Jump to content
Objectivism Online Forum


  • Posts

  • Joined

  • Last visited

  • Days Won


Everything posted by icosahedron

  1. Bukhari, your idea does not appear to me to add value over and above existing physical theories. I urge you to check your premises and be more deliberate in your inductions. So far, your posts appear to express more or less whimsical connections to observable reality. Now, if it is simply an issue with lack of familiarity with English, then maybe try writing your ideas in your native tongue and getting someone to help you translate them more faithfully. If English is your native tongue then this cannot explain your lack of coherence. - David
  2. Interested to hear how folk think about this -- clearly their is an intimate relationship between knowledge and information, but it is interesting and fruitful to explore it, IMHO. Thoughts?
  3. First, realize that at 25 you are still maturing philosophically (my sense is that most men don't fully mature until at least 35 if not 40 years old). At some point, if you are reasonably successful at living, you will find your confidence reaches a plateau which it never again falls below, even if circumstances get dire. This seems to me related to Ayn's idea of "pain going in only so far". Now, what you want to do is accelerate your development. My suggestions, in order: 0) Use downtime to read. Standing in line? Read. Commuting? Read. Walking down the street? Read. On the toilet? Read. Maybe not whilst driving tho'. 1) Read Peikoff's book. Over and over again. Until you understand it in your bones. 2) Read all of Ayn's writings; reread as the mood strikes you. 3) Find a few intelligent, articulate, driven people to share ideas with, and perhaps to associate with in a professional sense, too. 4) If you aren't married yet, wait. 5) If you don't have children yet, wait. 6) Wait to commit to an "ultimate lover" until you are mature enough, in your own estimation, to make a good choice. 7) Don't buy expensive things (such as cars or houses) on credit unless you are accounting them as investments intended to provide you more value than they cost -- and even then, be very careful. 8) Work for/with experts in your chosen field to refine your skills 9) When prepared, start your own thing, maybe with a few friends, and kick it. Hard. Work your dang butt off -- which will be easy to do if you pick a field you like and have talent for. 10) Once your success is more or less predictable, and your view has resolved into a long-range, clear perspective, start searching for your "ultimate lover" (no need to refrain from love in the meantime, just reserve the right to change your mind if you discover that you need a different kind of lover). Never compromise your principles to attract a lover ... if you are mature and rational, then in a world of 7 billion you'll have a pretty good chance of finding someone who agrees with you in principle -- or is interested to learn. 11) Now that you are happy, enjoy life as long as possible! Cheers, - David
  4. I applaud the Tea Party movement as a step in the right direction, but fear their principles are compromised enough, relative to the purity of Objectivism's political standard, that they may end up being yet another distraction from the root principle: individual life and the freedom necessary to live it successfully.
  5. A1. No disagreement that the traditional formalism is efficient for computations with pen on paper. In the age of computers it hardly matters. And the fact that I have to keep in mind more than necessary, including all the strange rules of multiplication (-1 * -1 = 1 ... wtf?), makes the paper gain not worth the conceptual headache, at least in my mind. As for the "algebraically unified way" you allude to, my system is no less integrated -- it is isomorphic. Finally on this point, if "unified" refers to the rays' endpoints fused to form a line, whether intentionally or not, then it is exactly that notion I am questioning. In reality, I observe visually by reference to my eyes as sensational centerpoint, and all distances are outward and positive relative to my eyes. A2. The inner product is defined as the Euclidean product of the fully reduced representatives with no loss of generality, i.e., it is a proper inner product on the elements of the vector space of equivalence classes. A3. Why should I stand aside while a less convenient model continues to co-opt the conceptual space? Linearly independent works just as well for colors as it does for spatial relations, with, e.g., green, red, and blue being orthogonal under an RGB inner product. Since I don't recognize any cognitive primacy of negative numbers as such (beyond some symbolic sleight of hand hiding the fact that they are vectors), neither do I feel it necessary to qualify what I have already, and clearly, shown to be a perfectly good vector space, together with a perfectly good inner product (which I am tempted to dub "displacement product"), and hence perfectly natural sense of linear independence -- plain old-fashioned linear independence, not needing any qualifier such as "positive" to give it cognitive significance. Cheers. - David
  6. There is a potential physical constraint with ejaculating overly often, assuming you have better uses for your energy than constantly regenerating semen. Someone who ejaculates too frequently over a long period of time may end up a bit tired, I guess. Masturbation often leads to ejaculation, so may contribute to lowering ones overall energy level. Is this an important effect? I haven't looked for studies, but guessing not so much. Sex trades, on the other hand, are dicier. While in principle not a violation of morality, participating in the sex trade as any of producer, model, prostitute, actor, broker, seller, or buyer, is fraught with moral peril in a society that is more or less irrational (or at least mixed) in its views of sex. In the context of a society whose members generally carried a rational relationship to the sexual act, its consequences, and its implications, the sex trades would be voluntary and legal ... and would not be particularly lucrative. Because of the perversion of society away from rational ethics and politics, however, the fact is that the lower level participants in the sex trades get the shaft (pun intended). The sex trades as currently constituted involve rampant deception, coercion, blackmail, theft, kidnapping, imprisonment, and rape. The sex workers, as a rule, do it for the money at the expense of their pride. I can't see how this equation can balance in reality, although I guess some sex workers claim to love their jobs. There is something going on here that is not yet well understood, and I don't think Ayn nailed it. Cheers. - David
  7. I wish it had never been raised; having said that, I am not about to let it stand unchallenged while speculators blather ... because I have time to spare at the moment. And I can't quite tell if you were referring to my post as support for the half-baked pea-shooter, so just to be clear, it wasn't. I was making the point that without gravity or something awfully similar in principle, you can't make physical structures (they would explode or evaporate). Amen to dropping this thread. (I like that you refer to past experience for its evidential merit, rather than as a matter of tradition.) Cheers, - David
  8. I agree that considering integers as scalars breaks the tie to reality, that is my point. Integers have magnitude and direction, are vectors. I map +1 to (1,0), -1 to (0,1), and then use the standard Euclidean dot product of 2-tuples. I represent +N as N*(1,0), or (N,0); I represent -N as N*(0,1), or (0,N). When adding, I reduce the result to simplest form using the equivalence relation (s,t)==(u,v) iff s+v=t+u. For example, I would write the equation (-7)+(+5)=(-2) as (0,7)+(5,0)=(5,7)==(0,2), with the latter step invoking the equivalence relation. Look, its a perfectly good mathematical invention. It may seem strange to describe a line this way, but it's consistent with reality and linear algebra, and when you go up to planes and volumes, the benefit only increases, conceptually speaking, in terms of simplicity of representation and computation (no negatives to worry about). I don't expect anyone to take this on faith, but can anybody else see that integers are vectors (have magnitude and direction)? That is all I ask that you grant; the rest of my framework follows logically from that. Cheers. - David
  9. The fact of conscious volitional entities implies that the future cannot be predicted with certainty. If one measures a precisely observable state variable of an isolated system regularly, one will obtain the same answer each time, independent of the frequency of measurement, because the state cannot change unless the system interacts with another system. In reality, it is not possible to isolate a system from the rest of existence. Over time, depending both on the level of interaction with impinging systems (such as photons) and the frequency of measurement, the state of a regularly observed system may change between measurements. And the Law of Entropy indicates that, statistically speaking, the system state is more likely to become less ordered than more due to haphazard "background" interactions. In everyday terms, i.e., macroscopically, this shows up as visible decay of longstanding structures. For example, your abode is more or less in the same state as you left it in the morning, when you return in the evening of a busy day. But over a span of years, the state of your abode will visibly and functionally decay (dang it!). Microscopically, decay of excited states is the rule. The only difference between the micro and macro cases is the scale of the system involved, and hence the lifespan of the observed state. For a house, the lifespan of its state is probably immeasurably tiny, so tiny that it appears impossible to construct a device to measure the state of the house twice in quick enough succession to obtain the same precise result. This is because the house is a high-frequency agglomeration of bajillions of atoms, each acting as a potential state change lever. For an electron, the lifespan of a polarized spin state (half-spin state) in a dark vacuum is of macroscopic duration, and one can make regular measurements at relatively low frequencies without observing variance in the result. Eventually, however, the electron will decay into its unpolarized (zero-spin) state (which latter is more entropically favorable, hence the state in which one usually finds electrons "in the wild"). The electron spin state is more stable than that of the house; the electron interacts less with its ambient environment, so its state is less likely to decay over any given period of time. However, it is not the precise micro state of the house that is important to the person who lives in it; rather it is the macroscopic average state that matters. In this sense, the house does indeed last longer than the electron remains in an excited spin state -- within the parameters of one's ability to discern with the senses, the result of measuring a property of the house's macro-state is more predictable than the result of measuring an electron's micro-state. Once a measurement is taken its result is a fact. By observing and understanding patterns among facts generated by measuring similar systems, one becomes more able to predict the outcomes of future measurements of such systems. Because the state of a system may change between measurements, however, perfect predictability is not an option. All that one can do as a limited being is learn to predict outcomes just precisely enough to enable one to engineer a future, i.e., to enable one to plan long range. Cheers. - David
  10. The idea of matter as a condensate of cosmic energy is not unwarranted per se (not certain that is a fair restatement of bukhari's conjecture, fwiw). However, what would cause the condensation? Are purely electrical forces, as are involved in a balloon floating in the wind rather than homing in on the Earth's center, sufficient to explain such global condensation? And, what would prevent the atmosphere from evaporating into the lower density of space if not gravity? As I see it, there must be at minimum two distinct, complementary types of force in dynamic balance to create stable structures: a global, scalar potential, tensionally contiguous (although not continuous) envelopment tending to force things together (which would be observed locally as an omnidirectional attractive force, e.g. gravity); and a locally repulsive "strutting" force that acts explosively to keep the global envelopment from collapsing completely (e.g., the electrostatic repulsion between electrons). I am not saying I can reduce the known physical forces to these two (yet). I am simply observing that to create volumetric (i.e. real) physical structures requires balancing globally integrated, omnidirectional tension (as in a shrink-net basketball sack) against locally isolated bi-directional compression (as in the strutting of a bicycle wheel). Point is, you can't get away without the omnidirectional tensional aspect ... and this one's potential is by nature dependent only on the relative locations of entities, not their motion (which is why physicists will never isolate a graviton ...). Cheers. - David
  11. It is your choice not to deal with me, or my ideas. Just in case you might still want to deal, I'll show you the formalism, because it is pretty simple. I find linear algebra, and especially the notion of inner product spaces, to be conceptually unimpeachable as a framework for analyzing problems that can be mapped into a vectorial framework. Orthogonal in linear algebra simply means that the projection of two vectors vanishes under the chosen inner product. The inner product is a matter of choice, but must be a proper inner product, i.e., it must satisfy the definitional axioms of an inner product. Formally, then, you ought to have no trouble granting the following vector space construct: Let P be the set of pairs of natural numbers. Let R be the equivalence relation defined, for (s,t) and (u,v) elements of P, as: (s,t)R(u,v) iff s+v==t+u Let X be the set of equivalence classes of P under the relation R. Define addition of elements x, y of X by reference to addition of their equivalence class representatives. The additive identity element is the equivalence class of (0,0). The equivalence class of (t,s) is the additive inverse of the class of (s,t), because (s+t,t+s)R(0,0) by the commutative nature of addition of natural numbers. Under this addition, X is a proper vector space. Note that the simplest, most reduced representative of each equivalence class has one component equal to 0. Thus, (N,0) is the prime representative of the class (N+K,K), and (0,N) for (N,N+K). In fact, all equivalence class prime representatives are of one of these two reduced forms. This is analogous to rational number formalism where 2/3==4/6 where 2/3 is the reduced, prime representation, but 4/6 is equivalent. (In fact, a similar process of considering pairs and forming equivalence classes is applicable when constructing the rational numbers -- but the addition rule is defined differently) X is seen to be isomorphic to the integers when the following correspondences are noted: +1 → (1,0) -1 → (0,1) Thus it is a perfectly legitimate representation of the integers under addition; it highlights the vectorial nature of the concept “integer”, which is obfuscated in the current +x/-x formalism. The representations are isomorphic, so if one is clearly vectorial, the other must be, too. So stop trying to multiply integers directly, you can’t do that with vectors! What you can do is scale vectors and add them. The problem is that the current formalism encourages conflation of the concept “2” as a scalar multiple, with the concept “+2” as a vector displacement. So, when one speaks like “-2 equals 2 times -1”, what one is really saying is “the vector (0,2) is equal to the vector (0,1) scaled by the scalar 2, i.e., (0,2) = 2*(0,1). My formalism makes it obvious what is going on here, it is a standard linear combination formalism. Now one can proceed to choose and define an inner product on X, e.g., for the purpose of considering the projection of one vector onto the direction of another. What inner product to use depends on the context one is analyzing. In the context of modeling spatial relationships, the key conceptual invariant is that spatial displacements are travelable, at least in principle. Since you can’t travel left by going right, I desire an inner product which respects the identification of displacements as travelable, including the fact that left and right are distinctly accessible directions. The inner product I choose is simply the natural dot product between prime class representatives, so that (1,0) and (0,1) are orthogonal (in the sense of linear algebra, not Euclidean geometry). This fits with observations of physical reality, wherein you can’t ever go left by going right. It seems strange only because you are used to thinking of lines as the basis of dimensionality. To represent a plane takes three rays towards the vertexes of a triangle. Cartesian coordinates appear to use only two lines, but they use four rays – three is more efficient, especially so since only positives are necessary to represent the plane in my framework. To represent a volume takes four rays towards the vertexes of a tetrahedron, not the redundant 6 rays found in Euclidean representations. It’s a matter of conceptual efficiency, as I said. If my sense of efficiency offends your sense of orthogonality, then one of us is wrong. Cheers. - David
  12. I do not model numbers as rays; I model quantitative spectra for specifically measurable properties as rays. For example, I model the concept of mass as a ray, with zero as the minimum. Clearly, negative mass is nonsensical. I model the concept "scalar, graded value potential of some measurable quantity" as a ray. Why not? How is that model lacking in representing mass, energy, length, time, or any other measurable quantity? Measurements are always of quantities, which are something, not less than nothing. Measurement often relates to operationally continuously variable quantities, such as a visually observable mass of gold (not always, cf. atomic spectra). The rays are arrays of numbers ... rational numbers in ascending sequence. Integers account the distance left or right of zero. -1 means distance 1 to the left of zero. Simple and clear. But this is a vectorial concept -- magnitude and direction is the hallmark of the abstraction mathematician's call "vector". To me, that is also simple and clear. If integers are indeed vectors, as I claim, then multiplying them is not well-defined. But adding them is. All I am really doing here is demoting the integers from a ring to a commutative group. And yes, I understand the repercussions in the organization of mathematical knowledge. But please note, this is a matter of representational truth and efficiency; I'm not saying computations with integers are wrong in appropriate context; I am saying that the concept as currently integrated into the institutional knowledge base is awkward at best, and anything awkward can be improved (otherwise we'd use the word "perfect"). Awkward systems of mensuration stunt the growth of conceptual knowledge by increasing processing time. Put positively, the computer has rendered tables of logarithms obsolete; how could humans have attained the current level of economic capacity using tables of logarithms? An efficient system of mensuration (and hence also representation of computational tasks) redounds forward in time on the cumulative knowledge of humans. That's why it matters even though the results for already considered cases will be identical. I am after the cases currently occluded by the awkwardness built in to the institutional mensuration conceptioning. For more fun, note my progression of spatial dimensionality: 1-D: ray 2-D: line (opposing rays) 3-D: plane (rays arrayed triangularly) 4-D: volume (rays arrayed tetrahedrally) The linear algebra is clear and simple if you consider it a bit. Cheers. - David
  13. Excellent! One must distinguish between constructing a concept, and using it (e.g., for the purpose of relating measurements of its properties). Fully agree: concept formation does not follow rationally from arbitrary focus, nor haphazard measurement omission. That sounds more like an Epicurean recipe for evasion. My point is that, with rational focus, it is possible to choose which properties are relevant in a given inquiry, which I think you also agree. Cheers. - David
  14. On reading my prior post, I think it appropriate to give a simple visualization for my model. Consider a discrete random walk where each step is of the same length and can be taken either left or right. Let (x,y) represent all paths that subtend x steps left and y steps right. Notice that (x,y) and (x+q,y+q) refer to the same endpoint in a walk starting at (0,0) (The number of steps is ignored for this purpose ... but should not be in general, e.g., when planning lunch dates. But that's another, more involved, story of mine; I'll fire it up on a new thread at some point.) So, I simply take (x,y) to represent the integer y-x, and accept the fact that there are multiple ways to refer to this integer, kinda like the case with rationals where both numerator and denominator can be scaled without changing the value. (Actually, I end up exploiting these multiple ways to model translational momentum, but that is part of the other story). Cheers. - David
  15. Hi Steve, thanks for noticing in both threads! Check the other thread for more detailed comments, but suffice to say here that your rebuttal was based on a premise I question, and ask that you re-examine your conclusions in light of my thoughts. For the sake of this thread, my argument in a nutshell: Any two distinct rays emanating from the same point are linearly independent in the simplest sense: linear displacements along one ray can never get you to a point on the other. The premise I question is that opposing rays somehow coalesce into linear dependence, whilst rays at a tiny angular displacement from opposition are linearly independent. Here's the link to the related thread: http://forum.ObjectivismOnline.com/index.php?showtopic=18245&st=60 My detailed response to your rebuttal, Steve, is #77 in that thread. Cheers. - David
  16. Mr. D'Ipolito, I respectfully ask that you check your premises. Any two distinct rays emanating from the same point are linearly independent in the simplest sense: linear displacements along one ray can never get you to a point on the other. The premise I question is that opposing rays somehow coalesce into linear dependence, whilst rays at a tiny angular displacement from opposition are linearly independent. The contradiction is due to context-dropping, and a bit of conflation, around the concept "linear". Linear means something different than Euclidean rectitude in the more general context of so-called "linear" algebra, and unfortunately leads to the common mistake of assuming that lines are the basis of dimension in volumetric space. Volumetric space is the conceptual context. What can be said with certainty about volumetric space? Well, it's finite, bounded, enclosed. In other words, it has an inside, and an outside (editorial note: people traditionally have underestimated the importance of the inside/outside complementation that necessarily adheres to volumetric spaces). Now, the thing about inside and outside is that, like any truly complementary pair, they are not merely symmetric opposites. This becomes clear from a simple observation: you can only go inward so far before you pass through a volume and thenceforth move outward indefinitely (assuming a ray-like trajectory). In volumetric space, in means in towards something, and directions are defined by reference to "inward-nesses". Mathematicians may play with more or less unrelated abstractions, but we humans live in volumetric space(s), and cannot assume things are where they are without seeing a signal indicating such facts ... and the signal must traverse volumetric space to reach us ... and two signals sent in different directions are clearly not gonna end up in the same place, clearly represent two distinct dimensions of information gathering. This is just another way of saying, as I originally stated: Any two distinct rays emanating from the same point are linearly independent in the simplest sense: linear displacements along one ray can never get you to a point on the other. Rays, not lines, are the proper basis of dimension when traversing motion is taken as the gold standard of proof of (potentially evolving) relative location. I challenge anyone to impeach this standard. Ergo, lines are 2-dimensional. And therefore, since a vectorial representation is required, I submit my representation of the integers as the most conceptually economic I could devise: pairs of counting numbers with equality defined by reference to equivalent net displacement when one imagines finite, constant-length steps taken to the left, or the right. I can keep trying to make it clear if you choose to pursue it -- it is worth the effort, I promise. Cheers. - David
  17. Here's a point: gravity won't let you escape. The javelin doesn't have unlimited energy, so cannot expand the boundary. As for a volitional being standing on the boundary, that is absurd. How did he/she get there, given the huge gravitational potential pulling back to the center? Context dropping is the only way this is a puzzle.
  18. The modern "concept" of numbers ignores context and conflates distinct concepts. Each class of numbers in modern use is a product of a specific process of accounting. Different accounting context cannot be dropped except in very special cases, e.g., when one is only interested in the quantity and not its units. Counting leads to counting numbers. The unit is the type of entity counted. Relating counting numbers leads to rational numbers. The unit is the denominator. Solving for quantities to fit equations leads to algebraic numbers. The unit is predetermined and then abstracted out of the process -- but the domain of applicability of the equation determines the unit. Complex numbers are solutions of equations, no more, no less -- and labeling the solution to x*x=-1 as "imaginary" is facetious. There is nothing imaginary about the solution to this equation, and physical examples of its reality abound, e.g., electro-magnetics in the context of conductors where phase frequency is complex; or in quantum mechanics where the operators are complex, but the observable eigenvalues are potentially directly measurable, i.e., "real". Complex numbers cannot be directly measured or observed, duh, it would take at least two observations to pin down the real and imaginary components. Taking the natural limits of sequences of rational numbers leads to the so-called real numbers. The unit is implicit in that the elements of the sequence must be commensurable. This allows one to model even the transcendentals. To my mind, the representation of numbers as limits of sequences, as used in constructing the real numbers, is the most general; but you can't just blithely manipulate sequences as if they were unitary numerals, esp. the non-terminating sequences become bothersome. And you don't get the complex numbers unless you use two sequences. Thinking of the solution of x*x=-1 as the square root of -1 is just people mistaking formalism for fact. It is just a number defined by the fact that it solves the equation, and should have a disparate name such as "freddie" to make sure folk don't conflate it with observable, measurable quantity. The problem goes deeper. The notion that -1 or +1 are numbers is false. They have distance and direction. They are vectors. The integers are naturally represented as pairs (x,y) with the equivalence relation (x,y)==(s,t) iff x-y==s-t, with normal vector addition used to construct a vector space from the equivalence classes under this relation across the set of pairs of counting numbers. So, if -1 is represented as (0,1), and +1 by (1,0), then what sense does it make to take second power roots of either of these vectors? multiply(-1,-1) = (0,1) * (0,1) = (1,0) How does that make sense? The conceptual flaw is that you can't get somewhere by going in the opposite direction, so scaling +1 by multiplying it with -2 has no basis in reality. Cheers. - ico
  19. Measurement may be defined abstractly as: the process of relating an observed value to a known (and commensurable!) value (the unit of value employed as the basis of measurement). Note that by nature a value MUST be unitized, i.e., it must be a definite definite quantity along a specific spectrum of possible outcomes (thus QM insistence on observables expressed as linear operators, with the eigenvalues determining the spectrum of possible outcomes of measurement). Note also that the observer's focus determines the unit of comparison, and one only needs to consider those facts about an entity that are relevant to the given context considered. So, if one is only interested in color of things, then the rest of their properties can be excluded from consideration as irrelevant. An actual entity has multiple observable properties, each with a specific unit; the nature of the entity can be described as the integrated set of observations of its properties and functions. So, e.g., shape is no problem, it just requires observation of multiple properties to pin down ... and one of those properties is the topology of the shape itself (topological forms are the available observables). So, measurement omission is the basis of conceptualization because it is not the actual value of measurement(s) that determines the entity's type/class; and emergent or unknown properties are no problem either ... at a higher order, the determination of a given entity's type, in terms of its known properties, does not preclude the discovery of other properties with experience. The type itself is an observable and evolves over time (concepts are contextual). In a sense, relative to the set of all observable properties of an entity, one's focus on a specific subset of properties involves measurement omission, yes; but it also involves property omission, because one must winnow out irrelevancies even in cases where one knows in principle all the properties of an entity. For example, the fact of a person's gender is not relevant when considering their character -- in the context of assessing character, we not only omit measurements (e.g., of one's actions), we also ignore properties that are irrelevant. So, concepts in all their glory come from measurement omission applied to similar entities; but the measurements one focuses on and omits are also a matter of choice, i.e., the observer's context masks out only the known properties relevant to current consideration. Finally, notice that the notion of class type in object oriented software languages provides a convenient and accurate means to represent a concept. Conceptual hierarchies can be directly and reliably modeled as object oriented class hierarchies. And, when one decides to add another property to one's concept of a given class of entities, then one need only add another member variable to the corresponding OOP class type. Now this is interesting. I'll be starting a thread on that soon. Cheers. - ico
  20. The concept "number" as used in phrases like "zero", "negative number", "rational number", "real number", "complex number", is a misnomer. The proper mathematical term for these cases is "scalar", i.e., an element of a mathematical field. Let me separate the conflated concerns a bit, for the sake of clarity. First, you get counting numbers by simply counting, this is the simplest notion of measure one can discover -- how many crows flew off when the man entered the field? Oh, and really you only need to count to two, and have a means for holding place, to express any other counting number as a binary string. Next, you get positive rational numbers, or fractional quantities, by using a unit that does not divide equally into the measured value. This is always the case when looking at statistically large agglomerations where the individual parts (e.g., atoms) cannot be resolved and one must resort to an approximation. In other words, one can form ratios of counting numbers. And the nature of these ratios is not the same as the nature of the counting numbers from which they are formed -- the conceptual "units" are different, the context of application is different: the fact that 4/2 corresponds to 2 does not imply any correspondence between the whole ticks ... just because we can put 2 and 2/3 on the same line does not mean they are commensurable -- all one can claim is the correspondence, one cannot count with rational numbers, nor approximate with counting numbers, per se. Then comes so-called irrational (but not transcendental) numbers (?!). But these, again, have different context than do counting or rational numbers ... they are solutions of algebraic equations, and are only nominally related to the other classes of numeric concepts, via the crude, conflationary context of the so-called "real" number line. Finally comes the transcendental numbers, which can always be approximated to arbitrary precision as finite sequences of rational numbers (so can irrationals, for that matter). The most general context is not some imagined "real" line, but rather the fact that each and every number can be approximated with a finitely generated sequence of rationals. Rationals and counters happen to be wholly finite sequences, whilst irrationals and transcendentals cannot be precisely determined (what a surprise ...). The other perspective is solutions of algebraic equations, but does not capture the transcendentals. Note that human sense perception reduces to counters and/or rationals ... nobody can observe an irrational number, nor a transcendental one, because measurements are by nature ratios to a unit. As for negative numbers, they aren't scalars, they are vectors! A vector is a directed line segment; -1 means one step to the left, +1 means one step to the right (dropping the '+' in '+1' is facetious and confusing practice that allows the inveigling of young minds with the false notion of negative "numbers" as scalars). Note that mathematicians have managed to cast negatives as scalars only with a particularly irrational assumption that adding two scalar quantities together can lead to zero. There is no such thing as a negative quantity: quantity implies something, as opposed to less than nothing (!?!?). But of course it is easy to do the math with vectors -- go left one step, then go right a step, and you are back where you started, your net displacement is indeed vanishing (albeit, time has passed -- that is another thread). So negatives aren't really numbers. The concept that mathematicians traditionally call "negative integers" is, properly analyzed, (isomorphic to) the set of equivalence classes of pairs (s,t) where s,t are counting numbers and the equivalence relation is defined, naturally, as: (s,t)==(u,v) iff s-u==t-v (in English, two pair are in the same equivalence class if and only if the absolute values of the differences between their respective element are equal). Best way to visualize this: Consider a Cartesian grid, and draw the "diagonals", i.e., all lines parallel to x=y and crossing grid points. All the points on any one of these lines are equivalent, and the set of lines represent the integers. The mapping is pretty easy: x=y corresponds to the integer '0', and x=y+N corresponds to the integer +N, and x=y-N to the integer -N. Hope this helps, but expect from experience it may light a fuse or two ... Cheers. - ico
  21. The geometric intuition is correct. Imagine a spherical mass is photon-ized in the absence of angular or translational momentum. Then, to conserve these, the photons produced must be ejected spherically from where the mass was. Measuring out one second in time from the inception of the disintegration of the mass into photons, one ought to have a spherical shell of photons in flight that has surface area of 4*PI*c*c, and an intensity proportional to the mass (assuming a reasonable mass and subsequent large number of photons radially and evenly distributed). In the case where the mass has translational momentum, the sphere becomes an "ice cream cone" shape, due to doppler effect -- this corresponds to the momentum-dependent term (E=m*c*c only works for rest mass). Other, much more complex forms of equi-time photon shells/embracements are obtained if the mass is not spherical, or if it has angular momentum. Now, I prefer not to use geometry to analyze micro-context phenomena -- geometry was designed for use in everyday context, and is not suited to explaining the behavior of entities whose translational momentum investment approaches the limit. That's where Einstein pops in, with non-Euclidean geometry as the norm, and space as a complex "stress" tensor depending on the gravitating bodies in proximity to the space considered. There are probably simpler ways to get to Einstein's point. While perhaps merely suggestive, and certainly not a proof, yet the geometric approach together with the assumption of proportionality between mass and energy yields a useful visualization: photons emanating radially outward as a consequence of explosive expulsion of energy when spherical, non-rotating, rest mass is disintegrated. Then, simply choosing to measure at the same time/distance, the intensity of photonic energy at the chosen radial shell will be proportional to the mass -- how could it not be? - ico
  22. "The mathematicians feel that they can do anything they want with their abstraction because they don’t relate it to reality. And, of course, they can really do anything they want with their abstractions, even though, like masturbation, it is irrelevant to the propagation of life." -- R. Buckminster Fuller The mathematical notion of line attempts to do more than abstract the notion of relationship (and the related concept of physical extent) -- it creates a concept at least partly divorced from reality: in reality, relationships between things (including the related parts of a body which determine its extent in any given direction) are in general neither linear, nor continuous, nor static ... and they have "diameter" in the sense of physical (or conceptual!) interference effects. In reality, relationships cannot be established except by exchanging information, at minimum in the form of little energy packets called photons. Their are no real lines, only interactions among related parts of larger wholes. Or so it appears from my perspective, consistent with the logic of my experience. Cheers. - ico
  23. My understanding of Ayn's use of the idea "unit" does NOT encompass that which can not be experienced or imagined, at least in principle. Ergo, infinitesimal ANYTHING is impossible -- including but not limited to infinitesimal points, lines, and planes. It is incorrect to assume the limit necessarily exists. The attempt by geometers to divorce their wares from the field of reality is traditional, but like any other attempt to vivisect mind from body, it only leads to a false dichotomy. Real things have finite extent in every observable property, at least as far as my experience goes -- have yet to sense or imagine something that is infinitesimal, i.e., has no measurable value for some unitizable property. Go figure ...
  • Create New...