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uncool

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Everything posted by uncool

  1. ...Mordred, I've said something like this to you before (about other people than Conjurer), but I think you are trying to tell Conjurer things at much too high a level.
  2. The answer...is a bit complicated. From the other thread, I'm guessing that your level is around a freshman or sophomore undergraduate with a new interest in math. At that level, and for a few more years, that is the approximate idea: learn the definitions, understand the postulates (there's less of a difference between those two than you might think), play around with them a bit, see if you can figure out patterns for yourself, figure out or read theorems, learn the proofs, etc. In time, you will be able to figure out some of the standard proofs for yourself. (One of my favorite/hated lines from Munkres, chapter 4: "Why do we call the Urysohn lemma a ‘deep’ theorem? Because its proof involves a really original idea, which the previous proofs did not. Perhaps we can explain what we mean this way: By and large, one would expect that if one went through this book and deleted all the proofs we have given up to now and then handed the book to a bright student who had not studied topology, that student ought to be able to go through the book and work out the proofs independently.") To a large extent, the "game" is "Here are our assumptions; what are our implications?" When you have gotten comfortable with this, usually around the time you graduate from a math or math-related major, things start to change a bit. By that time, you stop simply accepting definitions and start asking "Why was this definition chosen, rather than that one?" The "game" shifts; definitions become more fluid, although rigorous proofs stay, in some sense, the center. You may or may not have heard of Terry Tao; he's a very famous mathematician. He explains what I said above in much more detail here: https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ For now, it is a good idea to focus on learning the definitions and thinking in the "undergraduate" way. Getting to the "graduate" way takes a lot of investment and time that you haven't had the chance to put in yet, and knowledge that you haven't had a chance to learn yet. But you are well on your way through the first transition, from the sounds of it.
  3. I'm not sure I understand what you are trying to say. What's the difference between "rounding to the nearest real number" and truncation? The result on the hyperreals is also exact. The "standard" function is specifically defined to give a result in the real numbers, not the hyperreal numbers. There are two correspondences I can think of, but the one I think you are referring to is that the reals are a subset of the hyperreals (i.e. there is a natural embedding of the reals in the hyperreals). This isn't "super advanced algebraic structure theory"; it's just...a fact. I'm not sure what you think needs enlightening.
  4. I am "looking for" someone who cares whether what they say is correct. You apparently don't. I don't simply trust what people say, especially when I specifically know better. When we say you are wrong, consider the possibility that it is because you are wrong. To put it quickly: We are saying you are wrong because you are wrong. We are not correcting you because you seem to not be looking for nor appreciating correction.
  5. ...no. That's not even close to what I said. I don't even know how you read that from what I said. If anything, I'd say the problem is that you've talked too much. Specifically, you've confidently made assertions that are clearly wrong, and otherwise made statements that fail to demonstrate a deeper understanding of the things you discuss, when such a demonstration is clearly appropriate. For example: in this case, when challenged on your understanding of phase space, you superficially described uses of it. The challenge was about what phase space is. You eventually provided it, but only by copying and pasting from Wikipedia, without showing that you understood what you copied and pasted.
  6. Because your descriptions are asserted with much more confidence than is warranted. As well as very often being wrong.
  7. ...this description, if anything, only confirms what Strange said: you have no idea what "phase space" means.
  8. No, they do not. Hyperreal is not simply "not real"; it is a specific extension of the reals. One of the major properties is that it is an ordered field extension; that is, it allows addition, subtraction, multiplication, and division (except division by 0), and that these operations act nicely with respect to "Which of these is bigger". One result from that is that all squares are positive (as x^2 = (-x)^2, and one of x and -x is positive, and the product of two positive things must be positive). So if i were a hyperreal, then -1 would be positive - which it can't be. That said, it is possible to construct the hypercomplex numbers, in the same way the complex numbers are constructed from the real numbers.
  9. What's the difference, in your opinion? I'm not sure what "it" refers to here, but both hyperreals and dual numbers are rigorous (though the "dual number" has some trouble with properly defining higher derivatives); a version of "dual numbers" is actually used in algebraic geometry. That shouldn't be especially surprising; the hyperreals aren't really meant to be a fundamental change in outcome, only in outlook. As such, most formal processes should look essentially the same.
  10. Formally, no. Informally, spot the fuck on. And of course, with the new structures, you get new patterns and relationships, which need new structures, which ...
  11. Among many, many other reasons, because I have some basic knowledge from quantum mechanics: photons are bosons, not fermions. What does this mean? This is not a good argument.
  12. It may be worth noting, however, that that idea of infinitesimals is not the same as the hyperreal idea of infinitesimals.
  13. I assume you mean the equation for the derivative. The denominator is specifically assumed to be infinitesimal, and 0 is not infinitesimal.
  14. "Nonstandard analysis deals primarily with the pair [real numbers, hyperreal numbers], where the hyperreals are an ordered field extension of the reals, and contain infinitesimals, in addition to the reals." In addition to the reals.
  15. Once again: The fact that the set of hyperreal numbers contains infinite numbers doesn't mean it can't contain finite numbers. Not all hyperreal numbers are infinite. Infinitude is not a "requirement" for all hyperreal numbers. There are finite hyperreal numbers. Some hyperreal numbers are finite.
  16. The fact that the set of hyperreal numbers contains infinite numbers doesn't mean it can't contain finite numbers. Read that definition again: the hyperreals are an extension of the real numbers. The set of hyperreal numbers includes the set of real numbers. The statement "Such numbers are infinite" refers specifically to "numbers greater than anything of the form", not to all hyperreal numbers.
  17. It is a hyperreal number. It's also a real number. Every real number is a hyperreal number (more formally: there is a natural map embedding the real numbers in the hyperreal numbers).
  18. That's not how infinitesimals work. 1/pi is not infinitesimal.
  19. ... No, I'm not colorblind. No, I'm not arguing that "anything even has a color". https://en.wikipedia.org/wiki/Lyman_series (specifically, the n = 3 case) It sounds like you thinking of something akin to the Rutherford model of the atom, with electrons physically circling the nucleus; even the standard version of that is a century out of date. Further, "a frequency that is a multiple of a full wave wavelength" doesn't make sense - frequency and wavelength don't even have the same units.
  20. In other words, by completely changing what I said. My argument is not that there is no such thing as color. Electrons don't "have" colors. That would indicate that there is only one color that can be seen emitted by an electron. But, for example (and using a crude approximation) an electron in the 3rd shell can emit photons corresponding to two colors: one for when it drops to the first shell, and one when it drops to the second shell (without the approximation, there are many more colors). The color comes from the interaction of the electron and the nucleons around which it orbits.
  21. ...no, that's not what I said. Color is related to photon frequency.
  22. ...irrational numbers are not infinitesimals. They are real numbers.
  23. ...no matter what idea of "discovery of electrons" you mean, it wasn't done by Newton, and generally was done nearly two centuries after him. A reasonable choice: "In 1897, the British physicist J. J. Thomson, with his colleagues John S. Townsend and H. A. Wilson, performed experiments indicating that cathode rays really were unique particles, rather than waves, atoms or molecules as was believed earlier.[5] Thomson made good estimates of both the charge e and the mass m, finding that cathode ray particles, which he called "corpuscles," had perhaps one thousandth of the mass of the least massive ion known: hydrogen.[5] He showed that their charge-to-mass ratio, e/m, was independent of cathode material. He further showed that the negatively charged particles produced by radioactive materials, by heated materials and by illuminated materials were universal.[5][35] The name electron was adopted for these particles by the scientific community, mainly due to the advocation by G. F. Fitzgerald, J. Larmor, and H. A. Lorenz.[36]:273" https://en.wikipedia.org/wiki/Electron#Discovery_of_free_electrons_outside_matter Further, it's not that electrons have a color. Yes, light is emitted by the photons, but that doesn't mean the electrons themselves have that color. If any particle can be said to have a color (in the "you see it" sense, not the QCD sense), it is the photon.
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