<p>Oh god, the word physics makes my head hurt.</p>
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*. Still, I'm not sure that these sorts of numbers can be appropriately applied to the real world...
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<p>But they can, because you need a normative (idealized) idea of how things behave before you can apply corrections for the laws of physics. For example, a ball dropped from a certain height IS going to obey Newtonian laws even if things like air friction prevent it from ideally doing so - but you don't come up with a new model, you just add in the influence of air resistance. So there is value to having idealized ideas of how things work, even if they never really do that. </p>
<p>Is your problem that they don't really "exist"? That you could never, for instance, visualize pi? You don't really ever "see" anything except for the electrochemical neural impulses your brain interprets based on how reflected photons strike your retina, so maybe we need a better definition of what you mean by "perceiving" things like spacial motion.</p>
<p>edit: the philosophical implications of things like particle physics and cosmology is basically my major, so if I apologize in advance for posting 30,000 times in this thread...</p>
<p>barium:
Well, I could admit the triple 1.5-2-2.5, just because it's really the same thing as 3-4-5, just measured differently. What I do not accept is the notion of incommensurability.</p>
<p>jack4640:
That sounds like a very interesting major, indeed. Cool shturff!
Well, the air resistance thing is interesting. I still think that what I'm talking about is a little different from that, though; it's not having models accurately describe a plethora of events, but whether the way in which we have constructed our models is justified. I know there is a good correspondence between physics and observed phenomena - but does this imply that the assumptions made in our models have to be correct?</p>
<p>For example, consider the following situation. A ball is dropped where g = 10 m/s^2. How long does it take the ball to fall 10 meters? Well, solving x = 1/2 a t^2 yields t = sqrt(2). My question is whether the time is ever <em>equal</em> to this time; and hence whether the ball ever <em>technically</em> occupies <em>exactly</em> the space 10 m below where the ball was dropped. This is just an example and I made it up in my head just now.</p>
<p>Can the distance between your face and the monitor be divided infinitely, that is, can a <em>point</em> mass occupy an uncountably infinite number of positions between your face and the screen? Or is there some set of positions which it must occupy, such that transitions from adjacent positions are "all or nothing", and no existence is possible in between spaces.</p>
<p>Visualizing PI isn't a problem (I mean, I can't really do it, but that's not my concern). The problem for me is in believing that any two things in the physical world are actually in that proportion (exactly). I know also that talking about exact things in the real world is a little bit of a misnomer, but that's also part of this thread.</p>
<p>And as far as that whole "time is linear versus non-linear" thing, ... I don't know, it just seems that PI, sqrt(2), etc. correspond to geometric, rather than... oh dear, what's the word... arithmetic, algebraic, etc? ... quantities. I mean, PI has to do with circles, basically. What have circles got to do with time? And sqrt(2) has to do with squares, no? So what on earth has it to do with time? The golden ratio? e? etc.</p>
<p>I should also emphasize that I don't mean to question the validity of the models. I mean to question whether <em>truth</em> can be attained in this way. Maybe it depends on what I mean by truth? And maybe it doesn't make any sense to ask?</p>
<p>quicksilver, you really need to read up on some really pretty basic material. Your questions are very similar to what some of the Greeks asked about 2000 years ago, and they've been answered extremely accurately. Also, you have some very vague, non mathematical definitions of what it means for numbers to "exist." </p>
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but does this imply that the assumptions made in our models have to be correct?
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<p>No. The models we have are approximations and depend on many assumptions. Simply put, there are too many factors to consider to get a good estimate..</p>
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Can the distance between your face and the monitor be divided infinitely, that is, can a <em>point</em> mass occupy an uncountably infinite number of positions between your face and the screen? Or is there some set of positions which it must occupy, such that transitions from adjacent positions are "all or nothing", and no existence is possible in between spaces.
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<p>You are asking if physical space is discrete or continuous. As far as I know, all physical models describe it as continuous, but it is possible some disagree. Do some research into physics.</p>
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Visualizing PI isn't a problem (I mean, I can't really do it, but that's not my concern). The problem for me is in believing that any two things in the physical world are actually in that proportion (exactly). I know also that talking about exact things in the real world is a little bit of a misnomer, but that's also part of this thread.
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<p>Yes, pi is the ratio of the circumference to the diameter of any circle. There is no ambiguity about it.</p>
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I should also emphasize that I don't mean to question the validity of the models. I mean to question whether <em>truth</em> can be attained in this way. Maybe it depends on what I mean by truth? And maybe it doesn't make any sense to ask?
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<p>You need to start thinking about some philosophical concepts of what truth means. In any case, mathematicians (for the most part) don't concern themselves with this too much. Their job is to work under sets of basic assumptions called axioms. Whether these correspond or not to something in the physical world is irrelevant. Self-consistency is what counts.</p>
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While it's clear the imaginary ones don't have a physical existence
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<p>They certainly do. Imaginary numbers are identical to vectors, and thus certainly have a physical manifestation. A good example would be to look at phasors in AC circuit analysis.</p>
<p>"quicksilver, you really need to read up on some really pretty basic material. Your questions are very similar to what some of the Greeks asked about 2000 years ago, and they've been answered extremely accurately. Also, you have some very vague, non mathematical definitions of what it means for numbers to "exist.""</p>
<p>You need to not be condescending. No one else has been, and I doubt you are so much smarter than any of the rest of us. Perhaps you are simply too impatient to fully consider the scope of the sorts of questions I'm asking? And don't assume I haven't researched these topics.</p>
<p>"No. The models we have are approximations and depend on many assumptions. Simply put, there are too many factors to consider to get a good estimate.."
I've already attempted numerous times to make it clear that I'm not saying the models are bad, just that their being good may be something of a coincidence. I accept mathematical models as estimates for the physical world. So this statement of yours is more than a little redundant, and has been said before...</p>
<p>"far as I know, all physical models describe it as continuous, but it is possible some disagree. Do some research into physics."
Had you been paying attention, you would have realized that this is exactly my question. And I have researched it. Have you?</p>
<p>"Yes, pi is the ratio of the circumference to the diameter of any circle. There is no ambiguity about it."
Mathematically, yes. But do perfect circles exist in the physical world?</p>
<p>"You need to start thinking about some philosophical concepts of what truth means. In any case, mathematicians (for the most part) don't concern themselves with this too much. Their job is to work under sets of basic assumptions called axioms. Whether these correspond or not to something in the physical world is irrelevant. Self-consistency is what counts."
My question isn't about undermining mathematics. And it isn't intended to undermine the idea of mathematical models for practice's sake. I do a lot of work in numerical analysis, and those models are even more imperfect that the mathematical ones we have. My line of inquiry actually has to do with some philosophical notions of applying math to physical systems. While it certainly works in a practical sense, I wonder if it works <em>in principle</em>. And how would we design an experiment to ever know?</p>
<p>I like the idea of the planck length mentioned by... somebody... earlier. And planck time, too. Well, goodnight, folks.</p>
<p>[Careful with your tone, tetrahedron. Sorry if I overreacted to your comments.]</p>
<p>i'm not sure why you assume that there must exist some underlying truth about the physical world that can't be captured by a mathematical model. if we haven't observed it and can't describe it, why should we believe it exists? just because results are counterintuitive doesn't make them "less true"... it's no less absurd to ask "do perfect circles exist in the physical world?" than to ask "does santa claus exist in the physical world?" if you accept that "2" corresponds to some physical quantity, you can't reject "sqrt(2)" or complex numbers just because it's harder to think about.</p>
<p>Quicksilver, I am having a hard time figuring out what exactly you are arguing for. That "mathematical" models cannot be used to describe the world accurately, or that our current models may not describe the world accurately since we don't know if our underlying assumptions (e.g. continuity) are correct? </p>
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Well, I could admit the triple 1.5-2-2.5, just because it's really the same thing as 3-4-5, just measured differently. What I do not accept is the notion of incommensurability.
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Would you mind explaining how incommensurability applies to right triangles?</p>
<p>barium:
Alright, I'll try to explain this better.</p>
<p>What I am not trying to do is deny the usefulness or the accuracy of the mathematical models which approximate physical phenomena. I am not saying "if I am right, physics must be wrong".</p>
<p>What I am trying to say is that our approach, while useful, might not be exactly right. Consider the direction of current in a DC circuit. It is shown to flow from positive to negative; well, we know that this is not technically true, as it is electrons that are really moving. Therefore, while very useful and technically fine to do, we understand now that it's not really <em>right</em>.</p>
<p>And what of luminiferous aether? It was certainly a useful idea, as it helped people to understand electromagnetic radiation's propagation in the void. But that doesn't mean it <em>really</em> exists, no?</p>
<p>Dark matter and dark energy? Useful, useful, useful. Nevermind that it's not actually been seen.</p>
<p>And Newton's laws... the best example yet. Do they work? Yes. Do they always work? No. Therefore are they useful? Probably moreso than anything else in Physics. Do they provide a <em>true</em> model for our world? Not by my standards (classical mechanics is far and away my favorite field in physics).</p>
<p>So my things are really more of philosophy that mathematics. Try not looking at this as a mathematician. Look at it as a natural philosopher.</p>
<p>And as far as incommensurability and right triangles... well, any right triangle whose sides don't form a Pythagorean triple (in some measurement system) have sides which are incommensurable. Say, half of a square.
Now let me anticipate your argument that you could have a triangle with sides of length 3e, 4e, and 5e. No you can't, under my assumptions. Simply because you'd have to get something of length e first, and I would deny the source of that length as well.</p>
<p>iostream:
Assuming that something is wrong and asking others whether they agree that something might be wrong is a bit different. And it's not the difficulty of the ideas that makes me wary of their applicability... it is the seemingly non-physical way in which these objects are constructed that leaves me wondering how far we can use them. And why does accepting 2 preclude my accepting of sqrt(2)? They are fundamentally different ideas. I can show you 2 apples. If you can show me sqrt(2) of anything, this discussion will be resolved.</p>
<p>If you grant me any measuring system I want (especially with any number of dimensions that I want), no right triangle is incommensurable. And you know that because you won't let my measuring system include irrational numbers so that it cannot account for irrational numbers. </p>
<p>If a perfect circle is just be an illusion, why not an apple (where does an apple end on a sub-atomic scale? And who decides that it is "one" apple and not just the illusion of an object created by some uncountable structure on a sub-atomic scale?) Show me something that <em>truly</em> has the quantity 2 and I will show you something of quantity sqrt(2).</p>
<p>We cannot know what is <em>truly</em> correct because we do not live in reality but in our model of reality. I assume you would agree that colors and emotions do not really exist but are just our interpretation of reality. How about light, sound and weight? Do apples exist, or do we just interpret a certain structure as an apple?</p>
<p>All we do is create models, and we have to name the object we use for our model. + and - are labels, and so are 2 and sqrt(2). Note that a label - a name - does not make a model correct or incorrect. "Dark matter" is just another one of these labels. We just try to explain what we are seeing, that's all physics does.</p>
<p>Your seem to doubt one central assumption of our model: continuity. Well, our perception of reality tells us that time and space are continuous. Is it true? It might just be an illusion, in the same way that time and space are illusions and none of them necessarily have to exist. But as long as our models work with the assumption of continuity, there is no reason to abandon it (especially because continuous models are a lot easier to deal with than discrete ones).</p>
<p>Does anything that physics talks about (time, space, continuity, dark matter, electrons, apples...) truly exist? - Nope, because it's a model. It's the very nature of a model to be different from the system it describes. That's why there's no point arguing about the truth of our models. But we can argue about their coherence. Undoubtedly, physical models have become more accurate over the centuries. Will we ever get to the point where the model coincides with our perception of reality? Consider that our "reality" itself is just a model of the true reality, I don't see a reason why not.</p>
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You need to not be condescending. No one else has been, and I doubt you are so much smarter than any of the rest of us. Perhaps you are simply too impatient to fully consider the scope of the sorts of questions I'm asking? And don't assume I haven't researched these topics.
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<p>Sorry if I came off as condescending. That was not my intention, and I apologize for it. Your questions though, are extremely similar to questions that were asked by the Greeks. There is nothing wrong with that - the people who asked these questions were obviously the brilliant thinkers of the time. </p>
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just that their being good may be something of a coincidence.
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<p>I suppose this is my issue with what you're saying. We agree that all models are based on estimations. But why are these coincidences? </p>
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While it certainly works in a practical sense, I wonder if it works <em>in principle</em>.
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<p>I believe the whole point of a model is to work in a practical sense. This is probably where we disagree. The model attempts to explain how some phenomena works, under certain "perfect" conditions. Will these perfect conditions ever take place in the real world? No. Can we approximate them better and better and thus gauge just how good the model is? Yes.</p>
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I can show you 2 apples.
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<p>Can you? "2" is an abstraction, in the same way that sqrt(2) is. "2" is a more fundamental abstraction, so it's easier to see than sqrt(2), but it's just the same way an abstraction. It's not real. There is no "two" out there existing in reality.</p>
<p>Again quicksilver, I hope I'm not offending you. I'm interested in this thread and I like discussing questions like these, sometimes a bit too harshly.</p>
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Now let me anticipate your argument that you could have a triangle with sides of length 3e, 4e, and 5e. No you can't, under my assumptions. Simply because you'd have to get something of length e first, and I would deny the source of that length as well.
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<p>Earlier you talked about how there are no perfect circles in the world, so there is no "natural" quantity that has length or measure pi, a trascendental number. At the same time, I don't think you will be able to ever find an object that is exactly 1 unit long either, no matter how hard you try. Unless you start out with a definition of what 1 unit is, and ascribe it to a real object, you're still stuck in the same place. Is there a perfect 3-4-5 triangle out there? The concept is just as difficult as finding a perfect circle.</p>