Genady Posted March 17 Author Share Posted March 17 3 hours ago, Sensei said: Is that the only thing that matters to you? You didn't give enough thought and attention to what I said in my previous posts.. In general, no, this is not the only thing that matters to me. In this thread, yes, it is. Your previous posts are interesting on their own. We can discuss them in your thread. Link to comment Share on other sites More sharing options...

studiot Posted March 17 Share Posted March 17 9 hours ago, TheVat said: Measurements cannot even provide many rational numbers. No one can measure 1/3. Maybe you will get 0.33332. With some margin of error. That’s not 1/3 and it doesn't seem you can improve the accuracy of the experiment to make it so. 1/3 will always lie between error margins and beyond our observation. Is there any physical distinction that matters between a rational number and an irrational number if the two are closer to each other than the experimental error? 10 hours ago, Genady said: Are irrationality of a number or uncountability of a set used in physics to derive a result, to prove a theorem, to formulate a model, etc.? The point is that in order to draw a continuous line your pen must pass through irrational points to get from one rational to the next. Equally for (physics) fields to follow the equations of physics, the field lines must pass through irrational points to get from one rational point to another and in order to make a completely closed line or shell or surface or whatever a the closure must block the flux from escaping through irrational points on that closure. 2 Link to comment Share on other sites More sharing options...

Genady Posted March 17 Author Share Posted March 17 8 minutes ago, studiot said: The point is that in order to draw a continuous line your pen must pass through irrational points to get from one rational to the next. Equally for (physics) fields to follow the equations of physics, the field lines must pass through irrational points to get from one rational point to another and in order to make a completely closed line or shell or surface or whatever a the closure must block the flux from escaping through irrational points on that closure. I think I understand this intuition. To rephrase, if variables in equations were rational instead of real, some terms would be undefined. This is a good answer to the OP question. Thanks. +1 Link to comment Share on other sites More sharing options...

studiot Posted March 17 Share Posted March 17 17 minutes ago, Genady said: I think I understand this intuition. To rephrase, if variables in equations were rational instead of real, some terms would be undefined. This is a good answer to the OP question. Thanks. +1 Thanks. Intuition is the right word. Continuity and infinity are strange beasts. The thing is that in Physics we can shift the coordinate system any distance we like, including by a single point. This implies that in performing that shift we can overlay a rational point with an irrational one and vice versa - something we cannot do in maths. Link to comment Share on other sites More sharing options...

dimreepr Posted March 17 Share Posted March 17 1 hour ago, studiot said: This implies that in performing that shift we can overlay a rational point with an irrational one and vice versa - something we cannot do in maths. Isn't that the point of the imaginary number? As discussed by Rutherford and Fry on their podcast. Link to comment Share on other sites More sharing options...

studiot Posted March 17 Share Posted March 17 (edited) 4 minutes ago, dimreepr said: Isn't that the point of the imaginary number? As discussed by Rutherford and Fry on their podcast. I don't know about any podcast, but my answer to your question is a simple no it is not. By the way are you (and they) confusing imaginary numbers with complex numbers ? Edited March 17 by studiot Link to comment Share on other sites More sharing options...

dimreepr Posted March 17 Share Posted March 17 1 minute ago, studiot said: I don't know about any podcast, but my answer to your question is a simple no it is not. By the way are you (and they) confusing imaginary numbers with complex numbers ? I clearly am confused by something I didn't fully understand, but the podcast is worth a listen (they both have a doctorate). Link to comment Share on other sites More sharing options...

studiot Posted March 17 Share Posted March 17 1 hour ago, dimreepr said: I clearly am confused by something I didn't fully understand, but the podcast is worth a listen (they both have a doctorate). Well If I knew what yoy might not have fully understood, I might be able to help. The podcast was overlong and oversensationalist for my taste but they did make a clear distinction between imaginary and complex numbers. They also pointed out that 'imaginary' is a bit of a misnomer as imaginary numbers are every bit as rigorous and valid as real numbers. The thing is that the terms imaginary, virtual, borrow 1, are just examples of a technique used in both maths and science more generally. Children become comfortable with the idea of 'subtract 19 from 435' by saying 9 from 5 wont go so add (or borrow) 1 (when they really mean add 10), now 9 from 15 leaves 6 then add 1 back to the 1 from the 19 or take it off the 3 in the 435 to get the answer 416. A more advanced version of this is solution by substitution of variables, both in algebraic and differential equations. Engineers and physicists have long used virtual work, virtual displacements, as powerful tools for solving classical problems and more recently we have the virtual quantum paticle. The podcast also floated the idea that all these things are devices to help solve equations by the perhaps perhaps permanent temporary introduction of something new. Centuries after the controvery of imaginary numbers a similar difficulty arose describing angles and rotations in higher dimensions. This was eventually resolved (Hamilton) by the introduction of dyadics, quaternions and octonions. But none of these are directly involved with the op question of (my paraphrasing) "Why do we bother with the reals when the rationals are nearly as good" ? Link to comment Share on other sites More sharing options...

TheVat Posted March 17 Share Posted March 17 11 hours ago, Genady said: @TheVat, I think we can measure 1/3 (or any other rational number) in cases when we can measure by counting. I mean, if we have counted 1000 of something and we know that there are 3000 of them in total, we have measured 1/3, right? Yep. I should have mentioned that counting discrete objects would be a different situation. Just catching up with thread now. 4 hours ago, studiot said: ...fields to follow the equations of physics, the field lines must pass through irrational points to get from one rational point to another and in order to make a completely closed line or shell or surface or whatever a the closure must block the flux from escaping through irrational points on that closure. I also thank you. Link to comment Share on other sites More sharing options...

joigus Posted March 17 Share Posted March 17 (edited) I think the question could be translated into: Is there any way in physics that we can (experimentally meaningfully) ask the question of whether space/time, or either one of them --or any other variables for that matter-- has the power of the continuum? I don't think there's been any proposal to test that, and I don't think there can be. It is a very interesting question, though. When you report an experiment, you always do it by means of a finite string of numbers and/or characters. I think this says it all. It's probably an undecidable question, unless some connection is established --that we don't know of as yet-- between discrete mathematics and transcendent mathematics. It doesn't seem feasible that that will happen any time soon. Plus, 14 hours ago, TheVat said: Measurements cannot even provide many rational numbers. No one can measure 1/3. Maybe you will get 0.33332. With some margin of error. That’s not 1/3 and it doesn't seem you can improve the accuracy of the experiment to make it so. 1/3 will always lie between error margins and beyond our observation. Is there any physical distinction that matters between a rational number and an irrational number if the two are closer to each other than the experimental error? I also said somewhere before, On 7/21/2022 at 12:58 AM, joigus said: as the question whether, say, the speed of light, or the mass of a body, or any other physical quantity is a rational or irrational number (or prime, or has any other number-recursion connection) doesn't really make sense, for the very simple reason that any of those are measured numbers, and inevitably go with a range of experimental precision. So any claim that you would want to make in terms of number theory would immediately be whitewashed by the fact that all physical quantities are cut off by virtue of experimental uncertainties. Self-quotation from: Edited March 17 by joigus minor addition 1 Link to comment Share on other sites More sharing options...

martillo Posted March 17 Share Posted March 17 (edited) 47 minutes ago, joigus said: I think the question could be translated into: Is there any way in physics that we can (experimentally meaningfully) ask the question of whether space/time, or either one of them --or any other variables for that matter-- has the power of the continuum? I don't think there's been any proposal to test that, and I don't think there can be. It is a very interesting question, though. When you report an experiment, you always do it by means of a finite string of numbers and/or characters. I think this says it all. It's probably an undecidable question, unless some connection is established --that we don't know of as yet-- between discrete mathematics and transcendent mathematics. It doesn't seem feasible that that will happen any time soon. I agree. +1 5 hours ago, studiot said: The point is that in order to draw a continuous line your pen must pass through irrational points to get from one rational to the next. Except if the space be actually discretized in a grid with some imperceptible resolution. In this case places represented by irrational numbers would not really apply. Edited March 17 by martillo Link to comment Share on other sites More sharing options...

Genady Posted March 17 Author Share Posted March 17 9 minutes ago, martillo said: Except if the space be actually discretized in a grid with some imperceptible resolution. In this case places represented by irrational numbers would not really apply. In this case places represented by rational numbers would not apply too. Link to comment Share on other sites More sharing options...

martillo Posted March 17 Share Posted March 17 19 minutes ago, Genady said: In this case places represented by rational numbers would not apply too. I think it would depend on how the grid is defined. For instance could be a grid in steps of 1/3 or whatever 1/n with integer n. Some rational numbers could apply in a grid while others don't. But of course an entirely integer grid could in principle be possible too. As @joigus said may be is something undecidable for us. Link to comment Share on other sites More sharing options...

Genady Posted March 17 Author Share Posted March 17 7 minutes ago, martillo said: I think it would depend on how the grid is defined. For instance could be a grid in steps of 1/3 or whatever 1/n with integer n. Some rational numbers could apply in a grid while others don't. But of course an entirely integer grid could in principle be possible too. As @joigus said may be is something undecidable for us. I take it back. In an infinite discrete grid of finite cell size, all rational numbers can be represented. Link to comment Share on other sites More sharing options...

martillo Posted March 17 Share Posted March 17 4 minutes ago, Genady said: I take it back. In an infinite discrete grid of finite cell size, all rational numbers can be represented. Seems it all depends in how you define the steps or the resolution of the grid. Link to comment Share on other sites More sharing options...

Genady Posted March 17 Author Share Posted March 17 7 minutes ago, martillo said: Seems it all depends in how you define the steps or the resolution of the grid. Why? I don't see that my previous statement depends on the grid resolution. I see only two conditions for the grid: a) the steps have a finite size, and b) there are infinitely many of them. Link to comment Share on other sites More sharing options...

martillo Posted March 17 Share Posted March 17 (edited) 9 minutes ago, Genady said: Why? I don't see that my previous statement depends on the grid resolution. I see only two conditions for the grid: a) the steps have a finite size, and b) there are infinitely many of them. In a) which would be the finite size of the steps? In b) it would depend if the space is infinite or finite. Edited March 17 by martillo Link to comment Share on other sites More sharing options...

Genady Posted March 17 Author Share Posted March 17 2 hours ago, joigus said: I think the question could be translated into: Is there any way in physics that we can (experimentally meaningfully) ask the question of whether space/time, or either one of them --or any other variables for that matter-- has the power of the continuum? I disagree with this translation. It assumes that there are 'real' space, time, or other variables. IMO, all these variables are components of our models. They are what they need to be for our models to work. That's why my question rather is, is it important / used / implied anywhere in our current models if the variables have or don't have the power of continuum. 2 hours ago, joigus said: When you report an experiment, you always do it by means of a finite string of numbers and/or characters. I think this says it all. I am sorry, but it doesn't say much to me. 27 minutes ago, martillo said: In a) which would be the finite size of the steps? It does not matter for being able to represent all rational numbers. Any finite size will do. 28 minutes ago, martillo said: In b) it would depend if the space is infinite or finite. Yes, this is the necessary condition for being able to represent all rational numbers. Link to comment Share on other sites More sharing options...

martillo Posted March 17 Share Posted March 17 5 minutes ago, Genady said: It does not matter for being able to represent all rational numbers. Any finite size will do. ... 5 minutes ago, Genady said: Yes, this is the necessary condition for being able to represent all rational numbers. Fine if your intention is just to represent all rational numbers. A different story is to represent the physical Space of the Universe. Link to comment Share on other sites More sharing options...

studiot Posted March 17 Share Posted March 17 (edited) I recommend this book edited by Shahn Majid - Cambridege University Press It is dedicated to the issue of granularity v continuity in real physical space. Please note that a mathematical space is quite a different animal. Edited March 17 by studiot Link to comment Share on other sites More sharing options...

Genady Posted March 17 Author Share Posted March 17 20 minutes ago, martillo said: Fine if your intention is just to represent all rational numbers. This is what we are talking about for the last few posts, isn't it? 21 minutes ago, martillo said: A different story is to represent the physical Space of the Universe. Yes, different and completely imaginary story. Not interested. Link to comment Share on other sites More sharing options...

joigus Posted March 17 Share Posted March 17 54 minutes ago, Genady said: That's why my question rather is, is it important / used / implied anywhere in our current models if the variables have or don't have the power of continuum. The only way it could be important is if it had observable consequences, isn't it? Why care so much about the model? Not every complex space with an inner product has a numerable basis. Hilbert spaces do. They must be dense (or closed under limits). That's a very nice property, and so we demand it. Does it play any role? Not really, as far as I can tell. No feature that's impossible to measure really plays any important role in the physics. Calculational convenience, I suppose. 1 hour ago, Genady said: 3 hours ago, joigus said: When you report an experiment, you always do it by means of a finite string of numbers and/or characters. I think this says it all. I am sorry, but it doesn't say much to me. Ok. That's your take. To me it does. Link to comment Share on other sites More sharing options...

Genady Posted March 17 Author Share Posted March 17 6 minutes ago, joigus said: The only way it could be important is if it had observable consequences, isn't it? Yes, it is. The observable consequences are not necessarily immediate, they can be way down the road. This might hide the importance of some features. E.g., as has been concluded in this thread already, it is important for variables to be real rather than rational because otherwise some terms in equations would be undefined, in which case the model would not be self-consistent. Link to comment Share on other sites More sharing options...

studiot Posted March 17 Share Posted March 17 (edited) 54 minutes ago, Genady said: 1 hour ago, martillo said: A different story is to represent the physical Space of the Universe. Yes, different and completely imaginary story. Not interested. I seem to have completely misunderstood you intentions. I thought this was about the use of mathematical theory in Physics, which is why it was posted in the Phyiscs section. Your last answer seems to imply that it is nothing to do with physics at all ?? Edited March 17 by studiot Link to comment Share on other sites More sharing options...

Genady Posted March 17 Author Share Posted March 17 1 hour ago, studiot said: I seem to have completely misunderstood you intentions. I thought this was about the use of mathematical theory in Physics, which is why it was posted in the Phyiscs section. Your last answer seems to imply that it is nothing to do with physics at all ?? I think you have understood me correctly. It has everything to do with physics and nothing to do with metaphysics. Link to comment Share on other sites More sharing options...

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