cladking Posted April 29, 2016 Author Posted April 29, 2016 Not exactly arbitrary. However, there is another thread on exactly this subject already so I don't see any point in rehashing it here. Infinity is well defined in mathematics (even if that is based on choosing some "arbitrary" axioms). But that is getting off the topic of the the thread which was that infinity doesn't exist "in the real world" (whatever that means). It's so convenient to say infinity is real and then to extrapolate an infinity of worlds. In reality you can no more divide by zero than there is a world with no Lincoln. The reality is that we can only estimate the odds of an event within a few million orders of magnitude based on current knowledge and this means that real numbers like the odds of a specific atom in a specific place a hundred years out are far greater than our concept of infinity, at least to our ability to measure and predict. These are all metaphysical failures but it's not apparent because each sees the world in terms of beliefs and models. Need I even mention that this is no small consideration because it is the difference between an infinite universe or infinite universes and our virtually infintesimal knowledge? Models are OK when properly applied but they are never reality itself. Mathematics works OK when it's properly applied but it can't really be applied exactly in any circumstance. It can be applied within 99.999+% perfectly within our ability to measure and understand all the variables.
Strange Posted April 29, 2016 Posted April 29, 2016 Well, does math exist independent from our own construction of it to model reality? Who knows. What do you think? Can math ever perfectly model reality or will reality always have a discrepancy from our models? Models will always be approximations to reality. In reality you can no more divide by zero than there is a world with no Lincoln. No one said you can, so why bring it up? The reality is that we can only estimate the odds of an event within a few million orders of magnitude This is obviously wrong as you have already calculated the odds of coins tosses far more accurately than that. Models are OK when properly applied but they are never reality itself. Everyone knows that. Mathematics works OK when it's properly applied but it can't really be applied exactly in any circumstance. I'm glad to see you have moved on from your "mathematics never works" stance. There is hope for you yet. It can be applied within 99.999+% perfectly within our ability to measure and understand all the variables. I would love to know how you calculated that 99.999%. (There are a lot of models that can't achieve anywhere near that accuracy. As well as some that are a lot more accurate.)
BiotechFusion Posted April 30, 2016 Posted April 30, 2016 (edited) It's so convenient to say infinity is real and then to extrapolate an infinity of worlds. Wait I don't think many worlds theory is what we're talking about... Who knows. What do you think? I tend to say say yes, but on the other hand we continuously show ourselves that our predictions are wrong. I'ts not to say that maybe eventually we can create a perfect mathematical model that perfectly models the observable universe. So ultimately, I just don't like when people assume either way. Edited April 30, 2016 by BiotechFusion
ajb Posted April 30, 2016 Posted April 30, 2016 (edited) I am not going to reply to everything... this is just getting silly. Oh no, some precious *model* wasn't a picture-perfect image of reality? How expected. It is expected. We all know this... so why the attitude? It's actually still not "ok" in math, it's only ok to take a limit, And taking limits I would understand as part of mathematics. So? Oh, so then you do admit the fallacy that we measured an infinite amount of a dimensional unit? Please can you indicate where I have claimed that an infinite 'value' of anything has been measured. In fact, I keep on saying quite the opposite. We have never measured an infinite 'value' and we never expect to be able to do so. Do I have to keep saying this? You seem to like assuming you solved everything for the sake of it. There are actually things that I have solved... but I have not claimed to have solved anything in relation to this discussion. Only claim I have made is that physical theories can misbehave and give infinite results or other results that are just not seen in nature. This usually is seen as pointing towards some new physics. We need to make better models that cover a larger parameter space. Even if infinity does exist... What do you mean by exist? Infinity exists in the mathematical sence that I can extend the real numbers (and similar) to include an object called 'infinity'. I have some rules for dealing with this 'generalised number'. If you mean do we expect to see infinity in nature, say as a result of some measurement, then the answer is no. Or at least, nobody, or almost nobody, expects infinities to be realised in nature. The same can be said of complex numbers. We extend the real numbers to include a new 'number' which is the squareroot of -1. It is not a real number, but still we know mathematically how to deal with it. In this sense it exists. Moreover, complex numbers seem vital in quantum mechanics. Yet we never expect to be able to measure the value of some observable to be a complex number. In this sense, nature does not realise complex numbers. There are some even more exotic rings and algebras that I deal with that seem vital in building quantum field theories. However, they are more like a half-way house between the pure classical world and the quantum one. We do not expect that nature will realise these algebras in the sense that you could actually go out and measure something with values in these algebras. However, as I said, they seem vital to our overall understanding. ...we can never measure it as being so, we can only make assumptions from models wherein we take limits to it. As I have said, so many times already... this is correct. From our models a calulation of some expected observable could involve a limit, in this case we can and sometimes do encounter infinities. And again, we do not expect these infinities to be actually realised in nature in the sense explained above. See, the problem is you're not actually proving anything. What did I set out to prove? I have only pointed out that infinites can and so arise in physical theoreis and that this is usually taken as a signal that someithing is not quite right. For the simple example of electrons as point charges in classical electrostatics, we see that this is a good model, provided we are not 'too close' to the point charges. The infinity that we see in this model is due to the treating of electrons as point-like. This is okay in a sense, but we need quantum theory to make proper sense of this. You see that the trouble with a simple model pointed to new physics. ...that's where your issue comes in: you're using the assumption that infinity exists based on some model as a proof that infinity exists. I am not thinking of a proof that infinity exists. We know within standard mathematics how to handle infinites (with some care!) and we know that they can appear in our models. That is all I have claimed and all that I and claiming. And again, it is generally thought that infinities will not be physically realised by nature.... shall I say it one more time? ...but mathematically you cannot have infinity as a result of an operation on a finite number. You mean that you cannot 'reach' infinity by a finite number of algebraic operations on the real numbers? I think we all agree on this. Infinities don't actually "occur" anywhere in elementary mathematics... Okay, I see what you are saying... but physics does not just use 'elementary mathematics', by which I assume you just mean the field structure of the real numbers. "does infinity exist?" Well, infinity is just a mathematical object we made up... Mod arguments about if mathematics if invented or discovered... this is not really any different to any other mathematical concept. Also, it does not change the fact that we deal with such things in our models. and even though within math it's not perfect... Well lets not get drawn into what one means by perfect and lets not get too bogged down with mathematical logic here. We know how to deal with infinites in a reasonable way. That is usually enough for model building. "If there was no space, time, or anything that physically existed, would a number or mathematics still exist? Interesting question, but this is a question for philosophy. Well, does math exist independent from our own construction of it to model reality? Can math ever perfectly model reality or will reality always have a discrepancy from our models? Good questions... but again these seem more related to philosophy than anything else. Most mathematicans I think do not worry too much about these kinds of questions. I'ts not to say that maybe eventually we can create a perfect mathematical model that perfectly models the observable universe. So ultimately, I just don't like when people assume either way. In my experience, people tend to think of models as models and not actually nature. In that sense it is not clear what one means by a perfect model. The best one can hope for is a model that describes nature to some level of accuracy that we find acceptable. Maybe we could find models that take us right to the limit of what we could even in principle measure... who knows? Edited April 30, 2016 by ajb 3
Strange Posted April 30, 2016 Posted April 30, 2016 So ultimately, I just don't like when people assume either way. I'm not sure they do (or, if they do, it is rarely apparent). When I have seen discussions of this, people usually have reasons for thinking one way or the other. But this has now turned into a meta-meta-discussion! 1
studiot Posted April 30, 2016 Posted April 30, 2016 (edited) In defence of models, their very imperfection is their greatest strength. A perfect model matches every characteristic of that which it models in every detail, both discovered and yet to be discovered characteristics and possesses no unmatched details of its own. So the only perfect model of say an electron is an electron. Our imperfect models are much simpler and only match some (hopefully desirable) details over some limited range. But they also come sans the details that are not of interest and do not therefore need wasting effort on. As regards infinities, mathematics in this discussion has become more technical than the OP was able to support and I commend the discussion in appendix IV of Hardy's famous book (A Course of Pure Mathematics, Appendix IV, The infinite in analysis and geometry) where he discusses the Aristotelian idea of real and potential infinites in the light of 20th century mathemnatical knowledge. As regards infinities in the real world, I have a question for consideration. In quantum theory an electron has many but finite energy states available to it whilst it is part of an atom. According to quantum theory, these energy states become closer and closer together as energy increases until the energy reaches a point where the energies are contiguous and a 'continuum' of energy states arises. So the question arises what is the enumeration of the energy states within the continuum? Edited April 30, 2016 by studiot
Strange Posted April 30, 2016 Posted April 30, 2016 So the question arises what is the enumeration of the energy states within the continuum? This is an interesting point. There seems to be a difference between something like (I'm not sure how to describe it) range or extent, which can in principle be infinite, and the value of a particular measurement. So, in your example, the electron can have infinite number of energies, but the energy can never be infinite. Similarly, the universe may be infinite in extent but there will not be anything with infinite mass within it.
cladking Posted April 30, 2016 Author Posted April 30, 2016 Models will always be approximations to reality. No! Models are our estimation of reality based on logic and the effect of nature on experiment. There is no need for any model to approximate reality itself. I would love to know how you calculated that 99.999%. I actually said 99.999+%. This is merely reflecting the fact that as a general rule units are chosen in the lab that are accurate to three decimal points. If you choose to measure electron orbits in parsecs then this will break down a little. Wait I don't think many worlds theory is what we're talking about... It's merely an example of a hypothetical model much like reality at the beginning of the big bang or the "event horizon". Nobody expects to measure "infinity" but my understanding is that many people believe it actually exists in reality.
BiotechFusion Posted April 30, 2016 Posted April 30, 2016 (edited) It is expected. We all know this... so why the attitude? I don't think you expect it because you keep using some assumption in a physical model as a basis for a mathematical proof of some kind, not very logical. And taking limits I would understand as part of mathematics. So? And in that mathematics, a limit is explicitly the object or value which is apparently approached, not the value itself at the coordinate you are approaching which is why it works. Please can you indicate where I have claimed that an infinite 'value' of anything has been measured. I asked you multiple times if there aren't in fact infinite field tensors within any finite space, like those in the *finite* space of a gluon field, but you keep saying those objects are in fact infinite, it's your mess to clean up. I keep on saying quite the opposite. We have never measured an infinite 'value' and we never expect to be able to do so. Do I have to keep saying this? If that was true our discussion would have been over 8 posts ago. You're clearly a proponent that a physical infinity is measurably within our models, otherwise you would be a proponent the concept that we can only assume to have an infinity arise in reality if we start with the assumption of infinity to begin with, thus you wouldn't keep trying to debate with me. Only claim I have made is that physical theories can misbehave and give infinite results or other results that are just not seen in nature. This usually is seen as pointing towards some new physics. We need to make better models that cover a larger parameter space. There you go again, infinity cannot be the "result" of a model, you keep posing yourself as a proponent of a lack of infinity in nature, then you say its some result we find from a model of empirical data, even though we never ever ever ever do. Saying "we start out with infinity" to measure electro-static potential doesn't mean infinity is a result of any mathematical operation or measure of data. What do you mean by exist? Infinity exists in the mathematical sence that I can extend the real numbers (and similar) to include an object called 'infinity'. I have some rules for dealing with this 'generalised number'. Exist only in the sense that any other mathematical object can exist, which is to say it was defined and constructed by someone, that we have not discovered it existing on its own. There isn't a logical "extension" of a number line to infinity because you can't count to it, there is no process of succession, multiplication, exponentiation or any iterative operator that yields infinity as an actual result, but you can apply these operators to finite numbers to give you complex numbers. There is only the arbitrarily defined axiom for this mathematical object which we then attempt to derive properties of by assuming an isomorphism to number line or a set which can allow the object to be algebraically manipulated in some sense, like that infinity+1=infinity. There's no inherent proof infinity+1=infinity any more than there is a proof 1/0=infinity, it's just an axiom we define based on our own intuition of numbers, just a convenient way we like to think of infinity for taking limits and counting sets as cardinal numbers, even though again we can't actually count all the elements in an infinite set, at best we can just come up with a pattern that will give us an element for any finite input or a technical result from taking a limit. Everything about transfinite numbers is provisional, it's always "if one chooses this axiom of choice is true...then if one then chooses this axiom is true...then if one then chooses this other axiom is true...then infinity+1=infinity" we had to make an arbitrary choice of what is true and what is not along the way to deriving its properties. If you mean do we expect to see infinity in nature, say as a result of some measurement, then the answer is no. Or at least, nobody, or almost nobody, expects infinities to be realised in nature. Good, you're making progress. Now, I want you to take a big leap and say "infinity cannot be the result of any quantitative model of reality derived from empirical data." The same can be said of complex numbers. We extend the real numbers to include a new 'number' which is the squareroot of -1. It is not a real number, but still we know mathematically how to deal with it. In this sense it exists. In the sense that it exists in the realm of mathematics. Transfinite numbers have much different axioms than numbers on a number line, and again, even within that interpretation of infinity, you only have infinity when you start to infinity to begin with, you can't extend a line of finite numbers to a transfinite number. Moreover, complex numbers seem vital in quantum mechanics. Yet we never expect to be able to measure the value of some observable to be a complex number. In this sense, nature does not realise complex numbers. Well we never actually measure a number on its own to begin with in the first place, not even a natural number, we only measure an object that we choose to abstract the results to a mathematical system as representing a numbers, so in that sense we could be measuring complex numbers all the time without realizing it. There are some even more exotic rings and algebras that I deal with that seem vital in building quantum field theories. However, they are more like a half-way house between the pure classical world and the quantum one. We do not expect that nature will realise these algebras in the sense that you could actually go out and measure something with values in these algebras. However, as I said, they seem vital to our overall understanding. Vital to the accuracy of some of our predictions, not necessarily to our understanding, not yet anyway. And again, it is generally thought that infinities will not be physically realised by nature.... shall I say it one more time? You can say it as many times as you want, but it won't matter until you stop saying "infinity is a result of something we model reality with..." Okay, I see what you are saying... but physics does not just use 'elementary mathematics', by which I assume you just mean the field structure of the real numbers. Most of the most accurate models do in some sense, they typically use some kind of vector space. The standard model uses vector space for instance. I am not thinking of a proof that infinity exists. We know within standard mathematics how to handle infinites (with some care!) and we know that they can appear in our models. That is all I have claimed and all that I and claiming. Well, I guess that's better, but, starting with infinity with the assumption that you have infinity it is not a result, not a logical conclusion, not a deduction, not a trend fitted to empirical data, just an axiom. Transfinite cardinal numbers only have relevance if you have a set of infinite data...which we never supposedly never measure as having...thus transfinite numbers are often limited to things like abstract mathematics...combinatorics...set theory...not often physics. Interesting question, but this is a question for philosophy. You assume. This is why there is a debate about it, because it's not philosophy. We see nothing in the standard model that should limit the size of the universe. Does that mean the size of the universe is "infinite"? Does that mean that there is a physical manifestation of infinity in the same way that there is a physical manifestation that we assign as any finite value? We'll never measure it, yet according to our own physics, the universe should be should have no limit on its size, does that fit the definition of a transfinite cardinal number well enough? On top of all of that, changes in quantum states. Logically, if our models are correct, electrons bound to an atom (to say the least) cannot have intermediate energy states between energy levels, they can only have very specific quantum numbers. Now, this leaves us with the issue that: if an electron changes energy states, say from an S1 orbital to an S2 orbital, is it limited to only being measured in those two energy states because of the mathematical logicality that it cannot have intermediate energy states, because that would essentially nullify its existence. An electron, according to our models *cannot exist* in those intermediate energy states, so as we delve deeper into what makes the universe work, it starts to appear as though it works because of mathematical logic itself, because only certain *values* of quantum numbers, matter and energy are allowed to exist, it would be illogical for other values to exist. Even though we can't measure numbers themselves, it seems math and reality start to merge together, that there is a logical pattern to the universe, that it's not completely random. It seems like there is more evidence for that math and more evidence for the expansion to planck length and time in a way...you might say "oh, a free electron can travel at any angle and any velocity, accelerate at any angle for any amount of time to emit any photon of any value of energy..." but what you're forgetting is that according to all of our best models, photons can only have specific frequencies. So, if an electron is free, it could only have been freed with a quantized amount of energy in the first place. Then, if that free electron that is supposedly allowed to move at any angle with any speed gets recaptured by an atom, it must, absolutely must only emit photons of certain frequencies and fall into only quantized energy states around the atom, and thus we can extrapolate that the energy the electron had in free space must have been quantized as well. Or, if an electron accelerates, it could only have emitted a photon at quantized energy levels, thus we can again extrapolate that its acceleration must in some way be quantized, and since acceleration is a function of the distance an electron travels in a given time, space and time must be quantized. In other words, speed isn't just moving meters per second, it's moving multiples of planck lengths per multiples of planck seconds. You can't mix and match continuous space and a continuous spectrum of energies with a quantized system of energy, at least not without some weird and extremely impractical models. Even doing something as simple as converting a "quantized" summation to an integral on a continuous number line gives you something like the integral of x-1/2-arctan(tan(pi*x-pi*/2))/pi which is so impractical you might as well ignore it, or you'd still have to use a flooring function or a ceiling function anyway. Edited April 30, 2016 by BiotechFusion
Strange Posted April 30, 2016 Posted April 30, 2016 No! Models are our estimation of reality based on logic and the effect of nature on experiment. There is no need for any model to approximate reality itself. As all we can know of reality is our observations and the results of experiments, I think the only answer is: Yes, of course. Duh. This is merely reflecting the fact that as a general rule units are chosen in the lab that are accurate to three decimal points. Really? Citation needed.
cladking Posted April 30, 2016 Author Posted April 30, 2016 As all we can know of reality is our observations and the results of experiments, I think the only answer is: Yes, of course. Duh. Yes. This is a metaphysical truism. The problem is this truth blinds us to the nature of the model and, more importantly, to the extent of our ignorance. Really? Citation needed. Formatting for reality is as inconsequential as semantics. But the fact remains that constants, measurements, and variables can only be executed to a certain degree of accuracy and equations are not always applied properly. Even in the lab there is still "slop". How this is expressed is a matter of the means and context used to make the point which is largely semantics which I refuse to discuss.
Strange Posted April 30, 2016 Posted April 30, 2016 (edited) Yes. This is a metaphysical truism. The problem is this truth blinds us to the nature of the model and, more importantly, to the extent of our ignorance. Less of the "us" please. It may blind you, but I think other people are fully aware of this. Formatting for reality is as inconsequential as semantics. Colourless green sheep dream furiously. (In other words, that sentence appears to have no semantic content.) But the fact remains that constants, measurements, and variables can only be executed to a certain degree of accuracy and equations are not always applied properly. Again, duh. It is also a complete non-sequitur. You specifically said "units are chosen in the lab that are accurate to three decimal points". Do you want to admit that is just another of your false statements, or would you like to provide some evidence? Edited April 30, 2016 by Strange
ajb Posted May 1, 2016 Posted May 1, 2016 (edited) I don't think you expect it because you keep using some assumption in a physical model as a basis for a mathematical proof of some kind, not very logical. You have really lost me... are you just making stuff up (again)? All I have assumed, and this is not really an assumption, is that our mathematical models are based on mathematics. And in that mathematics, a limit is explicitly the object or value which is apparently approached, not the value itself at the coordinate you are approaching which is why it works. I know what limits are and that the limit in a given space can be outside that space. So what? I asked you multiple times if there aren't in fact infinite field tensors within any finite space, I am now wondering if tensor fields that take the 'value' infinity are actually tensor fields. I think not. Anyway we can have points of regions on a manifold where various objects are singular in exactly this sense. If we encounter such things in our physical models, then we should be careful giving phsyical meaning to them at these points of regions. like those in the *finite* space of a gluon field, but you keep saying those objects are in fact infinite, it's your mess to clean up. Please quote where I have said that the gluon field is infinite or finite? (Classically it is not even real valued, so I am confused by what you mean here anyway) You're clearly a proponent that a physical infinity is measurably within our models, Please quote where I have said that infinity is measurable. otherwise you would be a proponent the concept that we can only assume to have an infinity arise in reality if we start with the assumption of infinity to begin with, thus you wouldn't keep trying to debate with me. I do not follow this at all. here you go again, infinity cannot be the "result" of a model, you keep posing yourself as a proponent of a lack of infinity in nature, then you say its some result we find from a model of empirical data, even though we never ever ever ever do. Okay, you do not understand what we mean by a model in physics. You seem to be making some assumption that a given mathematical model will fit all the data obtained (to some level of accuracy) and that it wil do so for all possible ranges of the parameters in this model. But this is trivially not so. I have given you a simple model of electrostatics and we see that this gives infinity in the limit that r ->0. This shows that you are wrong, but I think this is based on your misunderstanding of mathematical modeling. There isn't a logical "extension" of a number line to infinity because you can't count to it, Yet we dealing with such things... Good, you're making progress. You seem to have this the wrong way round! Now, I want you to take a big leap and say "infinity cannot be the result of any quantitative model of reality derived from empirical data." But as I keep saying this is not true. I have given you a simple model that has this feature! Electrostatics of point charges! You can say it as many times as you want, but it won't matter until you stop saying "infinity is a result of something we model reality with..." What I have said is that infinities can and do arise in physical models. I have not said that these infinities are realised in nature. Most of the most accurate models do in some sense, they typically use some kind of vector space. Well, vector spaces are one of most basic algebraic structures you can have and are found behind lots more structure. But so what? You assume. This is why there is a debate about it, because it's not philosophy. It seems more like philosophy because these questions do not have methodologies that could solve them. I cannot construct some proof or find supporting evidnece via experiments. We see nothing in the standard model that should limit the size of the universe. Which standard model? The one from particle physics or the one from cosmology. Anyway, neither says anything about the global shape of size of the Universe. Okay, it is clear that you have missed some of my points with mathematical modeling and how the presence of singular objects is usually taken as a signal that the model is being pushed to far. Let us direct attention to something a little more specific: With the above comments in mind, can you tell me something about the curvature singularites we see in general relativity? These are technical things, but they seem well known in pop-sci, but the definitions are harder to understand. Does general relativity (under some technical assumptions) have curvature singularites? Can you give me an example of an exact solution to the Field Equations that has such a singularity? Can you tell me if people actually expect this singularity to be realised? P.S. A little off topic but... You have obtained -9 points (when I checked). You should think about why this is. It is not a good sign. Edited May 1, 2016 by ajb
BiotechFusion Posted May 2, 2016 Posted May 2, 2016 (edited) You have really lost me... are you just making stuff up (again)? Not surprised you are lost, is it's only as made up as when you say infinity is the result of a model. All I have assumed, and this is not really an assumption, is that our mathematical models are based on mathematics. I'd like to see where you stated that. I know what limits are and that the limit in a given space can be outside that space. So what? So, stop trying to divide by zero, stop trying to say we reached infinity on the real number line, stop saying infinity is a result. I am now wondering if tensor fields that take the 'value' infinity are actually tensor fields. I think not. Anyway we can have points of regions on a manifold where various objects are singular in exactly this sense. If we encounter such things in our physical models, then we should be careful giving phsyical meaning to them at these points of regions. Yes, we can have coordinates where the result is undefined... Please quote where I have said that the gluon field is infinite or finite? (Classically it is not even real valued, so I am confused by what you mean here anyway) A gluon field strength can be modeled as a tensor field and you already said tensor fields can have an infinite value, so I guess a gluon field can somehow have an infinite tensor. Please quote where I have said that infinity is measurable. possibily regulate the infinites that are found in semiclassical gravity, maybe not. How do you "regulate" something that isn't an actual value? BiotechFusion, on 29 Apr 2016 - 08:34 AM, said: ...infinity cannot be the defined result of any mathematical operation and thus it cannot be the result of any quantitative model. You are simply wrong. So, I'm wrong to say we can't have a model where a physical representation of infinity results? Seems like a direct contradiction to what you said earlier. I do not follow this at all. Not surprising you don't understand it, Infinity is never the "result" of a model, you never put in a piece of empirical, finite data and get in infinity as a result of predicting a value. You seem to be making some assumption that a given mathematical model will fit all the data obtained (to some level of accuracy) and that it wil do so for all possible ranges of the parameters in this model. But this is trivially not so. I have given you a simple model of electrostatics and we see that this gives infinity in the limit that r ->0. This shows that you are wrong, but I think this is based on your misunderstanding of mathematical modeling. Please quote me where I said a given model will fit all obtained data, even though I've been arguing just the opposite. Yet we dealing with such things... Are we? And, are "we?" Or just you? Maybe the universe is of infinite size, but so far, we don't have a way to prove that because the way we quantify models doesn't allow us to. You seem to have this the wrong way round! Oh, so you're not making progress, my bad. But as I keep saying this is not true. I have given you a simple model that has this feature! Electrostatics of point charges! You've giving a model where people assumed that you could indefinitely approach the boundary of the universe to infinity and that somehow an electron has physical relevance at infinite distance, which logically it can't. But, you still didn't didn't show that infinity is a result, you've only showed that it can be an *assumed* *abstraction* as a basis for a deduction of the pattern of seemingly organized data. What I have said is that infinities can and do arise in physical models. I have not said that these infinities are realised in nature. You can assume an infinite value of something which is more or less illogical to begin with when trying to model something physical in our terrestrial physics, but infinite anything can't be the "result." Well, vector spaces are one of most basic algebraic structures you can have and are found behind lots more structure. But so what? So, you can't count to infinity in a vector space, so stop saying infinity is the result of quantitative dimensional models. It seems more like philosophy because these questions do not have methodologies that could solve them. I cannot construct some proof or find supporting evidnece via experiments. Maybe we can't find proof, but I would argue about evidence. Since our standard models show no limit to the size of the universe, you could argue that is evidence for the universe somehow having an infinite size, depending on how you define infinity. Or, conversely, we could find some sort of hyper-dimensional curvature that would show the universe is a loop over some distance to show it is a finite size in a new model. Which standard model? The one from particle physics or the one from cosmology. Both. Neither standard model places an upper limit on what size the universe can be. That's why the early universe is often discussed in terms of density, not volume, because in both particle physics and our own observations of cosmology, we see nothing that limits how much space there can be. Anyway, neither says anything about the global shape of size of the Universe. Exactly. Okay, it is clear that you have missed some of my points with mathematical modeling and how the presence of singular objects is usually taken as a signal that the model is being pushed to far. It actually seems like I've seen so much of your points that you can't even keep track of all your own points when I discuss them. Let us direct attention to something a little more specific: With the above comments in mind, can you tell me something about the curvature singularites we see in general relativity? These are technical things, but they seem well known in pop-sci, but the definitions are harder to understand. Does general relativity (under some technical assumptions) have curvature singularites? Can you give me an example of an exact solution to the Field Equations that has such a singularity? Can you tell me if people actually expect this singularity to be realised? General relativity does not state what occurs at these singularities since the transformations that model such curvature results in a division by zero at those singularities. Some people expect these singularities to be manifested in the form of some infinite value of curvature or time dilation or length contraction, some people do not. For the people who assume it does, they are inferring that a division by zero is equal to an infinite value and thus there is much controversy about the role of infinity in our models, which, is why quantized models show promise as they get rid of a lower and upper limit that would allow an infinite value of a dimensional quantity to exist in finite space. P.S. A little off topic but... You have obtained -9 points (when I checked). You should think about why this is. It is not a good sign. And you should think about how a platform that encourages the arbitrary mixing of emotions with objective logic can actually be credible. It doesn't matter if the dumbest, smartest, kindest or meanest person in the world says it, 1+1=2 in elementary mathematics, it's just logic. As far as I'm concerned, any number of reputation points is a demonstration of this platform's lack of regulation of concise scientific standards. Edited May 2, 2016 by BiotechFusion -5
cladking Posted May 2, 2016 Author Posted May 2, 2016 I had said, The problem is this truth blinds us to the nature of the model and, more importantly, to the extent of our ignorance. Less of the "us" please. It may blind you, but I think other people are fully aware of this. I think this might provide a clue to the nature of the disagreement in this thread. People are seeing reality as the sum total of their knowledge and the models they use. As such the existence of infinity is patently obvious to them. What isn't obvious is that reality far transcends our knowledge and our ability to model it. What isn't obvious is that our models are woefully incomplete to understand the totality of reality. If all we see are our models then reality is an open book to us. We can find an equation for everything and every equation has/ is its own reality. Of course no infinity has been measured but this seems a minor point if you know some equations demand its existence. Colourless green sheep dream furiously. (In other words, that sentence appears to have no semantic content.) One half is point five. There's half a four in two. The first of two apples is half the apples. "Half" is real and the way anyone chooses to say it is irrelevant. It is a matter of taste, formatting, or semantics. A thousand meters is a klick and one doesn't measure parsecs in nanometers (not in the world I live). -2
studiot Posted May 2, 2016 Posted May 2, 2016 (edited) Cladking People are seeing reality as the sum total of their knowledge and the models they use. As such the existence of infinity is patently obvious to them. What isn't obvious is that reality far transcends our knowledge and our ability to model it. What isn't obvious is that our models are woefully incomplete to understand the totality of reality. If all we see are our models then reality is an open book to us. We can find an equation for everything and every equation has/ is its own reality. Yes seems a reasonable if pessimistic summing up, with the exception that many think the underlined words are obvious. What's more because reality is more that our (best) models (or to put it another way reality is our best models and then a whole lot more) it surely must include infinity since the models do so. Edited May 2, 2016 by studiot
Strange Posted May 2, 2016 Posted May 2, 2016 People are seeing reality as the sum total of their knowledge and the models they use. No one else thinks that. So stop saying "we" when it is only you that believes such nonsense. As such the existence of infinity is patently obvious to them. No one is claiming any such thing. By constantly saying "we" and "us" you pretend you are characterising what others think. You are not. You are just making repeated strawman arguments. Of course no infinity has been measured but this seems a minor point if you know some equations demand its existence. As you can see from the comments on this thread no one thinks that a model that returns infinity represents reality. I think you like to pretend that other people believe silly things like that so you can feel smugly superior. And then state the bleeding obvious as if it were some sort of insight. You shallow opinions are getting very tedious. 2
ajb Posted May 3, 2016 Posted May 3, 2016 BiotechFusion, I am not sure what I can say to you to make you understand that you are wrong on a lot of things. Your attitude is one of sticking your fingers in your ears! I am not really sure what you hope to contribute to this forum or gain from it. Anyway... I will answer one point to show that you clearly have no idea what you are talking about. General relativity under some reasonable physical assumptions is plague with infinites, you can consult the singularity theorems of Penrose and Hawking (there are also older theorems). That is singularites seem rather unavoidable in general relativity. We have some simple examples such as the Schwarzschild solution, but there are plenty of others. For the Schwarzschild solution you can examine the Kretschmann scaler (this is not actually the best way to 'detect' or define singularities, but it is okay for the class of solutions we are discussing right now) and as you approach r ->0 this scaler tends to infinity. This means that at r=0 we have to be careful with the physical significance of this point and what this infinity means (1/0 is not defined and so to make sense of this we need to think of r-> 0). Usually to keep the structure of a Riemannian manifold we remove this point. (The same is done when dealing with other space-times, one usually 'cuts out' these regions to keep the nice mathematical structure .) People do not actually expect these kinds of singularities to be physically realised. They are understood as being something to do with the very small length scale (or high energy scale) and that we should expect our classical smooth model of space-time to need modification. For example, if space-time has a granular structure then it maybe the case that these singularies become 'smeared out' and 'regularised'. This generically what one would expect in a quantum theory of gravity. The renormalised energy-monemtum tensor, of say a scaler field, shows similar behaviour as we approach a closed time-like curve. Everything seems okay until we look at the field close to a CTC and in the limit of reaching the CTC the tensor diverges and we get infinite energy etc. This is taken as meaning that CTCs cannot be realised in nature. This is the Hawkinng conjecture. Like the infinities in general relativity, these may get regulated by a quantum theory of gravity and time machines maybe okay! But without a proper theory it is hard to make clear and well stated conjectures. As far as I'm concerned, any number of reputation points is a demonstration of this platform's lack of regulation of concise scientific standards. In truth it seems to correspond to the scientific standards of this forum and in particular the quality of the posts of the members. However, collecting lots of negative points quickly is usually a sign of poor attitude and civility. I tell you this because we know that collecting negative points is an indicator of a member quickly becoming banned for breaking the rules. I do not wish that to happen to you. As you can see from the comments on this thread no one thinks that a model that returns infinity represents reality. This really is the key point and one I seem to have to keep making (but not for your benefit). It does not matter if a theory gives us infinity for a 'value' of something we would like to meassure, which is usually the result of taking limits either in terms like 1/r or in terms of power series, this is interpreted as 'pushing the theory too far'. Singularies can appear in very simple models, like electrostatics or much more complicated theoreis in quantum field theory. They just show that the theories we have are not complete and that physics beyond these models is needed. 1
studiot Posted May 3, 2016 Posted May 3, 2016 (edited) Strange, on 02 May 2016 - 6:39 PM, said: As you can see from the comments on this thread no one thinks that a model that returns infinity represents reality. ajb, This really is the key point and one I seem to have to keep making (but not for your benefit). It does not matter if a theory gives us infinity for a 'value' of something we would like to meassure, which is usually the result of taking limits either in terms like 1/r or in terms of power series, this is interpreted as 'pushing the theory too far'. Singularies can appear in very simple models, like electrostatics or much more complicated theoreis in quantum field theory. They just show that the theories we have are not complete and that physics beyond these models is needed. ajb, I would really value your comment on the situation posed at the end of my post#56? BiotechFusion, on 02 May 2016 - 4:43 PM, said: As far as I'm concerned, any number of reputation points is a demonstration of this platform's lack of regulation of concise scientific standards. ajb In truth it seems to correspond to the scientific standards of this forum and in particular the quality of the posts of the members. However, collecting lots of negative points quickly is usually a sign of poor attitude and civility. I tell you this because we know that collecting negative points is an indicator of a member quickly becoming banned for breaking the rules. I do not wish that to happen to you. Once again, +1 for displaying far greater tolerance and patience than I can. Edited May 3, 2016 by studiot
ajb Posted May 3, 2016 Posted May 3, 2016 (edited) ajb, I would really value your comment on the situation posed at the end of my post#56? On electronic band structure? There are some assumptions needed in the theory such as infinite size and the homogeneous nature of the material, and no interactions with phonons etc. For a lot of samples under the right conditions this is a good model and we see band structure of the electronic states. However, form small samples the surface states are important and inhomogeneities can also effect the band structure. So like all models and their results, care has to be taken with where one would expect the theory to be 'good'. Condensed matter physics is not my speciality, so I am sure you can find much more detailed explanations that I can offer here. Another similar example could be a single free quantum particle. The solutions to the Schrödinger equation are plane waves and the spectrum is continous. (Note that plane waves are not vectors in a Hilbert space, you need to extend the notion to 'rigged Hilbert spaces'). So, for an 'isolated' particle this is a good model, but we do not really have free particles and we have assumed that our physical space is flat and goes on forver. Again, this is not a big problem for some calculations, but we have to be careful giving real phsyical meaning to some of this. In scattering theory we assume that we have asymptotically free states and look at a local interaction. This works well for a large range of systems, but again we do not really start from free particles in our physical experiments: we just model them as if they were free 'far enough away' from the target. Doing so has lead to the standard model of particle physics being established as a 'good' theory. Edited May 3, 2016 by ajb
studiot Posted May 3, 2016 Posted May 3, 2016 (edited) Thank you ajb, I see I did not explain myself very well since you have picked up far more complicated cases than I envisaged. I was not thinking of anything so complicated as the band structures in solid state physics, just a single atom or molecule, possibly the simplest ie hydrogen. In any event band structures still have a finite number of states available. I was just thinking about the morse curve http://www.chemicool.com/definition/morse_potential.html and the ionisation energy, when the electron breaks free of a simple molecule to form an ion and a free electron, which is in a continuum of energy states. studiot As regards infinities in the real world, I have a question for consideration. In quantum theory an electron has many but finite energy states available to it whilst it is part of an atom. According to quantum theory, these energy states become closer and closer together as energy increases until the energy reaches a point where the energies are contiguous and a 'continuum' of energy states arises. So the question arises what is the enumeration of the energy states within the continuum? Edited May 3, 2016 by studiot
ajb Posted May 3, 2016 Posted May 3, 2016 (edited) and the ionisation energy, when the electron breaks free of a simple molecule to form an ion and a free electron, which is in a continuum of energy states. Okay, yes we have a continuum of energy states. This is not uncommon with potential wells. The question is if this is a good model for a pair of unbounded H atoms? Edit: As for 'counting' the number of states the closest I can think of is the density of states. Edited May 3, 2016 by ajb
swansont Posted May 3, 2016 Posted May 3, 2016 Not surprising you don't understand it, Infinity is never the "result" of a model, you never put in a piece of empirical, finite data and get in infinity as a result of predicting a value. That's not the only way for a model to be used. They can be used to predict results for which you don't have any empirical data.
studiot Posted May 3, 2016 Posted May 3, 2016 The question is if this is a good model for a pair of unbounded H atoms? Of course, but the original purpose of this model was substantially more exacting than the question we are discussing here. Surely it is enough to know that 1) An electron can be removed form a hydrogen atom 2) Whiilst bound within the atom the electron has a finite number of energy states available to it (neither the exact number, nor their exact values are relevent here) 3) Once the electron has left tthe atom, leaving a hydrogen ion behind, it is in a continuum and has surely therefore has an infinity of states availble to it. So is this not an example of physical realisation of infinity?
Strange Posted May 3, 2016 Posted May 3, 2016 3) Once the electron has left tthe atom, leaving a hydrogen ion behind, it is in a continuum and has surely therefore has an infinity of states availble to it. So is this not an example of physical realisation of infinity? It has an infinity of states available, but it will always be in a finite-valued state. So I'm not sure if that counts as a "physical realization of infinity" or not. (Those who claim that infinities cannot exist in nature will insist it isn't. )
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