - 20 lines] > and nobody has any > problems with it.</QUOTE> There is no argument here, except prejudice, besides a number of inaccuracies, lies and misunderstandings. Regards, G. Rodrigues > <QUOTE> > Oh no ! This is a completely _objective_ definition. _Other_ people than > me (i.e. Virgil, Rodrigues, Hughes), have said repeatedly that they find > any geometrical intuition _irrelevant_. There is a difference between what is relevant and what is binding. While limits are often suggested by geometrical considerations, once limits have a geometry-free definition, they are no longer bound by geometry. > Only the formal algebraic result > seems to count. Only the constraints of the formal definition count. To have it otherwise imposes hidden meanings whose effects cannot be known. > Considering them as representatives of mainstream maths, > this leads to the conclusion that common limits are developed without a > picture in mind, without seeing, therefore BLIND-ly. Whether a picture was /in mind/ during development is irrelevant unless that picture is /in words/ a part of the definition. In proper math, definitions are not allowed to have such implicit constraints as HdB wold impose. > Since geometry _is_ > an integral part of mathematics, since ancient times, blind limits _are_ > limited to the algebraic part of mathematics, and thus _are_ irrelevant, Except when doing algebraic mathematics, of course. > if mathematics is considered as a unification of algebra _and_ geometry, > for mathematics _as a whole_. There are all sorts of specialities in mathematics which are largely to totally independent of other specialities. This includes large parts of algebra which are independent of geometry. > But it seems that you are pissed, because "blind" sounds pejorative. It is meant to sound pejorative. But any such blindness is in the eyes of one beholder, HdB. > So does "false" in logic, and nobody has any > problems with it. If you were to call non-commuting iterated limits "false", a lot of people would have problems with it, as they should. > Horand.Gassm...@googlemail.com wrote: > > On Sep 22, 4:44 am, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> [quoted text clipped - 38 lines] > not commute are _blind_ limits. They are only formal, not geometrical, > hence irrelevant for mathematics _as a whole_. Your sense of self importance is astounding, if you even for a millisecond believe that what *you* think is relevant or irrelevant to mathematics (as a whole or even a small part of it) is relevant to anyone. > Not to speak about any > applications in sciences, such as physics. I've mentioned already the > example in Thermodynamics, where not a single exception can be found > on the rule that crossed partial derivatives always commute. Doesn't > that mean anything to anybody? One could at least ask: WHY oh WHY ?! It only means that people doing thermodynamics choose to model they physical systems using sufficiently smooth functions. Their choice is amply justified in that their measurements are never precise enough, so they may just as well assume their functions are smooth. But their choice does not imply in any imaginable way that "in thermodynamics partial derivatives always commute". -- m >>Horand.Gassm...@googlemail.com wrote: >> [quoted text clipped - 44 lines] > is relevant or irrelevant to mathematics (as a whole > or even a small part of it) is relevant to anyone. Oh no ! This is a completely _objective_ definition. _Other_ people than me (i.e. Virgil, Rodrigues, Hughes), have said repeatedly that they find any geometrical intuition _irrelevant_. Only the formal algebraic result seems to count. Considering them as representatives of mainstream maths, this leads to the conclusion that common limits are developed without a picture in mind, without seeing, therefore BLIND-ly. Since geometry _is_ an integral part of mathematics, since ancient times, blind limits _are_ limited to the algebraic part of mathematics, and thus _are_ irrelevant, if mathematics is considered as a unification of algebra _and_ geometry, for mathematics _as a whole_. But it seems that you are pissed, because "blind" sounds pejorative. So does "false" in logic, and nobody has any problems with it. >>Not to speak about any >>applications in sciences, such as physics. I've mentioned already the [quoted text clipped - 10 lines] > way that "in thermodynamics partial derivatives > always commute". My statements are, _always_, so explicit that they are quite vulnerable. The only thing you have to do is to find a counter example. If you can't then start asking yourself (like I did) why things in science are ALWAYS so different (and easier) than in mathematics. Start asking yourself why all those pathological "thinkable" special cases NEVER occur in physics, nor in any other of the applied sciences. Start asking, that's a start. Han de Bruijn > >>Horand.Gassm...@googlemail.com wrote: > >> [quoted text clipped - 51 lines] > this leads to the conclusion that common limits are developed without a > picture in mind, without seeing, therefore BLIND-ly. It is HdB who is criticizing blindly. No one said that the development of ideas could not be based on 'pictures'. What we say is that those ideas are not thereafter necessarily limited by the pictures which suggested them. Not all groups have to be 'pictured' as permutation groups. Not all vector spaces are finite dimensional real vector spaces. Not all geometries are Euclidean. > Since geometry _is_ > an integral part of mathematics, since ancient times, blind limits _are_ > limited to the algebraic part of mathematics, and thus _are_ irrelevant, So that in HdB's mind, even if mathematics as a whole need not be subservient to physics, he wants to force it to be subservient to geometry? What is it about mathematics' independence from HdB's artificial controls that upsets him so? The only limits to mathematics is what cannot even be imagined. If one person can imagine iterated limits which exist where joint limits do not, then that is a legitimate part of mathematics, and not all of HdB's objections can prevent it. > if mathematics is considered as a unification of algebra _and_ geometry, > for mathematics _as a whole_. But it seems that you are pissed, because > "blind" sounds pejorative. So does "false" in logic, and nobody has any > problems with it. When it is meant pejoratively, as HdB means it, it often sounds pejorative. > My statements are, _always_, so explicit that they are quite vulnerable. > The only thing you have to do is to find a counter example. If you can't > then start asking yourself (like I did) why things in science are ALWAYS > so different (and easier) than in mathematics. Because they impose limits that mathematicians can ignore in order to see what happens when they are ignored. > Start asking yourself why > all those pathological "thinkable" special cases NEVER occur in physics, > nor in any other of the applied sciences. Start asking, that's a start. Unless those allegedly pathological cases are considered and analyzed, how does anyone know they will never occur, or how to work out the consequences if one of them should occur? Mathematics is a world of "what if". And there is no compulsion, at least within mathematics to limit it to "what is". > > On Sep 22, 4:44 am, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> > > [quoted text clipped - 37 lines] > not commute are _blind_ limits. They are only formal, not geometrical, > hence irrelevant for mathematics _as a whole_. At most, such iterated limits are irrelevant for certain geometrical interpretations. Unless HdB can prove that they are also irrelevant for each and every non-geometrical application, it is HdB who is wrong. There is no bit of analysis that is inexorably tied to any any geometrical interpretation, not even those expressed in geometric terms. > Not to speak about any > applications in sciences, such as physics. Good! Don't speak of them at all, as they are irrelevant, at least to those who do not regard mathematics as subordinate to physics the way HdB does. > I've mentioned already the > example in Thermodynamics, where not a single exception can be found > on the rule that crossed partial derivatives always commute. Doesn't > that mean anything to anybody? One could at least ask: WHY oh WHY ?! It just means that physics and mathematics do not coincide. Why doesn't that mean anything to HdB? He could at least explain:WHY oh WHY ?! > > Nice excercise for the weekend. Thanks. > [quoted text clipped - 49 lines] > _both_ variables > approach infinity in whatever way. <snip mathematical irrelevancies> In "whatever way"? Look, to see geometrically why you are wrong, suppose you have a continuous function of two variables, e.g. f(x, y), as a function on R^2. If the function has a limit *no matter* how we approach infinity then by using stereographic projection, this is equivalent to a function on the sphere S^2 with a well defined value at the north pole (the "point at infinity"). But of course, this class of functions is much smaller than the class of continuous functions, even bounded ones, since there are *many* ways to approach "infinity" in R^2. The inequality of the iterated limits just means that the behaviour of f at "infinity" depends on how we approach it, which is fairly obvious by looking at a few surphace graphs of such functions. There is really nothing surprising in this. A student with only elementary calculus will have learned that in order to be able to interchange limits some sort of uniformity hypothesis are needed. It seems you have not, and, what is really damning, you do not want to. Your loss. Regards, G. Rodrigues. >>>Nice excercise for the weekend. Thanks. >> [quoted text clipped - 51 lines] > > <snip mathematical irrelevancies> Yes. Quite at your convenience. > In "whatever way"? Look, to see geometrically why you are wrong, suppose you have a continuous function of two variables, e.g. f(x, y), as a function on R^2. If the function has a limit *no matter* how we approach infinity then by using stereographic projection, this is equivalent to a function on the sphere S^2 with a well defined value at the north pole (the "point at infinity"). But of course, this class of functions is much smaller than the class of continuous functions, even bounded ones, since there are *many* ways to approach "infinity" in R^2. The inequality of the iterated limits just means that the behaviour of f at "infinity" depends on how we approach it, which is fairly obvious by looking at a few surphace graphs of such functions. Yes. Am I not saying exactly _that_ all the time? Now what's your point? Why must that broader class of functions _have_ such nonsensical limits? All I've shown is that _single_ (non iterated) limits are sufficient to describe the behaviour of your broader class of functions, _completely_. Iterated limits add _nothing_ to it; they are, essentially, redundant. Ah well, but you're snipping these "mathematical irrelevancies" of mine. > There is really nothing surprising in this. A student with only elementary calculus will have learned that in order to be able to interchange limits some sort of uniformity hypothesis are needed. It seems you have not, and, what is really damning, you do not want to. Your loss. So you _know_ what the remedies are. Why don't you just tell us? Han de Bruijn > > There is really nothing surprising in this. A student with only elementary > > calculus will have learned that in order to be able to interchange limits [quoted text clipped - 4 lines] > > Han de Bruijn One remedy would be to send HdB back to take a purely mathematical sequence of calculus/analysis courses, devoid of physics. > >>>Nice excercise for the weekend. Thanks. > >> [quoted text clipped - 86 lines] > Iterated limits add _nothing_ to it; they are, > essentially, redundant. Read again what I have wrote. > Ah well, but you're snipping these "mathematical > irrelevancies" of mine. Yes, they are irrelevant. > > There is really nothing surprising in this. A > > student with only elementary calculus will have [quoted text clipped - 5 lines] > So you _know_ what the remedies are. Why don't you > just tell us? There is no "us" here, there is only mr. de Bruijn that does not know this material. I have learned it in my first undergraduate year. And for your information my undergraduate studies were in physics in an engineering school. You want to know the details? Check any book on real analysis or advanced calculus. There are plenty decent books available for download on the internet. Regards, G. Rodrigues > There is no "us" here, there is only mr. de Bruijn that does not know this material. I have learned it in my first undergraduate year. And for your information my undergraduate studies were in physics in an engineering school. You want to know the details? Check any book on real analysis or advanced calculus. There are plenty decent books available for download on the internet. Pfff. And that's it. What a cheap answer .. Han de Bruijn > > There is no "us" here, there is only mr. de Bruijn that does not know this > > material. I have learned it in my first undergraduate year. And for your [quoted text clipped - 6 lines] > > Han de Bruijn But not so cheap as the question! |
Enable EMail Alerts
Start New Thread
Reply to this Message