



Willful IgnoranceReader comment on item: Musing on History Submitted by Michael S. (United States), Jul 12, 2015 at 10:52 Hi, Waz. You said, "Ouch... giving me a headache and reminding me of my incomplete understanding of imaginary numbers and quaternions." You seem to be younger than me. I was never even taught quaternions. My studies were all broken up by work, family and ministry. I did high school in the mid60s, undergrad study in the late 70s, and grad school in the early 90s for a year, then in the early 00s. Every time, I would pick up a new area of specialization, just to get variety; but it became incredibly difficult after a while  I was automatically expected to know everything about, say, optically active crystals, without having studied any math at all for over 20 years. Imaginary numbers are easy to visualize: Instead of graphing a function on "x" and "y" "real" axes, one uses, alternately, "x" vs. "ix" and "y" vs. "iy". The equation y=x², for instance, is an upwardfacing "U" above the xaxis in "real" space; but on the xi vs. y axes, it is a downwardfacing "u". Combining the two in your mind gives a 3d picture of the function with respect to the xxi plane: two parabolae, one up and one down, at right angles to each other and pointing in opposite directions. Seen from above, this would look like a rightangle cross. The equation x²y²=1 is intriguing. In "real space, this is represented by hyperbolae in the xy plane along the xaxis, with a space in betweeen because the square root of 1 cannot be represented in real space. In imaginary space, it becomes a cirlcle connecting the two hyperbolae, and at right angles to them. I was able to link to the web reference, because this is all commonly taught nowadays. I had to discover it myself in junior high school (around age 14) by playing with the numbers in my spare time. Both of the above functions are discontinuous, because they have parts that meet at right angles. We usually think of "real" phenomena as having continuous functions, like the unbroken graphs of circles and lines; but even in "real" space, the graph of x²y²=1 is discontinuous, consisting of two separated halves. Concerning the application of complex numbers in descriptions of real phenomena, there is an application in combining the wavefunctions of electrons in neighboring atoms (or, for that matter, in a whole array of atoms in a crystal). One adds or subtracts the wavefunctions, in order to see the probability density of the bonding and antibonding orbitals. This turns out to be different from the result one would get by simply adding the probability densities of the orbitals of the electrons, because those densities are described by the square of the wavefunctions. To find the wavefunction, one must use the square root of the probability density, and this has both a real and an imaginary component. Knowing the imaginary component is therefore vital to understanding the nature of the bonding and antibonding orbitals in "reality", even though those components themselves are partly "imaginary". The Materialist community, which includes our supposedly most "brilliant" geniuses, discounts the "reality" of things such as "imaginary" space, because it doesn't represent things that are "manifest". The electron density pattern around an atom, for instance, is "manifest". It produces a real "presence" that is felt by nearby atoms. When one of those nearby atoms is in your finger or a probe, that presence gives you the sense that you have "touched" the atom in question. The underlying wavefunctions that make up that energy density are invisible and "untouchable", so we dismiss them as "imaginary". This is the problem we encounter, when we think of reality as consisting or "wavicles" such as pions, muons, electrons and photons. Those waves/particles are simply manifestations of an underlying "reality" that we can't physically perceive. Nevertheless, that "unseen" world is real; and we cannot understand "real" phenomena such as electrons without understanding the underlying "unreal" phenomena of wavefunctions. That is why I don't consider this physical world, described by the "Standard Model" as consisting of particles (really "wavicles") as the fundamental "reality". That fundamental reality is UNSEEN. We can't see it, but we can see the effect of it. This, in part, is the world Jesus was talking about: John 3: In the Bible, the word "spirit" was used to describe things that are unseen, yet powerful. It is a generic term, and does not refer to "ghosts". It does, however, perfectly describe wavefunctions and other such phenomena. This is why the writer of Hebrews says, Hebrews 11: The writer of Hebrews (probably Paul) therefore anticipated Schrödinger, who came some 1900 years after him; and Paul was not a scientific genius. He simply understood the reality of faith, which the "wise" men in our universities are willfully ignorant of. Note: Opinions expressed in comments are those of the authors alone and not necessarily those of Daniel Pipes. Original writing only, please. Comments are screened and in some cases edited before posting. Reasoned disagreement is welcome but not comments that are scurrilous, offtopic, commercial, disparaging religions, or otherwise inappropriate. For complete regulations, see the "Guidelines for Reader Comments". Comment on this item 
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