Bob Kolker

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Everything posted by Bob Kolker

  1. In mathematical parlance "postulate" and "axiom" are the same and they both mean a statement which is not proven but from which other conclusions are proven. Just as undefined terms are the basic objects of a mathematical system, unproven propositions are the foundation statements of the system. Conclusions are reached through logic alone. It is considered must "unkosher" to appeal to diagrams. Diagrams can only be suggestive and aid understanding but they are not part of the proofs. They have a pedagogical function, but not a logical function. Bob Kolker
  2. The "Health Care" bill passes

    There are parts of the bill that are blatantly unconstitutional. Congress has no power to compel the purchase of any privately provided service or good. States have that power, but not the Central Government. Regulation of Interstate Commerce does not include compelling the purchase of anything including health insurance. Bob Kolker
  3. The Consequences of Socialized Medicine

    Yes. I have thoughts, mostly grim. Last night The Equalization of Opportunity Bill passed. How much longer until Directive 10-289? The only comfort I can offer is this. The vote was very close, given the majority the Democrats have in the house. That means a lot of Democrats are running scared. Now, if the Independent Voters pour out their wrath in November many Democrats will go down. There will be blood on the snow. If this puts the Fear into the Democrats the rest of us might just be able to save ourselves. On the other hand, if the Independents do not come through in November I am afraid night has fallen on America. Bob Kolker
  4. It is also full of logical holes which is why Hilbert fixed Euclid's Elements in 1899. Pleas read -Foundations of Geometry- by David Hilbert. Bob Kolker
  5. Ladyhawke (1985)

    I enjoyed the movie for its purposeful action and its up-beat theme. And besides, seeing Michelle Pfeiffer at her peak of beauty was worth the price of the ticket. I really liked Leo McKern performance as the not so sober monk. He was terrific. Ruthgar Hower was also in his prime for this part. It turns out that all of the not so heroic secondaries did their level best to produce a heroic ending. Evil was defeated by good hearted folk doing the Right Thing. Bob Kolker
  6. Beautiful guns

    The first two items of post #32 would make great back scratchers. Provided they are unloaded for that use, however. Bob Kolker
  7. In an axiomatic (or postulational) system, if the definition is nowhere used in the proofs it might just as well not be there. The term is effectively undefined, which mean that any properties it has comes from the axioms (postulates) and nowhere else. In Hilbert's reconstruction of Euclidean Geometry he lists six undefined terms right up front. The semantics of these terms comes from the assumptions using them. In formal systems definitions serve as means of abbreviation so as to make the proofs easier to read. One does not need (strictly speaking) definitions. One needs only a list of undefined terms and a set of unproved assumptions making assertions about the terms. At best, Euclid's "definitions" of straight line, curve, side, etc. were hints to the reader on how to imagine the items intuitively. Taken literally, Euclid's definition of a straight line as length with no width would apply just as well to the boundary of a circle. The "lying evenly" phrase simply suggests the straightness of a straight line (or line segment). By modern standards of rigor, Euclid's system is full of holes. However it was the first attempt at a comprehensive axiomatic, deductive mathematical discipline, so it has much virtue. Bob Kolker
  8. Choices for Educators

  9. Global Hot Air Hysteria

    You are mistaken if you think honest to goodness science is no longer happening. Applied physics is first rate. Chemistry, Molecular Biology, Genetics is first rate. Mathematics has never been better. There are some problem areas in fundamental physics but they are the kind of problems one gets when when reaches the boundaries of current science. We need a break through in unifying the physics of the very small (quantum physics) with the physics of the very large (gravitation). That is similar to the situation of physics at the end of the 19th century. Radioactivity was beyond the bounds of (then) classical physics, but the breakthroughs came in good time. There is no epistemological corruption. Physics is still driven by experimental results (which is as it should be). In any case we have enough applied problems in the pipeline to keep scientists busy for the next two hundred years. Then we will worry about running dry. Once the LHC is running up to speed enough New Stuff will be found to jolt fundamental physics. It will be the end of the 19th century all over again. New Stuff, some confusion and then the advances. Unfortunately the required advances we need in climate science are yet to come. We do not have climate science. We have a lot of statistical models that can be fiddled every which way. In dealing with highly non-linear chaotic processes we are at the bounds of our mathematical know how. Further advances will be required there. The kind of political tomfoolery found in climate studies is not happening in physics, chemistry and biology, at least, not yet. Keep smiling. Good times will keep on coming in science. Bob Kolker
  10. Defence of property.

    Leg hits are difficult in a running target. In combat training one learns to shoot at the torso which is a large target and one where a hit will take the target down. In general, when one has to shoot, one should shoot to kill, not to wound. That is standard procedure. Bob Kolker
  11. 1. What does "lies evenly with" mean? It is nowhere defined in Euclid's Elements 2. The above definition is not used, not even once in any of the proofs of theorems comprising Euclid's Elements. The only properties of straight lines and points used are those properties given in the postulates. The above "definition" (quotes intentional) is more a hint, an intuitive suggest of what a straight line is. You may find the following article useful to read: http://www.cut-the-knot.org/WhatIs/WhatIsGeometry.shtml The article discusses the role of undefined terms in mathematical axiomatic systems. Let us be historical. Euclid's Elements was among the earliest systematic axiomatic treatments of geometry. Euclid collected theorems from several sources, proved several himself and constructed a systematic way of deriving geometric theorems from a relatively few definitions and postulates. This was the first attempt at a grand axiomatic system in mathematics. You could justly call it Axiomatic Geometry Version 1.0. It turns out there are several defects or omissions in his system. For example in the proof of Theorem 1, Book I. He constructs a pair of circles each passing through the center of the other. He concludes that this circle pair produces points of intersection of the circles. But this does not follow from his definition of circle. Deficiencies such as this were finally removed from plane geometry by David Hilbert in -Grundlagen des Geometrie- (Foundations of Geometry), 1899. In this work Hilbert constructs a system of 20 geometric postulates and explicitly lists the undefined terms which are used in the postulates. Please see: http://userpages.umbc.edu/~rcampbel/Math30...ms/Hilbert.html You will notice that he gives six undefined terms each of which is used on one or another of the axioms (postulates) which follow. These terms acquire their meanings from the postulates in which they appear. Hilbert's system eliminates some of the implicit assumptions that Euclid made in his proofs (like the example given Theorem 1, Book I) and he makes perfectly clear what the undefined terms are. Hilbert was able to prove that his axiom system is economical in that no postulate can be proved from any of the others and his system is complete in that it produces all the analytic properties of a plane. Furthermore his system is unique (up to an isomorphism) that satisfies all his postulate. It took 2200 years to get version 1.0 upgraded to a system with "no bugs". Defects and all, Euclid's Elements served as a role model for deductive mathematical systems and his works never went out of publication or replication in the 2200 years follow their original publication. Euclid's Elements became the "gold standard" for axiomatic systems in mathematics (and rightly so, for there were none better until recent times). It turns out that the difficulty of the fifth postulate (it is complicated compared to the others) lead to the discovery of other systems of geometry (non Euclidean) geometries. Euclid himself was aware of the complicated nature of the fifth postulate and he put off using it for as long as it was logically possible. There is a very accessible book -The Non-Euclidean Revolution- by Richard J. Trudeau --Birkhuaser, 1986. In this book Trudeau gives a good elementary introduction to deductive axiomatic systems in mathematics and he follows Euclid's thinking. He also points out some of the omissions and uses of implicit assumptions that occurred early on ( version 1.0 of anything is bound to have defects). He goes on to show how the omission or modification of the fifth postulate leads to other geometries and he gives the first few theorems of hyperbolic geometry, a system where given a "straight line" and a point not on it there are an infinite number of parallels to the given line through the given point. Hyperbolic geometries are for saddle shaped curved surfaces. Enjoy. Bob Kolker
  12. Global Hot Air Hysteria

    There is plenty of True Science in the field of fundamental physics, applied physics, chemistry, medicine, geology, biology. Applied science and engineering are still alive and well and technology is being churned out at a glorious pace. But in the area of meteorology and climate where there is a political involvement the situation is not that good. In point of fact we have no "climate science" in the sense of a solid fundamental empirically grounded theory. Unfortunately weather (and ultimately climate) are the result of chaotic dynamics. It turns out that our handling of chaotic dynamics (dynamical systems which are non-linearly dependent on initial conditions) is not nearly as sound as physics based on non-chaotic dynamics or where the conditions are imposed to eliminate or minimize the non-linearity. To this day we do not have a complete mathematical technique for dealing with the Navier-Stokes equations, for example. What we do have are climate models which are statistical in nature and have far too many adjustable parameters. That means the models can be fiddled to produce desired results. That is not a happy situation. Perhaps one day we will get decent climate science, but that day is not here and now. Bob Kolker
  13. Defence of property.

    A simple risk/benefit analysis would yield the conclusion it is not worth injuring someone for a stolen pencil. If one did such an injury he would be open to both criminal and civil legal processes. On the other hand if the loss of property is substantial and there is no other alternative than to apply force at the time the property is stolen, damaged or compromised then either one uses force, or one suffers the property damage or loss. In principle the loss of little property or much property is the same thing/principle (abstractly) but one must factor in practical consideration as well. Principle sometime has to be constrained in the name of practicality. Sometimes. Not always. Bob Kolker
  14. The Insoluble Problem of Democracy occurs when the vote is wide spread and a majority of the voters in cahoots with some politicians raid the treasury of the nation and redistribute the income of the producers. It has been happening for over 150 years in this country and now proceeds at a quickening pace. Democracy cannot last. A republic can. Democracy was what you get when two wolves and a lamb vote on what to have for tonight's dinner. Bob Kolker
  15. Learning a second language

    English was never centrally planned or controlled. It evolved through historical events such as the Norman Invasion of England. English renews itself through borrowing from other languages as well as innovative locutions produced by its own speakers or writers. English is a "People's Language". It grows organically through usage and is sometimes modified by accident. Two major events gave English its current character: Shakespeare's inventing new words based on French and other languages without inhibition. Shakespeare's plays doubled the vocabulary of English. Samuel Johnson's dictionary gave English some uniformity of use. British Imperialism did a great deal to make English the second or third languages of countries that did not particularly like the British. This explains why English is the "lingua Franca" of modern times. It is used everywhere and by the Warsaw Convention of 1927 is the international language of navigation and aviation. If England and the the U.S. disappeared tomorrow, English would persist by its own inertia for several generations more. Prediction: English will pick up some Chinese terms in the near and not too distant future, just as English picked up some Japanese words like: honcho, hari-kari, banzai-charge and such. Some English speakers are gung-ho about picking up terms from Asian languages. English has absorbed a fair number of Yiddish locutions as well. Almost everyone knows what a potzer is, they know about kibbitzing and schmoosing and they know mazel-tov. adieu, aufwiedersehen, adios, ciaou, sei gesundt! Bob Kolker
  16. Leasing or Buying a Car - Which is better?

    How long do you want to keep the car? When I buy an auto I go for very long term ownership. I have a 1995 Honda Civic which still runs beautifully. I plan to keep it as long as I can get an inspection ticket for it and the repairs are not prohibitive. I have a 2006 Scion XB bought early in 2007. I plan to keep it as long as it lasts and is affordable to maintain and repair. In previous cars I have owned I have held them for something like 10-12 years before they died or became too expensive to repair. Bob Kolker
  17. Defence of property.

    If property is not defended by force, then soon no property will be safe which means people will be denied the right to hold property and derive benefits from their property. So I have to assume that whatever degree of force is required to protect property is o.k. to use. In the example, I would shoot for the legs first. Killing to protect property is generally not necessary, but using force is. I have no principled objection to using concealed leg hold traps to defend one's home, provided warning signs are posted. If someone is stupid enough to try breaking and entering after being warned of snares and traps, then maybe they deserve to lose a leg. Achtung! Minen! Bob Kolker
  18. Beautiful guns

    Lock and Load? It has occurred to me that in order to lock one should have a round in the chamber. So, the logical expression is load/chamber then lock. If one locks with an empty chamber one will have unlock, then chamber, then lock again. What a silly expression! Load and Lock! Bob Kolker
  19. The esthetics of modern motor vehicles.

    I find these designs a bit on the feminine side. I like machines that look like machines. These scooters look a bit "pimped up" to me. I like objects that are male-crude or plain. But what can you expect from someone who deliberately bought a 2006 Scion XB? I love my ugly box on 4 wheels. And who is going to try to steal it? Bob Kolker
  20. Beautiful guns

    I consider them clever and functional. I do not think they are pretty. I cannot get emotionally worked up about these tools. In combat I am interested in what works, not what looks good. As to confessing, I must admit that some of the instruments I saw in Edmond's Scientific Catalog did quicken my pulse by a beat or two per minute. There are some tools which have the same classical virtues as the Renaissance sculptures; symmetry, flow of line and proportion. Bob Kolker
  21. The esthetics of modern motor vehicles.

    Part of "right design" is ease of manufacture. That is very important for items mass produced in high volume. Bob Kolker
  22. Public education at work

    We have had public schools in the U.S. since the middle of the 19th century (due to the efforts of Horace Mann to make schooling a function of the State). That means they have lasted 160+ years which is considerably more than the 3rd Reich lasted (12 years). There is no sign of public school ending in the U.S. that I can see. Bob Kolker
  23. "We Aren't Lighting The Fires"

    No, they were not bored in the class room because we got the school authorities to let them do more advanced work for credit. The teachers knew they were dealing with forces of nature when my kids arrived in class. My grand-daughter can navigate both Windows and Mac with a mouse and can play the computer games and use the drawing tools. C++ is for when she turns five. My seven year old grandchildren (they are fraternal twins) can do Basic now. My son and their mother insist that they use brains only for their current arithmetic homework. Which I agree with. My father, who was a CPA, would not let me use a Frieden Calculator until I cold manually add the phone numbers in the telephone directory and do it accurately. I would not let my children use hand calculators until they showed me they could do good order of magnitude estimates using their brains only (batteries not included). Computers and calculators are tools, not substitutes for thinking. Bob Kolker
  24. The esthetics of modern motor vehicles.

    Because most people cannot afford to buy such items. What makes manufactured items affordable is their mass reproducibility and that works against elegance and beauty to some extent. My daughter is a head designer at Fisher Price Toys and she used to be a hands on designer from scratch* (she is a vice president in the company now). She liked to produce toys with a beauty appropriate for child's toys, but she also knew the manufacturing process stone cold. She had to sacrifice a bit of the beauty to make the toys manufacturable at a price that could enable large numbers of buyers to purchase the toys. The same principle holds for automobiles which are mass production and mass sales items. Sailboats are a different story. They are not a mass market item Bob Kolker * my daughter is the mother of the second generation of Fisher-Price Little People among other things. She was the lady that give Little People arms and hands.
  25. Beautiful guns

    I appreciate fire-arms as a tool. They are cleverly designed kinetic energy mass projectors. They are an application of the second law of thermodynamics and Newtonian mechanics. Any justice (or injustice) lies in how they are used, not in their physics and design. A well aimed rock or spear can deliver the same kind of justice (or injustice) as a kinetic energy chemically powered mass projector. Fire arms are among the best examples of applied Newtonian dynamics, and being a physics fan, I appreciate them as such. Eventually we will have light projectors (lasers, "ray guns" if you will) and they will be to quantum physics what fire arms are to Newtonian/classical physics and thermodynamics. No amount of physics and cleverness can guarantee justice, however. In the scheme of things, the first amendment is more important than the second amendment although both are important. Bob Kolker