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Are Christoffel symbols measurable? 
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#109
Feb2012, 10:01 PM

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You could argue that "all measurements can be reduced to all measurements that physicists are used to making." That's an extremely strong philosophical statement, and one that I'm inclined to consider to be false. There's one famous counter example called the "marriage meter." There is no set of physical measurements that you could make on me or my wife that could tell you whether or not we are married. By doing some sort of brain scan, you could establish that we *think* we are married. Since there is no such thing as a "marriage meter" then this means that things like marriage/divorce rates aren't physically measurable and you can put these things in vector spaces. I run into this sort of thing all the time at work. Two of the big, big questions write now is "how do you measure liquidity?" and "how do you measure risk?" Which quickly gets you into some philosophical questions "what is liquidity?" and "what is risk?" The relevance of this to the current discussion is that it seems that whatever liquidity and risk are, they somehow involve rather complicated vector spaces and the same sort of math that you find in GR. (Correlation matrices from hell.) Stock portfolios have similar issues. There are multiple equivalent ways of representing the dynamics of stock portfolio, but the dynamics exists independent of those representations. Vector spaces and the math associated with it comes in very handy when you have an "underlying reality" that's independent of the measurements. Relativity is one such example, but it's not the only one. There might be a physics reason to do this. In QM, to observe means to "collapse the wavefunction." 


#110
Feb2012, 10:09 PM

P: 6,863

The other thing to be careful here is "proof by lack of imagination." In order to prove nonexistence, you have to show that something really bad happens if something did exist. I can think of a lot of bad things that would happen if you had physical measurements that were none Lorenz invariant, or if quantum observations didn't reduce to a single number.
However, asserting that something is impossible because one can't think of counterexamples is a bad way of showing that something is impossible. This is particularly true because vector spaces are really useful, and can represent things that are pure fantasy (i.e. any first person shooter video game has vector space representations of all sorts of imaginary things). 


#111
Feb2012, 10:18 PM

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#112
Feb2112, 12:59 AM

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P: 1,588

However, this goes beyond merely spacetime scalars. Any measurement of a vector quantity (in the vector space sense, not in the computer science sense) must come by choosing a basis and projecting out components, thereby measuring a set of scalars. If you think about it for a moment, you will see that this statement is a trivial tautology. What I'm really saying here is that every measurement is a comparison. After all, that is what we really mean, isn't it? When we say something is "2 meters", all we're saying is it's twice as long as a certain metal bar in France. (Or, in modern SI, it's twice the distance light travels during so many oscillations of Cesium 133.) You could probably geometrize the idea of color space if you like, and make it a manifold, possibly with notions of parallel transport. Who knows, maybe there's a useful way to model some psychophysical process using a color space bundle over spacetime. At any rate, "color" is measured by first comparing incoming light against certain bands of frequency; i.e., taking an inner product in frequency space against certain basis vectors in order to form a collection of scalars. These scalars can then be used as a coordinate system on color space. This method works for RGB, CMYK, and color temperature type coordinates. Mapping physical observables to HSB coordinates is more complicated, and will require projecting the incoming spectrum onto several bands and doing some analysis with the results. Since frequency space is infinitedimensional, there is no real reason for color space to be 3dimensional; the ultimate reason is that we have 3 kinds of color receptors in our eyes (which project the spectrum onto 3 bands). Other animals have 2dimensional, or sometimes 4dimensional color spaces. 


#113
Feb2112, 01:25 AM

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In fact, I think I specifically mentioned that a collection of scalars can be used to construct a tensor quantity in the basis used to measure all the scalars. So, e.g., I can use a set of (scalar) measurements to deduce all 6 components of E and B in whatever basis I choose. The point is what constitutes a measurement. 


#114
Feb2212, 07:48 AM

P: 6,863

Axioms are axioms. I don't understand how "physically reasonable axioms" is a grammatical statement. Is "addition" physically reasonable? Now once you've defined addition and multiplication, you can then use them to make statements that are physically true or false. But the fact that you can make a statement that "the scalar multiplication of "red" by 2 gets us outside the set of physically valid colors" means that the axioms are defined. And it's also a *useful* vector space. Once you've defined the operations, then you can define a "norm" which then describes how "close" two elections are. You can then do matrix transformations from one set of coordinates to another. I define a C++ class RGB color, I define the operations of 2*Color and Color A + Color B. Once I've defined those operations, it's a vector space. I can even start do to tensor algebra. 


#115
Feb2212, 08:00 AM

P: 6,863

RGB numbers are physically bound within a range, but if I measure x, y, z coordinates of the earth, there are some values which are invalid. Also there *isn't* much of a difference between the mathematical concept of a vector space and the computer science one. All mathematics requires is that you have a defined addition and multiplication that has eight axioms. Once you have a collection of data for which you do that, then you have a vector space. 


#116
Feb2212, 09:18 AM

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I'm not sure if this makes any difference to your argument, but in this "electionresultspace" do you mean that the space is the discrete set of electoral districts, or do you mean the space is a continuous geographical/political map of the region?
Also, the "electionresults" are vectorlike, in the sense that they are multivalued arrays, but they are nonvectorlike in the sense that they do not have any direction. i.e. they don't point from one district toward another. 


#117
Feb2312, 01:21 AM

P: 177

The measurement of a tuple of values either entails many different comparisions to unit references, and thus many measurements, or the the comparison of a smaller number of measurements to tuples of unit references. In any case, each comparison must yield precisely one number. It is the combination of this number and the associated unit tuple that represents the measured value. No matter what mappings are done on the collection of numbers, if the appropriate inverse mappings are done such that the necessary unit tuples map back to themselves the measured values must be represented by the same numbers (this places restrictions on what is considered a valid mapping and/or the types of valid measurements (in the context of this thread I prefer to think of it as the former rather than the latter)). It is in this sense that all measurable quantities are collections of single numbers obtained from single measurements. If the units involve space or time references, then they must pick out a set of vectors in spacetime. Any such set of vectors can be used to construct a tensor of appropriate rank such that after any transformation, the evaluation of the resulting tensor on the images of the aforementioned set of vectors yields the same number as obtained originally (since one requires covarient tensors and vectors to transform in such a way that this is true). Sorry if the vagueness (or triviality) of the above was excessive, I was going for an abstract approach but may have overreached the bounds of my knowledge and/or conventional nomenclature. 


#118
Feb2312, 02:27 AM

P: 6,863

I'm located in a spot on the earth. By asking me yesno questions, you can figure out my latitude and longitude. Am I on land? Yes. Do I see taxicabs? Yes. Are they green? No. Are they yellow? No. Do I see water? Yes. With each question, you can eliminate parts of the vector space. The fact that I see tax cabs and they are not yellow, means that I'm not in Manhattan. Now if you can ask enough questions, you can figure out my location and convert to GPS coordinates. Note that you've figured out my GPS coordinates without actually measuring my latitude and longitude or doing any reference comparisons at all. You can show that no reference comparisons were done, because you can play this game without knowing anything about latitude and longitude at all, and it's the same game that you can play with things that are *not* vector spaces (i.e. words in a dictionary). One other way of thinking about it is that you can specify points in a vector space as the interaction of subsets of that vector space, which allows you to specify a point in that space without reference to basis vectors at all. 


#119
Feb2312, 02:42 AM

P: 6,863

One thing that got me started thinking along these lines is the fact that you can call functions in C++ "covariant" and "contravariant". So what does tensor calculus have to do with C++. Well, that got me into the world of category theory....... 


#120
Feb2312, 09:25 AM

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You cannot scale election results by arbitrary real numbers, nor even by arbitrary integers. *Vectors can be generalized to multiplication over other fields besides the real numbers, but the conclusion remains. There is an additive identity element, but no additive inverse in the space of election results. 


#121
Feb2312, 10:41 AM

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#122
Feb2312, 04:38 PM

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#123
Feb2312, 04:58 PM

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#124
Feb2312, 05:24 PM

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#125
Feb2312, 05:35 PM

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#126
Feb2312, 09:32 PM

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Once you pick a specific landmark, you now have a reference comparisonit's not latitude or longitude, but it is a reference comparison. 


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