icosahedron
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Posts posted by icosahedron
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One doesn't need to count them, only realize that change involves both what kind of change (the direction), and how much of it (the magnitude).
Direction answers the question: What change?
Magnitude answers the question: How much change?
Magnitude can be, and often is, discrete, as in counting pennies. In such cases, the natural numbers can be used to account magnitude.
If I owe 100 pennies, I don't say that I own -100 pennies. Why does mathematics try to say that?
Thinking that negatives are something in and of themselves leads to conceptual difficulties, eventually, no less in mathematics than any other discipline.
Fact is, only the natural numbers are necessary to form rational numbers and hence obtain the ability to relate reality to units, and measure quantities. Real numbers are irrelevant to reality because the irrationals permeate the reals, and an irrational number cannot be the result of a real measurement.
If folk would stop with the demand for compactness, then the rationals would be a perfectly sufficient measurement space for quantities. But it is hard to do calculus without the (at least implicit) notion of compactness.
And if folk would create and maintain proper conceptual hierarchies based on personal verification, they wouldn't be tempted to conflate disparate concepts (such as scalar and vector).
I'm trying to simplify things, believe it or not.
- David
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Magnitude is a multiple of a scalar unit.
Direction is towards or away from something, with respect to a focal point.
For example, in Euclidean 3-space, vectors have both magnitude and direction ... a vector may have any magnitude, and its direction correspond to one of the points on a sphere (the focal point is the origin).
- David
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A space is compact if, whenever a collection of open sets covers the space, then so does (at least) one of the available, finite sub-collections. Such spaces are very nice, analytically, and serves to generalize the notion of a bounded, open subset of the real line to higher order topological spaces.
Is that the notion of compact you had in mind? Compact spaces are continuous at any level of focus, and modern analytic methods rely on the fact that spaces are compact to derive all sorts of useful results.
In particular, modern physics uses the machinery of partial differential equations in its root formulations of physical law. Have you ever tried to think about PDE's outside the context of a compact space? Good luck.
- David
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Volumetric space is a concept very close to the perceptual level, don't you think? Much closer than Euclidean space ...
- David
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What is distance, physically?
Isn't it the time it takes a photon emitted at point A to be absorbed at point B?
What is rest mass, physically?
Isn't it the resistance to acceleration of a body at rest?
And what is translational momentum, physically (and hence velocity, the ratio of mass and momentum)?
Isn't it the cumulative resistance to being decelerated to a state of rest?
Now, if acceleration (in the absence of energy exchange, e.g., that due to gravity) is reduced to re-allocating the step frequency among the available directions of expression, as my system does, then the two ideas, rest mass and translational momentum, can be combined into a more general concept, of which rest mass and translational momentum become logical derivatives.
Thus I have my concept I (properly, IMHO) call inertia: inertia is the allocation of frequency across the available directions. If the allocation is even, then you have a rest mass. If the allocation has only one non-zero component, then you have a zero rest-mass particle such as a photon.
Representing the volumetric inertia as a 4-tuple of natural numbers, it can be decomposed into two addends, one of which is symmetric, and the other reduced to having at least one component equal to zero.
For example, (5,3,2,4) can be written as: (2,2,2,2) + (3,1,0,2)
The symmetric addend represents the portion of the motion which cycles back to its starting point, and the sum of its components is proportional to the relativistic mass.
The reduced addend represents the conventional velocity.
Thus, in this case, the conventional momentum is 8*(3,1,0,2).
And here is why both mass and momentum are not relativistically invariant in the usual formulation: because, if acceleration causes reallocation of the components of the inertial 4-tuple, then it alters both the symmetric and reduced addends.
Contrast that with their sum, the inertia (5,3,2,4). This is relativistically invariant, and can be used to generalize the notion of force, from causing a change in momentum (which it still does) to causing a change in inertia, which is an integral of mass and velocity distinctly different, and to my mind more holistically.
Check it out for yourself, and if you aren't willing to take the time to do that, I understand you might be under time pressure, but it doesn't give you the right to sweep my arguments aside with traditional mathematical bromides.
Cheers.
- David
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I wonder, when did the compact perspective on reality take hold? Certainly, not before Newton ... so, what did Aristotle think was going on when a body moved?
To me, space is the space between things, i.e., that which must be traversed to bring things together. Since there are only a finite number of things, with finitely many states and interactions, how can there be continuous, never mind compact, motion?
- David
- David
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In OPAR, Peikoff presents Ayn's conviction that it was too soon to begin reforming government revenue generating methodology.
I never understood this. Reform requires both a better system, and a means to convert to that better system.
Of course the conversion will be difficult, but the idea is correct, and if there is one thing I have learned in my experience, it is that nothing gets done by doing nothing.
Can someone explain the rationale behind Ayn's position to me?
I think the good ol' USA needs a new constitution, written to ensure the government is not allowed to take moral positions, nor initiate the use of force against normal (non-criminal) citizens.
- David
- David
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The primary political issue is: does she grant the initiation of force as a proper means to some ends?
If she does, or equivocates on this fundamental political precept in any way, you need to either set her straight or get the heck out of dodge. It took me 14 years to realize this was a dead-end road, and while things have worked out well for me in any case, I sure wish I had been less stubborn in evading the truth for so long, I'd be much better off in my health and career at least if I had been more principled in my primary relationship (the sad thing is, she was incredibly intelligent, and one of the few cases that, had I straightened my back sooner, might have worked out with me convincing her to change her principles ... and by accepting her bad behavior, I only encouraged it.)
If she says she doesn't and sticks to the principle consciously, but is not consistent in applying this principle, then you can leverage her recognition of the principle to correct her application of it. There WILL be arguments, but if you are consistent, either you will persuade her, or she will leave you (be prepared, this latter is the likely outcome if you attempt to persuade her to follow the logical consequences of the principal).
If she doesn't and her behavior is consistent with her claim, then you can, and likely will, work any issues out if you both are committed to the relationship.
- David
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So if your notion of "speed" is entirely useless for science, while the traditional notion of "speed" can be defined in terms of your notion of "speed", and can be of use to science, then why bother with your notion of "speed" at all? Does rewriting the equations to use your notion of "speed" make visualizing anything easier, or make certain calculations simpler, or lead to any scientific advancement?
For example, the concept of an "electron hole" makes many calculations simpler - does your concept of "speed" do the same?
If I thought it useless for science, I wouldn't bother wasting time on it. I guess I haven't earned the credit in your eyes for you to accord me this respect yet. Fair enough.
It is useful for science if it clarifies or advances understanding of the data. At very least, a wholly discrete approach should be tried, because our minds do not sample reality continuously. Compactness is a nice feature in mathematical analysis, but perceptual reality cannot be shown to be compact by any demonstration, because the demonstration itself must be discretely accomplished and described. Assuming analytic compactness as a feature of reality is speculative at best, and just asking for computational trouble (such as singularities).
If you want a discrete picture that does not contradict known macro and micro theories (and please don't forget to feed Heisenberg!), then you'll end up in my neighborhood.
- David
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If all the speed is in the same direction, how do frequency/wavelength of light remain defined? How can light have varying frequencies if "all the speed" is only in the direction the beam of light is traveling?
The problem is that speed is not an observable, it is inferred by comparing displacements to the time they take.
Forget about speed: if it is constant, then it is irrelevant.
The control variables are frequency and direction of motion.
For a photon, the direction of motion doesn't change, and frequency determines the number of cycles per unit time.
- David
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But no, speed does not "mean' speed of displacement.
It doesn't? Really? Would you mind explaining this? What else could speed mean than differential rate of linear displacement?
Interesting.
- David
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So if I take two electrons starting at point A, shoot them in the same direction, and later observe that one of them reached point B in half the time of the other, how did it get there sooner? Because its frequency was higher and step length was shorter? So with each step, the faster electron travels toward point B less distance than the slower electron, but it does this more often?
Couldn't the traditional notion of "speed" simply be considered the magnitude of the average velocity over one oscillation?
How would you go about calculating this "universal constant of speed"?
The electron that got there sooner was shot out with more polarized inertia, i.e., a less uncertain state of motion, with a higher fraction of its steps directed towards the endpoint. So it drifts more each cycle of motion. On average, its displacement is greater in equal time.
Consider motion constrained to a circular ring. In this case, the moving entity can be moving very fast, yet the net displacement on average is zero. Moving at any given speed says nothing about the direction of motion. And yes, the traditional notion of speed exactly maps to the magnitude of the average velocity over one cycle of motion.
As for the universal constant of speed, I'd look for an entity that only moved in one direction for a macroscopic period of time, and measure its displacement as a function of time. In other words, I'd measure it as the speed of a photon in a vacuum.
But then, I wouldn't really care what the speed was if I had no control of it; if it's the same for everything, it is irrelevant, conceptually. What is relevant is the observable displacement per unit time, i.e., the magnitude of the average velocity in the macro context.
Rather than thinking of it a speed, per se, think of it as the fact that the product of step length and frequency is constant. Don't even need to consider trajectories, a purely discrete formalism applies.
Since our minds cannot re-focus continuously, our reality results from a process of discrete sampling of the given. There is no basis (and never was!) for postulating that the given is continuous, rather than an agglomeration of discrete interactions. Continuous approximations are very useful, but sight should not be lost that they are approximations, and our minds do not work continuously, in terms of the ability to sample reality.
Speed in my system, at the micro level, is not a control variable. Frequency and direction of motion is all I need, conceptually (and all I have access to, perceptually).
- David
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Look, I said it in another thread and I will say it again ( well I implied it at least) : This approach to math is worse than futile, it is ridiculous. And not worth analyzing because you see math is the science of MEASUREMENT, which means you have to start with reality. I seriously doubt that this is where this vector of nonsenses origin point is.
I start with reality. In reality, every entity is interacting with its surroundings, causing every entity to change progressively, which we observe as relative motion of parts of the entity. The overarching entity, which contains all others, is Existence -- and Existence is not unitarily conceptualizable, because that would mean, in addition to pinning down the given, knowing the mind of every individual.
How do entities change? By interacting with other entities. Our measurements of the changes are discrete, and we can't assume we know what is going on between measurements, or else we'd have to answer to Heisenberg. That is reality based.
Speed means speed of displacement, but says nothing about the direction of displacement. That is reality based.
Now I notice that I can devise a system in which speed is irrelevant, only direction and frequency matter. And I don't have to think interpolatively, in terms of trajectories, which idea is known to be flawed when descending to micro level. This is inductive, but in no way contradicts my reality basis.
Finally, I realize that the resulting framework is more useful, integrates more facts, than prior ones -- but does not contradict them. Thus I justify my induction.
Is that real enough for you?
Or would you like to hit me with a theory vs. practice suit?
- David
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Is there any evidence-based reason to believe that the speed of fundamental particles is a universal constant?
Sure, photons move at the speed of light. Of course, you meant ALL the fundamental particles ... and I am claiming ALL entities of any stripe.
The trick is to realize that instantaneous speed, if applied to different directions in varying proportions, leads to net displacement in space as well as time. For photons, all the speed is involved in only one direction, so the net displacement the same as the length of the path; but if more than one direction is invested in, then the net displacement is necessarily less than a photon would accomplish in the same amount of time.
As long as your frequency of observation is much lower than the frequency of that which you observe, it will appear solid and moving in a trajectory. At the other extreme, it will appear to hop around with no definite trajectory, only an average blur. The apparent extent and solidity of the object is due to the relative slowness of our sense relative to the frequency of the observed object -- we identify the blur as the object, properly ... but in doing so we miss the micro-motion.
- David
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In case of an automobile, the step length if very tiny because the frequency-equivalent of the total energy is huge.
The question is, how come a car doesn't appear to move at constant speed? Because what we measure, macroscopically, is the magnitude of the average velocity of the car, not it's instantaneous speed. The car can be seen as a self-regenerative pattern of motion, like a bee buzzing around. When the car is accelerated, the drift motion is increased by, effectively, polarization of the self-regenerative pattern of motion that is the car. Up to you whether you see the pattern of motion as a tracery, or a now-you-see-it-now-you-don't quantum flashing, although I think the quantum flashing is conceptually purer -- the traditional notion of trajectory is an interpolation of actual observation, which latter is of course discrete insofar as our mental focus is only discretely tunable).
It takes work to accelerate the car because polarization of its motion reduces the number paths available to execute the motion.
Since the car occupies volume, the minimum number of rays to represent its translational motion is 4, towards the corners of a tetrahedron (note that the tetrahedron can take any shape without loss of generality; the symmetric case corresponds to flat space, but asymmetric coordinates could be used to model curved space).
An inertial state of the car is then a 4-tuple of natural numbers, with their sum equal to the car's frequency. Imagine the car's frequency is 4*N, and its inertia is balanced, i.e., its macroscopic velocity is zero -- e.g., when idling. It's inertia can be represented as (N,N,N,N).
Now imagine the car accelerates into the state (N+3,N-1,N-1,N-1), i.e., you accelerate towards a corner of the coordinate tetrahedron.
When idle, the number of accessible paths consistent with the car's inertia is:
(4*N)!
------
4*N!
Once accelerated to (N+3,N-1,N-1,N-1), the car's inertial path count is:
(4*N)!
-----------------
(N+3)! * 3*(N+1)!
At this point, the speed shown on the speedometer would be 4/N times the speed of light -- for the car, with N inordinately large, this is a trivial acceleration.
The ratio of the path counts measures how much work is required to perform the acceleration (albeit one may want to take logarithm of the ratio to make the measure additive); the work is necessary to overcome the reduction in the uncertainty of which path the car will actually take.
The ratio is:
(N+3)! * 3*(N+1)!
-----------------
4*N!
which reduces to:
(N+3)*(N+2)*(N+1)
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N*N*N
The ratio is larger than 1, so it takes work to reduce the uncertainty and perform the acceleration as expected.
Does that make sense to you, dream_weaver? Or is this not worth the time to analyze?
- David
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I have had this model for a long time, but it has been a while since I have discussed it.
Is it possible that every thing is moving at the same speed (including photons)?
In other words, what if speed (the magnitude of velocity) cannot vary, is not a fundamental control variable in modeling physical systems?
In fact, it is imaginable: what if an entity moves at constant unit speed, but changes direction with some frequency? Then, since speed is constant and frequency of directional is given, the distance traveled between direction changes is equal to the inverse of the frequency. Call this the step length, then what I am claiming is that product of step length and step frequency is constant. This is obvious for photons, except that a photon only has one direction available to it, so its steps are laid out along a ray; what I am suggesting is that the same step length/frequency relationship applies even for objects which appear to be motionless on average, a la de Broglie except that, unlike photons, more complex objects have multiple directions accessible at each step. The macroscopic concept "velocity" is then the average spatial drift accomplished by a sequence of steps of fixed frequency and concomitant length.
The simplest complex case involves two available directions of motion, and the inertia of an entity in such case can be represented as a pair of natural numbers, with their sum equal to the step frequency, and with each element giving the number of steps in the corresponding direction.
Interestingly, there are multiple paths associated with a given inertia state. For example, there are 6 ways to take 2 steps in each direction and end up back where you start after one cycle of motion. The number of paths associated with an inertial state is a measure of the uncertainty of the actual path taken, and the logarithm of the number of paths is what I call the inertial entropy. Changes in inertial state require effort if the new state has lower inertial entropy, but can happen spontaneously if the inertial entropy is increased by the change.
For example, (3,1) has less inertial entropy than (2,2), and a system in the state (3,1) has a lower available state consistent with its frequency; since you can't escape existence, over time (3,1) must "decay" to (2,2) ... and this decay happens in quantum fashion with cumulative probability as the duration of the (3,1) state increases.
Inertial entropy changes also accompany frequency changes, i.e., when a photon is absorbed and system energy increases, the inertial entropy will be altered. So absorption or emission of energy by a system involves changes in both energy and entropy; but it is the entropy change that drives things.
For example, given two entities, one with inertia (3,1) and the other with inertia (1,3), the number of joint paths is 16. If, instead, the entities had states of (1,2) and (2,3), respectively, then, even though the total frequency is the same, the number of paths is 30, or almost twice as much. If there is a means for the system to reconfigure and liberate entropy, it will ... but it may take a while, as the time to decay increases as the entropy liberated decreases.
I understand this will likely seem speculative and uncomfortable to anyone reading it for the first time; and it would take a volume to deal precisely with the ins and outs; but the gist of the idea is sound and does not contradict any known science -- and it provides enormous conceptual leverage, with the potential to bring relativistic quantum mechanical ideas into common parlance whilst obviating most of the computational complexity involved in traditional formulations.
Thoughts and/or questions? I know there is a lot more work to do to make this clear to anyone who hasn't seen it before, and I will be as patient as you are, if you choose to explore this with me.
Cheers.
- David
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3. You gave your model a notion of positive the second you associated one of the slots in your ordered pair with 1 and the other with -1. Just because you don't use the word doesn't mean that isn't exactly what's going on.
No I did not, you did. And you are wrong. I'm using plain old algebra here, not a special variant.
- David
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2. It is true that if you restrict the plane to the union of the positive x- and y-axes, the usual Euclidean inner product says that two nonzero numbers in that set are of "opposite sign" if and only if they are orthogonal. But in that case, you've essentially invented new notation for the negative sign that takes 3 to 4 times as long to write.
The conceptual leverage comes in two ways:
1. I get a means to represent time in addition to space, in the same framework, because the non-reduced pairs of natural numbers can be reduced to get the position, but also, the sum of the non-reduced components can be used to account the number of steps taken in reaching a location. In other words, (2,1) is the same spatial position as (1,0), but two steps later in time.
2. At higher orders, e.g., for planes, the efficiency really pays off because you have one less direction to worry about, and no negatives simplifies computations.
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1. Being isomorphic to the usual number system is not enough: two systems can be isomorphic without being conceptually similar. For instance, how would you rather deal with an object like a parabola-- as the graph of a quadratic function, or as intersections of certain planes with double cones, like Apollonius? Can you imagine doing trajectories being limited to such a framework? (Incidentally, how does this system of yours affect coordinate geometry?)
Exactly my point. The traditional notion of integers obfuscates the fact that they are vectors. You haven't directly challenged this contention, can I assume you agree that integers are vectors, conceptually, even if that is not how they are traditionally formulated?
The system has the effect of changing the coordinate system via a linear transformation, and so changes nothing about the results obtained so far. It makes conceptualization of space no harder, whilst opening up new frontiers of conceptualization due to the ease with which certain hard traditional ideas can be grasped (such as relativity) and allows conceptualization of what I term time-space, a space defined by motion at constant speed, where direction of motion is the only control variable.
I know it sounds strange at first blush, but this is what is going on: every iota of existence is moving at the same speed, but each can change direction. A seemingly static object, like my coffee cup sitting on the table next to me, is actually a tiny little mindless bee buzzing around at super high frequency such that the after-image of its tracery is my cup. This is why gravity is essential, to keep the bees from wandering off in all directions indefinitely.
Speed is not a control variable. What we observe as translational motion is the average of the buzzing around, like a weighted random walk.
I'll be starting a new topic on the idea that instantaneous speed is not a control variable, i.e., that all identifiable items of existence, at every level of time-size relation, including photons, are moving at the same speed.
Interesting?
- David
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Look, you can't get away with purely particle-mediated forces.
Existence is a sub-divisible whole You can't escape it. The essential feature of a whole is that it maintains its integrity in spite of any gyrations of its parts. Existence acts as a single entity, the purpose of which is at minimum to contain all other entities.
Containment is a fundamental concept, I would go so far as to suggest it is a "fourth axiom", implied in any attempt to dispute it or any of the other axioms. Why? Because, before you can even consider experience, you must be contained within it. I'll grant this is probably deducible from "existence exists", but haven't quite seen how yet.
Existence insists on containing that which exists. We feel the effect of existence containing its parts, because the effect acts to govern and limit our range of motion outward and away from the other parts of existence. For two parts to increase their separation requires effort against the effect. For two parts to increase their proximity goes with the flow of the effect.
Gravity is the "force" that implements containment by existence.
Without containment by existence, you are out of this world. You can't base a theory of physics solely on local, directional interactions, you must have a global containment to create systems. (This is true conceptually too, but that is a much more involved topic.)
- David
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"Where is the wisdom we have
lost in knowledge?
Where is the knowledge we have
lost in information?"
TS Eliot 1934
(Someone mentioned the hierarchy ...)
data-->information-->knowledge-->wisdom
Plus there's the bootstrap process: induction from hypothesis to theory via validation of predictions against data, that allows the formation of new abstractions -- which allow more robust interpretation of data.
- David
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Data are recordable results of measurement.
Information is data mapped into a conceptual object. The concept provides the mapping, allowing the user of the concept to interpret the data as an instance of the concept. Concepts are formally analogous to classes in object oriented programming languages, and information is like an object instance of a class. The mapping of data to information is like a deserialization routine. (Note for completeness that data is trivially information by the identity mapping, but that is only trivially useful, as you'd expect.)
Knowledge is information organized into concepts by measurement omission (that is how one materially grasps reality).
It's a boot-strapping procedure: start with no concepts, just the axioms; gather some data and use it to infer simple concepts, thereby converting data into information; and use that information to induce further concepts. And so on.
- David
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Better yet, one's mind acts as an information sieve. One's premises determine how one interprets the information. Knowledge is the logical product of correct premises held in the mind while observing existence, with existence being the field of validation for one's premises.
It's a boot-strapping procedure: start with no premises, gather some information and use it to create premises, that are used to more efficiently gather more information, and so on.
- David
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I'd like to have a dialogue, so you are correct, I must initiate.
But now that a few of you have had a chance to wade through some of my expository posts, and have some context on me to reference, I'd like to try a bit more conversational style.
So I'll throw out a starting point, and let's see if we can generate some knowledge.
Information, to be communicated, must be rendered concrete by the initiator, and this sensible form must be transported to the recipient, who must convert the sensations evoked into perceptual cues, and interpret them via a conceptual framework more or less similar to that of the initiator. Otherwise, the message will be garbled at best.
Knowledge both derives from, and facilitates, the accumulation and organization of useful information. It is a process of optimization with respect to the flow of information.
Knowledge in an individual's mind acts as an information sieve!
- David
- David
Handling mathematical concepts that have no relation to reality
in Physics and Mathematics
Posted · Edited by icosahedron
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