icosahedron

A concept is an object of individual consideration. A mind can consider at most one concept at a time, but has the power to choose both the concept and duration of consideration. Thinking is the act of deliberately choosing the sequence and duration of concepts that one considers. In particular, one can consider the relationship between an object and its surroundings, and observe that the relationship can change over time -- an object can move relative to its surroundings. An object's motion is accounted by the mind as a discrete sequence of relationships to its surroundings. Objectively, motion can be no more, and no less, than the mind's grasp of such sequences. Each element of such a sequence of motion is specified by two variables: 1) The state of the object as determined at that point in the sequence; and 2) The duration until the next determination of state. In other words, each element of an observed motion sequence has both state and duration. It is sufficient to consider the case where the duration is a fixed unit of time for every element of the sequence, because one can always find a unit of time that divides equally into any finite set of durations, so that a sequence with varying durations can be viewed, without loss of generality, as a sequence of unit durations such that every original element is modeled as a sequence of identical unit elements. Thus, a specific path of motion is naturally modeled as a discrete sequence of changes in position, with the time between changes fixed, i.e., with the frequency of change constant. So far I have been considering changes in position and ignoring the fact that material objects have mass. But, according to Uncle Albert, there is a radiative frequency proportional to the mass of an object. I assume that the radiative frequency is the natural frequency of (potential) change of direction of a material object. I model a photon as a set of line segments end to end along a given ray. I model a massive object by working back from its radiative frequency equivalent photon, changing the direction of segments until, on balance, the net effect of the steps is to bring about the discretely observable motion of the object. And by construction, it is special-relativistically invariant. The rest follows from considering the nature of volumetric space, for which a tetrahedron is the simplest enclosure and the most natural basis for erecting coordinate rays. Gravity appears to be closely related to the shape of the local coordinate tetrahedron, or as I call it, the motive tetrahedron: it is equilateral in a radially symmetric local gravity field, and distorted if the local gravitational field is skewed. What evidence would be sufficient to prove my hypothesis? How about correlation to spectroscopic data? I'm trying to explain the atomic spectrum of hydrogen with this framework, in the expectation that it will lead to a similar description for helium. That is my first goal, a better theory for predicting atomic spectra. And it really really helps me to discuss it, but I have no one nearby to discuss with. So thanks for the attention. - ico

Go to dictionary.com and look up the word "speed". Judge for yourself. What I propose is a model based on constant speed, but varying frequency and direction of displacement. I suppose I could use the phrase "instantaneous speed", or even "differential speed", or maybe "path speed" is the best phrase. I kinda like that last one -- but my interpretation of the existing definition of the word speed seems to be essentially the same as any of these. But I like path speed ... maybe, pathspeed? (paths peed? heh heh). To reiterate: What I propose is a model based on constant speed, but varying frequency and direction of displacement. Obviously I think it has legs or I wouldn't be yapping about it, but I am happy to have any and all challenges and objections, because I need to tighten up my arguments and render them palatable before I will be satisfied. - ico

Go to dictionary.com and look up the words "tax" and "voluntary". The problem is, the word tax itself almost insists on being involuntary. The phrase "voluntary taxation" puts together two ideas that substantially contend; the only way to understand this phrase is to form a compromise where the two words contend; but as Ayn taught, compromise between the moral and immoral serves the interest of the latter at the expense of the former. Whatever proper system is devised and implemented in the (hopefully near future) must dispense with the idea represented by the word "tax" if it is to have any chance of avoiding the same cultural malaise as has finally settled on the currently constituted United States after the "juice" of freedom ran out, ran up against the contradictions in what was, and was not, spelled out in the original constitutional documents. Why should I pay any individual, representing themselves or as agent of a group (including the government), for nothing? On the other hand, why should I not receive that which I pay for? And who is to decide which trades I make? In other words, give me greater value than you ask in return, and you have a deal. Otherwise, catch you later alligator. And the only way to invert the trading logic is to initiate the use of force to compel another to act against their will. The government, properly formulated, provides valuable services that cannot be provided by private individuals: military, police, courts. And those who use these services should pay for them. But I don't like the idea of the military or the criminal justice system being revenue generators because the idea of profit accruing to those who have the legal right to use retaliatory and/or preemptive force smells like a bad concoction. Which leaves the civil courts as the only governmental function that could be charged for as a service. In effect, this would amount to contract insurance (which would far swamp any bureaucratic fees): if the contracting parties don't pay the insurance premium, then the contract is not legally enforceable. Since contracts are agreements about how one or both parties will act, they are credit agreements, and the insurance premium is properly seen to be credit insurance. Instead of "voluntary taxation" how about "voluntary insurance"? Presumably the amount of the premium will be more for more complex contracts, but in any case no less than a certain fixed percentage of the contract's nominal dollar value, as determined by an independent contract auditor. The rate would be set by the revenue needs of the government, with severe prejudice against government borrowing in normal times. This would just work if implemented; the problem is, how do we get there from here? - ico

Taxation is immoral. Period. Any system based on involuntary "contribution" is a form of slavery. Period. - ico

What exactly is "the opposite of 1" anyhow? Opposite in what sense? In the sense that by "1" you mean "+1", i.e., opposite as vectors? In which case you haven't defined multiplication of vectors yet ... The dot product results in a scalar, not a vector, so is not a proper group-theoretic multiplication operation. The cross product is not commutative, so it doesn't meet the requirements. No, the naturals are vectors of order 1 in the sense you mean; to have a vector, you need an origin. And if the origin is the integer 0, then -1 and +1 lie on different rays from the origin, i.e., different vectors. Why are you surprised that my inner product works as intended? I have already made clear that I don't grant the possibility of moving left by moving right -- or, to put it more concretely, you can't get to the moon by heading towards the center of the Earth. Left/right, in/out are orthogonal conceptually and existentially in the simplest sense: displacements in one direction cannot be added up to form displacements in the other. I never claimed all numbers are vectors. I simply claimed that vectors are not numbers, and that integers are a form of vector by definition: magnitude and direction. I don't know how to define multiplication on vectors in a group-theoretic sense ... Not sure why you can't bring yourself to use +3 when you mean the integer, and 3 only when you mean the natural number. If you'd do that, you'd be representing the integers as the cross product of the naturals with the cyclic group of order two -- which is also a vector form, albeit not symmetric in its coordinates (which is where I started, btw -- but quickly realized the benefit of the symmetry in terms of simplicity and ease of use). In other words, instead of -3 and +3, you could write (-,3) and (+,3) -- what's the conceptual difference. Why do you insist on conflating the positive integers with the natural numbers? They are clearly different concepts, arrived at by distinct means, so why obfuscate their difference? Its magnitude AND sign that are needed to specify an integer. Sign has two states, which is no coincidence because the integers encompass two rays in distinct directions. And a number line certainly IS two rays point oppositely from the origin, why do you want to ignore the fact that the origin is special. Ah, maybe that's the problem -- you want the luxury of moving your origin along the line without loss of generality. But the origin is just a point of perspective, and with respect to any such point, the line is seen as a ray when one looks left, and a ray when one looks right -- two rays, yes. Oh, and the rays need not be at 180 degrees for my math to work, in fact the math works (and you wouldn't question it for any sequence of angles less than 180 ... but if 180 is somehow special, then you are claiming that the process of widening an angle is not uniform, so that the limit of the process can have properties unpredictable from the properties at any given angle less than 180. Where do you draw the line, so to speak? It takes two rays to define a plane, but three to span it, which is what I said. If I used "define" where I meant "span", mea culpa. Finally, Steve, regarding your repeated swerves towards dubbing my assertions "insane" without understanding my analysis in full, how about a little Shakespeare?: "m'lady doth protest too much, methinks." The conventional system uses one number plus one sign to specify an integer. This is one of my issues: when you conflate 1(natural) with +1(integer), and refer to them with the symbol "1", and when the magnitude of +1 is equal to 1, its too easy to lose sight of what is really going on. And, Steve, you have lost sight of it. Check your premises, and please refrain from attacking my character obliquely in the future. I don't appreciate it, nor do I consider it harmless. I have been respectful to you, but am beginning to accumulate evidence to support a suspicion that you consider my challenge of your premises to be a form of disrespect. If that is the case, let me know and I'll stop dealing with you altogether. Cheers. - David

Dante: I agree that character is the point, and especially the motives that the drive the character. And khaight said it very well, I thought: over time, if the political convictions are at odds with the character's moral premises, then the problem is one of logic, not character -- and can often be worked through if one puts in the effort with genuine love and caring for oneself (first) and therefore also for one's partner as an extension of one's happiness. Irrationality does have a tendency to bleed progressively into parts of the mind one thinks immune, like a gaseous fog of floating abstraction that occludes the mind and progressively degrades its ability to put two and two together. This is standard Objectivism AFAIK. Usually, it is the moral character that drives, with the political views derivative or parroted -- which supports your point that the bleeding is not automatically happen, the political-view-tail rarely wags the moral-character-dog. But, contradictions between personal and political convictions, especially if the contradictions are maintained in the face of a partner's protests, indicate a great deal about moral character, don't you think? - ico

Very well said.

Gotta agree with you on that point, claire. I suggest keeping most, if not all, off-topics out of the bedroom (where the topic ought to be celebration of one's life by sharing physical intimacy). Read over some of the love scenes in Atlas Shrugged to get a sense of the psychological flavor (Boys: Ayn did y'all a BIG favor by showing you the sense of life in this context that is most seductive to many rational women!) Then use your imagination to fill in the specific details for yourself and your partner, without losing sight of the style and point you are aiming for. - David

What do you mean by "treatment of loved ones"? Is this a special science, or should it flow naturally from one's assumptions and goals? This smells a bit like an evasion attempting to sever the tie between political views and personal views. And, once people end bad relationships, they don't usually speak much after that either. Which is their right -- the right of free association is also the right not to deal with folk you can't get along with. - David

That is not my experience. If someone believes force is an appropriate tool in some cases, then where do they draw the line. When they are under duress will they grab the "club" rather than work it out deliberately? And by "force", I am not thinking of gross physical violence per se; in relationships, force is usually more subtle, involving coercion, blackmail, threats, etc. Or they may invert the roles, and use tears of need to break down resistance. In most cases, such exhibitions of irrationality/mistrust/disrespect are not effective in and of themselves, and require the cooperation of the victim to be successful (as usual). What happens oftentimes is that one partner indulges the whims of the other, thereby encouraging the irrationality, rather than challenge it and attempt to bring things back to reality. This indulgence is also often a function of the confusion/ignorance/unawareness in principle of the victim. I have experienced this to some degree or other in all but one of my relationships, and all of my friends have had similar experiences (usually it's "all of them" or "all but one" because they ended up happy with the one who didn't try to force them). So I can say with certainty that this is not an isolated pattern. Refraining from initiation of force, including all forms of emotional blackmail, is necessary and sufficient to maintain respect in a relationship. If someone allows the initiation of force in some situations, then it's a slippery slope, and when their back is to the wall, all bets are off and even their primary relationship may not be safe, even if they have consciously insisted that they will never initiate force against their lover. Then, when the chips are down, the high position quickly gets down and dirty. If you haven't experienced a long term relationship with a high-power mind capable of wickedly subtle manipulation punctuated with threats, coercion, and emotional blackmail when their deceptions are discovered, then I claim you are not particularly qualified to dismiss the possibility. And I envy you. I kick myself for every time I put up with destructive irrationality in a partner. Appeasement is not a means to an honorable relationship. As Ayn has written, there are some violations perpetrated on one by another that are hard if not impossible to forgive, because even determining that the violator's remorse is honest is impossible. If someone does something to you that falls into this category, run for the hills. For minor disagreements, no big deal, neither partner should be walking on eggshells. But if someone cannot understand the logical consequences and moral implications of the initiation of force, then they are liable to transgress out of ignorance if nothing else, all the more so if they explicitly support emotionalism as a decision making method. And over time, the transgressions often get more blatant, or (what is equivalent in practice), the victim gets more and more fed up with the baloney. Either way, the relationship degrades with time because the partners are not operating from the same moral playbook. Please don't be cavalier on this topic, we are discussing the most intimate, and arguably most morally charged, aspect of life: finding a compatible mate and gaining/keeping their loving attention. Cheers. - David

Sorry, didn't mean to malign the past results obtained with continuum models; for conventional engineering, such models are so useful that even if a better model were out there, I'm not sure it would have mattered much until folk realized that continuum models were deficient in modeling small things. That's when the focus shifted to eigenvalue models, so that whilst the machinery was still continuously formulated, the observable outcomes became discretely spectral for precisely measurable properties. And, didn't the industrial revolution occur before the experimental debunking of the continuous trajectory models in the context of small things? Since then, the biggest advances have come from the quantum models, e.g., transistors and micro-electronics. But looking to the future, to spur future innovation, we need better models, IMHO. And, computing macro-level phenomenon from micro-states is only difficult because the computational framework is inefficient -- the given seems to navigate these computations with ease, eh? Like, where does Nature choose to round off PI in forming a soap bubble? Cheers. - David

Like, why can't an intelligent 12 year old understand the essentials of modern physics, and develop experience with the ideas? Because its too dang hard to explain with one foot in traditional formulations. I remember learning Schrodinger's equation and it's significance, and then using that formulation to solve problems -- what a pain in the butt! Then the next semester, the professor rolls out matrix mechanics and everything became so simple by comparison. This is typical in traditional schooling at all levels: teach one thing in an old-fashioned way, then show the easy way to do the same thing, or recast the idea in a larger context that requires different assumptions than before Done for good reason, this is a useful pedagogical device (rarely, IMHO -- just show folk the latest and greatest, like, none of us needs to know how to pilot a horse and buggy). However, it's usually done as a sequence of "Aha"'s and "Gotcha"'s that leave most students wondering how the conceptual framework they spent so much time learning is going to be pulled out from under them in the future. If the symbolic representation of ideas is inefficient, then the leverage of those ideas is limited relative to what it could be. The assumption of continuity of the physical substrate is precisely such an outmoded, inefficient idea -- and the formulation of physics in terms of PDE's is suspect in terms of conceptual efficiency, if not practical results. In developing software, one learns that when two bugs affect the same system, it can be hard to see one of them because the other occludes it (e.g., by preventing the system from reaching the code path with the other bug in it). Similarly, if the software is "smelly", then even if it works today, the cost of maintaining it, or improving it, can become prohibitive. Then you can either refactor it or junk it and start from scratch. The point is, good developers spend significant time refactoring their codes to keep the smell to a manageable minimum. I strongly suspect that humans are being held back in their intellectual and technical progress by inefficient means of representing and solving problems. Cheers. - ico

So you are claiming that real analysis provides THE objective standard for the correctness of computations? I challenge that. Mathematics is a result of humans thinking, and hence just as prone to vogues and pressures as every other discipline. True, mathematicians as a rule embrace logic and causality, so they are way ahead of many others from the get go; but they are people in society, and the history of mathematics shows how often the accepted patterns of representation have needed a refresh. Like anything man-made, real analysis didn't have to be the way it is. And I say it floats away from reality a bit, and this leads to awkward, complex formulations that reduce the ability of young minds to come up to speed. - ico

It all comes down to whether she would sanction the initiation of force, morally; because, if she does so sanction, then she may at some point feel justified in initiating force against YOU! - ico

I don't think that will work. I think we need to make sure the government has the cash to fund it's current obligations, but switch the source of revenue so that the wind down can occur in a rational legal context (and I advocate changing the Constitution to ensure no slide backs). Understood there must be an orderly transition, but if a system for charging for civil court services, including contract guarantees, can at least be conceptualized and worked out in enough detail to determine what the funding overhang would be at different cost levels, then there would be a basis for beginning to make the necessary changes. The first step, IMHO, is to separate the concerns of revenue generation and (the moral equivalent of) spending that have become so conflated in the current tax rules. The problem is that with these concerns intertwined as they currently are, government prejudice in allocating the backdoor "spending" hidden in the tax code is rampant and almost impossible to control or account transparently: credits, deductions, etc., are the moral issue, because they single out specific economic and political groups for special treatment; but the practical issues of inefficiency and uncertainty in planning are also huge problems. Now, if I told you it was possible to convert to a graduated flat tax system with minimal impact on the net income of individuals and corporations (btw, why should corporations pay tax?), would you believe me? Here's how: in the year of the switchover, each tax paying entity would use a rate equal to the ratio of the actual tax paid to the gross income (perhaps averaged over 3 years to smooth out the data). In the second year, the tax rate would be obtained by computing the best fit function for the rate-vs-income data, and from then on, the tax for any payer is just the result of the best fit function with the considered gross income as input. Done. But that's just the first step. After that, I think removing the notion of corporate profit from the system is imperative. A corporation is a collection of individual owners, and its profit cannot be anything other than the sum of the profits of its owners -- so write the rules to reflect that, i.e., make sure all profit is partitioned to the owners so they can pay the taxes, and eliminate corporate taxes. Eliminate all other taxes, except property taxes (which are actually the one form of tax I think is morally justifiable under certain circumstances): dividend, capital gain, sales, VAT, etc. Raise the flat income tax rate to make up the revenue. Once all that is done, the tax code will be simple, Simple, SIMPLE, and the actual rate of taxes paid will be clear and transparent and predictable. Then we can consider how to reduce the tax rate by converting to a pay-to-play system. - David

Erizo, what would you like to produce that is of greater value to others than your cost to produce it? How do you get from where you are to the point where you can produce what you like to produce, and sell your products for a profit? Do you need to learn a trade from someone else? Working as an employee of folk who are in the business you like might be an option. Can you learn what you need to know without help? Great! Go for it. The process is one of induction: define your rational long term goals, including personal goals and professional goals, and then connect the dots from where you are today to the realization of those goals. In the process, your goals will need to be in harmony with one another, and form a hierarchy, but don't be dismayed if you can't see the whole road before you start -- none of us can or will, the fact that other volition beings share space with you is incontrovertible. Just make sure that at each major decision point, you move in the right direction, and don't let minor decisions erode your resolution. Cheers. - David

That is the argument, yes. Do you think it holds water? Seems like we are caught in a chicken and egg situation, because without the political will, how can we wind down the illegitimate government programs? In my experience, most people don't change their political attitudes and behavior until they have little or no choice (one way this happens is when a rational individual is shown incontrovertible evidence and/or logic -- being rational, the individual has no choice but to accept rational conclusions based on factual evidence -- that's called "learning" in my lexicon). So, how are the problems going to disappear unless the law is laid down first? In other words, the government's job is to enforce the law, but the current law allows the government to finance itself via taxation, so how will the individuals in the government be incented to change things, to reduce unwarranted programs, unless they are told to by law? - David

And you reference the existence of a conceptual framework called "information theory", which purports to talk of physical things, as evidence that the word "information" means what I call data. I suppose information was undefined prior to physicists promulgating such an elegant theory. Look up the word "information" on dictionary.com. The first definition given is: knowledge communicated or received concerning a particular fact or circumstance. That is what I mean by "information". - David

Nate T., or anyone else who wants to handle it: If, as I claim, magnitude and direction are the definitional attributes of the concept "vector", then: 1) Integers are vectors 2) What sense does it make to multiply vectors? 3) How does (-1)*(-1) = (+1) make any sense? Said in English: how does adding -1 copies of -1 to yield +1 make any sense?. Huh? - ico

I consider the space of solutions to algebraic equations to be perfectly fine, don't question it per se. My problem is, why does it need to be conflated with a one's metric space, just because the two correspond at some points? Why must the second root of 2 (obtained originally as a solution to an algebraic equation) be considered on the same basis as PI, which is not algebraic? I think the conflation is an historical artifact from the days before fast computers, which can't handle floating abstractions. - ico

As Nate T. has pointed out, I mean continuous, not compact. Sorry for any confusion. It isn't a gain in simplicity on its own. But it makes it easier to take the next conceptual steps towards one of my goals, a wholly rational mathematics where unreal ideas such as "less than nothing" and "continuously variable" are left by the wayside. - ico

Meaningful as vectors, yes. Why not? That's what they are (words like "below" indicate a direction).

Nate T, thanks for the correction. I'll substitute continuous for compact in my future writing when I mean continuous (which you were astute enough to discern; my points pertain to the continuous assumption). It's been too many years since I cracked an analysis text. I guess it's simpler anyhow to show that continuous spaces do not correspond to the discretion of measurement, especially if, as you claim, my space is compact. You're probably right, but I will have to verify for myself. Now where is that confounded analysis text? As far as PDE's, I too have used computers to "solve" them numerically. Interestingly, computer proofs used to have a big stigma attached to them, which demonstrates the inversion of fact versus invention in mathematics. Of course, a computer cannot represent irrationals, as it operates with finite sequences of bits. Which only emboldens me ... Thanks, - David

Nate T said: "Ah, I see, icosahedron. Your argument is that we have nothing but (positive) rationals to measure with in reality (which is true), and therefore we should have no need of real (or negative) numbers in mathematics?" In a nutshell, yes. Given that every other facet of our society is less than wholly rational, why would one expect mathematics as it stands to be pristine? Now, if one chooses to identify limits of non-terminating sequences of rationals with special symbols, fine. Not sure what the point is of identifying them for metrical purposes, however. On the other hand, if one discovers solutions to equations in algebra, and then insists on conflating them with rational measurements from reality, to form the so-called real algebraic numbers, then one will have to drop some context to get the scheme to work in practice. And that's not good, long range -- it makes the math muddy at best, harder to learn and teach, and harder to use, forcing humans to "fly by instruments" instead of using their conceptual peepers. The context dropped is whether or not the idea measurably exists in reality. As usual in any invalid compromise, it is the reality basis that gets cut off in such compromise. Perhaps one wants to take the ratio of a circle's circumference to its diameter, in the limit of an unreal, perfect circle (measuring any real "circle" will yield a rational approximation to PI, of course). In this case, even beyond the irrationality of PI, I question the premise that a curved segment is commensurable with a straight segment, except for the (rational) measurement of how much stuff they are made of. In this case, the context dropped is that angles have units (they are fractions of a cycle), and one should properly say that a one meter radius circle has a circumference of one meter-cycle. Cheers. - David

a) (N,N) represents 0 for any natural number N See response to point (a) c) No. One of my tenets is we only need natural numbers, and ratios of them, in order to record measurements. d) Yes and no. Depends on what you are doing. For example, when subtracting 4 from 3, the steps would be: (3,0) - (4,0) = (4,1) - (4,0) = (0,1) The non-reduced representation, (4,1), is useful in doing computations without encountering numbers less than zero. This is analogous to how addition of 1/3 and 5/6 goes like: 1/3 + 5/6 = 2/6 + 5/6 = 7/6 e) (3,2) is a non-reduced version of (1,0) - David